DSTQN

Design and Simulation of Traffic in Queueing Network 1

Design and Simulation of Traffic in Queueing

Network

Md. Shahid Hossain,

B.Sc. (Computer Science)

A thesis submitted in partial fulfillment of the requirements of the degree

of

Master of Science In

Computer Science

From

IBAIS University Dhaka, Bangladesh

April, 2003


Certificate This is to certify that the thesis titled Design And Simulation of Traffic in Queueing Network in the

bona fide record at thesis is done by Mr. Md. Shahid Hossain bears registration number IBAIS-624 as

a partial fulfillment of requirements of Master of Science in Computer Science degree from

International Business Administration & Information System (IBAIS) University, Dhaka, Bangladesh.

The thesis has been carried out under my guidance and is a record of the bona fide work carried out

successfully. No part of this thesis consists of materials copied or plagiarized from any published or

unpublished works.

Signature of Supervisor

(Md. Imdadul Islam) Assistant Professor Dept. Of Electronics and Computer Science Jahangirnagar University, Savar, Dhaka Bangladesh


Declaration

I hereby do solemnly declare that the work presented in this thesis has been carried out by me and has

not been previously submitted to any other University for an academic degree.

This paper is being submitted as a partial fulfillment of the requirement for the Degree of Master of

Science in Computer Science of session April, 2003 of The International Business Administration &

Information System University is the result of my own research and thesis work. That no part of this

research and thesis consists of materials copied or plagiarized from published or unpublished work of

the writers and that all materials borrowed or reproduced from other published or unpublished source

have either been put under quotation or duly acknowledged with full reference in appropriate places(s).

Signature of Author

(Md. Shahid Hossain)

Master of Science in Computer Science

Registration No. IBAIS-624

International Business Administration & Information System (IBAIS) University

Dhaka, Bangladesh


Acknowledgements

First of all I like to thank Mr. Md. Imdadul Islam, Assistant Professor, Dept. Of Electronics and

Computer Science, Jahangirnagar University, Savar, Dhaka, Bangladesh. His keen interest in the field of

Telecommunication has encouraged me to carry out my thesis work on Design and Simulation of

Traffic in Queueing Network. His constant supervision, constructive criticism, valuable advice,

scholarly guidance and suggestion at all stages of my work have made it possible to complete this

research.

I would like to thank Prof. Dr. M. Maniruzzaman Mia, Vice Chancellor IBAIS University and Dr. Zakaria Lincoln, President, IBAIS University, for their moral encouragement and help.

I would also like to acknowledge with sincere thanks the all-out co-operation and services rendered by

the faculty members and staff of IBAIS University. Special thanks to Mohammad Nurul Huda, Head,

Department of CSE, Mohammad Abu Hassan, Senior Lecturer, Department of CSE and Mr. Noman

Kabir, Assistant System Administrator, IBAIS University for their encouragement.

I would also like to give special thanks to my parents and other family members who gave me

unconditional support and encouragement throughout my work.

Finally, I would like to express my deep appreciation and sincere feelings to all my friends and

colleagues who helped and supported me during this time with their love and understanding.


Abstract Performance of a network is measured based on call/packet loss and average delay of packet/cells. In circuit switching network there is no provision of storage. Hence any arrival when all channels are busy/engaged will be lost. But in packet switching network any arrival of packet/cell will enter in a queue when all channels are busy. Therefore frame/packet delay is a common phenomenon of packet switching network becomes a parameter of network performance. In this paper a 2-D traffic model with limited queueing capacity is proposed to evaluate network performance. Any arrival after all storage of queueing fills up yield loss of packet. Here probability of blocking/loss, entering queue, utilization of nominal channel etc was evaluated theoretically. A simulation program is developed to compare with theoretical result derived from 2-D Markovian chain.


Objective of the thesis For last few years there is a tremendous development done in the field of telecommunication. In early days the theory of loss system was implemented successfully to deal with the overloaded traffic. But during recent years the theory of queueing system is introduced to handle the overload traffic. In general a queueing system allows a greater utilization of servers. We designed a queueing model and tried to prove our proposed model superior to other queueing model by mathematical derivation and simulation program throughout our research work. Also we explained the basic teletraffic terminologies and theories like Poisson, Erlang, Binomial and Engset models in case of one dimensional and two dimensional teletraffic in our research paper. The simulation programs are developed in a way as simple as possible such that the coding complexity does not overshadow the original mathematical algorithm. If required, the pictorial representation can be implemented with least labor using the simulation programs. Thus our design and simulation proceeds from elementary concepts to design tools then frame working, which is followed by, complete mathematical and logical algorithm development. The Simulation Program is developed using the C programming language. The Pseudo code for the simulation program is also provided.

Chapter 1 mainly deals with background materials and terminologies involving traffic estimation, traffic

characterization, traffic measurement and traffic dimensioning.

Chapter 2 deals with multidimensional traffic. Here we generalized the classical teletraffic theory to deal

with different types of traffic streams according to arrival process, service time distribution, bandwidth

requirement and quality of service requirement.

In chapter 3 we discussed about the classical queueing theory. And hence propose a queueing model.

We derived some equations and tried to solve the equations using different tools.

In chapter 4 we describe simulation basics and different types of simulation models and strategy. Also

different types of Simulation Languages that are specially developed for Teletraffic Simulation. And

finally we described the simulation program that we developed for our research project to explain the

effect of using our proposed queueing model practically.

In chapter 5 we discussed about various results and curves obtained by the simulation program as well

as other means and we tried to draw a conclusion of our thesis work.


CONTENTS (+14) Page #

Chapter 1 Basic Teletraffic Engineering Theory...1 - 25 1.1 Introduction..1

1.2 Basic Traffic Terms..1

1.3 Teletraffic Performance Parameters...11

1.3.1 The Grade of Service (GoS).11

1.4 Traffic Characterization.....12

1.5 Traffic Estimation......13

1.5.1 The Traffic Source Model Overview...14

1.5.2 Network Traffic Model....15

1.6 Traffic Measurements15

1.6.1 Measurement of Traffic Flow..15

1.6.2 Busy Hour Measurements16

1.6.2.1 Time Consistent Busy Hour Intensity...16

1.6.2.2 Average of the Daily Peak Hours Traffic..16

1.6.2.3 Fixed Daily Measurement Period..17

1.6.3 Determination of Normal Load and High Load Levels..17

1.6.3.1 Yearly Continuous Measurements18

1.6.3.2 Yearly Noncontinuous Measurements..18

1.7 Traffic Dimensioning ....19

1.7.1 Dimensioning Principles..19

1.8 Poissons Distribution in Teletraffic..20

1.9 Erlangs Distribution in Teletraffic....22

1.10 Concluding Remarks..25

Chapter 2 2-Dimensional Teletraffic Theory...26 - 42

2.1 Introduction....26

2.2 State Transition Rate Diagram.......26

2.2.1 Node Equations..28

2.2.2 Cut Equations.28


2.3 Multi-dimensional Poisson Process ......30

2.3.1 Multi-dimensional Erlang-B Formula.33

2.4 Multi-Dimensional Loss Systems..40

2.4.1 Class Limitation......40

2.4.2 Generalized Traffic Processes ....41

2.4.3 Multi-slot Traffic ........41


Chapter 3 Queueing Model..... 43-89 3.1 Introduction........43

3.2 Basic Concepts of Queueing .....43

3.3 Classification of Queueing Models44

3.4 Queueing Organization (Strategy, scheduling algorithm)..45

3.5 M/M/n Traffic Model.46

3.6 M/M/n/K Traffic Model.....55

3.7 Proposed 2-D M/M/n/K Traffic Model..62


Chapter 4 Traffic Simulation.90-119 4.1 Introduction....90

4.2 Why Simulation..91

4.3 Simulation Model...91

4.4 Simulation Life-Cycle92

4.5 Simulation Model Taxonomy....94

4.6 Discrete Event Simulation......95

4.7 Parallel Discrete Event Simulation.....96

4.7.1 Conservative Synchronization....99

4.7.2 Optimistic Synchronization..100

4.7.3 Issues and Optimizations......102

4.8 Simulation Languages..104

4.8.1 GPSS....105

4.8.2 SIMSCRIPT.....106

4.8.3 GASP....107


4.8.4 SIMULA...107

4.8.5 SLAM.......108

4.8.6 RESQ....108

4.9 Simulation of Poissons Arrival Process..109

4.10 Simulation of Proposed Traffic Model.113

4.10.1 Simulation Model.113

4.10.2 Simulation Program..114

4.10.2.1 Main Program.....115 4.10.2.2 Traffic Generation Part..115 4.10.2.3 Call Event Execution Part..116 4.10.2.4 Measurement and Data Analysis Part...116 4.10.2.5 Parameters and Statistical Counters..116

4.10.3 Simulation Flowchart...117 4.10.4 Simulation Results....118

Chapter 5 Result and Discussion 120-120

Appendix....121-147

Appendix I Matrix used for State Transition Diagrams........121

Appendix II Sample Matlab Program.........................123

Appendix III Pseudo Code and Output of Simulation Programs.135

Appendix IV Output of Simulation Programs..139

Appendix V Data of Poissions Distribution..........145

Bibliography .........148-150


List of Figures (+14) Page # Figure 1.1 Call arrival, call holding, blocked call, traffic volume

2

Figure 1.2 Offered traffic, Carried traffic, Lost traffic

5

Figure 1.3 Pure Loss System

8

Figure 1.4 Pure Waiting System

9

Figure 1.5 Mixed System 10

Figure 1.6 Poissons probability distribution

20

Figure 1.7 Markov state transition diagram for Poissons probability Distribution

21

Figure 1.8 Erlangs distribution

23

Figure 1.9 State transition diagram for Erlangs distribution

23

Figure 1.10 Sample output of simulation program where offered traffic was 6 Erl

24


25

Figure 2.1 State transition diagram of 2-dimensional traffic with offered Poisson traffic A1 and A2. Where the system has n= servers

27

Figure 2.2 Kolmogorovs criteria: a necessary and sufficient condition for reversibility of a two-dimensional Markov process is that the circulation flow among 4 neighboring states in a square equals zero: Flow forward path = reverse path

29

Figure 2.3 Two-dimensional state transition diagram for a loss system with n channels offered two PCT-I traffic streams

35

Figure 2.4 2-dimensional traffic states with offered Poisson traffics A1 and A2, where system has n=4 servers

37

Figure 2.5 2-D traffic in tabulated form 39

Figure 2.6 2-D traffic in normalized form 39 Figure 2.7 Structure of the state transition diagram for two-dimensional 41


traffic processes with class limitations (cf. 2.4.1). When calculating the equilibrium probabilities the state (i; j) can be expressed by state (i; j - 1) and recursively by state (i; 0) and finally by (0; 0) (cf. (2.2.3)).

Figure 2.8 Generalization of the classical teletraffic model to BPP-traffic and multi-slot traffic. The parameters i and Zi describe the BPP traffic, whereas di denotes the number of slots required.

42

Figure 3.1 State transition diagram for M/M/n model 47

Figure 3.2 Variation of probability of delayed service against number of channel

52

Figure 3.3 Variation of probability of immediate service against number of channel.

52

Figure 3.4 Variation of probability of immediate service against arrival rate

53

Figure 3.5 Variation of probability of delayed service against arrival rate

53

Figure 3.6 Variation of probability of immediate service against termination rate

54

Figure 3.7 Variation of probability of delayed service against termination rate

54

Figure 3.8 Stare diagram for M/M/n/K traffic model 55

Figure 3.9 Variation of probability of blocking against number of channel

58

Figure 3.10 Variation of probability of blocking against arrival rate 58

Figure 3.11 Variation of probability of blocking against termination rate 59

Figure 3.12 Variation of probability of immediate service against number of channel

59

Figure 3.13 Variation of probability of immediate service against arrival rate

60

Figure 3.14 Variation of probability of Immediate service against termination rate

60

Figure 3.15 Variation of probability of delayed service against number of channel

61


Figure 3.16 Variation of probability of delayed service against arrival rate 61

Figure 3.17 Variation of probability of delayed service against termination

rate 62

Figure 3.18 Proposed 2-D M/M/n/K traffic model 62

Figure 3.19 State transition diagram with 2-D queueing model with number of channel=1 and number of queue=1

63

Figure 3.20

Variation of blocking probability against arrival rate lamda1 65

Figure 3.21 Variation of blocking probability against termination rate mu1 65

Figure 3.22 Variation of probability of using nominal channel against arrival rate lamda1

66

Figure 3.23 Variation of probability of using nominal channel against termination rate mu1

66

Figure 3.24 Variation of probability of entering queue against arrival rate lamda1

67

Figure 3.25 Variation of probability of entering queue against termination rate mu1

67

Figure 3.26 Variation of blocking probability against arrival rate lamda2 68



69


69


70


70

Figure 3.32 State transition diagram with 2-D queueing model with no of channel=1 and no of queue=2

71



Figure 3.35 Variation of probability of using nominal channel against 74


arrival rate lamda1


74


75


75




77


77


78


78

Figure 3.45 State transition diagram with 2-D queueing model with no of channel=2 and no of queue=3

79




84


84


85


85





87


87


88


88

Figure 4.1 System study method alternatives 91

Figure 4.2 Simplified version of the modeling process 93

Figure 4.3 Simulation model taxonomy 94

Figure 4.4 Execution of a simulation viewed over the space-time plane 98 Figure 4.5 The causality principle can be upheld in a conservative

synchronization scheme by always selecting the message with the lowest time-stamp for processing, provided that there is at least one message in each input channel

99

Figure 4.6 Example of rollback in optimistic synchronization 102

Figure 4.7 State saving methods 103

Figure 4.8 Poissons arrival process 109

Figure 4.9 Call arrival probabilities against number of calls 112

Figure 4.10 Poissions Distribution 112

Figure 4.11 The Simulation Model Diagram 114

Figure 4.12 Complete flowchart of the simulation program 117

Figure 4.13 Variation of arrival rate against blocked calls for mu=7,8,9 and success call=1000

118

Figure 4.14 Variation of termination rate against blocked calls where lamda=7,9,11 and success call =1000

118

Figure 4.15 Variation arrival rate against calls in queue where mu=7,8,9 success call=1000 and size of queue=199

119

Figure 4.16 Variation termination rate against calls in queue where lamda=6,7,8 success call=1000 and size of queue=199

119


Chapter 1 Basic Teletraffic Engineering Theory

1.1 Introduction Traffic in a communication network refers to the aggregate of all user requests being serviced by the

network. As far as the network is concerned, the service requests arrive randomly and usually require

unpredictable service times. The first step of traffic analysis is the characterization of basic traffic terms

and theories. In this chapter we describe the basic traffic terms, traffic estimation, traffic

characterization, traffic measurements and traffic dimensioning.

1.2 Basic Traffic Terms The following are some commonly used terms when dealing with telecommunication traffic theory.

Arrival Call Rate

The demand for telephone calls may arise randomly at any hour of the day or night. Although the arrival

call rate varies significantly with time, the rate associated with peak demand is the most important.

Dividing the number of call requests during a measured time interval by the interval gives the mean call

rate .

Call Duration/ Holding Time

Once a call attempt is successful and a channel is assigned, the period of time during which the user

occupies the channel is called the call duration or the channel holding time Tc. The reciprocal of the call

duration is called the termination rate, the service rate, = 1/Tc. Because calls occupy the channel for a random length of time, we are usually interested in the mean holding time.

Mean Holding Time It is reasonable to assume that different types of call attempts will have different holding times. The total

volume of each type of call attempt can be expected to have an average holding time values that are


typical of the types of call in question. This average holding time is defined as the mean holding time

and is calculated as below.

S= T / Y

Where S = Mean holding time.

T = The traffic flow.

Y = Number of calls per time unit.

6 5 4 3 2 1

traffic volume time

time

Blocked call

No of channels occupied

call arrival time

call holding time

channel-by channel occupation

chan

nels

no o

f cha

nnel

s

6

1 0

2

4 3

5

Figure 1. 6 Call arrival, call holding, blocked call, traffic volume

Traffic Volume

To express the sum of all holding times carried by a group during a given period, the unit erlang-hours is

used. This is called traffic volume.


Number of Simultaneous Occupation

If we consider a single switching device, it is easily understood that the occupation of this device

depends on how many calls it receives and how long each call will occupy this device.

If we now consider a group of devices where every device can only handle one call at a time, we

understand that the number of occupied devices may vary.

Traffic Unit

The traffic unit is called the ERLANG. ERLANG: The mean number of simultaneous occupations during

a specified period of time (often implied to be one hour)

For example:

Equated Busy Hour Call (EBHC). 1 EBHC = 1/30 erlang Century (Hundred) Call Second (CCS). 1 CCS = 1/36 erlang

The unit CCS (Hundred Call Seconds) is generally used in the practical traffic work in USA.

Traffic Flow

Traffic flow is the traffic volume per time unit. Traffic flow can be calculated as follows:

T= Y* S

Where T = Traffic flow

Y = Number of calls per time unit.

S = Mean holding time (also can use H)

Traffic Demand (AD)

The traffic demand is the traffic flow which would be offered to an idle traffic system, i.e. the Traffic

Demand is the traffic that the subscribers would like to realize if no obstacles were at hand, such as

congestion, technical faults or busy B-subscriber. The traffic demand can also vary with the cost of the

calls, so one has also to distinguish between traffic demand at fixed rate and when the telephone rate is

varied. The traffic demand is a hypothetical quantity that only can be estimated but not measured.


Offered Traffic (Ao) The Offered Traffic is the traffic offered to a group in accordance with a defined theoretical description

of the traffic case. Depending on the traffic demand, a certain traffic flow is offered to the telephone

system. The traffic offered can be estimated as the sum of the number of occupations and congested

calls multiplied by the mean holding time of the occupation.

To make the concept clear in case of cellular mobile system, let us discuss a simple example; Consider a

simple cell of radius R and user density Du and propagation coverage factor Ccover for the cell area.

Suppose the average call rate attempted by each user is u with mean call duration Th. If the server has N channels then approximating the hexagonal to a circle of the same area, the circle should have radius of

Rc=0.9R.Thus the offered traffic to the BS is:

A0 = .Rc .Ccover. u Th

Carried Traffic (Ac) The Traffic carried is the traffic that is handled by a group. The inlet of the telephone system accepts a

certain part of the traffic offered; that will be carried. The carried traffic can be expressed as the mean

number of simultaneous occupations during a specified time interval.

For all channels or devices that are congestion free, we accept that certain proportions, B, of calls are

lost. This means that the carried traffic is (1 - B).Ao = Ac.

Offered traffic: A0 Carried traffic: Ac

Lost traffic: (A0-Ac) = BxA0

N channel

Where B=Blocking Probability

Figure 1. 7 Offered traffic, Carried traffic, Lost traffic

Traffic Lost (Al) The traffic lost or rejected is that part of the offered traffic, which was not conveyed by the telephone

system due to congestion or other failures. For all channels or devices that are congestion free, we


accept that certain proportions, B, of calls are lost. If we offer the traffic, Ao (in erlang) to a device, B x

Ao = Al will be the traffic that is lost.

Charged Traffic

The Charged traffic is the part of the traffic carried that is charged to the subscriber. The traffic carried

and the charged traffic differs since in most cases the subscriber only pays for calls that are answered.

For long distance calls and for international calls, the charging may start a little earlier, but not before

the call reaches the first transit exchange. Consequently, statistics of charged traffic will only give a part

of the real carried traffic.

Conversation Traffic Conversation traffic is the through-connected portion of the traffic offered. The unsuccessful portion of

the traffic carried can be called "Busy-tone traffic and "No-answer traffic".

Call Intensity It is the total number of conversations per time unit, which makes up the call intensity.

The setting-up procedure for a telephone call involves consequently a series of requests for the call to be

processed. The total number of such requests to a switching unit - or subscriber - per time unit is the call

intensity. Consequently, the first switching unit in the selection chain receives calls from subscribers, the

following switching units receive calls from the previous one, and finally, the called subscribers receive

calls from the last link in the switching equipments. One can therefore define the call intensity for each

distinct part of the switching chain from calling to called subscriber.

Congestion/Blocking When allocation of a new connection is impossible due to the unavailability of channels, congestion

occurs. This means that the call cannot be accepted for the moment. Depending on the system used, such

a call may either be rejected (loss system) or be allowed to wait (delay system). In the former case, the

calling subscriber receives busy tone and must make a new trial. In the latter case, delayed calls will be

served as soon as any of the devices become free. The probability of the occurrence of congestion is also

called congestion. The design of most digital switches is such that, if no internal congestion occurs, only

so-called "Route congestion" will be handled. Route congestion occurs when a connection cannot be

established due to lack of idle devices along the given route. Two types of congestion are as follows:


Call Congestion The Call Congestion is the part of the calls that are rejected or forced to wait in a delay system. When a

call arrives and is not served immediately due to lack of devices, a congestion situation has occurred.

This is called "Call congestion" or Blocking. It is measured as:

Call blocking Bc = probability that an arriving call finds all n channels occupied

= fraction of calls that are lost.

Time Congestion The Time Congestion is the part of the time additional calls cannot be served. When the last idle device

is seized, there will be no more idle devices available to serve any new calls. From the time the last idle

device was seized until a free (idle) device is obtainable, "Time congestion" will be in effect. It is

measured as:

Time blocking B t = probability that all n channels are occupied at an arbitrary time.

= the fraction of time that all n channels are occupied.

The two blocking quantities are not necessarily equal. If calls arrive according to a Poisson process,

then Bc=Bt. Call blocking is a better measure for the quality of service experienced by the subscribers

but, typically, time blocking is easier to calculate

Traffic Congestion Another type of congestion known as traffic congestion is the fraction of offered traffic that is not

carried, possibly despite several attempts. It is measured as:

Traffic congestion BTraffic = c

c

AAA 0

Where _X is the carried traffic.

Blocking Probability of New Call Request

The new call blocking probability is defined as the probability that a new call will be unable to access

the network due to equipment unavailability, either due to no free radio channel being available, or due

to no link through the fixed network being able to established as like [7]. The new call blocking

probability (PB) is given by


no

nbB N

NP =

Where Nnb is the number of new calls blocked.

Nno is the number of new calls offered.

Loss Systems This is a switching system in which a call attempt is immediately rejected because there is no free device

to serve the required connection. The subscriber will have to make a new attempt to establish the desired

connection. Lost calls in a "Loss system" are the calls that cannot be served because of congestion.

A loss system is a system where the subscriber is rejected if an idle channel or device can not be found.

An example of a loss system in AXE is the Group Switch. For most loss systems in AXE the erlang first

formula (the erlang B formula) can be used . Erlangs (a Danish traffic theorist) B-model is based on the

most common assumptions used. These assumptions are:

no queues, i.e. a pure loss system number of subscribers much higher than number of traffic channels no dedicated (guardd) traffic channels Poisson distributed (random) traffic, i.e. the number of calls is large and the calls are independent of each other

blocked calls abandon the call attempt immediately. Erlangs B-model relates the number of traffic channels, the GoS and the traffic offered.

l

n

Figure 1. 8 Pure Loss System.

Pure loss system:

No waiting places (m = 0), the system is full (with all n servers occupied) when a customer arrives, he

is not served at all but lost.

From the customers point of view, it is interesting to know e.g. the blocking probability


Delay Systems

A delay system is characterized by the fact that it has a queue (buffer) where calls are put if no free

channel or device is available. Calls that arrive when there are queued calls are said to meet congestion

and are themselves forced to wait. Contrary to the loss system, these unsuccessful calls are not discarded

but only delayed. The time that elapses between the instant of arrival of a call and the instant at which a

call is allocated to a channel or device (i.e. the time spent waiting in a queue) is known as the waiting

time. The formation of a queue is the main feature, which distinguishes waiting systems from loss

systems.

For most delay systems the erlang second formula (the erlang C formula) can be used, based on the

following additional assumptions (over the erlang first fomula):

calls arriving when all channels or devices are busy form a queue and wait in the order of their arrival for free channels or devices.

Pure waiting system:

Infinite number of waiting place (m=), if all n servers are occupied when a customer arrives, she occupies one of the waiting places, no customers are lost but some of them have to wait before getting

served.

l

n

Figure 1. 9 Pure Waiting System.

Mixed system:

Finite number of waiting place (0


l

n

mm

Figure 1. 10 Mixed System.

Traffic Intensity

The traffic intensity is frequently simply called traffic. The unit for traffic is erlang. Consequently, the

number of Erlangs is simply the average number of simultaneous occupations for a defined time

internal.

The number of Erlangs (A) is usually calculated as: A = y.Th

Where

y = the number of calls per time unit or the number of new occupations per time

unit;

Th = the average duration of these occupations, expressed in the same time unit.

Busy Hour

Three types of busy hour defined by CCITT in its recommendations

Busy Hour: Continuous 1 hour period lying wholly in the time interval concerned, for which the traffic volume or the number of call attempts is greatest.

Peak Busy Hour: The busy hour each day, it is usually varies from day to day, or over a number of days.

Time Consistent Busy Hour: The 1-hour period starting at the same time each day for which the average traffic volume or the number of call attempts is greatest over the days under

consideration. Based on [2]

1.3 Teletraffic Performance Parameters The teletraffic performance of a cellular mobile system can be assessed in terms of different parameters.

These may be include the grade of service, the traffic carried by the network, the channel utilization, the

system capacity and the spectrum efficiency. We now briefly discuss these parameters.


1.3.1 The Grade of Service (GoS) Grade of Service How many traffic one cells can carry depends on the number of traffic channels available and the

acceptable probability that the system is congested, which is referred to as the Grade of Service (GoS).

The grade of service refers either to the blocking probability in the case of a loss system, or the

probability of delay in the case of a delay system. The GOS of cellular mobile could be defined as a

combination of the following specifications:

i) Call Set-up delay or loss

In a mobile cellular network, a control channels is dedicated for setting-up the calls and is never

used for speech transmissions. When a MS wishes to attempt a call he must access the network

first through the set-up channel which is also called the access channel or the request channel. If

the MS verifies that the channel is free, by checking the busy/idle bits of the access channel, it

then transmits its seizure message requesting a voice channel. The access channel may handle

the arriving traffic on a delay basis or in a loss basis.

ii) Voice radio channel blocking

The voice channel blocking probability represents the probability that a call attempt that

succeeded in accessing the set-up channel is not assigned a voice channel because of the

unavailability of free channels.

iii) Land line network blocking

More than 90% of the mobile calls pass through the land- line network (PSTN) to or from the

land users. A blocking on the PSTN lines affects the GOS of the cellular network. It is usually

assumed that the blocking probability of PSTN is much smaller that that of the cellular network

due to the shortage of the mobile radio channels.

We can consider the fraction of calls, which will not be completed as a unified measure of the GoS. The

incompletion of calls may be because of the following: Blocking on the access channel PBa. Blocking on the radio voice channel PBv. The blocking in this stage is (1 - PBa) PBv. Blocking on the land-line network PBl. The blocking in this stage is (1 - PBa) (1 - PBv) PBl.


Blocking during the handover process resulting in forced termination PF. In this stage, the blocking is (1 PBa) (1 PBv) (1 PBl) PF.

Therefore we may express the GOS as we found in [3],

GoS = PBa + (1 - PBa) PBv + (1 - PBa) (1 - PBv) PBl + (1 - PBa) (1 - PBv) (1 - PBl) PF .

1.4 Traffic Characterization: Because of the random nature of network traffic, it requires certain fundamentals of probability theory

and stochastic processes. The intent of traffic characterization is to provide an indication of how to apply

results of traffic analysis not to deliver deeply into analytical formulations.

Based on the estimation method, the traffic characterization has to compute the spatial traffic intensity

and its discrete demand node representation from realistic data taken from available databases. In order

to handle this type of data, the complete characterization process comprises four sequential steps:

Step 1 Traffic model definition: Identification of traffic factors and determination of the traffic parameters in the geographical

traffic model.

Step 2 Data preprocessing: Preprocessing of the information in the geographical and demographical data base.

Step 3 Traffic estimation: Calculation of the spatial traffic intensity matrix of the service region.

Step 4 Demand node generation: Generation of the discrete demand node distribution by the application of clustering methods as

we found in [12].

1.5 Traffic Estimation: Designing modern telecommunication networks has become a very demanding engineering task. Large-

scale mobile communication networks with nation wide coverage are encompassing thousands of radio


transmitters and hundreds of controllers and switching facilities. An efficient configuration of such

large-scale systems can only be obtained by a systematic and requirement-oriented engineering concept.

Such an approach ensures that the obtained network configuration can be justified and that the quality of

the solution obeys the specified bounds.

The application of requirement-based procedures claims to start the design process with an analysis of

the design constraints. Additionally, it forces the specification of the design requirements and the

definition of a functional model of the network. Therefore, automated tools are needed to support the

design engineer. CUTE, the Customer Traffic Estimation Tool, was developed to support the design

engineer during the specification and design task. CUTE enables a demand-based system engineering by

introducing a new simple and efficient spatial traffic model. It is able to estimate the parameters of a

spatial traffic model using publicly available geographical databases.

1.5.1 The Traffic Source Model Overview A widely used single cell model was first introduced by Hong and Rappaport (1986). Their model

assumes a uniformly distributed mobile user density and non-directed uniform velocity distribution of

the mobiles under this premise, local performance measures like the mean channel holding time, the

average call origination rate, fresh call blocking probability or handover blocking probability can be

derived from the mobility pattern. Additionally, these models can be used to calculate the subjective

quality-of-service values for individual users.

A more accurate modeling of the calling behavior of users in a single cell was proposed by Tran-Gia and

Mandjes [4]. The model considers a base station with a finite customer population and repeated

attempts. The appealing characteristic of the model is the assumption of a small, finite users population.

This is the typical case in real networks. However, the model is limited to a single cell and does not

consider the spatial variation of the teletraffic within the service area.

As described in [1] El-Dolil characterized the mobile phone traffic on vehicular highways by assuming a

one-dimensional mobility pattern. They derive the performance values by applying a stationary flow

model for the vehicular traffic. The traffic orientation is non-directed and uniformly distributed.

A limited directed two-dimensional mobility model was investigated by Fosichini, Gopinath, and

Miljanic (1993). The model assumes a spatially homogenous distribution of the demand and isotropic


structure. Chlebus (1993) investigates a mobility model with a homogeneous demand distribution but

assumes a non-uniform velocity distribution. The traffic orientation is non-directed and equally

distributed.

The application of these traffic source models in real network planning cases is strongly limited. Some

models, like the highway model proposed by Leung, Massey, and Whitt, give a deep insight on the

impact of the terminal mobility on the cellular system performance; however they are rather complex to

be applied in real network design. Other models, like the one suggested by Hong and Rappaport, due to

their simplification assumptions, can only be applied for the determination of the parameters in an

isolated cell.

1.5.2 Network Traffic Model The offered traffic in a region can be estimated by the geographical and demographical characteristics of

the service area. Such a demand model relates factors like land use, population density, vehicular traffic,

and income per capita with the calling behavior of the mobile units. [6]

1.6 Traffic Measurements The measurement information can be divided by measurement processing time to statistical information

and reas-time information. Statistical information is a stochastic process of long term measurement

results which the same measurement process in different time will give the similar results. Real-time

information is the nonprocessing data which the measurement period is in a few of minutes or seconds.

1.6.1 Measurement of Traffic Flow The traffic carried in the base station can be measured by scan the channel at regular interval and record

the number of occupied channels at that time. The accumulated number of channel busies divided by

number of scans can give the mean carried traffic in the BS.

The scanning error should be taken into account if the scanning period is large. The mean of scanning

error equal to zero. The variance of scanning error is shown below [13]

{ }

+ 2

11._

ee

TTX

errorscanningVar h


hT =

where X = carried traffic

Th = call duration (for the case of cellular system tm is channel holding time).

= scanning interval.

1.6.2 Busy Hour Measurements One important parameter to be measured is the busy hour traffic flow in each cell, which determines the

provision of equipment. Since the busy hour may vary, it is necessary to record the traffic during a

longer period than 60 min. to ensure that the busy hr. traffic is measured [14]. CCITT divides the busy

hour measurement process to Daily continuous measurements and Daily non-continuous measurements.

Daily continuous measurements take traffic measurements continuously over the day throughout the

measurement period. The daily continuous measurement process recommended by CCITT are Time

consistent busy hour (TCBH) and Average of the daily peak hours traffic defined on quarter hour or on

full hour restrict measurements to a few hours or only one hour per day. The time of fixed daily traffic

profile for every cell. The measurement can cover several periods daily as well. The daily non-

continuous measurement processes recommended by CCITT are fixed daily measurement period

(FDMP) and Fixed measurement hour (FDMH).

1.6.2.1 Time Consistent Busy Hour Intensity For a number of days, carried traffic values for each quarter hour for each day are recorded. The values

for the same quarter hour each day are averaged.

The four consecutive quarter hours in this average day which together give the largest sum of observed

values from the TCBH with its TCBH intensity. This is sometimes referred to as post-selected TCBH .

1.6.2.2 Average of the Daily Peak Hours Traffic To find the average of daily peak quarterly defined hour (ADPQH) intensity, the traffic intensity is

measured continuously over a day in quarter hour periods. The intensity values are processed daily to

find out the four consecutive quarter hours with the highest intensity value sum. Only this daily peak

hour traffic intensity value is registered. The average is taken over a number of working days peak

intensities. The timing of peak intensity normally varies from day to day.


To find the average of daily peak full hour (ADPFH) intensity, the traffic intensity is measured

continuously over a day in full hour periods. Only the highest of these intensity values is registered. The

average is taken over a number of days peak intensities.

1.6.2.3 Fixed Daily Measurement Period With this method measurements are taken within a fixed period (e.g., of 3 hours) each day. This period

should correspond to the highest part of the traffic profile, which is expected to include the TCBH.

Measurement values are accumulated separately for each quarter hour, and the busiest hour is

determined at the end of the measurement period, as for the TCBH .

1.6.3 Determination of Normal Load and High Load Levels Teletraffic performance objectives and dimensioning practices generally set objectives for two sets of

traffic load conditions.

A normal traffic load can be considered the typical operating condition of a network for which

subscribers service expectations should be met.

A high traffic load can be considered a less frequent encountered operating condition of a network for

which normal subscriber expectations would not be met but for which a reduced level of performance

should be achieved to prevent excessive repeat calling and spread of network congestion.

The measurement data, which is used for normal load and high load calculation, can be divided to the

data from yearly continuous measurements and data from yearly non-continuous measurements.

1.6.3.1 Yearly Continuous Measurements Traffic statistics should be measured for the significant period of each day of the whole year. The

significant period may in principle be 24 hours of the day.

The measurements for computing normal traffic load should be 30 highest days in a fixed 12-month

period. Normally these will be working days. Normal traffic load calculated by average of the 30 highest

working days during a 12-month period. High traffic load calculated by average of the five highest days

in the same period as normal load calculation.


1.6.3.2 Yearly Noncontinuous Measurements This method consists in taking measurements on a limited sample of days in each year. Limited sample

measurements will normally be taken on working days.

An approximate statistical method for estimating normal and high load levels from limited sample

measurements is provided below.

Measurements are taken on a limited sample of days, and the mean (M) and standard deviation (L) are

given by

L = M+ k.S

Different values of the factor k being used for normal and high load levels (CCITT Rec. E.500)

( ) 21

2

111

= = MxnS i

n

i

where xi is the time consistent busy hour traffic measured on the ith day.

i

n

ix

nM

==

1

1 is the sample mean, and

n is the number of measurement days.

If the measurement period is less than 30 days then the estimate will not be very reliable. It is important

to determine the basic period since the length of this period influences the values assigned to the

multiplication factor k.

The base period is the set of valid days in each year from which measurement days are pre selected. The

base period may be restricted to a busy season provided that the traffic is known to be consistency

higher during this period than during remainder of the year. Measurement days should be distributed

reasonably throughout the base period. If the base period extends over the whole year then the

measurement sample should include some days from the busiest part of the year, if there are known. The

limited sample should comprise at least 30 days to ensure reliable estimates.

1.7 Traffic Dimensioning The number of traffic channels available to serve to certain number of subscribers is one of the key

factors when designing a cellular network. Dimensioning the traffic means determining the required


number of channels to meet a specified grade of service under a specific teletraffic demand.

Dimensioning the traffic is also called engineering the network or seizing the network. The traffic for

each user is defined by the calling rate and the average holding time.

The number of channels needed for a given traffic is calculated according to an Erlang table, with

respect to a certain Grade of Service. Typically 2% GoS is used in the dimensioning of cellular traffic.

1.7.1 Dimensioning Principles Due to the structure and design of telecommunication systems, the equipment provision cannot be made

to match all short time variations of the traffic. The various parts of the system must, therefore, be

extended in steps large enough to last for a certain period, usually at least 6 months up till 5 years. The

extension steps must be so large that disturbing congestion can be avoided during the whole period.

For such an extension period, a traffic forecast is made which should describe how the traffic will vary

and grow. The provision depends on the following facts:

i) The revenue from telephone traffic will depend on the traffic carried, usually on the number of

conversations established.

ii) The expenditure depends on the grade of service it is desired to give to the subscribers under

peak traffic conditions.

iii) Most of the carried traffic volume is handled at times when there is practically no congestion

at all.

iv) Really high congestion occurs only on a few occasions.

v) These occasions of high congestion occur more often at the end of an extension period than at

the beginning.

vi) It is not economically feasible to dimension a telephone system so that congestion never

occurs.

vii) An improved grade of service usually results in a certain growth of traffic when the

subscribers notice that it has become easier to get through (service improvement jump or traffic

stimulus).

viii) Human factors always cause more failures than the telephone system. By human factoris

meant: mistakes by calling subscriber, called subscriber engaged, and no answer.

ix) The individual subscriber does not react much to congestion below, say, 10%.


x) The peak traffic is usually comparatively unaffected by minor changes of rates. This is

because the bulk of the traffic during peak hours comes from business subscribers and not from

private persons. As mentioned in [2]

1.8 Poissons Probability Distribution in Teletraffic In this distribution number of channel n & number of user N are unlimited where call arrival rate is and call termination rate is as per [17]

Figure 1.6 Poissons traffic model

Figure 1.7 Markov state transition diagram of Poissons probability distribution.

Applying cut equation in P0

10 PP = or,

01 PP =

We know,

= A

APP 01 = ---(1.1) Applying cut equation in P1

P0 P1 P3P2

2 3

Poisson n = N =


221 PP = or,

212PP =

APP .21.12 = or, AAPP .2

1.02 =

!2

2

02APP = ---(1.2)

Applying cut equation in P2

332 PP = or,

323PP =

3.

!2

2

03AAPP =

!3

3

03APP = ---(1.3)

: :

: : Applying cut equation in Pn

!0 nAPP

n

n = ---(1.4) For entire space

1............210 =+++ PPP

1........!

............!3!2 03

0

2

000 =++++++ nAPAPAPAPP

n

1....................!3!2

132

0 =

++++ AAAP

10 = AeP or, AeP =0 ---(1.5) Probability state

!0 xAPP

x

x =


Ax

x exAP =

! ---(1.6)

Known as Poissons probability distribution

1.9 Erlangs Distribution (Ek) in Teletraffic Erlangs Distribution is normalized truncated version of Poissons probability distribution. In this

distribution number of channel n is limited, number of user N is unlimited, call arrival rate is and call termination rate is . So any call arrival after the state Pn is lost.

Figure 1.8 Erlangs distribution

Figure 1.9 State transition diagram of Erlangs distribution.

Applying cut equation in P0 we found from equation 1.1

APP 01 = ---(1.7) Applying cut equation in P1 we found from equation 1.2

!2

2

02APP = ---(1.8)

Applying cut equation in P2 we found from equation 1.3

!3

3

03APP = ---(1.9)

: :

P0 P1 PnP2 2 3

Lost

n

Poisson n = limited


: : Applying cut equation in Pn we found

!0 nAPP

n

n = ---(1.10) For entire probability state

1............210 =+++ PPP

1!

............!3!2 03

0

2

000 =+++++ nAPAPAPAPP

n

1!

.............!3!2

132

0 =

+++++

nAAAAP

n

+++++

=!

.............!3!2

1

1320

nAAAA

Pn

---(1.11)

Probability State

+++++

=!

.............!3!2

1

!32

nAAAA

xA

Pn

x

x ---(1.12)

Known as Erlangs Distribution

Call Blocking Provability

=

=n

i

i

n

n

iA

nA

B

0 !

! ---(1.13)


Helsinki University of Technology (HUT) developed a simulation result (Simu-a) program of one-

dimensional traffic for Erlangs (limited channel) case. Two sample outputs are given bellow:




1.10 Concluding Remarks A substantial amount of research has been devoted to teletraffic theory, with many books written in this

subject [3]. One of the early books on telephone traffic by Syski [5], which presents a mathematically

comprehensive self-contained introduction to the congestion theory. Book by Bear [14] is concerned

with general principles of teletraffic engineering. Coopers queuing theory [18] dealt with the teletraffic

concepts and the main formulae.

The widespread deployment of telephony network all over the world has led many researchers to study

on teletraffic performance. To design a telephony network and to plan for the continuous increase in

service demands, teletraffic engineering formulae are required to determine the level of channel

provision. Significant researches have been undertaken to extend the classical teletraffic theory.

Chapter 2 2-Dimensional Teletraffic Theory

2.1 Introduction Traffic offered to a system is called multidimensional when it is categorized into different types or

class according to arrival process, Service time distribution, bandwidth requirement, Quality of service

requirement, priority, Serve differentiation etc.


Normally traffic components have different state probabilities, and individual performance measures.

The traffic in the system is observed from many view angles:

- Individual traffic type

- Multidimensional traffic (x1, x2, x3, .,xk) indicates states (xi) of all traffic types

- Total traffic, e.g. total used bandwidth (xi).

In this chapter we generalize the classical teletraffic theory to deal with multiple traffic cases, where

every class of service corresponds to a traffic stream. Several traffic streams are offered to the same

trunk group.

In sec.2.2 we consider the Poisson distribution for multidimensional traffic and in Sec.2.3 we consider

the classical multi-dimensional Erlang-B loss formula with an example.

2.2 State Transition Rate Diagram If there are two traffic types simultaneously in the system, it is convenient to arrange states x1 and x2 to

grow in x-,and y-directions. For K-dimensional traffic case, K- dimensional coordinate system may be

used.


X1

4

3

2

1

0

0 1 2 3 4 X2

0,0 0,1 0,2 0,3 0,4

1,4 1,3 1,2 1,1 1,0

2,0 2,1 2,2 2,3 2,4

3,4 3,3 3,2 3,1 3,0

4,0 4,1 4,2 4,3 4,4

2 1

2 1

2 22 32 42 52

1 1 1 1 2 2 2 2

22 32 42 52 1 1 1 1

1

1

1

1

21 2

2

2

2

31

41

51

1

1

1

1

1 1

1

1

1 1

1

1

1 1

1

1

1 22

22

22

32

32

32

42

42

42

52

52

52

21 21 21 21

31 31 31 31

41 41 41 41

51 51 51 51

2

2

2

2 2 2 2

2 2 2 2

2 2 2 2

2 2 2 2

A1

51

A2

Figure 2.6 State transition diagram of 2-dimensional traffic with offered Poisson traffic A1 and A2. Where the system has n= servers.

2-Dimensional traffic state is (x1, x2), probability P(x1,x2). P(x1) and P(x2). Are marginal traffic

distribution of traffic components x1 and x2 .Total traffic x= x1+x2 has distribution p(x).Balance

equations formulated using

-Node Equations (global equations) or

-Cut Equations (local equations)

2.2.1 Node Equations In statistical equilibrium the number of transitions per time unit into state [i,j] equals the number of

transitions out of state [i,j]. Thus we get the following equilibrium or balance equation:

P00(1+2)=P10.1+P01. 2


P10(1+2+1)=P00. 1+P20.21+P112 P01(1+2+2)=P00. 2+P11.1+P0222 P11(1+2+1+2)=P00. 2+P21.21+P01. 1+P12.22 etc

2.2.2 Cut Equations As shown in previous section for an increasing number of traffic streams the number of states (and the

equations) increases very rapidly. However, we may simplify the problem by using cut equation. Cut

Equations can be used if Markov system is reversible. System is reversible if probability any closed path

calculated in forward direction is equal to the probability of the same path calculated is reverse

direction (Kolmogorovs criteria), found from [16] and [17].

Assume that 4 nodes in the next figure are a part of a bigger state space. Probability of forward path

(i,j).(i,j+1), (i+1,j+1). (i,j+1) should be equal to reverse path(i,j), (i,j+1), (i+1,j+1) (i+1,j).State of Traffic

components 1 and 2 are x1=i,i+1 and x2=j,j+1.Notations for transition states are

Traffic(Type) Arrival rate(From node i,j) Departure rate(From node i,j)

x1 1(i.j) 1(i,j) x2 2(i.j) 2(i.j)

1(i,j)

2(i,j)

2(i+1,j)

1(i+1,j)

2(i+1,j+1)

1(i,j+1)1(i+1,j+1)

2(i,j+1)

i,j i,j+1

i+1,j+1 i+1,j

Figure 2.7 Kolmogorovs criteria: a necessary and sufficient condition for reversibility of a two-dimensional Markov process is that the circulation flow among 4 neighboring states in a square equals zero: Flow forward path = reverse path.


Thus equating the forward and reverse path equation we get:

Pi, j.1(i,j). Pi+1,j.2(i+1,j). Pi+1,j+1.1(i+1,j+1). Pi, j+1.2(i, j+1) = Pi, j.2(i, j). Pi, j+1.1(i, j+1). Pi+1,j+1.2(i+1,j+1). Pi+1, j.1(i+1,j) (2.1)

We can reduce the expression by removing state probabilities and then obtain the condition given below:

1(i,j). 2(i+1,j). 1(i+1,j+1). 2(i,j+1) = 2(i,j). 1(i,j+1). 2(i+1,j+1). 1(i+1,j) (2.2)

If this condition is fulfilled, then there is local or detailed balance. A necessary condition for

reversibility is thus that if there is an arrow from state i to state j, then there must also be an arrow from j

to i. we may locally apply cut equations between any two connected states. Thus from fig.(2.2) we get:

P(i,j).1(i,j) = P(i+1,j).1(i+1,j) (2.3)

We can express any state probability P(0,0) by choosing any path between the tow states (Kolmogorovs

criteria). We may e.g. choose the path:

(0,0), (0,1),..(i,0),(i,1),.,(i,j)

and we then obtain the following balance equation:

)0,0(.).,(.).........2,().1,().0,()....0,2().0,1(

)1,(........).........1,().0,().0,1(........).........0,1().0,0(),(222111

222111 pjiiii

jiiiijip = (2.4)

We find P(0,0) by normalization of the total probability mass.

The condition for reversibility will be fulfilled in many cases, For example the following rates

(Poisson/Erlang and Binomial /Engset) fulfill the requirement.

Rate (examples) Poisson/Erlang Binomial /Engset

1(i,j)= 1((i) 1 (N1-x1) 1 2(i,j)= 2(j) 2 (N2-x2) 2 1(i,j).= i. 1 i. 1 i. 1 .2(i,j)= j. 2 j. 2 j. 2


If we consider a multi-dimensional loss system with N traffic streams, then any traffic stream may be a

state-dependent Poisson process, in particular BPP (Bernoulli, Poisson, Pascal) traffic streams. For

N{dimensional systems the conditions for reversibility are analogue to (2.2). Kolmogorov's criteria must

still be fulfilled for all possible paths. In practice, we experience no problems, because the solution

obtained under the assumption of reversibility will be the correct solution if and only if the node balance

equations are fulfilled. In the following section we use this as the basis for introducing a very general

multi-dimensional traffic model as mentioned in [15].

2.3 Multi-dimensional Poisson Process In this section we shall consider the multi-dimensional Poisson process. The derivations are based on the

graphical model as described in the previous section. Thus for 2-dimensional Poisson traffic cut-

equations are valid.

For example we consider a 2-dimensional traffic with offered Poisson traffic A1 and A2. System has

n= servers. Where A1=1/1, A2=2/2, A=A1+A2. For traffic type1 cut equations do not depend on state x2 of traffic 2,valid for any certain value x2=0,1,2

22222 ,01,011,11,11,0/ xxxxx PAPPPP === (2.5)

22222 ,0

21

,111,21,21,1 22/2 xxxxx P

APPPP === (2.6) and generally for any x1

2

1

21212121 ,01

1,1

1

11,11,1,1 !

/x

x

xxxxxxxx PxAP

xPxPP ===

The result is Poisson distribution, x1=0,1,,

2

1

21 ,01

1, ! x

x

xx PxAP = (2.7)

For x3 state distribution is solved according to exactly same procedure, the result is as well Poisson

distribution:

0,20,2

21,21,20, 11111 xxxxx

PAPPPP === (2.8)

and so on

0,2

2, 1

2

21 ! xx

xx PxAP = (2.9)


Combining these two distribution the 2-dimensional distribution as a function of P(0,0) is solved:

P(x1,x2), where x1 is variable, and x2=0, gives

0,01

10, !

1

1P

xAP

x

x = (2.10)

Inserting (2.10) into (2.9)

== 0,0

1

1

2

2,0,

2

2, !!!

12

211

2

21P

xA

xAPP

xAP

xx

xxx

x

xx (2.11)

Normalization:

0,02

2

1

1

0,0,

0,0 !!1

21

21

21

21

PxA

xAP

xx

xxxx

xx==

==

== (2.12)

!!!!1

2

2

01

1

00,00,0

2

2

1

1

00

2

2

1

1

21

21xA

xAPP

xA

xA x

x

x

x

xx

xx

=

=

=

=== (2.13)

21210,00,01

AAAA eePeeP == (2.14)

2-dimensional Poisson distribution:

2121

21 !! 22

1

1,

AAxx

xx eexA

xAP = (2.15)

Where x1 and x2=0,1,.,. Conclusion based on previous formula is that 2-dimensional distribution is product of component distributions as given in [9].

2121 , xxxxPPP =

K-dimensional Poisson distribution:

==

= iiKKK A

i

xi

K

i

AA

K

xx

xx exAee

xA

xAP

!! 12

1

1,....,

11

1 (2.16)

is also product of component distributions.

=

=K

ixxx iK

PP1

,...,1 (2.17)

Marginal distribution in K-dimensional Poisson

Distribution of traffic component i


ii

i

A

i

xi

x exAP = (2.18)

Distribution of total traffic x

If K=2, total load is x=x1+x2. To calculate P(x) all state probabilities P(x1,x2|x1+x2=x) are included (this

is a summation along diagonal direction is state space).

( ) 2111

1

121

1!! 1

2

1

1

0,

0

AAxxxx

xxxxx

x

xx eexx

AxAPP

==

===

( )( )

11

1

2111

1

2121

101

2

1

1

0 !!!

!!xxx

x

x

AAxxxx

x

AAx AAx

xx

exx

xxA

xAeeP

=

+

=

== (2.19)

( ) ( )xAAx AAxeP 21!

21 +=+

Ax

x exAP =

! (2.20)

Where for K traffic components:

=

==K

ii

K

ii xxandAA

11 (2.21)

2.3.1 Multi-dimensional Erlang-B Formula

When multidimensional Poisson traffic is offered to a limited number of servers (n


The state transition diagram is shown in Fig.2.1. Under the assumption of statistical equilibrium the state

probabilities are obtained by solving the global balance equations for each node (node equations), in

total (n + 1)(n + 2)/2 equations.

This diagram corresponds to a reversible Markov process, and the solution has product form as

mentioned [9]. We can easily show that the global balance equations are satisfied by the following state

probabilities, which can be written on product form:

p(x1,x2) = p(x1).p(x2)

= !

.!

.2

1

1

1 21

xA

xAQ

xx

(2.23)

Where p(x1) and p(x2) are one-dimensional truncated Poisson distributions, Q is a normalization

constant, and (x1,x2) fulfill the above restrictions (2.22). As we have Poisson arrival processes, which

have the PASTA-property (Poisson Arrivals See Time Averages), the time congestion, call congestion,

and traffic congestion are all identical for both traffic streams, and they are all equal to P(x1+x2 = n).

By the Binomial expansion or by convolving two Poisson distributions we find the following, where Q

is obtained by normalization:

p(x1+x2 = x) = !)( 21

xAAQ

x+ (2.24)

=

+=n AAQ

0

211

!)(

(2.25)

This is the Truncated Poisson distribution with the offered traffic

A =A1 + A2 :

We may also interpret this model as an Erlang loss system with one Poisson arrival process and hyper-

exponentially distributed holding times in the following way. The total Poisson arrival process is a

superposition of two Poisson processes with the total arrival rate:

= 1 + 2 and the holding time distribution is hyper-exponentially distributed:


f(t) = tt ee 21 .... 221

21

21

1

+++ (2.26)

We weight the two exponential distributions according to the relative number of calls per time unit. The

mean service time is

m1 =21

21

221

2

121

1 1.1.

++=+++

AA

m2= A

(2.27)

which is in agreement with the offered traffic. Thus we have shown that Erlang's loss model is valid for

Hyper-exponentially distributed holding times, a special case of the general insensitivity property of

Erlang's B-formula.


0,0 1,0 2,0

2,1 1,1 0,1

0,2 1,2 2,2

2 1

2 1

2 22 32

1 1 2 2

22 321 1

1

1

21 2

2

1

1

2 2

1 22 32

21 21 2

2 2

2 2

(n-1)2 n2

2 1

n-1,0 n,0

n-1,1 (n-1)2

1

1 2

(n-1)1 (n-1)1

n1

0,n-1

0,n

1,n-1

X2

X1

Figure 2.8 Two-dimensional state transition diagram for a loss system with n channels offered two PCT-I traffic streams.

We may generalize the above model to N traffic streams:

!x..............

!x.

!x.

!xQ. ) x., x, x,p(x

N

1

3

31

2

21

1

11

N32111

3

1

2

1

11111

NxN

xxx AAAA= (2.28)

,021

NXx

Nx 21 , =

N

xnX

x1

12

2

which is the general multi-dimensional Erlang-B formula. By a generalization of (2.24) we notice that

the global state probabilities can be calculated by the following recursion, where q( 1X ) denotes the

relative state probabilities, and p( 1X ) denotes the absolute state probabilities:

q(i) = =

=N

x

x qxqxA

11

12

2 1)0(),1(. (2.29)


Q(n) = =

N

xxq

01

1

),(

p(i) = nxnQxq 11 0,)(

)( (2.30)

This is similar to the recursion formula for the Poisson case, where

A = =

N

xxA

122

The probability of time congestion is p(n), and as the PASTA-property is valid, this is also equal to the

call congestion and the traffic congestion.

An Example (2-Dimensional case):

We consider 2-dimensional traffic with offered Poisson traffics A1 and A2. System has n=4 servers. All

states outside are truncated. In the state space all Poisson probabilities outside feasible state limited

with rule )4(0 21 =+= nxxx are assigned probability 0.All the Poisson terms inside the feasible area are normalized in usual way to find P00 as inverse of the sum of included terms.


X1

4

3

2

1

0

0 1 2 3 4 X2

0,0 0,1 0,2 0,3 0,4

1,3 1,2 1,1 1,0

2,0 2,1 2,2

3,1 3,0

4,0

2 1

2 1

2 22 32 42 52

1 1 1 1 2 2 2 2

22 32 42 52 1 1 1 1

1

1

1

1

21 2

2

2

2

31

41

51

1

1

1

1

1 1

1

1

1 1

1

1

1 1

1

1

1 22

22

22

32

32

32

42

42

42

52

52

52

21 21 21 21

31 31 31 31

41 41 41 41

51 51 51 51

2

2

2

2 2 2 2

2 2 2 2

2 2 2 2

2 2 2 2

A1

51

A2

P=0

N=4

Figure 2.9 2-dimensional traffic states with offered Poisson traffics A1 and A2, where system has n=4 servers.

Sum can be calculated using the column sums. The following formula collects row sums.

=

=

== 1

2

2

1

1

0 2

24

0 1

10,0 !

.!

.1xn

x

xn

x

x

xA

xAP (2.31)

Then state probability is


=

=

=

==1

2

2

1

1

21

21

21

0 2

24

0 1

1

2

2

1

1

2

2

1

10,0.

!.

!

!.

!!

.!

. xn

x

xn

x

x

xx

xx

xx

xA

xA

xA

xA

xA

xAPP (2.32)

Loss states for both traffic components, and for total traffic x are

B=B1=B2=Bx=P{x=4}=P40+P31+P22+P13+P04 (2.33)

This includes all states where x=x1+x2=n(=4).Erlangs loss formula in 2 dimensional case is then

=

=

=

=

= =1

2

2

1

1

1

1

1

0 2

24

0 1

1

1

24

0 1

1

!.

!

)!(.

!xn

x

xn

x

x

xnn

x

x

xA

xA

xnA

xA

B 2-dim Erlangs loss formula (2.34)

1.!4

)1(!3

)!2

1(!2

)!3!2

1.()!4!3!2

1.(1

!4!3!2!2.

!3.

!44

12

31

22

2

21

32

22

21

42

32

22

2

412

31

22

21

321

42

AAAAAAAAAAAAAA

AAAAAAAA

B++++++++++++++

++++= The same

writing row sums

!4.1

!3)1(

!2).

!21().

!3!21(1).

!4!3!21(

!4!3!2!2.

!3.

!442

32

1

22

21

12

31

21

1

41

31

21

1

412

31

22

21

321

42

AAAAAAAAAAAAAA

AAAAAAAA

B++++++++++++++

++++=

(2.35) Numerical example in tabulated form: Let us give an numerical example in tabulated form where

A1 = 2 Erl A2 = 3 Erl n = 7

7 0.025397 0.025397 6 0.088889 0.088889 0.266667


5 0.266667 0.266667 0.8 1.2 4 0.666667 0.666667 2 3 3 3 1.333333 1.333333 4 6 6 4.5 2 2 2 6 9 9 6.75 4.05 1 2 2 6 9 9 6.75 4.05 2.025 0 1 1 3 4.5 4.5 3.375 2.025 1.0125 0.433929 1 3 4.5 4.5 3.375 2.025 1.0125 0.433929 0 1 2 3 4 5 6 7

Figure 2.5 2-D Traffic in Tabulated Form Normalized form:

7 0.000197 6 0.000691 0.002073 5 0.002073 0.00622 0.00933 4 0.005183 0.01555 0.023325 0.023325 3 0.010367 0.0311 0.046649 0.046649 0.034987 2 0.01555 0.046649 0.069974 0.069974 0.052481 0.031488 1 0.01555 0.046649 0.069974 0.069974 0.052481 0.031488 0.015744 0 0.007775 0.023325 0.034987 0.034987 0.02624 0.015744 0.007872 0.003374 0 1 2 3 4 5 6 7

B 0.120519

Figure 2.6 2-D traffic in normalized form

Here sum of shaded cells gives blocking probability B = 0.120519 2.4 Multi-Dimensional Loss Systems In this section we consider generalizations of the classical teletraffic theory to cover several traffic streams offered to a single channel/trunk group. Each traffic stream may have individual parameters and may be state-dependent Poisson arrival processes with multi-slot traffic and class limitations. This general class of models is insensitive to the holding time distribution, which may be class dependent with individual parameters for each class. We introduce the generalizations one at a time and go through a small case-study to illustrate the basic ideas.


2.4.1 Class Limitation In comparison with the case considered in Sec. 2.3 we now restrict the number of simultaneous calls for

each traffic stream (class). Thus, we do not have full accessibility, but unlike gradings where we

physically only have access to specific channels, then we now have access to all channels, but at any

instant we may only occupy a limited number. This may be used for the purpose of service protection

(virtual circuit protection). We thus introduce restrictions to the number of simultaneous calls in class j

as follows:

Nnnx xx ......,.........3,2,1 x2 , 10 22 = (2.36)

Where .2

2 nnN

xx > , n is the number of channel or trunk and N is the number of traffic streams.

If we don't have the latter restriction, then we get separate groups corresponding to N ordinary

independent one-dimensional loss systems. Due to the restrictions the state transition diagram is

truncated. This is shown for two traffic streams in Fig. 2.7

We notice that the truncated state transition diagram still is reversible and that the value of p (i,j)

relatively to the value p(0; 0) is unchanged. Only the normalization constant is altered. In fact, due to the

local balance property we can remove any state without changing the above properties. We may

consider more general class limitations to sets of traffic streams so that any traffic stream has a

minimum (guaranteed) allocation of channels.


j

n2

(i,j)

X

1 n n1 i 0

0

2

i+j=n

- blocking for stream 2 X blocking for stream 1

Figure 2.7 Structure of the state transition diagram for two-dimensional traffic processes with class limitations (cf. 2.4.1). When calculating the equilibrium probabilities the state (i; j) can be expressed by state (i; j - 1) and recursively by state (i; 0) and finally by (0; 0) (cf. (2.2.3)).

2.4.2 Generalized traffic processes We are not restricted to consider PCT-I traffic only as in Sec. 2.3. Every traffic stream may be a state-

dependent Poisson arrival process with a linear state-dependent death rate. The system still fulfils the

reversibility conditions given in section (2.2.2).

Thus, the product form still exists for e.g. BPP traffic streams. If all traffic streams are Engset -

(Binomial-) processes, then we get the multi-dimensional Engset formula (Jensen, 1948 [19]). As

mentioned above, the system is insensitive to the holding time distributions. Every traffic stream may

have its own individual holding time distribution found from [15].

2.4.3 Multi-slot Traffic In service-integrated systems the bandwidth requested may depend on the type of service. Thus a voice

telephone call requires one channel (slot) only, whereas e.g. a video service may require d channels

simultaneously. We get the additional restrictions:

where ij is the actual number of type j calls. The resulting state transition diagram will still be reversible

and have product form.


N,Zn,CN

2,Z2,C2

1,Z1,C1 n1

n2

nN

n HT

Local exchanges Transit exchange Destination exchange

::

Figure 2.8 Generalization of the classical teletraffic model to BPP-traffic and multi-slot traffic. The parameters i and Zi describe the BPP traffic, whereas di denotes the number of slots required.

2.5 Concluding Remarks The above theory is exact when we consider charging of calls and measuring of time intervals. For

stochastic computer simulations the traffic process in usually stationary, and the theory can be applied

for estimation of the reliability of the results. However, the results are approximate as the theoretical

assumptions about congestion free systems seldom are of interest. In real life measurements on working

systems we have traffic variations during the day, technical errors, measuring errors etc. Some of these

factors compensate each other and the results we have derived give a good estimate of the reliability,

and it is a good basis for comparing different measurements and measuring principles.

Chapter 3 Queueing Model


3.1 Introduction Till now we have considered classical queueing systems, where all traffic processes are birth and death processes. The theory of loss systems has been successfully applied for many years with in the field of telephony, whereas the theory of delay or queueing systems has only been applied during recent years within the field of computer science. The classical queueing systems play a key role in queueing theory. Usually, we assume that either the inter-arrival time distribution or the service time distribution is exponentially distributed. As mentioned in [15]

3.2 Basic Concepts of Queueing Queuing analysis is one of the most important tools for those involved with computer and network

analysis. It can be used to provide approximate answers to a host of questions as given in [10], such as:

What happens to file retrieval time when disk I/O utilization goes up? Does response time change if both processor speed and the number of users on the system are doubled?

How many lines should a time-sharing system have on a dial-in rotary? How many terminals are needed in an on line inquiry center, and how much idle time will the operators have?

The number of questions that can be addressed with a queuing analysis is endless and

touches on virtually every area in computer science. The ability to make such an analysis is an essential

tool for those involved in this field. Although the theory of queuing is mathematically complex, the

application of queuing theory to the analysis of performance is, in many cases, remarkably

straightforward. A knowledge of elementary statistical concepts (means and standard deviations) and a

basic understanding of the applicability of queuing theory is all that is required. Armed with these, the

analyst can often make a queuing analysis on the back of an envelope using readily available queuing

tables, or with the use of simple computer programs that occupy only a few lines of code.

We may consider following two cases:

1. Poisson arrival process (an unlimited number of sources) and exponentially distributed service times (PCT-I).

2. A limited number of sources and exponentially distributed service times (PCT-II).

In this paper we have considered

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