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Design and SimulationTRANSCRIPT
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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
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Design and Simulation of Traffic in Queueing Network 2
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unpublished works.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
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Design and Simulation of Traffic in Queueing Network 3
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research and thesis consists of materials copied or plagiarized from published or unpublished work of
the writers and that all materials borrowed or reproduced from other published or unpublished source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
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Design and Simulation of Traffic in Queueing Network 4
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.
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Design and Simulation of Traffic in Queueing Network 5
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.
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Design and Simulation of Traffic in Queueing Network 6
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.
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Design and Simulation of Traffic in Queueing Network 7
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
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Design and Simulation of Traffic in Queueing Network 8
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
2.5 Concluding Remarks..42
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
3.8 Concluding Remarks..89
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
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Design and Simulation of Traffic in Queueing Network 9
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
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Design and Simulation of Traffic in Queueing Network 10
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
Figure 1.11 Sample output of simulation program where offered traffic was 4 Erl
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
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Design and Simulation of Traffic in Queueing Network 11
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
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Design and Simulation of Traffic in Queueing Network 12
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
Figure 3.27 Variation of blocking probability against termination rate mu2 68
Figure 3.28 Variation of probability of using nominal channel against arrival rate lamda2
69
Figure 3.29 Variation of probability of using nominal channel against termination rate mu2
69
Figure 3.30 Variation of probability of entering queue against arrival rate lamda2
70
Figure 3.31 Variation of probability of entering queue against termination rate mu2
70
Figure 3.32 State transition diagram with 2-D queueing model with no of channel=1 and no of queue=2
71
Figure 3.33 Variation of blocking probability against arrival rate lamda1 73
Figure 3.34 Variation of blocking probability against termination rate mu1 73
Figure 3.35 Variation of probability of using nominal channel against 74
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Design and Simulation of Traffic in Queueing Network 13
arrival rate lamda1
Figure 3.36 Variation of probability of using nominal channel against termination rate mu1
74
Figure 3.37 Variation of probability of entering queue against arrival rate lamda1
75
Figure 3.38 Variation of probability of entering queue against termination rate mu1
75
Figure 3.39 Variation of blocking probability against arrival rate lamda2 76
Figure 3.40 Variation of blocking probability against termination rate mu2 76
Figure 3.41 Variation of probability of using nominal channel against arrival rate lamda2
77
Figure 3.42 Variation of probability of using nominal channel against termination rate mu2
77
Figure 3.43 Variation of probability of entering queue against arrival rate lamda2
78
Figure 3.44 Variation of probability of entering queue against termination rate mu2
78
Figure 3.45 State transition diagram with 2-D queueing model with no of channel=2 and no of queue=3
79
Figure 3.46 Variation of blocking probability against arrival rate lamda1 83
Figure 3.47 Variation of blocking probability against termination rate mu1 83
Figure 3.48 Variation of probability of using nominal channel against arrival rate lamda1
84
Figure 3.49 Variation of probability of using nominal channel against termination rate mu1
84
Figure 3.50 Variation of probability of entering queue against arrival rate lamda1
85
Figure 3.51 Variation of probability of entering queue against termination rate mu1
85
Figure 3.52 Variation of blocking probability against arrival rate lamda2 86
Figure 3.53 Variation of blocking probability against termination rate mu2 86
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Design and Simulation of Traffic in Queueing Network 14
Figure 3.54 Variation of probability of using nominal channel against arrival rate lamda2
87
Figure 3.55 Variation of probability of using nominal channel against termination rate mu2
87
Figure 3.56 Variation of probability of entering queue against arrival rate lamda2
88
Figure 3.57 Variation of probability of entering queue against termination rate mu2
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
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Design and Simulation of Traffic in Queueing Network 15
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
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Design and Simulation of Traffic in Queueing Network 16
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.
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Design and Simulation of Traffic in Queueing Network 17
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.
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Design and Simulation of Traffic in Queueing Network 18
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
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Design and Simulation of Traffic in Queueing Network 19
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:
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Design and Simulation of Traffic in Queueing Network 20
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
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Design and Simulation of Traffic in Queueing Network 21
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
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Design and Simulation of Traffic in Queueing Network 22
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
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Design and Simulation of Traffic in Queueing Network 23
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.
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Design and Simulation of Traffic in Queueing Network 24
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.
-
Design and Simulation of Traffic in Queueing Network 25
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
-
Design and Simulation of Traffic in Queueing Network 26
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
-
Design and Simulation of Traffic in Queueing Network 27
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
-
Design and Simulation of Traffic in Queueing Network 28
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.
-
Design and Simulation of Traffic in Queueing Network 29
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.
-
Design and Simulation of Traffic in Queueing Network 30
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
-
Design and Simulation of Traffic in Queueing Network 31
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%.
-
Design and Simulation of Traffic in Queueing Network 32
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 =
-
Design and Simulation of Traffic in Queueing Network 33
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 =
-
Design and Simulation of Traffic in Queueing Network 34
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
-
Design and Simulation of Traffic in Queueing Network 35
: : 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)
-
Design and Simulation of Traffic in Queueing Network 36
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:
Figure 1.10 Sample output of simulation program where offered traffic was 6 Erl
-
Design and Simulation of Traffic in Queueing Network 37
Figure 1.11 Sample output of simulation program where offered traffic was 4 Erl
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.
-
Design and Simulation of Traffic in Queueing Network 38
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.
-
Design and Simulation of Traffic in Queueing Network 39
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
-
Design and Simulation of Traffic in Queueing Network 40
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.
-
Design and Simulation of Traffic in Queueing Network 41
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
-
Design and Simulation of Traffic in Queueing Network 42
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)
-
Design and Simulation of Traffic in Queueing Network 43
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
-
Design and Simulation of Traffic in Queueing Network 44
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
-
Design and Simulation of Traffic in Queueing Network 45
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:
-
Design and Simulation of Traffic in Queueing Network 46
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.
-
Design and Simulation of Traffic in Queueing Network 47
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)
-
Design and Simulation of Traffic in Queueing Network 48
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.
-
Design and Simulation of Traffic in Queueing Network 49
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
-
Design and Simulation of Traffic in Queueing Network 50
=
=
=
==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
-
Design and Simulation of Traffic in Queueing Network 51
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.
-
Design and Simulation of Traffic in Queueing Network 52
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.
-
Design and Simulation of Traffic in Queueing Network 53
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.
-
Design and Simulation of Traffic in Queueing Network 54
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
-
Design and Simulation of Traffic in Queueing Network 55
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