CENTER FOR COMMUNICATIONS AND SIGNAL PROCESSING

Optimization of Mean and Variance of End-to-EndDelay in Interconnected Networks

By

Manu Khanna

CENTER FOR COMMUNICATIONS AND SIGNAL PROCESSING

Department of Electrical and Computer EngineeringNorth Carolina State University

Raleigh, NC 27695-7914

December 1984

CCSP-TR-84/21

ABSTRACT

KHANNA, MANU. Optimization of mean and variance of end-

to-end delay in interconnected networks. (Under thef

direction of ARNE A. NILSSON).

In this research different strategies to optimize

mean and variance of end-to-end delay in interconnected

networks, where messages are passed between two networks

of identical architecture through an intermediate network

of different architecture, have been investigated. In

particular two schemes 'strip-and-pad' and 'pad-and-pass'

have been compared on the basis of end-to-end delay

incurred. Analytical modeling has been incorporated to

find the trade-oifs between the two schemes. Simulation

techniques are used as a tool to validate the analytical

models. The choice of either 'strip-and-pad' or 'pad­

and-pass' has been found to be a function of processing

times at gateways, mean message lengths to be taken care

of and the number of hops to be traversed through the

intermediate network. Parallel to Kleinrock's

'independence assumption' the concept of 'partial

independence' has been introduced.

11

BIOGRAPHY

Manu Khanna was born in Chandigarh, India on October 16,

1960. He received his S.Se. Engg. from Punjab Engineer­

ing College, Chandigarh, in Electronics & Electrical Com­

munications, in 1982. After working for about half an

year with Uniscans & Sanies Ltd., he came to the U.S.A. to

pursue his Master's degree at North Carolina State Univer­

sity, Raleigh.

His major area of interest is Computer Communica­

tions.

111

ACKNOWLEDGEMENT

I wish to express my heartiest gratitude to Dr. Arne A.

Nilsson, Chairman of the Advisory Committee, for his

invaluable guidance, advice and persistent encouragement

at every step of this work. My special thanks are also

extended to Dr. Harry G. Perras for his constructive cri­

ticism and the many papers he provided me for this study.

I would also like to extend my thanks to Dr Dharma P.

Agrawal for his encouragement during this work.

Finally, I would like to thank the Center for Commun­

ications and Signal Processing for making available the

computing facility and the funds for this research.

1v

TABLE 0 f CONTENTS

1.

2.

t nt r oduce i on ••••...••••..•••••••••••.••••..•...

1.1 Computer Networks -- motivation •••••••.•.•

1.2 Network Interconnection -- some issues ••.•

1.3 Problem Definition •••••••••••..••.••.•••••

Review of Some Existing Tools •••••••••••..••...

2.1 Kleinrock's Independence Assumption .••••••

2.2 J.W. Wong's Representation of a

1

2

5

14

19

20

Netwo r k ..••..••••••.•••••••••.•••.....•..• 26

2.2.1 Model - some assumptions and

notation •••••••••••••..•••••••....• 26

2.2.2 Summary of Results •.•.•.•..•....•.• 27

3.

4.

Verifying Wong's Analysis ••••••••••••••••••••••

3.1 Validation of Wong's Model ••••.•.•..•....•

3.2 Conclusion ••••••••••••.•••.•••.•••••••••.•

Modeling the Network •••••••••••••••••••••••••••

4.1 Some Assumpt ions ••••••••••••••••••••••••••

4.2 A Tentative Model •••.•••••••••••••••••••.•

4.3 Network-Modeling based on Wong's

30

31

69

70

71

7S

Ana1ysis •••••••••••••••••••••••••••••••••• 81

5. Towards a Refined Model •••••••••.••••••• •••••••

5.1 Verification of Partial Independence

for ./D/1 type gateways and 2 classes of

messages ••.•.•...••..••.•••••..•.•...•••.•

99

100

v

5.2 Comparison of the two Paths •••..•.•.••••.• 124

6 • Fina 1 Mode 1 ••••.•..•••••••••••••••••••••••••••• 141

6.1 Verification of Partial Independence for

D+M type of message distribution .•.•..••.. 142

6.2 Comparison of path 1 and path 2 •..••..•..• 154

7 • Can c 1us ion ••••••••••••••••••••••••••••••••••••• 170

7 • 1 Ove rv i ew •••••••••••••••••••••••••••••••••• 170

7 • 2 Fu t ure Wot: k ••••••••••••••••••••••••••••••• 173

8. List of References •.....•.••...•...•...•..•.... 176

CHAPTER 1

Introduction

'Analytical Engine', the world's first general­

purpose digital computer was developed nearly a century

and half ago by Charles Babbage. He also built a

prototype of the world's first special purpose digital

computer, which he called Difference Engine, in 1812 - ten

years after the invention of the steamboat!

We have come a long way since then. Mechanics of

Babbage's computer have been replaced by electronics.

Most of the modern ideas, many of which were first

conceived by Babbage(and remained just that because of the

state of the art at that time), have been made possible by

the tremendous breakthroughs in solid state Physics and

other sciences.

And so continues the man's quest for more, more and

more!

2

1.1. Comouter Networks -- motivation:t

Maximum utilization of resources has always been the

major area to be explored. In 1960 this led to the

multiprogramming and time sharing systems. And then began

an era of computer networks. The motivation for having

computer networks is many-fold. The first objective is to

have resource and load sharing among computer sites ~hich

may be situated thousands of miles apart. The tenn

'resource' may imply specialized hardware, software or

databases. In a computer network it becomes possible for

a user to access these resources housed at any of the

connected sites. Big organizations which have branches

allover the world can exchange information within a

matter of seconds.

A second goal is to provide high reliability by

having alternate sources of power (computing). If a

computer system breaks down at some site, its users can be

accommodated somewhere else until the service is =estored.

Another factor which influenced the evolution of

computer networks ~as the dramatic fall in processi~g

costs as compared to communication costs. Prior to 1970,

computers were relatively expensive as opposed to

communication facilities. So in some applications data

were collected at different sites and sent to a central

office for processing. But nov it is very cheap to

3

provide a computer at every individual site, and so data

can be processed right where it is collected.

Consequently, only occasional exchange of information

takes place, thus saving in communication costs.

Apart from above, the computer networks have emerged

as a strong medium of communication. Tanenbaum [3] claims

that it will have a strong impact on the society of the

future. Home-banking, automated newspaper, computer aided

education, teleconferencing will be commonplace. It will

then truly be an age of information. 'The information

revolution may change society as much as the industrial

revolution did.'

ARP~~ET, ETHERNET, TELENET, TYMNET, TRANSPAC,

DATAPAC, EURONET are a few of the examples of networks

working at local, national or international level.

A step ahead in the same direction is the

interconnection of computer networks themselves (the topic

on which the present thesis is based). Basically, the

motivation for interconnecting computer networks is the

same as for having computer networks, namely, to make the

best use of resources existing in different computer

networks, and exchange of information among remotely

situated sites. As more and more computer networks come

into existence, interest in interconnecting them will also

increase. Also operators of local area networks have

4

vital interest in having access to long haul networks.

Local area networks, in general use different t~ansmission

media(e.g. coaxial cable or optical fiber), to optimize on

delay and cost. Long hauls on the other hand use leased

line or satellite channel. So these two types of

networks, employing different technologies, are there to

exist, and so is the desire to interconnect them.

The legal, technical and political issues, concerning

interconnection of computer networks or communication

net~orks have been described at length in [1] and [2]. In

~hat follows we will give a brief account of technical

issues involved in interconnection of networks.

1 . 2 . Network Interconnec~ion --~ issues:

5

As a simple example, suppose net~orks A and B are to

be interconnec~ed, where A and B may be similar or

dissimilar net~orks.

fig 1.1: A simple ne~~ork interconnection

7he interface bet~een them may be though: of as a black­

box, which makes messages originating in net~ork A,

presentable to network B, and vice versa. This black-box

has been called 'gateway' by INWG (Internetwork Working

Group) of IFIP (International Fedration of Information

Processing). A gateway may be an actual processor

6

connected to both networks, or may be additional software

implemented in existing processors on one or both

networks. It may be noted that while designing a gateway

it is of utmost importance to keep in view the

independence of the individual networks. That is, if the

interconnection demands drastic changes in the hardware,

software or protocols of the existing networks it is in no

wayan acceptable proposition

There are a number of issues which must be looked

i~to before any interconnection can be designed. The

important among them are the following:

1) level of interconnection;

2) addressing and ~outing;

3) flow and congestion control;

4) accounting;

5) access control:

6) internet services;

The level of interconnection refers to at which level

in the hierarchy of network layers the individual networks

will be connected. T~o identical networks may be

connected at packet level. In this case gatevay consists

of only software routines which may provide readdressing

and accounting functions. Each node or user will see the

interconnected network as a single network but vith

modified add~ess space.

7

If the networks to be interconnected are dissimilar

the interconnection may be made at network layer level.

The function of the gateway now is to make the packets

originating in one network, presentable to the other

network by attaching appropriate headers and trailers

along with the addressing information. The individual

networks are supposed to take care of the higher level

protocols. The interface may provide 'datagram' or

'virtual circuit' interface.

Another alternative for interconnection is protocol

translation at the gateways. In certain instances when

the protocols of one network cannot be modified, protocol

translation is the only choice if interconnection is to

take place. But this becomes really difficult and

impracticable if the interconnection has to be done over

several networks.

The second important issue concerning

interconnections is addressing and routing. Addressing

and routing are in fact very closely related to each

other. An address tells where to go and routing tells how

to reach there. There are different approaches to the

problem of addressing and routing. An internet packet may

be wrapped in the header and trailer of local protocols.

Subsequently, at the gateway it is extracted and passed on

to the next network after examining the internet header

and t:-ailer,

a

~hich contain the required addressing

information.

/,

Local- Internee Local-ne: DATA !nt:ernet nee

headerheader r::-ailer trailer

Local-nee: text

t ,..... !~~e~-net packe: e~bedded in local nee ?ackec

~h:s app~oach has the advantage in that i: prese~ves the

':~cepencence' of the incividual networks.

In another approach the add=ess space may be modified

and an individual. user may di~ect~1 acdress the

ces:ina~ion i~ a hie~a~chical fashion as in t:aditional

tele;hone sys~ems. A pa~t of the acc:-ess ~ill tell the

local ne~~o~k ~o be accessed and the rest ~ill tell the

destination in that particular network.

Exic:t.Qc:aL-ner: Desr:inac:ion

gateway adcir!ss in Local-~ec

Address :ieLd

fig 1.3: Hierarcbi~al Addressing

9

A nice thing about hierarchical addressing is that a

source does not have to specify the whole route to

destination, and so it may not know about the routing

policies in intermediate networks.

A nice account of different addressing techniques has

been given by Carl Sunshine in [1].

Flow and Congestion control:

Flow control implies the regulation of traffic from

source-to-destination. Congestion control refers to the

procedures by which network resources such as bufferspace,

channel bandwidth, CPU capacity etc. are protected from

overload conditions, generated by 'all' sources of traffic

in a network. In general, successful operation of flow

control for all source destination pairs may not imply

that the system is uncongested.

Flow and congestion control problem is very complex

for individual networks even, and becomes all the more

complex for interconnections. The networks connected to a

gateway may have highly varying speeds (e.g. local area

networks connected to long haul) and may be dealing

differen~ly with congestion produced in them (e.g. some

networks may just drop the congested packets, while others

may rule out the dropping out of packets and stop entry of

new packets in the network to overcome congestion). All

this adds to the complexity of the problem and makes it

10

tough to deal with.

Functions like accounting and access control are also

supposed to be performed by the gateways. Accounting is

necessary for billing and revenue purposes. Access

control is a mechanism by which networks can prevent

traffic entering or leaving them. e.g. a public network

may allow some networks to be connected only during a

certain part of the day.

Another problem to be taken care of is that of

segmentation and reassembly. Each network imposes some

maximum size on its packets which may be governed by

factors like hardware (e.g. the width of a TOM

transmission slot), operating system (e.g. all buffers are

512 bytes), compliance with some national standarcs or a

need to prevent one packet from holding the channel too

long [4]. The need for segmentation and reassembly arises

vhen relatively large packets are transmitted over

subnetwo~ks which support relatively smaller packets.

Segmentation may be done at the source before a packet is

entered in the net~ork, or it may be done at the gateway

before a packet is passed to the next network which does

not support the packet size as carried by the original

packet. It may be possible to reassemble the packets at a

point prior to the destination, although the best

mechanism to do this is still a research topic.

11

Two different approaches to the reassembly of packets

can be found. The first approach called 'transparent

fragmentation' requires reassembly of packets as soon as

they leave the network for which they were fragmented. In

the second approach reassembly is done at the destination.

The second approach requires reassembly processing only at

the destination. But the messages travels in the form of

many small pieces, thus resulting in overheads in terms of

headers and trailers for each separate piece. In the

former approach overheads show up in the form of

reassembly to be done at every exit gateway (fig. 1.4).

Thus the two approaches have their pros and cons, and

before anyone can s~ttle on one of these, a great deal of

research is required to be done.

llicket I

o

[3J 00J Pacl<et I G 1 ~

0°o

fig 1.4:

oo

illr Packet 1---CSJo

(i) Transparent fragmentation

o°0_.-. .. _~ .... -DO(]

(ii) Non-transparent fragmentation

Oifferent fragmentation approaches

., Pacl(et I

o00

00

o

......"-l

13

A proposed solution to handle the problem of

segmentation and reassembly is to define some global limit

of packet size, which must be supported by all networks.

The choice of this limit may be difficult and it may not

find favor with the operators of the individual networks.

Other relavant work in the area of interconnection of

networks can be found in [5,7,9].

Finally, it is worthwhile to point out that efforts

are being made to standardize the interconnection

procedures, although not much success has come by so far.

X.7S, recommendations of the CCITT is one such standard

[6]. It was first defined in 1978 for the interconnection

of X.25 networks.

14

1.1. Problem Definition:

We have described above the concept of

interconnection and the related issues. Of this wide

gamut of problems and issues, we have taken up a very

specific p~oblem for our research. It is one of

optimizing delay and its variance in a configuration of

network interconnections given below. (fig. 1.5)

NetworkA

NetworkB

Net\lorkAt

fig 1.5: Interconnection of networks A, A' (identicalarchitecture> and network 8 (different architecture)

~

U1

16

Networks A and A' are identical in the sense that

they support the same set of protocols. Network B is

different from network A or A' • Gateways G1

and G2

interconnect A and S, and 8 and A' , respectively.

A packet 'm' originated in network A is to be taken

to network At (identical :0 network A) via network B. (The

same holds for a packet originated in A' and destined for

network A. But, as is obvious, the two cases are exactly

same. So from now onward we will concentrate only on the

first case i.e. traversal of a packet from network A to At

The interconnection of networks is at Network layer

level. The packet 'm' has to be provided with headers and

trailers in accordance with the protocols of the network

in ~hich it may be at a particular instant. Thus, a~ the

source (in A or At) packet 'm', before entering the

network, is appended with header and trailer as requi~ed

in network A. The packet 'm' traverses its path th~ough

the network A and reaches the gateway G1•

At this moment

gateway G1

accounting,

will take

verifying

ca:-e

access

of usual

rights

func~ions

etc. aut the most

important function of gateway G1

is to make packet 'm'

suitable for entry in network S by providing it with

header and trailer as per ~equirements of protocols in

that netW'ork. 3efore gateway G1

can do this it has two

17

alternatives before it:

i)Embed the packet 'm' directly in header

and trailer for network B. Subsequently at

gateway G2,

the original packet will be

extracted by stripping off the header and

trailer of network S. This extracted

packet can then be passed on to the network

A' without any further action (because this

packet started with the header/trailer of

network A/A' and retained them throughout

its journey in network B).

ii) As a second alternative, at gateway Gl

,

header and trailer of network A are

stripped off from the packet 'm' and new

header and trailer are attached as per

requirements of network B. At gateway G2

just the reverse action takes place. The

header and trailer of network B are

stripped off and those of network A' have

to be attached to packet 'm' (because 'm'

did not retain its original header and

trailer with which it started unlike in

approach 1).

Now, obviously, we are confronted with the problem of

selecting out of these two approaches the one which gives

18

better performance with regards to overall delay and its

variance. If we take first approach, the packet length

that traverses network B is comparatively larger (implies

more delay) but processing times at gateways are

comparatively less (implies less delay). On the other

hand, with approach2, we have more processing times at

gateways but less packet length to carry through network

B. The relative effects of these two factors, i.e. packet

length and processing times, dictate which approach to

take. Their effect in turn, is controlled by va~iables

like gateway traffic, internal traffic of the networks,

numbe~ of hops traversed through the network E etc.

The pr~sent research is based on finding the

t~adeoffs between the two approaches i.e. which one :s

better in relation to overall delay and variance of delay,

and under what environment (where environment may be

defined to mean the values of parameters like gateway

traffic, traffic internal to the networks, gate~ay

?rocessing times, packet length to be encountered in two

approaches and number of hops traversed through network

E).

19

CHAPTER 2

Review of Some Existing Tools

In the last chapter, we described the motivation for

interconnection of communication networks and the related

issues. In this chapter we take time out to treat two

very important subjects on which the analytical treatment

of our problem is based. The first one is Kleinrock's

Independence Assumption. The importance of independence

assumption may be judged from the fact that it is the

single most important factor which makes the application

of 'Queueing Theory' practicable to communication

networks. Without it no analytical treatment of any

communication network is tractable.

The second subject to be discussed in this chapter is

a model of communication network as suggested by J.W. Wong

[14].

20

2.1. Kleinrock'! rnde~endence Assumotion[lO):

Consider a simple tandem of t~o service stations ~ith

exponential service time dist~ibution, and infinite

\Waiting room. Customers enter at one end and afte~

~eceiving se~vice at both stations leave the system. The

interar:-ival time of customers is exponen~ial.ly

cist~:bu:ec(with mean such that uti~ization of ei~her

customer does not exceed 1). !n:erarrival time, service

ti~e at fi~st serve~ and service time at se~ond se~ve~ are

a" independen~ random variables i.e. ~no'Jledge of

i~~era~~ival ~ime of a custcme~ does no~ dete~ine its

service ~e~uirement at first se~vice station. Simila:-ly,

service re~uirement at second station is not dete~~ined by

:he se~~ice re~u:~emen~ a~ fi~s: station. The study of

such ~ueueing networks ~as :irs~ unde~taken by R.R.?

Jackson [ll]. He discoverec :ha~ the joi~t p~obabil:~y 0:hav:ng xl cus~ome~s in fi~st queue a~d x2 custorne~s i~

~ 0- --~ 0fig 2.1: 7andem of t~o queues ~i~h exponencial se~vice time

disc~ibucion and poisson arrivals.

21

second queue is given by

( 2 . 1 )

where P(xl) and P(x2) are the probabilities of having

xl and x2 customers in respective queues if they had

Poisson inputs at a rate equal to the external input.

Also the symbol '*' here means multiplication and not

convolution.

The result implies not only the independence of the

states of the two systems but also that the second system

has the same distribution as it would have if the input to

~hat system were Poisson with the same intensity as

external input. This in turn poses the question: is the

output process from a MIMII queue in fact Poisson? The

issue was settled by Burke [13] and Reich [12]. They

discovered that the output of a MIMic is indeed Poisson.

The above result put R.R.P. Jackson's result on more

general and firm footing. The joint state distribution

for a tandem consisting of k-exponential service stations

having c(i) servers can now be given as

( 2 . 2 )

where P(xi) is the state probability of the i-th

queue if it were having Poisson arrivals with the same

rate as the external arrivals.

22

Now let us turn our attention to the behavior of a

communication network. In a communication network it is

quite reasonable to assume that interarrival times for

messages for a given source destination pair have

exponential distribution. Similarly, the message length

distribution can be well approximated as exponential. The

servers ccrrespond to the transmission links between

adjacent nodes and service time is the time required to

transmit a given message of specific length on a link

having specific capacity. (For the time being we ignore

processing times at nodes, and the propagation delays

involved). 2.2 shows a specific route through the

network. Messages enter at the node A and

destination as node 8.

have their

At first glance, the above path seems to resemble the

'tandem' discussed earlier. But, as we know, a message

preserves its length as it t=averses the net~ork. So at

all nodes, except the first one, interarrival times a~e

dependent upon the service requirement. Consequently, the

given path cannot be t~eated as a tandem of JacKson's

networks. This 'dependency' ~akes analytical treat~ent of

even the simplest communication net~orks almost

intractable. The only exception where an exact solution

exists for a tandem is when the constant sized messages

are under consideration Yith no interfering traffic on any

23

\, , , intermediate nodes

fig 2.2:

node (18,19].

A path through the network

A great deal of research has gone into getting around

the problem of 'dependency'. In his doctoral thesis,

Kleinrock [10] first gave the 'Independence Assumption',

according to which at every node a 'new' length is

selected from the same exponential distribution. Under

this assumption, we can treat each individual link

independently as M/M/l queue and find associated delays

without much efforts. But now the question arises: How

far the independence assumption' is justified or ho~

closely we are approximating the behavior of the network

by invoking 'independence assumption' ?

24

In our earlier disc~s~:on we assumed that a single

traffic s~ream ar~ives at a given node and depa~ts as a

single traffic stream. 9ut in any real life net~ork, on

any link traffic ente~s f~om multiple sources and leaves

:or multiple destinations. Thus ~henever a message is

se:ec:ec for t~ansmission, though its length has some

depe~dency on the ar~ival st~eam to whic~ it belongs, the

dependence on t~e overall a~~ival proc~ss tencs to :ace

o~t.

Simulations were run by KleinrocK for- va=ious

c~nf:'gu=at.icr:s to ve:-ify the reascnability cf

'independence assumption' in approximating the ~e~~ork

delay [lOj. !t was discovered that ~henever the~e is a

mul:i?licity of tra:::c streams enteri~g anc leaving a

l:nk, :nce?endence assumption gives fai~ly good results up

depar::ng :ra::ic s:~earns

~

~en~e~i~g traffic s:reams

fig 2.3: A typical arri~al node l~ ~he nec~o~~ ~i~~ mul:i­ar~ival and ~ul~i-de?a~:ure screams.

25

to a link utilization of around 0.7, if applied to

calculate delay. (In a later chapter the validity of

'independence assumption' to calculate variance will be

dealt with).

We close our discussion about the 'independence

assumption' with the final comments that we can invoke it

with confidence for any network of 'moderate connectivity'

(where 'moderate connectivity' implies that most of the

nodes have more than one traffic streams entering and more

than one traffic st~eams leaving).

26

2.2. J.W. Wong'! ReDresentation of a Network:

Eased on the famous ECMP theorem [15,16,17] and

Kleinrock's independence assumption, J.W. Wong [14] have

suggested a model for calculating mean delay and higher

moments of delay in a network.

2.2.1. Model - some assumctions and notation:

The delay experienced by a message is assumed to be

ccmprised of queueing time and data transfer time on the

transmission links or channels. The processing times at

nodes and the propagation delays are assumed to be

negligible.

Let M be the number of channels and c·1 be the

capacity of channels i~1,2,3 •• M. Each of these M channels

can be represented as a single server queue. The queueing

discipline at each of the queues is FCFS. The chan~els

are ass~~ed to be error-free and buffer space is unlimited

i.e. infinite waiting room at each queue.

Messages are classified according to source-

destination pair. A message belongs to class (s,d) if it

originates at source '5' and has destination 'd'. Thus

there are total of R=N(N-l) classes in a net~ork of N

nodes. The message classes can be numbered f~om 1 to R.

The a~~ival process of class 'r' ~essages from outside the

27

network is assumed to be Poisson with rate 7(r). The

message length for all classes is assumed to be

exponentially distributed with mean l/~. Thus data

transmission time of all messages at node ' i ' is

exponentially distributed with mean l/~C .. Both fixed and1

random routing have been treated in the original paper

[14], but here we concentrate only on fixed routing. In

fixed routing a unique path exists for each message class

'r', denoted by a(r).

2.2.2. Summary of Results:

let Air (i=1,2 ..•.M;r=l,2 ..... R) be the mean arrival

rate of class 'r' messages to cha~nel 'i', which can be

given as:

Air = 7(r)

=0

if channel i,a(r)

otherwise ( 2 • 3 )

Similarly, we can define p.lr'

as utilization of

channel 'i' by class 'r' messages,

p.lr

= A. / ( J.LC. )lr 1

( 2 .4)

The total utilization of channel 'it can then be

written as:

RPi:S 1: Pir

r::l

28

( 2 .5)

Pi < 1 for all channels if no saturation is to take

place.

Let t (x) be the probability density function of ther

end-to-end delay of class 'r' message.

Laplace transfo~ is given by:

*Then T (5), itsr

( 2 • 6)

The above result is a direct consequence of our

ability to trea~ delay at each channel independently

because of the 'independence assumption' and the output 0:a M/M/l queue being identical to the input process.

*T (5) can easily be inverted using par~ial fractionsr

technique. Mean and variance of delay can also be

*obtained directly from T (5):r

and

( 2 • 7 )

= 1: 1

it:a(r)f ILC (l-p )1 2L~ i i J

( 2 .8)

29

In our analysis of the problem we have taken the

above model as a guideline. But this model is very

idealistic in the sense that it allows only exponentially

distributed message length and no other service

distribution is assumed to exist on a source to

destination path. As ~e know, message length is rather a

combination of fixed-length header (trailer) plus a

randomly chosen length (which may be approximated as

exponentially distributed). Also we may encounter some

servers with constant service time, as we go from one

network to the other. Thereby, we will have to deviate

from Wong's model.

30

CHAPTER 3

Verifying Wong's Analysis

It was pointed out in the last chapter that

validation of 'independence assumption' if used to compute

delay has been established beyond doubt. The model

proposed by Wong (14] also incorporates 'independence

assumption' and can be used .to calculate even higher

moments of delay. In this chapter our stress ~ill be to

verify the reasonability of Wong's model when applied to

compute the higher moments of delay. As a by-product ~e

will also be able to ve:ify the 'independence assumption'

for computing mean value of delay.

31

3.1. Validation of Wong'~ Model:

For our purpose we need to compare the values of

higher moments of end-to-end delay as obtained by two

different methods viz. by invoking 'independence

assumption' and by not invoking 'independence assumption'.

Computations using 'independence assumptions' are rather

trivial and can easily be carried out. But when we turn

to 'non-independence' case, it is not so easy to handle,

and the only feasible tool available is 'simulation'. The

simulation model used to carry out the above computations

is briefly explained below.

We know that the reason that independence assumption

is a good approximation is the entry and departure of

multiple streams of traffic at all nodes. We can

represent a specific path as a tandem of service stations

(fig. 3.1). Messages originating at A are to be carried

to B over this path. The traffic from A to B is referred

as 'internal' or 'tandem' traffic. The traffic streams

which are not a part of A-to-B traffic can be ~epresented

as a single traffic stream (because of their Poisson

nature) called external traffic. External traffic

enters a particular node and leaves it without going over

to any other node in the tandem. (refer fig. 3.1) The part

of non A-to-B traffic which does go to the nodes other

than the one where it entered in the tandem, can be

32

assumed to be a part of 'external traffic' for those

nodes. The message length distribution is exponential

with equal mean for tandem and external traffic.

Configurations (i) and (ii) given overleaf are identical.

We have simulated configuration (ii)8 We also assume that

at all nodes we have infinite buffer space and there is no

balking. The se~vice time consists of only t~ansmission

~ime on ~he links between the nodes.

A --~~ B

33

fig 3.1:

(ii)

Representacion of a path in the net~ork.

34

The simulation program has the capability to simulate

the required path vith or withou~ independence assumption,

but it was used to collect data only for 'non-independence

assumption'. For 'independence assumption' the analytical

model was used to avoid excessive CPU time.

Several runs were made for 'independence' and 'non-

independence' cases by varying external traffic and

internal t:-affic. In the discussion to folloW' ·Je assum.e

throughout that our- tandem path consists of three nodes

only. All links have a capacity of 9600 bps. Data 'Jere

collected for- mean and variance of delay at individual

nodes as TJell as for end-to-end case. The detailed

discussion for diffe~ent runs is given belo~:

(i) The exte~nal tra:fic .as maintained at 3000 bps.

Internal t~a!fic varied from 125 bps to 3500 bps. Hence,

the range of internal traffic varies f~om a point ~here

internal traffic is negligible compared to the exte~nal

traffic, to the point where internal traffic is comparable

to the external t~affic. The data collected were for mean

of end-to-end delay, variance 0: delay at individual nodes

or links and variance of end-to-end delay (~eferred as

varianceE). A comparison of these val~es for

'independence' and 'non-independence' has been depicted i~

the table 3.1 and fig. 3.2 to 3.7

35

The following points may be established from the

collected data:

(a) The 'independence assumption' for delay is quite

reasonable and computation by two different methods

match pretty closely.

(b) variance of delay at individual nodes, computed

using 'independence assumption', also seems to be a

good approximation as long as tandem traffic is not

very high in comparison to external traffic.

(c) Although variance of delay at individual nodes,

in t~o cases, is almost equal at low tandem

utilization, the variance of end-to-end delay has an

error ranging up to 50% at these low values of tandem

utilization.

36

Table 3 • 1 : Comoarison of independence and non-independence (exter-nal traffic 3000bps)

,I Ena-to-endtanaem

DelayVar'. a~

traffic individual I Yare

(bits)(ms) nodes (ms sms ) I (ms*ms)

Iy I N t y I N I y I N I

I I I , I I

125 2321 2321

59631 5921 178891 335921250 2361 241

162001 6386 186001 3715°1

3i5 2411 243 1 64521 6451 193561 36947,500 2461 253

167191 7051 201571 39049,

251 1t

625 2561 7003i 6962 21009i 385741750 256 1 2641 7305

17411 21915

1414631

875 262 1 2721 76281

7720 228841

4245611000 268 1 2i61 7972

18117 23916

1438521

I2aa

l I, , -"'.-27~1 83401 8551 250201 443371

__ i.:;:)

300 11250 2801 87341 9197 262021 50252,1375 2871 299 1

91571 9397 274711 5075511500 2941 319l 96121 10178 288361 55008,

! t

30303\1625 302' 3171 10101i 10341 5500411750 309 1 3231 10628

110401 31884

1543641

1875 318 1 3261 111981

10476 335941

5604812000 326 1 3461 11815

112125 35445

1637051

I , I

2125 335 3531

124841 12170 37452' 6l~3512250 345 369

1132121 14028 396361 725741

2375 353 3641

140051 12214 420151 6457312500 366 388, 148721 14330 446161 72803,

!

2625 377 4051 15822i 15105 47~66i 8459512750 390 4101 16866

116031 50598

1778061

2875 403 4211 180171

16842 540511

8166113000 417 4381 1929°1 17872 57870

1893901, t

3125 432 4561

207031 17895 621u91 91~5913250 448 454

1222771 18053 668311 89~6~

3375 "*65 4871

240371 21342 721::1 105314 1

3500 484 4981 260151 2280-* 780451 110460:

~Y=Independence Assumption invoked;N=Non-i~dependence.

.-"'\.

37

oo...,

Legend:C -> independenceo -> non-independence

o:::-L-----.,.-----r-----....,..-----,.------,c

fig 3.2:

D.l a.J 0.3 o.~ 0.5

tanden utilizationComparison of 'independence' and 'non-independence' fo~a path shown in fig 3.1.

38

:r..,...

Nr..

........"--c

\Jr =.- "':~.O

D

•L.ot: \I!

...." c:,>.0r:

~ C Cc CCCcO ~ cC C a

C ~ c C c eCce CCae C a

"!J-----.,.-----r----~----,__---..,ca

fig 3.3:

D.l Q.J 0.3 o.~ 0.5

t~nden utilizationDelay compu:ed under independence has been no~ali:edwith ~espect to delay found by non-i~dependence.

39

o~N

111NN

~

CO~ 0

cr• ...,

"l-E.. U'I

"U1~

e"D0c 0

0r N-'

L Legend:~

> o -> independencea -> non-independence

inaJ

a~+-----....------~---~----..,..------,

0.5

fig 3.4:

....

Nus.o

lit

C.,..----.,....----,~---~----------

40

fig

a

3.5:

0.1 a.J 0.3

tanden utilizationVariance computed unde~

normalized with respect toindependence.

0.1' 0.5

independence ~as

variance foundbeen

by non-

41

InLegend:

M C -> independence...a non-independence->

&It......

"0... UI• ~

"DE

• U1p) ...E

vlJJ

1)

LJc

IIID &11....L~

>

U'IIf'

~-l-:=-----r-----r----~-----r------,a

fig 3.6:

D.l O.J 0.3 o.~ 0.5

~ande~ utilization

Comparison of 'independence' and 'non-independence' forend-to-end variance.

42

ro.

UIIII,

A 0

"\)D..~ ...

D ~

&.

0LaeC~ ...LI.IY:

&) 00eI

I- rD :-:> c

,.'".

c cC

C a aCeQ acacrzc c a

c

c

c

IIIr4o.,..----......---.....----~----...------.

D.l a.J 0.3 0.1

tancen utilization0.5

fig 3.7: End-to-end variance computed unde~ independence hasbeen normalized with respect to delay found by non­independence.

43

(ii) Another set of 'simulation-runs' were made for

external traffic of 6000 bps at each node. Internal

traffic was varied from 125 bps to 2000 bps The results

obtained are shown in table 3.2 and fig. 3.8 to 3.13, and

are very much like the previous case.

44

Table 3.2: Comparison of indeoendence and non-independence (external traffic 6000bps)

Delay(ms)

N I!

a124218435°192877199486,

110486'117227112560811494961

I

2197401268066131638412802151

153S05,171513,180481,236023,"

621091668311721111780451

End-to-endVar.

(ms*ms)

1922761

1

219138,252048

12929681

84: 7411

1

923371010011110946

1

1224361135807115149il1700671

N

224-ii23180238352666829000320003330039641

43000459004637563662

55663741938100078956

y I1

Var. at Iindividual I

nodes (ms*ms) I,

408121452691504991566891

207031222i71240371260151

28247 1

130779133667,

369821

640921

1

730461840161976561

N 1,

5161539155115911

6311

432,451

14651

486r

767185018861931

1

y II

4321448146514841

6061638167417141

I

125250375500

625750875

1000

11251250137515001625175018752000

tandemt:affic

(bits)

·Y=Inde~endence Assumotion invoked;N=Non-lncepencence. ·

45

orr

~

• 0KI ,..

E~

Legend:C -> independenceo -> non-independence

c:.U1

0.2D.DS 0.1 O.!S

tande~ utilization

Compariso~ of 'independence' and 'non-independence' fora path shoYn in fig 3.1.

ao..L.---~---.,..----r----r----~~ 0.2S

fig 3.8:

46

N·

a:I~ 0LJ ....

'oJD....

...... CI

UI ~ w c ae rr C a c a.. · c aD 0 0e~

N~·

LC:

0-4-----~----,----...,.----_r_---___,

fig 3.9:

D.05 a.l o,~ 0.2 O.2~

tanden utilizationDelay compuced under independence has been normalizedwich respect to delay found by non-independence.

0P"1....

Legend:o -> independence0 -> non-independence

U"0-

r:c:..,.-I QI( aT

""I)faIt

illP) Lae...."

l)

0CIJ 0

&II....LJ;I,.

U1",

~+-~---""-----.,r------r-----,------,

47

D

fig 3.10:

D.DS D.l 0.15 0.2 0.25

t3nden utilizationComparison of 'independence' and 'non-independence'.

48

Nfit...

CN

I...~-0 =e Q aI

D ~ C

C1I

CE ct, UI a ce ~ c Cc I C D~ C'-'.

CC

CC Cc: ~IJ ~... Q!..a:>

I"t~

I

0

UI

o~----..------"'!'-----,..----------.a D.a~ a.l OILS 0.2t~nden utilization

fig 3.11: Variance computed undernormalized with respect toindependence.

independence has beenvariance found by ~on-

49

00m

~

0 /10

LaN

I,,IIt

0;/&0~ 0

N• N

"""1:1~E

• Q

IP'J IZI-4

E 0'V"

UJ

IL)0u 0c 0

J.J :r )1~....Legend:L

~

r/~ C -> independence)-

a -> non-independence0Q...

~/

~

o...l-...Q;=---...-----,r-----,.-----,-------,u:a'D 0.250.05 a.l O.L~ 0.2

tande~ utilization

fig 3.12: Comparison of 'independence' and 'non-independence' forend-to-end variance.

50

llIIlJ.c

UJ&JCC :faD ,..

.... Q

l-II>

cC a

C ec

c

c

C a

a

c

e

ao

D.QS a.l o.~~ 0.2

tanden utilizstiona

U'lJ- ...- -,. ..,.. -r- --,

o O.~5

fig 3.13: End-to-~nd variance computed unde~ independence hasbeen normali%ed with respect to delay found by ncn­independence.

51

(a) 'Independence Assumption' is again found to be

reasonable over most of the range of tandem

utilization for calculating mean end-to-end delay.

(b) Variance of delay at individual nodes behaves in

a manner similar to the last run.

(c) Variance of end-to-end delay, again, does not

seem to be a good approximation with independence

assumption. Even in the range where variance at

individual nodes is almost same in two cases,

varianceE may be in error by as much as 25%.

This, really, is an intriguing situation. On one

hand ~e find close match between 'independence' and 'non­

independence' variance at individual nodes, (implying that

it is reasonable to assume independence), at the other

hand we may get an error even up to 50% when the same

assumption is invoked to calculate the variance of end­

to-end delay. Or put in other words, 'independence

assumption' seems to work at individual nodes but not when

the system (tandem path) is considered as a whole. So

there must be some factor which causes some kind of

dependence among different nodes. And this factor cannot

be anything else but the length of tandem messages, which

is kept same all through the tandem. But then the

question arises how do we get around this factor ~hile

52

calculating variance at individual nodes ?

Eefo~e ~e ans~er this question, another system which

shows some~ha~ similar behavior is considered below. (:ig.

3 .14 )

.twt : ./M/l ./M/l

fig 3.14: Tandem of three ex?onen~ial queues wit~ Poissonar~ivals and ucili:acion approaching :ero.

~he exte~nal input to the sys~em is Poisson and a"

se~vice s~a~ions have exponentially dis~=ibuted service

time ~i:h ~ean equal to l/~. Also ass~~e ~~e util:zat:on

of the serve~s is close to zero. So any time a c~s~ome~

comes it will almost neve~ wait to get into service. The

service ~equir~ment of the cus~cmers stays cons:ant as

they go from one server to the next. Hence, this

situation is parallel to 'non-independence' case. The

mean delay at any service station is l/~ and its variance

is clea~ly 1/~2 Sut what about the mean and variance of

end-to-end delay ?

53

Now, if xl is random variable representing the

service requirement of individual customers, then mean of

end-to-end delay is given by, mean[3x ] = 3x and variance1 1

is given by var[3xl

J = 9var[xl

J , . rather than 3var[xl

J.

Hence, although, by using 'independence' we can correctly

find the mean and variance of delay at individual service

stations, it really does not help when we turn to end-to-

end delay.

Coming back to our original system, ~e find that

because of external traffic, the overall conditions like

queue length, mean delay, variance of delay at individual

nodes are controlled by external traffic. So when a

tandem customer arrives at a node, the above measures of

performance for it can be well approximated by

'independence assumption'. But once a tandem customer has

entered service, its service time is not at all under the

control of external traffic and it will stay 'same'

(assuming equal capacity of all links on the path) at the

subsequent nodes.

~~1

r~

><

t~

><

t~

><

fN

><

1'-- 1

..~

CQQ.

eou

eft::e....

..~.......,

54

55

Here we introduce the term 'partial independence' to mean

that only delay encountered in queues (excluding service

time) at individual nodes are independent random

variables. The original 'independence assumption' will be

referred as 'complete independence'. Consider below, a

system consisting of three nodes only. Random variables

x. are waiting times or delays as shown in the fig. 3.151

As argued earlier, if, the queue characteristics are

controlled by external traffic we can consider Xl' x3' xs,

to be independent random variables. But x2' x4' xs' are

not independent random variables (because the message

length remains same). Under these conditions, the mean of

end-to-end delay is given by mean[xl+x3+xS+3x2] i.e. the

same as with complete 'independence'.

given by

The variance is

Based on above argument varianceE was calculated for

different traffic conditions. A comparison of 'non-

independence' , 'complete independence' and 'partial

independence'

3.3 and 3.4.

is given in fig. 3.16 to 3.19, and tables

56

Table 3.3: Comcarison of comlete,partial and non-independence cases (external traffic3000bps)

N 1r

3359213715°136947139049,

4~3371

50252150755155008,

845951778061816611893901,

61435ii2:7~1

64573172803,

3857';1414631424561438521

I

550041543641560481637051

I

91~59\

894641la53l~

1':"0460:

p II

7838St831061883871943201

34i651348761356311364321

637~311

66875\70328,74146

1

537281559111582911608291

41296142479143748145111146578:4816°14987°15172°1

37284138191

139159140192,

y jI

Ena-to-endYare

(ms*ms>

25020j262021274711288361

621091668311721111780451

47~661150598

15405115787°1

21009 1

121915122884,

239161

178891186001193561201571

t

37452139636[420151446161

30303131884

1335941354451

N

59216386645170516962741177208117

178951805321j4222804

15105160311684217872

855191979397

l0178

12170140281221414330

1034110401104i612125

y I,

83~OI

873419157196121

I

70031730517628,

79721

59631620016452167191

207031222771240371260151

15822:16866

11801711929°1

Var. at Iindividual I

nodes (ms sms ) 1

I

124841132121140051148721

10101i10628

1111981118151

3531

3691

3641

388 1

3171323132613461,

2501264127212761

I288'300 1

299 1

319l

2321

241 1

243 1

253:

y 1I

1Delay I

(ms) J

IN I,

27~i

280129712941302:

3351345135513661

I

432144814651484

1

2321236124112461

251'

125250375500

2125225023752500

3125325033753500

2625275028753000

1625175018752000

1125125013iS1500

625750875

1000

* Y=Complete independence: P:Partial independence:N=Non-independence.

57

Legend:C -> complece independenceo -> non-independence~ -> partial independence

"o--41!!• rr

...,J)

e

• U1l. ,...e

vlLJ

1)

DI:r: ~....

:1-t------r-----.,...------,-----...----.....D.l D.2 0.3 o.~

tanden utilization0.5

fig 3.16: Comparison of 'independence', 'non-independence' and'partial independence' as used to find end-co-endvariance (external traffic 3000 bps).

58

Legend:C -> complete independence~ -> par:ial independence

a

cc cC

C ca cC a D CC CD C c a

c

,..It!.

...L&.i ~

1.) <:)

IJc~

L. rD r> c

v0

~l~,..lorQ .. J... .. ~

"~.A

•'"

4,A~

III ~ ..... ~ ~

"'" Q .. .. .£~ 4-C ..~U

~.... AD,...'" .. CD P-

6.

C)

1- C0

C cc.....,C CCC

'".....~ .....----......----,.....---......-----,,-----..

0.1 Q.J 0.3 O.~

~~ndc" utilizationo.s

fig 3.17: End-to-end variance compuced underindependence' and 'partial independen~e'

normalized with respect to va~iance foundindependence'.

'compLetehas beenby 'non-

59

Table 3.4: Comoarison of comlete,partial and non-independence cases (external traffic6000bps)

tandeml IDelayVar. at I End-to-end I

trai- I I individual I Var. 1fie I (ms) I nodes (ms sms ) I (ms*ms) I

(bits) Iy I

I I II ;

I 1I N I y N Y P N II 1

4321

, 1 I I125

I

432\ 20703 22447 621091 783851 81242,250 4481 451 1 22277 23180 668311 831061 84356,375 4651 465 1 24037 23835 721111 883871 928771500 4841 486

126015 26668 780451 943201 994861

t 1 !

1010161625 504 1 5161 28247 29000 84741i 1104861750 526' 5391 30779 32000 92337 108612

11172271

875 550 1 5511 33667 33300 1010011 1172781

12560811000 577 1 5911 36982 39641 110946

1

1272231

14949611

6311

153805:1125 6061 40812 43000 1224361 13871311250 6381 652

1 45269 45900 1358071 1520841 171513,1375 6741 672 1 50499 46375 1514971 1677721 180481,1500 7141 754 1

56689 63662 1700671 1863441 236023,I1625 760 1 7671 640921 59663 1922761 2085531 21974611750 811 1 8501 73046

1

74193 2191381

2354141

26806611875 870 1 8861 84016

181000 252048

1268324

1

31638412000 938 1 931

197656

1

78956 2929681

3092451

2802151I

*Y=Complete independence: P=Partial independence:N=Non-independence.

60

~#0

o~.1/

o /aI ~!I./. Legend:

~C -> ~omplete independence

. 0 -> non-independence4 -> partial independence

~/

~oJ....c::....--..----..,------..,r------,-------,UI

oom

C'U2N

J!0 e::-..-4 N• N

~

rte• Q

~CD~

E,~

a.&J

"0c: =D ~....1-~

>0Q,..

a D.a~ a.l O.lS o.~ o.=~

tandcn utilization

fig 3.18: Comparison of 'independence', 'non-independence' and'par:ial independence' as used to find end-eo-endvariance (external traffic 6000 bps).

,.. Legend:N

.... C -> complete independence~ -> partial independence

...J:.....c......

~ =L' rrP:' . ...... 0 A

-- ~ ~ AU Jj 4 ..E ~

'- •0 &Q ~

~II! CC I

.,J 0 ~ .a-D

~ 0c []Lj C C C C ~

U 0 Cc ~ c CI:J r: c..... Q

CL.[J')0

t'4~

0

61

fig

a D.O~ 0.1 0.15 0.2

t~~den utilization

3.19: End-to-end variance computed underindependence' and 'partial independence'no~malized with respect to variance foundindependence' •

'completehas beenby 'non-

62

The following points can clearly be established from

these curves:

(a) At low tandem utilization, 'partial independence'

gives results ve~y close to simulation data for 'non-

independence' • In this range of low tandem

utilization the error in varianceE may be brought down

from 50% to almost nil.

(b) At higher tandem utilization, the curves for

'partial independence' and 'non-independence' start

drifting apart, 'non-indepencence' curve being always

above the 'partial independence' curve. (This poses a

little problem because we are underestimating the

varianceE at these points. This can be overcome by

incorporating an appropriate safety-factor). The

region where 'partial independence' and 'non-

independence' curves a:-e not close by, is, in fact th.e

region where 'independence assumption' in general

starts fading out and even variance at individual

nodes is not well approximated by 'complete

independence' assumption.

At this point of our discussion, we should seek a

modification to the results obtained by Wong (14]. As

described in chapter 2, if tr(x) be the probability

density function of end-to-end delay of class 'r'

*messages, and Tr (s), be its Laplace Transform i.e.

63

= n s + ~C.(l-p.)ita(r) 1 1

( 3 • 1 )

The above result is under 'complete independence'

assumption. From this we can find the Laplace Transform

of delay encountered waiting in queue only ( i . e.

excluding service time), at all the nodes on path a(r).

*Let this Laplace transform be denoted by Tr q (5). Then

,uC. (l-p . ) s+,u.C.T *(s) = n 1 1 * 1 (3.2)

r q i 1: a ( r ) S + J.LC i ( 1-Pi) ,ue i

*Let T (s) denotes the Laplace transform for thers

*service times at all the nodes on path a(r). Then T (s)rs

is given by:

xli -s L c}

T *(s) = E[e l i e a I r ) iJ]rs

=( 3 .3)

*Now we can write Tr (5) as

* * *Tr

(5) = Tr q (5) X Tr 5 (5)

( JJ.C. ( 1-P • ) s +~C · 1= I n 1 1 * 11

lit: a ( r ) 5 + ~C i ( 1-P i ) ~c i J

{3 • 4 }

64

X t'

11Is. 1: C l + ~

i e a t r ) iJ

End-to-end delay and its variance can be calculated,

for class 'r' messages, along the same lines as:

and

1= . 1: ( ) J.'C. ( 1-p . )

1£ a r 1 1

(

= t I 1

i c a ( r ) t r~C. ( 1-p . ) 12II 1 1 J

(same as before)(3.5)

( 3 • 6 )

To further validate the above results a few more runs

were made 0 In one case we consider a tandem of 8 nodes

Yith different link capacities. The specifications of the

path and obtained ~esults are given in table 3.5. The

comparison of 'complete', 'partial' and 'non independence'

clearly shows the superiority of 'partial independence'

over 'complete independence'.

65

Table 3.5: Comparison of 'partial', 'complete' and'non' independence cases for a tandem of 8links (external traffic 3000bps, mean mes-sage SOObps) ·

I Ilnk IDelay

Var. at I delayN 1

I capacityl individual I square II (bits) I (ms ) nodes (ms*ms) I II I I (ms *rns) II I y N Y I N I II ! I'

6766:I

I 9600 82 82 6724 I 6724 II 4800 385 390 148225 I 1411651 152100 II 6400 172 181 29584 I 31016, 32761 II 9600 82 88 6724 I 6948 1 7744 II 4800 385 394 148225 I 1303311 155236 1I 6400 172 193 29584 I 333461 37249 II 9600 82 94 6724 I 7621

1

8836 II 4800 385 403 148225 I 118243 162409 II II End-to-end delay: II Y = 1745 II N = 1824 II End-to-end variance (varianceE): II y = 524015 1

I p = 861912 ,I N = 927116 I

I 1

*Y=Cornplete Independence; P=Partial Independnce;N=Non-independence.

66

The following table (table 3.6) shovs the same

compa~ison for calculation of third moment of end-to-end

delay, in a tandem of 2-nodes. Each node has an external

traffic of 3000 bps and link capacities are 9600 bps. The

curve for the same data is given in fig. 3.20

67

Table 3.6 Comcarison of 'partial' , 'complete' and 'non'independence cases as applied to compute thirdmoment of end-to-end delay(external traffic 3000bps, mean message SOObps)

Third moment oftandem I 1trafficl end-to-end 1(bits) 1 delay I

1 I1 y I p I N II 1 t I125.0 0.0110511 0.0152601 0.0150821

250.0 O.Ol1717r 0.0159751 0.0172441375.0 0.0124371 0.0167471 0.0173821500.0 0.0132171 0.0175811 0.018996,625.0 0.014: 064 1 0.018-*83: 0.0201911750.0 0.014985

10.019463

10.0214471

875.0 0.0159881

0.0205261

0.02464511000.0 0.017083

10.021685

10.0244571

r1125.0 0.0182801 0.0229481 0.02442511250.0 0.0195911 0.0243291 0.03020811375.0 0.0210311 0.0258421 0.03052711500.0 0.0226161 0.0275031 0.0314791

I !

1625.0 0.0243641

0.029331i 0.03005311750.0 0.026296

10.031348

10.0412711

1875.0 0.0284391

0.0335791

0.03346812000.0 0.030821

10.036055

10.0411641

I2125.0 0.0334771 0.0388091 0.04621112250.0 0.0364461 0.0418831 0.044499

12375.0 0.0397781 0.0453261 0.05143612500.0 0.0435281 0.0491931 0.05981°1

I

2625.0 0.047765, 0.053555, 0.07203312750.0 0.052570

10.058493

10.0749131

2875.0 0.0580421

0.0641071

0.06598113000.0 0.064300

10.070517

10.0791851

I

3125.0 0.0714921 0.0778711 0.116923,3250.0 0.0797971 0.0863511 0.094326\3375.0 0.0894401 0.0961821 0.117847,3500.0 0.1007021 0.1076471 0.110200,

*Y=Complete Independence: P=Partial Independnce;N=Non-independence.

68

o

.....

>­D =... ~IJ 0

"La

~Q

c CcEc&t ~

-00L. Q.-

No.o

complece i~dependence

non-independencepa~tial independence

o ....----~----,.----......-----.,.._---...,a 0.1 D.J 0.3 o.~

tandcn utilizationo.~

fig 3.20: A comparacive look ac the three approaches when used tocompute the chird moment of end-co-end delay.

69

3.2. Conclusion:

In this chapter we studied the validity of

'independence assumption', especially when applied to

calculate the higher moments of end-to-end delay.

'Independence assumption' was found not to provide good

approximation for higher end-to-end moments even when the

individual nodes behaved as if they were having no

dependence between incoming traffic stream and message

length. In this context the concept of 'partial

independence' was introduced and was found to give better

results than the 'complete independence'. Wong's [14]

results were accordingly modified.

Finally, in the light of above results, in all our

analytical treatment to follow, we will use 'partial

independence' rather than 'complete independence'.

70

CHAPTER 4

Modeling the Network

In this chapter we will develop a suitable model to

represent the network interconnection as given in chapter

1. We start with rather an oversimplified model to get

the first feel of what is going on. Then ye go on to a

more realistic model for our representation of the network

interconnection base on Wong's [14] model and the analysis

carried out in the last chapter. This model is used to

find delay and its variance under different conditions of

internetwork traffic, intranetwork traffic, pathlength

through the networks, gateway processing times and message

lengths. Later on, the delay and variance in case of

different alternatives (as described in chapter 1) a~e

calculated based on this model. All this gives us a

suitable tool to measure the supe~iority of one

alternative over the other.

71

4.1. Some Assumptions:

For recapitulation fig. 1.5 is reproduced overleaf.

As described in chapter 1, the two alternatives which

gateway G1

has, to deal with the packets originating in

network A and destined for network A', are the following:

i) The first one (hereafter referred as pad-and­

pass) is to embed the incoming packets (from A) in

header and trailer as required by network B, and

transmit.

ii) The second approach (hereafter referred as

strip-and-pad) is to first strip the header and

trailer from a packet which were attached to it

for its journey in network A, and then attach new

header and trailer as required by network B.

Net"orkA

NetworkB

fig 4.1: Interconnection of networks A, A' <identicalarchitecture} and network 8 (different architecture).

.......f\..)

73

Assumptions:

For all the models to be described the~e is common

set assumptions which will be applicable to all of them.

This set is given below:

i) Partial independence assumption is invoked

invariably (with the possible exception of

simulations where we do not have to face the

problem of analytical tractability).

ii) In calculating end-to-end delay error

probability is not taken into account. Although

any real network will have some error probability,

it still is a good approximation to assume error

free channels.

iii) Arrival rate associated with any traffic

stream is Poisson and is stationary.

iv) All queues have FCFS discipline.

v) Any message that enters the network has a

single destination, and will make to it sooner or

later without defection. This necessitates that

all nodes be having infinite waiting room. In a

real network buffer space is sufficiently large to

minimize loss of messages.

vi) A message is fully received before its

74

transmission can begin.

vii) Propagation delay and nodal processing time

are taken to be zero unless other~ise specified.

vii) The maximum packet length supported by

netvo~k B is a~ least as much as supported by

networks A and At. This assumption implies that

we will never have to face the problem

fragmentation and reassembly of packets9

In what follows, whenever we need any assumptions

other than those described above, they ~ill be given unde~

the heading 'Additional Assumptions'.

4.2. A Tentative Model:

As a first approximation (a

interconnection of fig. 4.1

following network of queues.

75

rather crude one) the

can be represented by the

- 6<

~I,I

_J -, II I .-- C

N IN

0N::..

n~....

~J

wU

N 1 a.JCC'.j

I e0

I u\II

1_, IQ.I...J,- "..-

I I .:I.~

r 0r , :I...

'N ~

=-1 J=- ~

c

=:r] J ~J:,

I 4t.J

I,....

I 0, I e0

_I J ..,-, a:Ie.... I N •...I 0

:L .. s..~

n I nQ.c.~

to:)I ..,

a'tI w

I :&.

N,IN .

~I I. '4'~ w ~"I r.c. Q. .....

-< nt

76

77

Here path 1 corresponds to pad-and-pass and path 2

corresponds to strip-and-pad.

In this model every delay element (gateways, networks

A/A', network B etc.) has been modeled as a single server

queue with exponential service time distribution. In fig.

4.2 either gateway has been shown as two queues. In fact

there will be only one queue at a gateway. The two queues

represent the fact that there are two alternate paths for

a message. Once a particular path (shown as dotted lines)

is chosen, all messages will travel through the same path.

In addition, complete independence is assumed among

different queues. Hence all our queues can be treated as

independent M/M/l queues.

Some notations:

A :

DP. :1

Mean number of messages originated per second

in network A and destined for network A'. This

process has Poisson distribution.

Service rate at gateTJay 'i' (i=1,2) if path j

(j=1,2) is selected.

End-to-end delay if path 'i' (i=1,2) is chosen.

Processing time at gateway 'it

path j (j-l,2) is taken.

(i=1,2) when

u . :1

Service rate in network S, if path i (i=1,2) is

chosen. It may be noted that to optimize delay

78

by choosing any of the two paths, delay or

message length through networks A and At is not

affected at all. So in subsequent discussion

we will concentrate only on delay encountered

at gateways and network B.

The delay on path 1 and path 2 can now be represented

as:

1 1 1(4: .1)DP l = A

+ A + - A.&0'11 - ~l - .u21

DP2 = 1 + 1 + 1 ( 4 .2)"'12 - A ""2 - A .u22 - A

A:so ~e know, processing time for pad-and-pass is

less than for strip-and-pad at both the gateways. i.e.

Also let,

( 4 .3)

The message length for strip-and-pad is less than for

pad-and-pass. Hence average service time at network 8

~ill be more for pad-and-pass. Let the excess mean

service time at network B for pad-and-pass be given by

'y'. Then ~e have,

79

1 = 1 + PI~12 ~ll

1 1= + P2

~22 ~21

1 = 1 - Y)J.2 ~l

( 4,. 5)

( 4 • 6 )

( 4 .7)

Special case:

Assume that gateways G1

and G2

have the same traffic

characteristics. Also it is reasonable to assume that

1.u11

= 1~21

( 4 .8)

1 1= (4.9)

}L12 . ,u22

Then from eqn. we get the delay on path 2 is given by:

DP2 = 2 + 1~12 - A ~2 - A

2 1= 1 + 1

1 - A 1 - A (4.10)-- + p - yJ.L 11 }Ll

Now if p=O, meaning, gateway delay is same

irrespective of the path chosen, then

2 1< DP 1DP2 = A + 1}Lll - (4.11)1 - A

,ul- y

The implication of the above result is rather

trivial. It simply means that if delay at the gateways is

80

not a function of the path chosen, then we must do strip­

and-pad, so that shorter messages travel through net~ork B

and encounter less delay.

Without going anv. further in this oversimplified

model, we turn to a little mo~e realistic model for our

network representation.

81

4.3. Network-modeling based ~ Wong'~ Analysis:

In this section we model our network based on Wong's

model given in chapter 2 and the analysis carried out in

the last chapter.

Additional Assumptions:

i) For the time being we assume that the only

traffic to be carried through the gateways, is the

one going between networks A and A'. i.e.

gateways are fully devoted to the traffic under

consideration.

ii) The processing times at gateways are

exponentially distributed random variables and do

not depend on the length of the message to be

processed. This assumption will be removed in the

successive refinements.

iii) The traffic internal to the network S, is

assumed to have exponential distribution with the

same mean as tandem traffic. In practice, the

mean of internal traffic of network E does not

depend on the mean of tandem traffic. Thus while

traversing network B, We may encounter two

different message classes with different mean

message-lengths. In the

Wong [14], there is no such

original model due to

distinction and all

82

messages have the same mean. In the present case

we assume that there is only one message class and

will remove this assumption later on.

iv) The message lengths on two paths differ by a

constant length, implying if on st~ip-and-pad we

assume exponential distribution, then on pad-and­

pass it must be a combination of constant plus an

exponential component with the same mean as on

strip-and-pad. Eut we assume that message length

on pad-and-pass too has exponential dist~ibution

but with higher mean.

Under above assump~ions the network can be modeled as

the following tandem of queues:

G1

:=J o(

fig 4.]:

Network 8

0 0 \( ,,;~

Representation of a path in the network.

G2

~Q-.-

oow

84

External traffic to all nodes has been represented as a

single traffic-stream. We have not taken into account, in

the above model, the external traffic to the gateways.

End-to-end delay calculations are carried out using

?artial independence and mean and variance are given by

(4.12)1Tr = . 1:( ) j.l.C. ( l-p . )

ll:a r 1 1

and

=(I 1

11 I

}r,uc.1 21l 1 J J

(4.13)

Having devised the above model, the delay

=alculations were done by varying the different t:affic

pa~ameters like gateway processing times and their

difference on t~o paths, mean message length of tandem

traffic and its difference on tvo paths, arrival ~ate of

tandem messages and the number of hops traversed through

network 8. ~he exte~nal t~affic on all the links was

assumed to be 3000 bps. All links have a capacity of 9600

bps.

The folloying notation has been used to represent the

various parameters:

pathl+:

path2-:

delay:

variance:

A :tandem

85

Indicates the curve corresponding to path 1.

Plus (+) sign indicates the fact that longer

messages will traverse the network if path 1

is chosen.

Same as for path 1. Minus (-) sign denotes

the presence of shorter messages on path 2.

Mean end-to-end delay on path 1 or path 2.

variance of end-to-end delay on path 1 or

path 2.

Internetwork t~affic (messages/sec).

differencex: Difference in message lengths on the two

paths.

m len.:1

n-hop:

Mean message length on path i (i=1,2).

Gateway processing time at gateway on path i

( i =1,2) .

Number of hops traversed through network B.

The results obtained as a comparison of path 1 and

path 2 are given in fig. 4.4. through fig. 4.15. These

curves clearly depict how the variation in different

86

parameters affect the mean and variance of end-to-end

delay. One thing clearly stands out by a mere look at

these curves: path 2 is to be preferred over most of the

range of controlling parameters. Path 2 has less delay

and variance for most part of 'difference in bits' for two

paths. Only close to a difference of 32 bits in message

lengths, path 1 seems to perform bette~. The same way

path 2 appears superior over most of the range of number

of hops to be traversed through the netvork a, the mean

message length to be taken care of and the input rate of

tandem traffic. In fact as we go from left to right on

these curves (i.e. as we increase the value of tnese

parameters) path 1 gets worse and ~orse.

The only cross-over points whe~e pa~hl gets better

are ~hen gateway processing time becomes excessive for

path 2 as compared to path 1.

Finally, based on the above model we can say that

path2 is to be preferred 'most of the time' (the phrase/

'most of the time' not to be confused with 'always') and

we postpone the furthe~ interpretation of t~e curves

obtained, till we go a step ahead to refine our model in

succeecing chapters.

87

In'-,....

Legend:~ -:> A = 3

tandem0 -> A = 1

tande..rn

;path2-

-z... pathl+

// --.r- pathl+

/ path2-

fCD ...--....--....- .......Q.

o

........~

IS

o-0 ~1

a-.0

't16128 12-.

di+f~r'ence

&It

Q ....------.----..,.------,-----,~J

Variation of delay with difference in message lengthson two pachs (P1=O.1, p =O.ll,n hoo=3,m len = 5002 _. - 2bics).

88

e­N.o

.."o......._---..----.....---..,...-----,o ,:2 12B J2'1 320 'tl~

diffe~ence {bits)

fig 4.5: Variation of variance ~ith difference in messagelengths on two paths (P1=O.1, p =O.ll,n hop=3,m ten =

2 - - 2500 bits).

L'1,...,...

...

Legend:L1 -> ,.\

tandemo -> ,.\

tandem

= 5

= 1

89

L.,....>-

".....LJ /'1J ~,

r~

/...

...

U1

o1 3 't

no. of hops5 6

fig 4.6: Variation of delay with number of hops traversedthrough network B (Pl=O.l, P2=O.11,difference Inlengths=160 bits, m_1en2=SOO bies).

o

90

Legend:d -:> Atandem :2 5

a -> Atandem = 1

./

~ pachl+

,..o~ ....---.....----.....----,....----..---~

1 J It

no. of hopes

fig 4.7: Variation of variance with number of hops traversedthrough net~ork B (Pl=O.l, P2=O.11,dif:erence Inlengths=160 bi~s, m_len2=SOO bics)o

91

Legend:d -> Atandem = 3

o -> Atandem = 1

0 •.1.6

path2-

1~»>:~

~ pathl+

~

4~' O.1~ 0.15

98t~,-,sf pr'ocess.

Variation of delay wich gateway processing cime forpath 2. Gateway processing time is held constant at0.1 sec (difference in message lengths =160 bits,m_1en2=SOO bies, n_hop=l).

LI')e-

Q ....-----~=-""...-=~---....------------..

fig 4.8:

94

L.agend:o -> differenceX = 224 bits

N.

L)

ce rII •.... 0

i,g:>

C -> di::erenceX = 160 b:'cs

o-L----.......---~----_r_---..,..---__,10 SS loa J.~~ 1 ~fJ

mea8sge 1ensih on psth2 {bi~a)

fig 4.11: Variacion of variance wiCh mean messag~ lengch on pach2 (Atanaem=l,Pl=a.l, P2=O.11).

95

'l)

c-:0

Legend:o -> Pl = 0.13 sec \c:~ -> Pl = 0.10 sec

tQ

path2-

,,) JPco.Q

~ /~\

c pathl+>-G /0-o~

~c

0.48a.~~ d.09 O.D~ a.DSproce~8in9 time di~f.

~co ......----...----...-----....,...---~----..----...a.~ D.D'

fig 4.12: Varia~ion of delay ~ich gaceway processing cimediffe~ence on t~o paths (Atandem=l, n_hop=l,differencein message lengchs=160 bits, m_ Len2=500 bits).

96

.....o

c­....o

n...00o&:II...t. ...G~

,.~

Legend:o -> Pl = 0.13 sec

d -> Pl 2 0.10 sec

pach2-

"pachl+

f

D.ar~.D' 0.0' 0.03 O.4aproce~sin9 time dj~~.

fig 4.13: Variation of variance with gateway processing cimedifference on two paths (Acandem=l, n_hop~l.diffe~encein message lengths=160 bits, m len~=500 bits).- ..

97

... Legend:~ -:> n_hop =3

LIt. o -:> n_hop =1...

55

pach2-

~ ,......-e~athl+

pathl+~~/

3 't

tand~m tra.f1=icJ1

fig 4.14: Variation of delayP2=O.11,difference Inm_1en2=SOO bits).

with tandemmessage

traffic (Pl=O.l,lengths=160 bits,

98

6

pathl+

5

pach2-

pachl+

3 '+

tandem ts-a-f.f:ic

Legend:d -> n_hop =3

1

N~.

...N

Q

1)

fJ ,..C m15 •.... 0LI,.

fig 4.15: Variacion of varianceP2=O.11,difference inm_1en2=500 bics)o

with tandem t~affic (Pl=O.l,message lengchs=160 bits,

99

CHAPTER 5

Towards a Refined Model

In this chapter we try to make a few refinements in

our model by removing some of the unrealistic assumptions

made earlier. Tvo of the assumptions made in the last

chapter can now be removed. The first one that

processing time at the gateways is an exponentially

dist~ibuted random variable independent of the length of

the messages to be handled, is not so reasonable. In

practice, only the latter part of the above statement i.e.

processing time at the gateways is independent of the

length of the messages to be handled, is true. But so far

as the distribution is concerned, it tends to be constant

rather than exponential. It depends only on the path

chosen.

The second assumption to be removed is that of single

class messages in the network B. Previously, we assumed

that the mean message length of the traffic internal to

the net~ork B is the same as that of tandem traffic. In

real life traffic internal to the network B and tandem

traffic may have different means.

5.1. Verification of Partial Indeoendence for ./2/1

100

tv~e~

gatewavs and ~ classes of messages:

We noted previously that the backbone of Wong's model

is Kleinrock's independence assumption coupled with the

fact that output of an MIMic system is identical to input

process. So it does not allow any cons~ant-time service

stations or multiple classes of customers. We, therefore,

first verify the effect of constant service time sta~ions

(in our case gateways) and multiple classes of custome~s

under pa:tial independence assumption.

To investigate the behavior of the network -i~h

constant service time servers we consider the falloYing

tandem of queues. The two extreme servers are our

gateways with constant service time. The remaining th~ee

se~vers represent the intermediate nodes .ith

exponentially distributed service time (message lengths

being assumed exponentially distributed). The external

traffic to the gateways is assumed to be 3 messages/sec.

The service time at the gateways is 0.1 sec. Exte~nal

t=affic to the other nodes is 3000 bps. Mean message

length for external as well as tandem traffic is 500 bits.

Capacity of all the links is 9600 bps.

101

~

.........Q

..........~

~

t123~....(If

0.0

QJe

en

~...

~

~

~

~

~

~

(J0.

0-

>

....

I.CIJ

--.....

02

%.........

..,

.e

nG3

*-J~

C0U

~"-

~.u....:teQ,)

~

e

n~

~

......

<;

.'"00.....

c.....

nt'--

102

The analytical model used differs a little from

Wong's model because of the presence of constant service

time stations. Mean delay at these stations was

calculated using M/G/l results. End-to-end variance was

calculated under partial independence assumptions. The

following M/G/l recurrence for~ula due to Takacs [21] was

used to find mean and variance of delay at individual

queues (not including the service time).

kw = 1 - p

where ;K is the k-th moment of delay in the queue.

"0w = 1. is the i-th moment of service time

distribution. From the above equations we get,

2 Ab 3var of queueing delay = (w) + 3(1-p)

( 5 • 3 )

End-to-end delay and its variance for the non-

independence case were found using simulations and were

compared with analytical results computed unde~ partial

independence assumption. The tandem traffic rate yas

va~ied from 0.25 messages/sec to 6.0 messages/sec. The

results obtained are given in table 5.1 and figs. 5.2

through 5.9. End-to-end delay calculations using

analytica~ methods are pretty close to the simulations

103

results with non independence. As gateway utilization is

increased these two curves start drifting apart. But even

in the worst case error in delay calculation using

analytical methods does not exceed -5%.

At low gateway utilization, there is very little or

no queueing at the first gateway. So any time a tandem

customer comes it will join the service with high

probability without waiting. Thus under low utilization

conditions the first gateway can be approximated as an

M/G/~ system whose output process we know 'is identical to

the input process[22,23]. Hence we get almost overlapping

curves in figs. 5.2 and 5.4 at low gateway utilization.

As we increase the gateway utilization the output process

of first gateway starts drifting towards one of a constant

interdeparture interval. But the decomposition of the

output in two streams, which are, tandem traffic and

external traffic, and the fact that this forms only a

small portion of the total traffic to the succeeding nodes

helps to keep the delay calculations within tolerable

limits of error. In fig. 5.6 through 5.9 the curves

showing the delay using the above mentioned two approaches

has been plotted against utilization of the links due to

tandem traffic alone. These curves clearly depict that

our constant service time gateways do not affect the

analytical results in any appreciable manner.

104

~able 5 . 1 : Compa~:son of pa~tial-independence and non-independence ~ith constant se rv : c e timegateyays(exter-nal traf:ic 3000bps)

I i End-to-endtandem DelayI gateway I Var.trafficl utiliza-l (ms) (ms*:ns)

I I(bits) I tion I ? N

fp I N I

125 0.325 1 ~ao 4791 38534: 390321250 0.350 I 490 4921 39916

1408351

375 0.375 I 501 5021 414311

423151500 0.400 I 513 5201 43099

1467491, I ,

625 0.425 525 527 1 44943' 46172i750 0.450 538 ~~~I 469931 50910,875 0.475 553 ~b 492841 54365,

1000 0.500 568 574 518591 541621

1125 0.525 Se4! SS9 ~~~~~l 6354811250 0.550 603 1 614 6791911375 0.575 622 1 632 61920 7062811500 0.600 644 1 657 66362 776511

IaS90S:1625 0.625 6681 687 71579

1750 0.650 6951 706 77787 90317,1875 0.675 7251 747 85285 966-i 4 t2000 0.700 7601 782 94499 1133~4!

2125 0.725 799: 836 106057 13677012250 0.750 845 1 878 120913 13962012375 0.775 SOOI 927 1~aS78 16280912500 0.800 966

1

975 167563, 18763il

1040:,

2625 0.825 10491 2062991 ~33"''''·it. :0'1

2750 0.850 11561 116°1 2652171 27ia6~12875 0.875 13031 1321

13620091 3724921

3000 0.900 15171

1429, 5391761 502102,

"?:Par~ al IndependenceN=Non- ndepencence.

105

o... rot0'"....•~ Q

• ce ...E~

~II~ 0

LJ =:I-C

:'egend:a -> non-independence~ -> ?ar~~al ~nQe?endence

D.'i a.! 0.5 0.7

sate~sy utilization

fig 5.2: Comoa~isvn

i.ndependence'gateways.

~f 'non-independence' and ?ar: al~n a canaem navlng constant ser~~ce : me

~

~

DD lSI ~~ ..a 4Il ~

OS..t... . .A ~ A .. A

~ 0 ~ .. ~ ~

-6 •

..IJ

•L.0C g...., =.

106

a.~ O.S O.~

s.te~ay utilizatian0.8

fig 5.3: Delay computed under partialnormalized wich ~espect coindependence.

independence hasdelay found by

bee:'\non-

107

0."o.~D.'1

Legend:o -> non-independencej -> ?arcial independence

D.~ O.S 0.7

S8te~ey utilization

. f" d d' fComparlson 0 non-in epen e~ce and partialindependence' on a tandem having constant service cimegaceways.

~ ..ll~!:::=-~-----r-----,-------,r-----....,..----.,D.~

0=:r

"'='..., 0cr-• rtt

"~E

• Q

J\ '=',..,e

'-"

1)

IJc: '='D --1

N

'-J:>

<:)r-4--t

fig 5.4:

r,.,...

,..N...

108

.--..g:IJ •! 0L-ae....,

..

0.8

fig 5.5:

a.5 0.5 0.1

9~t:~sy utili%s~:an

Variance ~omputea unde~ parcial independence has be~n

normaLi%ed ~ich ~especc ~o vari.nce found by ncn­indepenaence.

.. 109

o~...

_ .... Q

• C1(....

E....,

i..eg~~(J:

o -> ~on-:nde?endence

~ -> parcial i~de?endence

oJ- ...- -,.. ..,... ,.- -,~

a D.1 a,J 0.3 o.~

tanden utilization

fig 5.6: Com~arlsc~

:':lae?enc:e~c~

saceways.

~t '~on-inde?endence' ana ?art~a.~~ a tandem havin~ conscanc ~e~vice :i~e

110

~n·

..·,..

"1)

1 =~~--'-0I:El­eC 1:1~ ~·>.0r

0.1 a.J 0.3 o.~

t3nden utilization

&II J..----....----,....-----,r-------.,r-------,o O.S

fig 5.7: Delay compuced ~nder ~artial

nor~ali%ed wi~~ ~espect ~o

independenc:eo

independ~~ce ~as

delay touna :;y :'10n-

111

oP­ut

J:'=r

"Q.... 0c:r• ",

"f'E

• 0Lege~ci:1.\ '='M

E 0 -:> non- nc:ependence,"'

.l -> part al inde?e~cie!":ce0oc: '='~~,.~...

L,;:'-

0,...--1

~ .....----.....----~---.....,~-----,----....,a 0.1 a.J 0.3 o.~

t3nden utilization0.'5

fig 5.8: Comcarisonindep~ndence'gat evays .

ot 'non-lndependence' and parclal~n a ~andem having constanc service cime

....

...·"

.."g

D...c::I-'" ~

D · .~.. .Ite 0 ~

1- '" ~

0 .a. ..c...., ~

UI .a ... ...G

U · ~... .. • ••1) <:)

e 4E' It.,.L. ..J! ~> r:

c::a

112

'"oa D.l a.J 0.3 o.~

tanden utilizationo.s

fig 5.9: Va~iance computed unde~ par:ial independence ~as bee~

normalized wich respect to variance found by non­independence.

113

Now we try to find out how the presence of multi­

class customers in the network affect our results. For

this purpose we used the same model as described before.

We don't have any M/M/l queues in our tandem any more. So

results under 'partial independence' were found using

M/G/l analysis. Two different runs were made with mean

tandem message length 1000 bits and 2000 bits. Mean

message length of the exte~nal traffic was maintained at

500 bits. Tandem traffic was varied up to 3000 bps. The

results obtained with s:mulations and analytical model a~e

compared in fig. 5.10 to 5.17. For the case of 1000 bits

mean tandem message length, delay obtained for simulation

and analytical models match pretty closelya But for

variance the situation deteriorates as we increase the

tandem traffic. For the case of 2000 bits mean message

length, the identical observations can be made though the

drift between 'partial independence' and 'non-

independence' is somewhat apparent, especially for

'variance' calculations. But for low tandem utilization,

in both cases we get tolerable errors in mean delay and

variance using analytical models.

In brief, we can say that presence of multi-class

customers as well as constant service time stations is not

going to hamper our analytical results as long as the

tandem traffic is not very high. In practice ~e can

114

expect tandem traffic to be about 300 bps (with link

capacity 9600 bps). We have seen that in this range we

indeed get good results analytically. So, Wong's results

can now be extended to include our tandem consisting of

constant service time gateways and multiple classes of

messages, though they must be applied with utmost caution.

Let us denote by 'x' the random variable representing the

message length of the 'overall arrival process' (i.e.

tandem as well as external traffic) and by 'x ' the randomt

variable representing the message length of the tandem

t:-affic alone.

encountered in

*The Laplace transform, T (5), for delayrq

queues only can be ~ritten as (following

M/G/l results and the assumption that waiting times in the

individual queues are completely independent):

1 - Pi= n

i e a t r ) 1 _ r1 - a*(s)l,p.C.I-----1 1 L sx J

( 5 .4)

Here i is the mean message length for the overall arrival

process to link i (i.e. the mean taken over tandem and

*external traffic). B (s) is the Laplace transform for

service time distribution of the overall a~rival process.

The multiplicands on the right hand side are just the

expressions for the Laplace transform of queueing delay on

individual links represented as M/G/l systems.

115

*Let T (5) denotes the Laplace transform for thers

service times at all the nodes on path a(r). Then T *(s)rs

is given by:

( x t 1I -s 1: -}* . () c.T (5) = E[e l i e a r lJ]

rs

*Now we can write T (s) asr

( 5 • 5 )

T *(s) = T *(5) X T *(s) (5.6)r rq ~s

End-to-end delay and its variance can be calculated,

for class 'r' messages by inverting the above transform or

by using Takacs results and partial independence:

;2 x tP.) + . L( )c:-

1 i e a r 1

( 5 • 7 )

and

a 2 =r

( 5 .8)

116

:'e~e!'1C::

o 0 -> ~on-~ndepende~ce

: ~ -> ?ar:iai inde?enaence

0.. Q

Q ...~•'"• C

D rB~

>-D-- 0D =~

o,..

'/P

/:i-+-----.....---------".-----.....-----..a D.l a.J 0.3 o.~

tanden utilization0.5

fig 5.l0: A com~ara:~~e .JO~ It non-:nde?endence anc ~a~::aL

inde~encen~2 :~ ~ao:~13~~ ~elay ~n a :Ance~ ~l:~

c~ns:a~c ~e~v~:e ::~e za~e~ays a~Q 2 ~~3ssas j:message. l)ne :~aS3 ~as 500 bi:s mean ~eng:~ a~a ~~e

oc~e~ a mean or :000 oi~s.

117

NN...

...·~....-.LJ

1:1aJ ED .A..... ~ ~· A-. C) • .AD .A

~.a-

e ~ ~

LDr: La....,. =·Q~.,~

~

-C .,.r-:Q

0.1 a.J 0.3 o.~

tanden utilizationa

"'-L-----,.-----.-----,-----..,------.,o 0.5

fig 5.11: Same as fig 5.:0. Delay compuced using ?a~:ialindependence has been normalized wi:h ~espect to delayunde~ non-inde?endence.

118

oC).. Legend:

o -> non-independe~ce

~ -> partial independence

"o.... 0.~

eoc '='D 2...1­JJ>0

<:>II!...

a

o~ ....----.....---~----...,..----.,...-----.,

0.05~.l a.~ 0.3 o.~

tanden utilizationfig 5.12: A comparative :ook ~c non-lnde?en~ence an~ partial

inde~encence :0 calculata va~iance on a tandem withconstant 5e~vice cime gat~~ays and 2 ciasses ~f

message. One :lass ~as 500 bits mean length and ~he

othe~ a mean oi :000 bics.

119

...·"""

..."IJI)........ CDrr

D · ..E 0

i,

e ~

c:"" aD

L)I: ~· ~

1j 0t: .A.,

·rtL.

.6IS ~ ..:> r: •Q

0.1 a.J 0.3 o.~

tanden utilizationa

III

0 ....----...-----....----.-,------.,,------.0.5

fig 5.13: Same as f:g 5.:2. Variance :ompuced using parcial:ndependence has been normaliz~ci ~ich ~espect co~ariance unae~ ~on-independence.

120

I

Legend:o -> non-independenced -> partial independence

'".NQ...

~ ....----~----,.----...-----,.---- .....a D.1 a.J 0.3 o.~ o.~

t3nden utilizationfig 5.14: A com~arative look at non-independence and par:ial

independence to calculate delay on a tandem wicheonstant service time gateways and 2 classes ofmessage. One class has 500 bits mean leng:h and theother a mean of 2000 bits.

121

NN....

......~

-00 ....I) ell.... ~

~· .A~ 0 ... ..D .a .A

EL0c LD

",' =·0>-D.....L)

" ~

r:Q

N'II·C

0.5D.l D.J 0.3 o.~

ta~de~ utilization

5.15: Same as fig 5.14. Delay computed using partialindependence has been normalized wich respect co delayunder non-independence.

a~J..._---..,~---~----r------r----,C>

fig

122

eo~

er

.,.Q 0~ Q• a:I

'"DII

• c'-' ...Ii

'-'

Ll0 Legend:e

0D uz o -> non-independenceL. 4 -> parcial independenceQ)-

caIII

a 0.1 a.J 0.3 o.~

tandcn utilizationfig 5.16: A comparacive look at non-independence and partial

independence to caleulace variance on a tandem withconstant service time gateways and 2 classes ofmessage. One class has 500 bies mean length and t~e

other a mean of 2000 bics.

123

....

....~

...."Lj

D......... a:!~

" .! 0

L.~0

c" ~

La

0as.

D C> ~~

..cIJ~

L-a ~

" r:Q

NLa.o

&II

~~----...------....----....,~-----r----....,a D.1 a.2 0 •3 0 • 't 0 .5

t3nden utilizationfig 5.17: Same as fig 5.16. Variance computed using partial

independence has been normalized with respect tovariance unde~ non-independence.

124

5.2. Comparison of the two Paths:

We devoted the last section to establish that even

with constant service time gateways and multi-class

customers we can apply analytical techniques in the range

of interest. In this sec~ion we investigate our refined

model for network-interconnection representation. In this

model both gateways spend constant time to p~ocess any

message. The processing time depends only on the path

chosen. End-to-end delay and variance were calculated by

varying the different traffic parameters like gateway

processing times and their difference on two paths, mean

message length of tandem traffic and its difference on two

paths, arrival rate of tandem messages and number 0: hops

traversed through network 8. The external traffic on all

the links was assumed to be 3000 bps. All links have a

capacity of 9600 bps. So the network characteristics are

essentially same as for the model in the last chapter. In

figs. 5.18 through 5.29, is shown the comparison of the

two paths for different traffic conditions. Most of the

curves have been drawn for mean message length on path 2

as sao bits and tandem message arrival rate as 1

message/sec, thus giving tandem traffic of 500 bps. Also

most of the time we consider only one hop through the

network B. In the following we take up discussion of all

these curves one by one.

Leg~:1d:

.l ->.~ tandem = 3

a -:> Atandem = 1

125

=Q

.,£J~ pathl+

~~-"30---~--~_~ pa th2-

5~

lit

o32 1211 22'1

di1=fcrence'tlPS

fig 5.18: Var:ac~on oi 1eiay -len diffe~ence in ~essage ~eng:hs

on :wo ?acns (p~=O.i, ?~=O.11,n_hop=3.~_Le~~= 500bi:3).

126

rN

o

~....-.z- pathl+

Leg~nd:

.l ->~~anaem = 3

a -> \:anaem = !.

pathl+

4-z- pach2-

NN·

r...I) ·QL) -:c:D

/---z......4

L /lit

/II ...> ·c

~--~--'i!i---o£l~~ pach2-·

'"oo~----.....----,.----.,...-----,

32 1.21 :2'1 :0 '115

dl+fe~ence (bits)

fig 5.19: Variacion ~i ~ariance ~i~h difference in messa~~

Len~t~s on :~o pachs (Pt=O.:. ~2=O.11,n_hop=3.m_~en:=500 bi:sJ.

127

In figs. 5.18 and 5.19,we have plotted delay and its

variance against the difference in message lengths on two

paths. The two sets of curves are for tandem traffic 3

messages/sec and 1 message/sec. The message length on

path 2 is 500 bits and gateway processing times for path 1

and path 2 are as shown in the fig. As expected if we

increase the difference in lengths of path 1 and path 2,

the delay and variance on path 1 start getting worse.

After a while path 1 moves beyond delay and variance for

path 2, and keeps getting worse as we further increase the

difference in lengths. The cross-over point moves towards

right as we increase the tandem traffic. This implies

that at higher tandem traffic, to offset the excess

processing times at gateways on path 2, the difference in

lengths on two paths has to be higher.

Next, we look at how the number of hops to be

traversed through network B affect our choice of a

particular path. As the number of hops is increased delay

and variance for both paths increase but with unequal

rates because with every additional hop, path 1 because of

longer message lengths, suffers more additional delay than

path 2. Thus we see that although initially delay for

path 2 , for tandem traffic equal to 5 messages/sec, is

higher than on path 1, v i th increase in number of hops,

1ftr:·....

&IIo

I) III-eN·

...

\It,.·

fig 5.20:

128

:"eg~nd:

.1 -> A:andem = S

0 -:>~:andem = 1

pachl+

3 't

no. o~ hops

Va~iation of delay wich number of hops erave~sed

:hrou~h network 8 (p~=O.l, ~2=O.11,diffe~ence ~n

leng~hs=:60 bies, m_1en2=30o bits).

129

m&It·'='

-Z- pathl+

,.,~·<:)

l) / Legend:1)

'"~

E: m ~ -:>~:a:ldem = 5

IS ·... e\. IJ -> \:ancem = ,g:>

'"~Q

55

pach2-

1

o

m....

,.,oo

3 't

no. of hOp3fig 5.21: Variation at ~arlance ~lth ~umber of hops :~ave~sed

cnrough ~e:~orK 3 (o~=O.l. D~=O.ll.diffe~ence In. ~

len~~ns=:bO OlC3. ~_len~=SOO bics).

130

delay increases faster on path 1 and we see a cross-over

at n hop = 3 • Hence if we concentrate on delay and

variance for Atandem

=-5 messages/sec, path 1 appears

super'ior for n_hop<=3. But this high traffic is very rare

to come across in p~actice. For A d =1, path 2 is atan em

better candidate both on the basis of delay and variance.

Another point to be noted is that cross-over point moves

towards the right as tandem traffic increases.

Fig. 5.22 and 5.23 show variation of delay and

variance ~i~h gate~ay processing times for the parameters

shown i~ the respective figures. Quite naturally the

delay anc variance on path 2 inc~ease as gate~ay

processing time is increased for path 2 (for path 1 it

being held constant). Hence after a certain value of

gateway processing time, path 1 becomes p~eferable. Eut

as a~gued earlier sao bps is a more likely situation in

practice and for this path 2 performs better up to around

0.125 sec of gateway processing time yhen processing time

for path 1 is 0.1 sec.

Figs. 5.24 and 5.25 show how the delay and variance

vary with mean message length. For the given set of

parameters path 2 is better throughout.

The next set of curves is for difference in gate~ay

processing times. Initially path 2 seems to perform

131

~agend:

In .l -.:>.4.:anciem = 3c0-....

0 -:> A.. d = 1~an em

0 •.113

pathl+

f

d.~' O.l~ O.l~

sate",., p,...occse.D•.1:2

fig 5.12: Va~iaclon or deLay ~i~~ gaceway ?~OCesslng cime Eorpac~ 2. Gac~~ay ?~ocessing cime is heid conscant ac0.1 sec (dif:ere~ce In message le~gths =160 bies,m_len2=500 bi~s. n_hop=l).

132

Leg~nd:

.l -> A = J~andem

o -> \ = 1:andem

o0 ...c N~ .

..... 0C-o>

r.....Q

pachl+:QJ.-~'-':::::::'_-----_---.l~-----------"'"

~ path2-

4.1' O.l~ 0.15 o.!e9ate".r pr-ocea8.

fig 5.23: Variacion of variance ~ith ~ateway ?rOCesslng ~ime :orpath 2. Caceway p~ocessing ~lme is neld c~nstanc at0.1 sec (diffe~ence in messag~ Lengchs =160 bi~s,m Len =500 bics, ~ hoc=l).

- 2 - ·

133

N

~egenci:

o -> di£ferencel = ,., I,--- :>l:SU'1r-....

&n.

U'!....

C -> di::erenceX = :50 bi:s

.} pathl+

1ft

~ ....----....-----.,.----~----.,...---~l~ 55 loa l~~ L~O

meassge length on pa~h2 {bi~s)

fig 5.24: ~ariacion ot Je~ay ~ltn mean ~essag~ Lengch on path 2, l. _. p -.1' ~ -0 . 1 )'"':andem-·· L-v ..... ~2-·J. •

134

~egend:

0 .. > cii:ferencel .:: 214 oics....C -> di:fe:-~:lc:el = 160 bi~s

~.o

cue UIII •... 0

L.

";

O+-----...----......----r-----.....,.---~

pathl+

10 SS loa

mea.age length onJ..-'S 1.'0

peth2 (bi cs)

:i& 5.25: Variation of yariance ~i:h ~ean messag~ ~eng~h on ?ach2 (A~andem=l,?l=O.l, P2=a.ll).

135

r..2ge~ci:

o -> ~, = O.~J 5ec- .~ -> ?l = 0.10 sec

1.1)e-.Q

pathl+

!«).Q

....

0.1e~-L------.----....,----r----.....-----r-----,'='0.0.1 D.al'a.o~ 4.0' O.D~ o.as

proccasing time diff.

fig 5.26: Variacion ot delay ~ith gateway processing ~lme

diffe~ence on cwo pachs (A:andem=l, n_hop=l.diffe~~ncein message Lengchs=:60 bies, ~_len2=SOO bits).

It)-e.Q

ro0.0oeG....to __

a~

"0

N<>.Q

:ig 5.27:

136

pathl+

~

pachl+

path2-

:'e~enc::

a -> ~, = 0.13 sec. -

a.o~ 4.D' 0.0' o.asproce•• lns time dLFf.

Va~ia:lon ai ?arlanCe ~i~~ ga:e~ay ?r~cess:n~ c~me

ciiference ~n t~c ~achs (A~ande~=l, n_ho?:l.dii:e~~nc~~n messa~e Lengths=L60 bies, m ien~=500 bi:sJ.- '"

137

Leg~nd:

~ -> n_hop =3

o -:> n_hop =1

L>U

IJ',

o1 3 't

tandem tra-f.J=ic

path2-

6

fig 5•28 : Va r i a c ion 'J t" ,:e Lay

P2=O.ll,diif~~e~ce ~n

m_Len2=500 btts).

~ich ~andem :raffic (0,=0.1,:ne5 5 age l eng t hs= ~ 6a '. b i cs ,

138

N'&1

:1~e~enc::

.l -> ~ nop =3

0 -> n hop ="!-I:' 1-,. path2-

N

I~·0

I)

0N

paChl+c mII ·.. 0

LaPJ>

r-4

~~

N...·

N

0J.. ~---...,..----~----~---_..,o1 3 't

~andc:m ira.f~ic

....,fig 5.29: Variation of varian~e

o sO.ll,diffe~ence in~2~en =500 bics).

- 2

wi:h tandem :ra:fic (? =0.1,message lengc~s=i5a lbi~s,

139

better. But as we increase the difference path 2 starts

getting closer to path 1, ultimately crosses over and

performs worse than path 1.

Finally, mean delay and variance have been plotted

against tandem traffic (messages/sec). With increasing

traffic the delay and variance on path 2 increase faster

than on path 1. The cross-over point moves towards right

as number of hops through network B are increased. Path 2

outperforms path 1 in the region ~hich is most likely to

occur in p~actice.

In brief we can say that path 2 is more likely to be

a better choice than path 1, although ultimate choice of

any path depends on the overall traffic environment

(gateway processing time, message lengths to be taken care

of on tvo paths, tandem traffic, external traffic etc.)

specific to any ne~work.

It will be a nice idea to compare our previous model

with the one discussed in this chapter. For both models

~e concluded that path 2 is a better choice than path 1.

But model described in the last chapter is biased more

towards path 2 and pLedicts it to be better even while it

is just the reverse if we use our present model.

140

Finally it may be commented that there is still some

room for further ,efinements and we make an effort to do

so in the next chapter.

141

CHAPTER 6

Final Model

In this chapter one last attempt has been made to

refine our model for network interconnection. We know

that message length distribution in a network is not

truly exponential. In fact any message will consist of a

fixed header and a randomly chosen length. Moreover, the

difference in lengths over two paths will be only in the

constant part of the message. In what follows we make an

attempt to incorporate these features in our model. We

start with the validation of independence assumption for

the new model and then find the trade-offs between the two

paths.

142

6.1. Verification of Partial rndeoendence fo~ D+M tyoe of

message distribution:

We assume that randomly chosen part of a message is

exponentially distributed. To investigate the behavior of

the network ~ith constant plus exponential distribution ~e

consider the tandem of queues shown in fig. 6.1. The two

extreme servers are constant service time gateways. The

remaining three se~vers represent the intermediate links.

The external traffic to the gateways is assumed to be 3

messages/sec. The service time at the gateways is 0.1

sec. External traffic to the nodes is 3000 bps. The

capacity of all the links is 9600 bps.

143

~

........Q

.........

(J....~

""'"'=

en

w

QJ

u

:sQJ

~

~

Q3

c-

c

nw

...QJ

.........

J.J

Q

>C

+

~

:z:.........

.&:

.~"-

~....3

eQI

~C

n~

i-

.......<;

-.&J

eo....\.w

nt'-

144

The analytical model is again based on M/G/l results

due to Tackcs (21] and partial independence assumption.

Tackcs recurrence :ormula is reproduced below:

kw

A k k! bi+l ~:I 1 - P .Eli!(k-i)!·(l+l)·~11

1=

( 6.1)

kwhere w is the k-th moment of delay in the queue.

-0·11 = 1. b·1 is the i-th moment of service time

districutlon. From the above equations we get,

w = ( 6 • 2 )

var of queueing delay = (w)2( 6 • 3 )

End-to-end delay and its variance for the non-

independence case were found using simulations and were

compared with analytical results computed under pa~tial

independence assumption. The tandem traffic ~ate ~as

varied from 0.25 messages/sec to 6.0 messages/sec. TMO

different cases for message length distribution Ye~e

considered. In one case exponentially distributed part of

messages had a mean of 500 bits and cons~ant part

consisted of 96 bits. In second case exponentially

distributed part of messages had a mean of 500 bits and

constant part consisted of 224 bits, thus fOLming almost

50% of the exponential part of the messages. A comparison

of non-independence and pa~tial independence assumption

145

has been given in figs. 6.2 through 6.9.

The results obtained are really g~atifying and we get

a close match bet~een analytical and simulation results,

especially at low tandem utilization (defined as

utilization of the links due to tandem traffic alone). As

we increase the tandem utilization the error using

analytical results start increasing. But in any real

network we can expect a tandem traffic of about 500 bps

(tandem u~ilization =.05) and the analytical results are

quite close to simulation results in that low range of

tandem traffic. So the Laplace transforms presented in

the last chapter and equations 5.7 and 5.8 can directly be

applied to our new model.

and

x2+p . )

1

X t1: -

. () C .1 t: a r 1

( 6 . 4 )

a 2r

(r a , ~ 12

P. j 1I 1 X I + 1. X \

= i£;(rd Ixc. '2(1 - Pi)J1 xc? 3tl-Pi)1l l 1 1 J

2( 11

+ i L c-} var(x t )li£a(r) iJ

( 6 • 5 )

146

o......

Leg~nd:

o -> nOn-lnae?enaenCed -> par~iai inJepenaence

~Q...o ...•....,

~';"':~---""----"""------:"-----r------,0.5a.l 0.2 0.3 O.'t

t8nde~ ucili:stion

fig 6.2: Compar1.son of 'non-i~depe:"1de~cev and 'parcialindependence' used to compuca end-eo-end delay.Message len~th consists of a neade~ of 96 bi:s and anex?one~tially distributed ~arc ha~:n~ mean sao bics.~e gace~ays have constanc se~vice c~me.

147

..,~

""-0 JItC ~ A0 qz.... rr ....... 0G&C-OC ~

'" ~.>.0(7.....•-0 ~

r-:Q

hasnon-

0.50.'1'a a.l 0.2 0.3

tande~ ueilizationSame as fig 6.2. Delay for partial-independencebeen normalized with respect to delay forindependence.

~J-----.,------.,~-----r-----r------,o

fig 6.3:

148

&It

'II.

.,.Q ~~ at• ~

'"6• &II

• .0

('lit

1ft•""G

o '"ea::--a:-t)-

III

G'I..

Legend:a -> non-independenced -> partial independence

a a.l 0.2 O~ O.'t

tandCB u~ili%ation

fig 6.4: Compa:-ison of 'non-independence' dr:d ~ar:iaL

indepencence' ~sed ~o compu~e ~~~-:~-~~c ':a~iance.

Messag~ ~e~~:h ~onsists of a heade~ ~t 96 O~~S i~d a~

expcnenc i a l l y distributed ;:a:-t ~a'/ln~ me an SOO 01.':5.

The ga t eva y s have ccns eanc sere/lee ci me,

...·'"

..-0

C)

"... rzs.,.G ·2- 0C-OC~

GI

0lSI·0 e

CG...c,G ~

)- r-:Q

D D.I 0.2 0.3 o.~

tende~ u~ilization

149

fig 6.3: Same as fig 6.4.been normalizedindependence.

Variance for partial-independenc~ haswich respect to va~iance for non-

150

ooC't

'"....c

... '"0"--'•

Legend:a -> non-independence~ -> partial inde?endence

et ..... ...... .-.,. ..,.- p-. -,

WID.l G.2 0.3 OG~

tandeR ucili%stiono.s

fig 6.6: Comparison of 'non-independence' and 'partialindependence' used :0 compute end-~o-end deLay.Message length consists of a heade~ Ot 224 bits ana anex?onencially dist~ibuted part havi~g mean sao bics.!he gace~ays have conscant se~vice cime.

151

...."-0

(,

o ~.... .".

~Q

GI:~oC -=...,. cz;

>-cG

0.50.1 Q.~ 0.3 O.~

tande~ utilizationa

Ln

o~-----r-----""'-----""'---"""-----

fig 6.7: Same as fig 6.6.been normalizedinde?endence.

Delay~ich

for partial-independence hasrespect co delay for non-

152

Legend:a -> non-inde~e~de~ce

~ -> part:ai :nce?e~dence

&It~ ..s::::::;;:.._..... ,... ,...- ..., ...,

o.~

1ft.•III

,.0 ~.-1 ..• :r

"6S III• .0 fSI~

Sv

0Q 1ftC 0

a &ItC4...

"a>'"~..

6.3: Comuarison of '~on-i:lc:e?ende!"lce' and 'parcialindep~ndence' used co c~m?uce ~~d-~~-en~ variance.Message length conS1SCS of a header vi :2~ bics and anexponentially distr:ouC!d ?ar: havin~ mean 500 bics.Thega CetJa y 5 havee 0 ns : ar, C s ~ :"' "/lee : irne •

153

...·,..... ...-0 ..00........ ID~e ·! 0 .. .641

..C-o ~ ..C ... ...~

.A

CZI

CaD .. ~ .....·0 0 ..

CG

"r1

C.

" ~

)- ~Q

o

o.sa.l 0.: 0.3 o.~

tende~ utilizationa

III~ .. .....----.,...----,.----~r-------'

fig 6.9: Same as fig 6.8.been normalizedindependence.

Variance for parcial-independence has~ith respect to variance for non-

154

6.2. Comoarison of path 1 and path ~:

Having investigated the effect on our analytical

model, of the messages with fixed length header we move on

to find out the t~ade-offs between the two paths (pad­

and-pass and strip-and-pad). The network configuration is

essentially the same as considered previously. External

traffic to the gateways has a mean interarrival time of

0.3 sec. Service time for external traffic is constant

(0.1 sec). External traffic on the links is assumed to be

3000 bps with mean message length of 500 bits, of which 96

bits form a fixed length header and the rest is

exponent~ally distributed with mean of 404 bits. All

li~ks have a capacity of 9600 bps. Tandem traffic passing

through network B has to have a header of 96 bits to

comply ~ith the protocols in this network. So for pa~h 2

(strip-and-pad) all tandem messages will have a fixed

length heade~ of 96 bits and the rest of the leng~h will

be chosen randomly with exponential distribution. On path

1 (pad-and-pass) fixed part of the tandem message consists

of the header for net~ork 3 plus the header with which it

started its journey in network A. To find the trade-of:s

between the two paths their behavior was investigated for

varying traffic environments as in the last chapte~. The

curves obtained have been sho~n in fig. 6.10 to 6.21.

155

...

en Legend:N... d -:> A = 3tandem

0 -:> A = 1tandem--....

>--rL) ~ pathl+-C~.

0

..e-::.-- paCh2-

~pathl+

mQ

III ~.......- ......--~--~~path2­IQ.

U'I

oSJ US J21

di1=,cerence't16

fig 6.10: Variation of delay ~ith difference:n message lengchson t~o paths (Pl=O.l, P2=O.11.n_hop=3,m_1en2= saobits).

156

rN

o

r..\) <;)1)

I:D...L ,.~IS ...

> ~

Legena:.1 -> Atandem = 3

o -> Ata.ndem =a 1

-z. path2-

pathl+

i ~~

12! 22'1

di1=fer-enee

:

~re:) ~i----""'-'-----r-----r------,~ .,1'5

=:IQ

e

N...

fig 6.11: VariationLeng~hs onsao bies).

of ~ariance ~i:h diffe~ence in messa~e

t~o paths (Pl=O.l, P2=Ooll.n_ho~=3,m_ten2=

157

...~.

"!.z.. ~...... ...,... ,..... -,. ---,o

U'I.Legend:~

>- d -:> ,.\ = 5IS tandemo pathl+

0 -:>"tandem :: 1."

653 't

no. of hope

Variation of delay wich number of hops traversedthrough network a (Pl=O.l, P2=O.11,difference Inlengths=160 bits, m_len2=SOO bits).

1

fig 6.12:

158

Legend:~ -> \:andem = 5o -> " = 1:a~de:n

1ftUI.cr

'"~.o

oo '"emI ·

o..e CLJ:!:>

'"....Q

~ pat~l+

1ft...,

'"oo 4.t:=:=:--...,..----r------r----.,.------,

3 't

no. ot hope

fig 6.13: Variation oi variance wieh number ~f hops :~ave~sed

through nec~orK 3 (Pl=O.l, P2=O.11,dif:erence inlengths=160 bies, m_Len2='OO bits).

159

In Legend:e-

~ -> A = 3.... candem0 -:> A = 1

candem

~pat:h2-

...

pathl+

1

O.J.S

~path2-

a•.1~

It)e-c ...~....... ..... ...

~10-+------.-----..-----..,-.-----,.----,..-.---..,a.~ D.~~d.~' 0.1' 0.1'

98te~8T process.

fig 6.14: Variation of delay wich gateway processing time forpath 2. Gateway p~ocessing cime is held constant ac0.1 sec (difference in messa~e ~engths =160 bies,m len =500 bics, n hop=l).

- 2 -

160

/path2--.../

//

//

pachl+

I:

~~andem :: 1o ->

Lagend:~ -> ~t:andem = 3

".

-­.Q

......

:t ..J.' 0 •1'1 0 .15 e•.1.es-to'W.y p....oc:c:s ••

fig 6.15: Variacion of va~iance ~i~h ga~e~ay processin~ cime forpath 2. Cateway processing time is held ccns~anc ac0.1 sec (difference In message tengc~s =too bits,m len~=500 bies, n hop=l).- - -

162.

N

Legend:0 -> differenceX = 224 bits

U1r- C -> differenceX. = 160 bit:s...

} pathl+LIt....

'"o~--- ......----...,...-----r----..,-----,10 S~ lOa ~~5 ~,o 235

~ea8ege length on peth2 {bi~8) -ta!fig 6.16: Variation of delay with mean message length on path 2

(Atandem=l,Pl=O.lt P2=O.11).

162

Legend:o -> dif:erenceX = 224 bics

C -> differencaX = 160 bics

} pathl+

O~----..----..----......----....----~10 55 loa l~S l~O 235

me •• _ge length on pe~h2 (bits) -ta!fig 6.17: Variacion of variance with mean message length on pach

2 (Atandem=l,PlzO.l, P2=O.11).

163

\

-.path2-

1

""C!0

Legend:0 -:> PI = 0.13 secfJ:!

Q

;,l -:> PI = 0.10 sec

II')e-.Q

C-011'1~

C> ~ -.,.~ --.:,. - ......-----------.:J....

/,/

pathl+

f

0.46a.o~ d.O' O.D' O.DS

procc88ing time diff.

~..L..._--.....---...,...---..,....---...,-----r---.,Q D.enQ.Q1

fig 6.18: Variation of delay ~ich ~ac~~ay p~oce~St~g :i~e

difference on ~~o pachs (A~anciem=l, ~_~o~=i.aiffe~encein message lengchs=160 bies, ~_L~n2=500 ~l~S).

164

Le~end:

a -> 0 = 0.13 sec• T-

.1 -:> P. = o. :0 secJ.

~oOcoCG

D.OTa.~~ t.a, O.D' o.~s

procea.ing time dLf~.

O-t,----....---~--- ......~--- ..,.1----......----...a~

fig 6.19: Variacion of ~arlance wi:h ~acayay ~~ocessing timedifference ~n t~o pa:hs (A:andem=l, ~_~oc~i.differ~nce

in message len~:~s=:~a bic5. ~_~en1=SOO OlCS).

N

:.Itr:.

Legend;~ -.> n_hop =3

o -> n_hop =1

3 't ~

tandem trs-ffic

165

path2-

fig 6.20: Variation of delayP2=O.11,difference inm_1en2=500 bits).

~ir:h tandemmessage

c~aific (Pl=O.l,lengchs=160 bits,

166

J 3 ~ S

'tandem tr-9.f~ic

Va~iaciQn of variance wich candem cra:fic (p,=O.l,~2=O.11,difference in message lengchs=160 ·bi:s,m_1en2=SOO bits).

1

Noo

N...

NLDQ

Legend:d -> n_hop =3

NIII

C' a -> n_hop =1

~,I} ~ath2-

~. /'10

IJIJ NI: m., ..... Q

L.g:"

N

~Q

fig 6.21:

167

Fig. 6.10 and 6.11 show the variation of delay and

variance as the difference in message lengths on the two

paths is increased. The two sets of curves are for tandem

traffic 3 message/sec and 1 message/sec. The other

traffic parameters (gateway processing time etc.) are

sho~n in the figures. For A = 1 message/sec, path 2tandem

seems better both with regards to delay and variance

though the difference in variance for the two paths is not

1pathmessage/sec,3=appreciable. For Atandem

outperforms path 2 with respect to delay for difference in

length less than 64 bits. with regards to variance path 2

is better only when the difference in length is more than

160 bits.

Similar interpretations can be sought for other

curves. On the whole path 2 again appears to be superior

than path 1 for the traffic conditions often met in

p~actice. But here we would like to address another

important issue. Starting with a very modest model for

our network interconnection, we have brought it into the

current form. But do we really gain anything by adding

more and more complications? Does it really affect our

choice of a particular path? To answer these questions it

will be sufficient to compare the current model with the

one described in the last chapter.

168

In the previous model all message lengths were

assumed to be exponentially distributed. The difference

in lengths for the two parts was considered as the

difference in the mean of this exponential distribution.

But in our present model, this difference lies only in the

constant part of the message length which really is the

case in practice. So the coefficient of variation for

path 1 message lengths is less in the present model than

in the previous one. This facto~ has an important impact

on the relat:ve performance of path 1, especially on the

variance, due to decreased randomness. Path 1 is no

longer that muc~ of a villain now though we still get

better perfo~ance on path 2 most of the time. The slope

of the variation for delay and variance with cifferent

parameters has decreased because of fixed length header.

Eu~ as expected this dec~ease in slope is more for ~ath 1

~han for path 2. If we compare the variation of variance

with difference in lengths on the two pa~ts, the previous

model shows clear superiority of path 2 over path 1, but

with the present model path 1 appears better for A =3tandem

and almost as good as path 2 for A d =1.tan em

The similar observations can be made in fig. 6.12 and

6.13 where ~e have plotted delay and variance against

number of hops t~aversed through network B. The margin

between path 1 and path 2 is less than in the previous

169

model. The same type of behavior can be seen in the rest

of the curves. In all these we see that the cross-over

point has moved in favor of path 1. This movement is

rather substantial for variance curves (compare curves

6.17,5.25; 6.19,5.27; 6.21,5.29). All these factors

indicate that the present model gives better insight into

the network behavior than any of the previous model.

This concludes our discussion of the present model.

It may again be stressed that no fixed boundary lines can

be drawn between the relative performance of path 1 and

path 2. The network parameters must be studied carefully

before any decision can be taken.

170

CHAPTER 7

Conclusion

As noted p~eviously, this thesis is devoted to a

somewhat specific configuration in interconnection of the

communication networks. But some of the tools developed

and the conclusions ~eached are quite general in nature

and may find wide spread application in the area of

computer communicationo In what follows ~e present an

overview of the work done and the conclusions reached.

7.10 Overview:

In chapter 1, the issues related to interconnection

of networks (homogeneous as well as heterogeneous) were

discussed and a detailed discussion of the thesis problem

was presented. Chapter 2 ~as devoted to an introduction

of Kleinrock's 'independence assumption' (10] and J.W.

Wong's analysis of computer networks (14]. It was

followed by verification of the same in chapter 3 • We

:ound in chapter 3 that Kleinrock's 'independence

assumption' was quite a good approximation when invoked to

find mean delay. The results computed analytically match

very closely v i th those obtained using simUlations. It

vas discovered that if the same assumption is applied to

find the variance of delay, it works fine if ye a~e

171

concerned with variance at the individual nodes, but in

computing variance of end-to-end delay we may make an

error as much as 50%. An explanation to this behavior ~as

sought and the concept of 'partial independence' was

introduced. By 'partial independence' we mean that the

delay encountered in queues alone (excluding service time)

is calculated the same way as under 'independence

assumption', but the total service time (transmission time

on the links) for the messages is calculated with the same

message length traversing across the path. A comparison

of 'complete independence' and 'partial independence' with

simulation results was convincing enough to adopt 'partial

independence' in our subsequent analytical treatment.

,In chapter 4 a comparison of the delay on two paths

described in chapter 1 (viz. pad-and-pass and strip-and-

pas) was carried over. Strip-and-pad was found to be

better than pad-and-pass for most of the traffic

conditions. But for the modeling purposes we had made

some assumptions which are not so reasonable. In chapter

5 we introduced constant service time gateways in place of

the gateways with exponential distribution. Moreover we

allowed the situation where internetwork traffic and

intranetwork traffic for net~ork B could have different

mean message lengths. Thus two classes of messages had to

be treated in the new model. For these new conditions

172

independence assumption was analyzed and gave good results

as long as 'tandem' traffic is low as compared to the

'external' traffic at the individual nodes. Again a

comparative study of pad-and-pass and strip-and-pad vas

made based on the refined model.

Chapter 6 ~as the final step towards netwo~k

modeling. Here we considered messages to be consisting of

a fixed length header plus a randomly chosen length having

exponential distribution. Independence assumption ~as

again put to verification fo~ this type of message length

distribution, and was once again found to be sat:sfacto~y

as long as the 'tandem' traffic is low compared to

'external' traffic. For this last model a comparative

study of the two paths showed that pad-and-pass is not

that bad as shown by the models in chapter 4 and 5. In

fact a careful analysis of t~affic environment is required

to be done before one can settle on either of the t*o

paths.

173

7.2. Future Work:

In all the analytical treatment done so far a number

of assumptions were made - some of these for the. sake of

mathematical tractability and others so that we don't

expand our problem too much (still taking care that our

model does not misrepresent the real system). Some of the

aspects where further investigation can be carried out are

suggested below.

We concentrated all the time on the case ~here

network B supports at least as large packets as supported

by networks A and At. This saves us the trouble of

dealing with packet fragmentation and reassembly.

Investigation of a situation where we may have to break

messages into smaller packets before feeding in network S,

will be highly desirable.

It was assumed that all channels are noiseless. This

is not true. The probability of error is also a function

of message length. In pad-and-pass longer messages travel

across the network and as such will be more prone to

errors than the relatively short messages on strip-and­

pad. The probability of error has a direct effect on the

delay (caused by retransmissions). A comparison which

takes this factor into account should be looked into.

174

Finally, the buffer occupancies on the two paths is

likely to be different. It is affected by the tyO

factors:

i)On pad-and-pass service time ~ill be more than

on strip-and-pad, thus resulting in higher link

utilization on path 1. This in turn implies that

on the average there will be more messages

waiting in queues at the store-and-for~a~d nodes

on path 1.

ii)Not only the queue lengths ~ill be more O~

path 1, but also the buffer space occupied by

individual messages ~ill be larger because of

increased message-length on path 1.

Hence we will have relatively less buffe~ occupancy

on path 2 (strip-and-pad). It will be interes~ing to

study the network behavior taking into account t~e fini:e

buffer size and the possible buffer overflow. ~he

probability for blocking for path 1 and path 2 may then be

compared. A nice account of net~orks with blocking may be

found in [24,25].

Lastly, different parameters of performance (i.e mean

delay, variance, probability of blocking, error

probability etc.) can be assigned certain weights and a

ney measure of performance which takes all the above

parameters in:o account can be defined. Then this measure

175

can be ~sed to find which of the two paths performs

better.

176

List Of References

[1] Carl A. Sunshine, "Interconnection Of ComputerNetworks," Computer Netvorks 1 (1977) pp. 175-195.

[2] Vinton G. Cerf, Peter T. Kirstein, "Issues InPacket-Network Interconnection,ft Proc. IEEE, Vol. 66,No. 11, Nov. 1978, pp. 1386-1408.

[3] Andrew S.Tanenbaum, "Computer Netwo~ks,ft PrenticeHall Inc. Englewood Cliffs, New Jersey 07632.

(4] John F. Shock, "Packet Fragmentation In Inter-networkProtocols," Computer Netwoks 3 (1979) pp. 3-8.

[5] Jonathan S. Postel, "Internetwork P~otocol

Approaches,ft IEEE Trans. Commun., Vol. COM-28, No.4,April 1980, pp. 604-611.

(6] A.M. Rybczynski, M.S. Unsoy, "X.7S Internet~orking

Flow Control Considerations," ICC 1981, pp. 50.1.1­50.1.6

(7] Ross Callon, "Inte~network Protocol," Proc.Vol. 71, No. 12, Dec. 1983, pp. 1388-1393.

IEEE,

(8] A. Rybczynski, "X.25 Interface And End-to-end VirtualCircuit Service Characteristics," IEEE Trans. Commun.Vol. COM-28, No.4, April 1980, pp. 500-510.

(9] D.R. Boggs, J.F. Shock, E.A. Taft, R.M. Metcalfe,·PUP: An Internetwork Architecture," IEEE Trans.Commun., Vol. COM-28, No.4, April 1980, pp. 612-624.

[10] L. Kleinrock, "Communicatlon-Het~~,asticMessageFlow And Delay," New Yo~k, McGraw Hill;-!364.

~

[11] R.R.P. Jackson, ~Queuing Systems With Phase TypeService," Opere Res. Quart., Vol. 5, 1954, pp. 109­120.

(12] E. Reich ,"Waiting Times When Queues Are In Tandem,"Ann. Math. Stat., 1957, pp. 768-773.

[13] P.J. Burke ,"Output Processes And TandemProceedings of the Symposium onCommunication Networks and Teletraffic,N.Y., April 4-6., 1972, pp. 419-428.

177

Queues,"Computer

Ney York,

[14] J.W. Wong, "Distribution ofMessage Switched Networks,"(1978) pp. 44-49.

End-to-end DelayComputer Networks,

In2

[15] F. Baskett, K.M. Chandy, R.R. Muntz, F.G. Palacios,"Open, Closed And Mixed Networks Of Queues WithDifferent Classes Of Customers," J.A.C.M. 1975, pp.248-260.

[16] F. Baskett, F.G. Palacios, "Processor Sharing In ACentral Server Queuing Model Of Multiprogramming WithApplications," Proc. 6th Ann. Princeton Conf. onInformation Science And Systems, Princeton, Ne~

Jersey, 1972.

[17] K.M. Chandy, "The Analysis And Solutions For GeneralQueuing Networks," Proc. 6th Ann. Princeton Conf. onInformation Science And Systems, Princeton, NewJersey, 1972.

[18] B. Avi-Itzhak, "A Sequence Of Service Stations WithArbitrary Input And Regular Service Times,"Management Science, Vol. 11, Ne. 5, March 1965, pp.565-571.

[19] Izhak Rubin, "Message Path Delays In Packet SwitchingCommunication networks," IEEE Trans. Commun., Vol.COM-23, no. 2, Feb. 1975, pp. 186-192.

[20] M.C. Pennotti, M. Schwartz, "Congestion Control inStore-and-forward Tandem Links," IEEE Trans. Commun.,COM-23, No. 12, Dec. 1975, 1434-1443.

[21] L. Takacs, "A Single Server Queue With PoissonInput." Opere Res., 10 (1962), pp. 388-394.

(22] N.H. Mirasel,poisson," opere

"Theres.,

Output of A M/G/~

1963, pp. 282-284.Queue Is

(23] G. F. Ne~ell, "The M/G/- Queue," SIAM J. Appl. Math.,Vol. 14, No.1, Jan 1966, pp. 86-88.

178

(24] A.G. Konheim, M. Reiser, ftFinite Capacity QueuingSystems With Applications In Computer Modeling," SIAMJ. Computing, Vol. 7, no. 2, May 1978, pp. 210-229.

[25] F.G. Foster, E.G. Perros, ·On The Blocking P~ocess InQueue Networks," European Journal of OperationalResearch 5 (1980), pp. 276-283.

92

Legend:~ -> ACandem = 3

a -> Atanciem :: 1

/par:hl+

a ..1~ -J •.1, 0.1" 0 .1S O.le D•.1.1'

sete1'eT pr-oceaa.

Variation of variance with gateway ?rocessing ci~e forpath 2. Cateway processing time is held constant at0.1 sec (diffe~ence in message lengchs =160 bits,m_1en2=SOO bits, n_hop=l).

....... ...--~ ........_-------.....----------~~--- ..CJ

~

~-=+----..",..----....----~----.----....---....

D•.u

fig 4.9:

93

224 bits

160 bitsC -> differenceX =

Legend:o -> di£ferenceX =

In.....

&II~

I

o

anC;; .....---....,..---~,.....---..,...----....----..l~ S5 10D ~~5 ~'O

mea8~9c length on p~th2 {bits)

fig 4.10: Variation of delay with mean message length on path 2(Atandem=l,Pl=O.l, P2=O.11).