TSSN_CHAP4

Traffic Engineering

C. U. Shah College of Engineering & Technology, Wadhwan city

Introduction Switching elements are not the only parameters which decides

the blocking probability of a network.

There are many parameters like digit receivers, interstage

switching links, call processors and trunk between exchanges

along with, which decide the blocking probability of a

network.

Traffic engineering analysis enables one to determine the

ability of telecommunication network to carry a given traffic at

a particular loss probability.

It provides a means to determine the quantum of common

equipment required to provide a particular level of services for

a given traffic pattern and volume.

2

Network Traffic Load & Parameters

3

busy hour (BH)


In a day, the 60 minute interval in which the traffic is the

highest is called the busy hour (BH).

1) 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.

2) Peak Busy Hour : The busy hour each day; its usually

varies from day to day, or over a number of days.

3) 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.

For ease of records, the busy hour is taken to commence on the

hour or half-hour only.

4


A call attempt is said to be successful or completed if the

called party answers.

Call Completion Rate (CCR) is the ratio of the number of

successful calls to the number of call attempts.

The number of call attempts in the busy hour is called busy

hour call attempts (BHCA), which is an important parameter

in deciding the processing capacity of a common control or a

stored program control system of an exchange.

Networks are usually designed to provide an overall CCR of

over 0.70.

A CCR value of 0.75 is considered excellent and attempts to

further improve the value is generally not cost effective.

5


A related parameter that is often used in traffic engineering

calculation is the busy hour calling rate, which is defined as

the average number of calls originated by a subscriber during

the busy hour.

Another useful information is to know how much of the days

total traffic is carried during the busy hour is day-to-busy

hour traffic ratio which is the ratio of busy hour calling rate

to the average calling rate for the day.

Typically, this ratio may be over 20 for a city business area

and around six or seven for a rural area.

6

Example 1

An exchange serves 2000 subscribers. If the average BHCA is

10,000 and the CCR is 60%, calculate the busy hour calling

rate.

Answer :

Average busy hour calls = BHCA x CCR

= 6000 calls.

Busy hour calling rate =

Average busy hour calls/ Total number of subscribers = 3.

7


8

For analytical treatment, all the common subsystems of a

telecommunication network are collectively termed as servers.

In other publications, the term link or trunk is used.

The traffic on the network may then be measured in terms of

the occupancy of the servers in the network.

Such a measure is called the traffic intensity which is defined

as

Where A0 is dimensionless and called Earlang (E).

0

Period for whicha server is occupiedA

Total period of observation

Example 2

9

In a group of 10 servers, each server is occupied for 30

minutes in an observation interval of two hours. Calculate the

traffic carried by the group.

Answer

= 30/120 = 0.25 E

So total traffic carried by the group = 10 X 0.25 = 2.5 E

0

Period for whicha server is occupiedA

Total period of observation

Example 3

10

A group of 20 servers carry a traffic of 10 Earlangs. If the

average duration of a call is three minutes. Calculate the

number of calls put through by a single server and the group as

a whole in a one hour traffic period

Answer

Traffic carried per servers =

10/20 = 0.5 E (i.e. server is busy for 0.5 x 60=30 minutes in

one hour)

Number of calls put through by one server =

Duration for server is busy in one hour/ Average call duration

= 30/3 = 10 Calls.

So total number of calls put through by the group =

= 10 X 20 =200


Traffic intensity is also measured in another way. This

measure is known as centum call second (CCS) which

represents a call-time product.

One CCS may mean one call for 100 seconds duration or 100

calls for one second duration each or any other combination.

CCS as a measure of traffic intensity is valid only in

telephone circuits. For the present day networks which support

voice, data and many other services, erlang is a better measure

to use for representing the traffic intensity.

Sometimes, call seconds (CS) and call minutes (CM) are

also used as a measure of traffic intensity.

1 E = 36 CCS = 3600 CS = 60 CM.

11

Example 4

A subscriber makes three phone calls of three minutes, four

minutes and two minutes duration in a one-hour period.

Calculate the subscriber traffic in erlangs, CCS and CM.

Answer :

= (3 + 4 + 2)/ 60 = 0.15 E

Traffic in CCS = (3 + 4 + 2) x 60 / 100 =540/100 = 5.4 CCS

Traffic in CM = 3 + 4 + 2 = 9 CM.

12

=


Whenever we use the terminology "subscriber traffic" or

"trunk traffic", we mean the traffic intensity contributed by a

subscriber or the traffic intensity on a trunk.

As mentioned above, traffic intensity is a call-time product.

Hence two important parameters that are required to estimate

the traffic intensity or the network load are

Average call arrival rate, C

Average holding time per call, th

And load offered to the network in terms of these parameter

A = Cth

C and th must be expressed in like time units.

For example, if C is in number of calls per minute, th must be

in minutes per call.13

Example 5

Over a 20-minute observation interval, 40 subscribers initiate

calls. Total duration of the calls is 4800 seconds. Calculate the

load offered to the network by the subscribers and the average

subscriber traffic.

Answer :

Mean arrival rate C =

=40/20 = 2 calls/minute.

Mean arrival time th =

=4800/ (40 X 60) = 2 min/call.

Therefore, offered load =

= 2 X 2 = 4E.

Average subscriber traffic =

=4 /40 = 0.1 E14


We have calculated the traffic in two ways: one based on the

traffic generated by the subscribers and the other based on the

observation of busy servers in the network.

It is possible that the load generated by the subscribers

sometimes exceeds the network capacity.

If overload traffic is rejected than it is Loss system. i.e.

Automatic telephone exchange.

If overload traffic is delayed until the resources are available

then it is Delay system. i.e. operator- oriented manual

exchange.

In the limit, delay systems behave as loss system.

15


The basic performance parameters for a loss system are the

grade of service and the blocking probability, and for a delay

system, the service delays.

Average delays, or probability of delay exceeding a certain

limit or variance of delay may be important under different

circumstances.

The traffic models used for studying loss systems are known

as blocking or congestion models and ones used for studying

delay systems are called queuing models.

16

Grade of Service and Blocking Probability

In a loss system, the amount of traffic rejected by the network

is an index of the quality of the service offered by the network.

This is termed Grade of Service (GOS) and is defined as the

ratio of lost traffic to offered traffic.

Offered traffic is the product of the average number of calls

generated by the users and the average holding time per call.

A =Cth

Actual traffic carried by the network is called the carried

traffic and is the average occupancy of the servers in the

network

17

0

Period for which server is occupiedA

total period of observation


And

Where A =Offered traffic

A0 = carried traffic

A- A0 = lost traffic

Smaller the value of GOS, better the service is.

In India GOS = 0.002, which means 2 calls in every 1000 calls

may be lost.

The GOS of the full network is determined by the highest

GOS value of the subsystems in a simplistic sense.

18

0A AGOSA


Blocking probability PB is defined as the probability that all

the servers in the system are busy.

In a system with equal number of servers and subscribers , the

GOS is zero as there is always a server available to a

subscriber.

On the other hand, there is a definite probability that all the

servers are busy as a given instant and PB is non zero.

The fundamental difference is that the GOS is a measure from

the subscriber point of view whereas PB is a measure from

network or switching point of view.

GOS is defined by observing the number of rejected subscriber

calls, where PB is defined by observing the busy servers in

switching system.

19


GOS is called call congestion or loss probability and the PBis called time congestion.

In the case of delay system, overload traffic is queued, so GOS

is not a useful measure for the same.

So, delay probability, the probability that call experience

delay is a useful measure.

If queue lengthy becomes very large and system becomes

unstable, and easy way to stable the system is to operate

system as loss system until the queue is cleared.

This technique of maintaining the stable operation is called

flow control.

20


Quality of service (QOS) is used in more recent times and it

includes the quality of speech, error-free transmission capacity

etc.

Summary

Subscriber viewpoint

GOS = call congestion = loss probability

Network viewpoint

Blocking Probability = time congestion

21

Modelling Switching System

Subscribers generate calls in a random manner, so call

generation is a random process.

A Random Process or a Stochastic Process is the one in

which one or more quantities vary with time in a such away

that the instantaneous values of the quantities are not

determinable precisely but are predictable with certain

probability. This quantities are called Random Variables.

Figure 8.2 shows typical fluctuations in the number of

simultaneous calls in a half-hour period. The Pattern signifies

a typical random process.

22


23


The values taken on by the random variables of a random

process may be discrete or continuous.

In the case of telephone traffic, the random variable

representing the number of simultaneous calls can take on only

discrete values whereas a random variable representing

temperature variations in an experiment can take on

continuous values. Similarly, the time index of the random

variables can be discrete or continuous. Accordingly, we have

four different types of stochastic processes:

1) Continuous time continuous state

2) Continuous time discrete state

3) Discrete time continuous state

4) Discrete time discrete state.

24


A discrete state stochastic process is often called a chain.

Statistical properties of a random process may be obtained in

two ways:

Time Statistical Parameters, by observing its behaviour over

a very long period of time. Fig 8.3(a)

Ensemble statistical parameters, by observing

simultaneously, a very large number of statistically identical

random sources at any given instant of time. Fig 8.3(b)

25


26


Random processes whose statistical parameters do not change

with time arc known as stationary processes.

The random processes which have identical time and ensemble

averages arc known as ergodic processes.

In some random processes, the mean and the variance alone

are stationary and other higher order moments may vary with

time. Such processes are known as wide-sense stationary

processes.

We model and analyse telephone traffic in segments when they

can be considered to be stationary. In our modelling we use

discrete state stochastic processes.

27

Markov Processes

A.A. Markov proposed a simple and highly useful form of

dependency among the random variable forming a stochastic

process. A discrete time Markov chain, i.e discrete time

discrete state Markov process is defined

Equation says that, the duration for which a process has stayed

in a particular state does not influence the next state transition.

28

1 1 1 1 1 1

1 1

[{ ( ) } /{ ( ) , ( ) ,... ( ) }]

[{ ( ) } /{ ( ) }

n n n n n n

n n n n

P X t x X t x X t x X t x

P X t x X t x

Markov Processes

There are only two distribution function that satisfy this

criterion.

One is the exponential distribution, which is continuous , and

The other is the geometric distribution which is discrete.

Thus, the interstate transition time in a discrete time Markov

Process is geometrically distributed and in a continuous time

Markov process, it is exponentially distributed.

In view of above described property of these distributions,

they are said to be memoryless.

29

Birth-Death Processes

If we apply the restriction that the state transition of a Markov

Chain can occur only to the adjacent states, then we can obtain

Birth- Death (B-D) Proceses.

The number in the population is a random variable and

represents the state value of the process.

The B-D process moves from its state k to state k-1 if a death

occurs or moves to state k+1 if a birth occurs, and stays in the

same state if there is no birth or death during the time period

under consideration, as shown in Fig. 8.4 .

30


31

At Time t + t

At Time t

k +1

k

k -1

k

Death

No Change

Birth

Fig. 8. 4. State transitions at a Birth Death process


A telecommunication network can be modelled as a B-D

process, where the number of busy servers represents the

population, a call request means a birth and call termination

means a death.

In order to analyse a B-D process, we consider a time interval

t small enough such that:

1) There can almost be only one state transition in that

interval.

2) There is only one arrival or one termination but not both,

and

3) There may be no arrival or termination leaving the state

unchanged in the time interval t.

32


We further assume that,

The probability of an arrival or termination in a particular

interval is independent of what had happened in the earlier

time intervals and.

The probability of an arrival is directly proportional to the time

interval t .

Let ,

Pk(t) = The probability that the system is in the state k at time

t, i.e k servers are busy at time t.

k = Call arrival rate in state k.

k = Call termination rate in state k

33


Then, probabilities in the time interval t :

P[exactly one arrival] = t

P[exactly one termination] = t

P[no arrival] = 1 - t

P[no termination] = 1 - t

Probability of finding the system in state k at time t + t

Pk(t + t )

= Pk -1(t) k-1t + Pk +1 (t) k+1 t + (1 - k t )(1 - k t )Pk(t)

The first term on RHS represents the possibility of finding the

system in state k-1 at time t and a birth or a call request

occurring during the interval t to t+ t .

34


The possibility of finding system in state k+1 at time t and a

death or a call termination occurring during the interval t to t+

t is given by the second term.

The last term shows the no arrival and no termination case.

Expanding the equation and ignoring the second order t term

Pk(t + t ) =

= Pk -1(t) k-1t + Pk +1 (t) k+1 t - (k + k )Pk (t)t +Pk(t)

Rearranging the terms

35

1 1 1 1

( ) ( )( ) ( ) ( ) ( )k k k k k k k k k

P t t P tP t P t P t

t


In the limit t0, we get

For k =0, i.e. no calls in progress, there can be no termination

of a call so 0 = 0. Further there can be no state with -1 as the

state value. So

Under Steady state condition

36

1 1 1 1

( )( ) ( ) ( ) ( )k k k k k k k k

dP tP t P t P t

dt

01 1 0 0

( )( ) ( )

dP tP t P t

dt

( )0k

dP t

dt


So B D process becomes stationary. Therefore, the steady

state equation of B-D process are

37

1 1 1 1

1 1 0 0

( ) 0 1

0 0

k k k k k k kP P P for k

P P for k

Incoming Traffic and Service Time Characterisation

When subscriber originates the call, he adds one to the number

of calls arriving at the network.

To model originating process, we define zero death rate. This

is also known as a renewal process.

It is a pure birth process in the sense that it can only add to the

population as the time goes by and cannot reduce the

population by itself.

So considering k = 0 in B-D process

38

1 1

00 0

( )( ) ( ) 0

( )( ) 0

kk k k k

dP tP t P t for k

dt

dP tP t for k

dt


As soon as a birth occurs at t = t1, it is impossible to find the

system ins state 0.

Poisson process is used to find the probability of k births in a

given time interval.

1)

Assuming at t =0, the system is in state zero, i.e. no births

have taken place.

39

1

00

( )( ) ( ) 1

( )( ) 0

kk k

dP tP t P t for k

dt

dP tP t for k

dt

1 0(0)

0 0k

for kP

for k


With these condition we get the solution for probability of 0

arrival in the time interval t.

2)

For k =1

Solving this equation

For k = 2, the solution is

40

0 ( )tP t e

11

( )( ) t

dP tP t e

dt

1( )tP t te

2

2

( )( )

2!

tt eP t


By induction, general solution

3)

Equation (3) is the Poisson arrival process equation. It

represents the probability of k arrival in the time interval t.

41

( )( )

!

k t

k

t eP t

k

Example 6 A rural telephone exchange normally experiences four call

origination per minute. What is the probability that exactly

eight calls occur in an arbitrarily chosen interval of 30 seconds

?

Answer

= 4/60 = 1/15 calls per seconds.

When t = 30s, t = 2

Then probability of exactly eight arrivals is given by

0.00086

42

8 2

8

(2)( )

8!

eP t

Example 7 A switching system serves 10000 subscribers with a traffic

intensity of 0.1 E per subscriber. If there is a sudden spurt in

the traffic ,increasing the average traffic by 50 %, what is the

effect on the arrival rate ?

Answer:

Number of Active subscribers during

A) Normal Traffic = 1000. B) increased traffic =1500

Number of available subscribers for generating new traffic

during

A) Normal Traffic = 9000 . B) increased traffic = 8500

Change in the arrival rate = (500/9000) X 100 = 5.6 %

43


44

Fig. 8.5 Relationship among different Markov Process in the form of Venn diagram.

Poisson Process

k =

Renewal Process

k = 0

B-D Process

k 0

k 0


Venn diagram, we may describe the Poisson process as

1) A pure birth process with constant birth rate.

2) A birth-death process with zero death rate and a constant

birth rate.

3) A Markov process with state transitions limited to the next

higher state or to the same state, and having a constant

transition rate.

45


Example of Poisson process

1) Number of telephone calls arriving at an exchange.

2) Number of coughs generated in a medical ward by the

patients

3) Number of rainy days in a year.

4) Number of typing errors in a manuscript.

5) Number of bit errors occurring in a data communication

system.

46


Let us consider a pure death process.

Considering pure death process

47

1

01

( )( ) ( ) 1

( )( ) 0

kk k

dP tP t P t for k

dt

dP tP t for k

dt

1

01

( )( ) ( ) 1

( )( ) 0

kk k

dP tP t P t for k

dt

dP tP t for k

dt


Assuming t = 0 and assuming suitable boundary condition

Solving these equations,

4)

48

1

01

( )( ) ( ) 0

( )( )

( )( ) 0

kk k

NN

dP tP t P t for k N

dt

dP tP t for k N

dt

dP tP t for k

dt

1

0

( )

( )( ) 0

( )!

( )( ) 0

( 1)!

t

N

N kt

k

Nt

P t e for k N

tP t e for k N

N k

tP t e for k

N


Equation (4) expresses the probability of no termination or

death in a given interval as the initial level. It is the probability

distribution of the service times or the holding times in the

case of calls in a switching system.

The holding times of conventional voice conversations are

well approximated by an exponential distribution. Let tm be the

constant holding time. Then, the probability of finding k busy

channels at any instant of time is merely the probability that

there are k arrivals during the time interval tm immediately

preceding the instant of observation

49

( ) exp( )( )

!

k

m mk m

t tP t

k

Blocking Models and Loss Estimates

The behavior of a loss system is studied by using blocking

models and that of the delay systems by using queuing models.

Three aspects of telecommunication system

1) Modelling the system : B-D process

2) Traffic arrival Model : Poisson Process

3) Service time distribution: Considering for holding time as

an exponential or constant time distribution.

50

Blocking Models and Loss Estimates

There are three ways in which overflow traffic may be

handled:

1) Lost Call Cleared (LCC) : The traffic rejected by one set of

resources may be cleared by another set of resources in the

network.

2) Lost Calls Returned (LCR) : The traffic may return to the

same resources after sometime.

3) Lost Calls Held (LCH) : The traffic may be held by the

resources as if being serviced but actually serviced only after

the resources become available.

51

Documents

TSSN_CHAP4