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1998 Pr Karlsson, Department of Telecommunications and Mathematics,

University of Karlskrona/Ronneby

Teletraffic theory

Laboratory exercise 3:

Queuing Theory

Name Program Email Box no. Approved

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Teletraffic theory, Laboratory exercise 3: Queuing Theory, ITM, HK/R, 1998

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2 Home exercises

2.1 System description

During this exercise, we will study the following, imaginary, communication system. The system,

which is packet based, transports two types of traffics, each with its own properties and requirements.

We will for the remainder of this exercise focus on a single node in this communication system, see

Figure 1. In order to keep things simple we assume that all packets arriving to the node are destined

for the same outgoing link. The bandwidth of the outgoing link is denoted B Mbit/s. Users that

generate traffic are connected to this node with access links that have a much higher bandwidth than B.

The total number of users is M.

Figure 1: System overview

The first type of traffic is a data communication service. This is a non real-time service so it can

tolerate rather large delays of individual packets. Hereafter this service is denoted "data". Assume that

each user generates data packets according to a Poisson process with intensity d. (Note: this

assumption is probably somewhat unrealistic.) Including lower layer headers, each data PDU (ProtocolData Unit) is ldbytes (octets) long.

The second type of traffic comes from a voice communication application that has stringent delay

requirements. Short samples of voice are packetized and sent individually as packets over the system.

Voice packets from each user arrive according to a Poisson process with intensity v. (Again, this issomewhat unrealistic.) The size of a voice PDU is lv bytes.

In front of the outgoing link, buffering of packets is needed, for our analysis we can assume this buffer

to be of unlimited capacity.

During the exercise we will compare two different solutions to the internal structure of the node. The

first, and simplest, approach treats the data and voice packets identically. As packets arrive to the

node, they are placed in a buffer. The packets are then served in FIFO order, see Figure 2.

Figure 2: Design with common buffer

Outgoing link

Access links

Communication node

1

2

Data and

voice buffer

Voice packet

Data packet

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2.1.1

Explain why the data and the voice service differ in their delay requirements.

2.1.2

Explain why a buffer is needed in front of the outgoing link.

2.1.3

What advantages and disadvantages can you think of with the suggested design?

2.1.4

Given the definitions above, determine the time it takes to transmit a data and a voice PDU over the

outgoing link. Call these times xdand xv, respectively.

xd xv

2.1.5

Determine the probability that an arriving packet is a data or a voice packet.

pd= P(data packet) pv = P(voice packet)

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2.2 M/G/1

Let us now consider a theoretical modeling of the system described above. What we mainly are

interested in is the queuing process in the buffer.

2.2.1

What is the arrival process of packets into the node?

2.2.2

What is the service time distribution for sending packets over the outgoing link?

2.2.3

Determine the first and second moment of the service time distribution.

x2

x

2.2.4Determine

2

xC , the squared coefficient of variation of the service time distribution.

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2.2.5

Considering your answers to 2.2.1 and 2.2.2, suggest a queuing system that can be used to model the

system.

2.2.6

Determine - the utilization of the system, as a function ofM, d, v, ld, lv, and B.

2.2.7

Use your answers to 2.2.1 - 2.2.6 to derive the mean number of packets in the node expressed as a

function ofM, d, v, ld, lv, and B.

2.2.8

Determine the mean waiting time in the buffer for an arbitrary packet as a function ofM, d, v, ld, lv,and B.

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2.3 Priority queuingThe second approach to the internal design of the node is presented in Figure 3.

Figure 3: Design with separated buffers

Here data and voice packets are separated and placed in different buffers. Once the transmission of a

packet has finished over the outgoing link a new packet is fetched from the voice buffer and

transmitted. If no voice packet is available, a data packet is transmitted. If no packet is available at all

in the system, the first packet to arrive of any type will receive service. The packets are served in a

non-preemptive manner.

2.3.1

Compared to the first design, what advantages/disadvantages does this design display?

2.3.2

Explain the meaning of a non-preemptive manner in the context of the system described above.

2.3.3

Explain why a preemptive service of packets (as opposed to a non-preemptive) is less desirable in the

case of transmission of packets.

Voice buffer

Data buffer

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2.3.4

Considering the design presented above, suggest a queuing system that can be used to analyze the

performance.

2.3.5

Determine the first and second moments of the service time distributions for voice and data packets.

v

xd

x2

v

x2

d

x

2.3.6

Determine v and d, the offered load of data and voice traffic as a function ofM, d, v, ld, lv, and B.

2.3.7

Determine the mean delays caused by buffering for packets belonging to both services (voice and data

packets) as functions ofM, d, v, ld, lv, and B.

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3 Laboratory Exercises

3.1 M/G/1

Using your results from the home exercises, we will now further study the performance of our

proposed system. Unless otherwise stated, use the following parameters in the investigations below.

B 10 Mbit/s

v 10/s

d 5/s

lv 100 bytes

ld 1500 bytes

3.1.1

Complete the following table.

xv

xd

P(voice packet)

P(data packet)

2

xC

3.1.2

Considering design approach number one, determine the maximum number of users (Mmax) that can be

connected to the node before it becomes overloaded.

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3.1.3

Would you recommend connecting Mmax users to the node? What can be said about the expected

waiting times?

3.1.4

Using your result from 2.2.8, write a Matlab function that returns the average buffering time as a

function ofM, d, v, ld, lv, and B:

I X Q F W L R Q : Z G H V L J Q 0 O D P E G D G O D P E G D Y O G O Y %

where : and 0 are vectors, and O D P E G D G , O D P E G D Y , O G , O Y , and % are scalars.

Test case:

Z G H V L J Q > @ H

D Q V

3.1.5

Using your function, plot the mean buffering time as a function of the number of users. (Attach the

plot as an appendix to this report). (Hints: a good idea is to use a logarithmic scale for the y-axis, see

the Matlab command V H P L O R J \ . To annotate your plot you can use [ O D E H O , \ O D E H O , W L W O H .)

3.1.6

Let us assume that the maximum tolerable mean buffering delay for the voice packets is 0.5 ms. What

is the maximum number of users now?

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3.1.7

Let us consider what happens if the data PDU size (ld) is decreased 10 times and d is increased 10times. This corresponds to keeping the offered traffic from each user constant. Note: in real life this

would probably lead to an increase in the traffic since we usually have a fixed amount of overhead in

each PDU.

What do you expect to happen with the average buffering delay with this new setup (keeping all other

parameters constant)? Explain why. (Hint: calculate2

xC .)

3.1.8

Using Matlab, plot the mean buffering time as a function of the number of users. (Attach the plot as an

appendix to this report).

3.1.9

Let us again assume that the maximum tolerable mean buffering delay for the voice packets is 0.5 ms.

What is the maximum number of users now?

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3.2 Priority queuing

3.2.1Considering the second design approach, determine the maximum number of users that can be

connected to the node before the queue of voice packets grows unbounded?

3.2.2

How is then the situation for in the data buffer? What is the maximum number of users if both the

voice and data queue should be stable?

3.2.3

Using your result from 2.2.8, write a Matlab function that returns the average buffering times for data

and voice packets as a function ofM, d, v, ld, lv, and B:

I X Q F W L R Q > : Y : G @ Z G H V L J Q 0 O D P E G D G O D P E G D Y O G O Y %

Z K H U H : Y : G D Q G 0 D U H Y H F W R U V D Q G O D P E G D G O D P E G D Y O G O Y D Q G % D U H V F D O D U V

Test case:

> Z Y Z G @ Z G H V L J Q > @ H

Z Y

Z G

3.2.4

Using your function, plot the mean buffering times for data and voice packets as a function of the

number of users. (Attach the plot as an appendix to this report).

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3.2.5

Let us assume that the maximum tolerable mean buffering delay for the voice packets is 0.5 ms. What

is the maximum number of users now? How does your answer compare to the answer in 3.1.6?

3.2.6

The average delay an arbitrary packet (voice or data) experience can be defined in different ways. Plot

the average delay (as a function of the number of users) produced by weighting the individual delays

on voice and data packets with pv, pdor v / (v+d), d/ (v+d). Comment on the result. (Hint:compare the plots with the one produced in 3.1.5).

3.2.7

Our study so far has only been concerned with the mean delay caused by buffering. What other

parameters, besides the mean, can you think of that might be of importance?

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