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NH – 67, Karur – Trichy Highways, Puliyur C.F, 639 114 Karur District

DEPARTMENT OF INFORMATION TECHNOLOGY

COMMUNICATION SWITCHING TECHNIQUES

UNIT 4

TRAFFIC ENGINEERING

Contents

Network Traffic Load and Parameters

Grade of Service and Blocking Probability

Modeling Switching Systems

Loss Systems

Delay Systems

Overview

The telecommunication system has to service the voice traffic and data traffic.

The traffic is defined as the occupancy of the server. The basic purpose of the traffic

engineering is to determine the conditions under which adequate service is provided to

subscribers while making economical use of the resources providing the service. The

functions performed by the telecommunication network depends on the applications it

handles. Some major functions are switching, routing, flow control, security, failure

monitoring, traffic monitoring, accountability internetworking and network management.

To perform the above functions, a telephone network is composed of variety of

common equipment such as digit receivers, call processors, interstage switching links

and interoffice links etc. Thus traffic engineering provides the basis for analysis and

design of telecommunication networks or model. It provides means to determine the

quantum of common equipment required to provide a particular level of service for a

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given traffic pattern and volume. The developed model is capable to provide best

accessibility and greater utilization of their lines and trunks. Also the design is to provide

cost effectiveness of various sizes and configuration of networks.

The traffic engineering also determines the ability of a telecom network to carry a

given traffic at a particular loss probability. Traffic theory and queuing theory are used to

estimate the probability of the occurrence of call blocking. Earlier traffic analysis based

purely on analytical approach that involved advanced mathematical concepts and

complicated operations research techniques. Present day approaches combine the

advent of powerful and affordable software tools that aim to implement traffic

engineering concepts and automate network engineering tasks.

In this unit, the traffic design requirements, various probability distributions, loss

systems and delay systems are described.

4.1 Network Traffic Load and Parameters

In a telephone network, the traffic load on a typical working day during the 24

hours is shown in figure. Obviously, there is little use of the network during 0 and 6

hours when most of population is asleep. There is a large peak around mid-forenoon

and mid-afternoon signifying busy office activities. The afternoon peak is, however

slightly smaller. The load is low during the lunch hour period i.e. 12.00-14.00 hours. The

period 17.00-18.00 hours is characterized by low traffic signifying that the people are on

the move from offices to their residences. The peak of domestic calls occurs after 18.00

hours when persons reach home and reduced tariff applies. In many countries including

India the period during which the reduced tariff applies has been changed to begin later

than 18.00 hours and one may expect the domestic call patterns also to change

accordingly. During holidays and festival days the traffic pattern is different from the

shown in figure. Generally there is a peak of calls around 10.00 hours just before people

leave their homes on outings and another peak occurs in the evening.

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

busy hour (BH). In figure the 1-hour period between 11.00 and 12.00 hours is the busy

hour. The busy hour may vary from exchange to exchange depending on the location

and the community interest of the subscribers. The busy hour may also show seasonal,

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weekly and in some places even daily variations. In addition to these variations, there

are also unpredictable peaks caused by stock market or money activity, weather,

natural disaster, international events, sporting events etc. To take into account such

fluctuations while designing switching networks, three types of busy hours are defined

by CCITT in its recommendations:

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 it 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 considerations.

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

only.

Not all call attempts materialize into actual conversations for a variety of reasons

such as called line busy, no answer from the called line and blocking in the trunk groups

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

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called party answers. Call Completion Rate (CCR) is defined as 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. The CCR parameter is used in dimensioning the network

capacity. 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. A related parameter that is often used in traffic engineering

calculations is the busy hour calling rate which is defined as the average number of

calls originated by a subscriber during the busy hour.

The busy hour calling rate is useful in sizing the exchange to handle the peak

traffic. In a rural exchange, the busy hour calling rate may be as low as 0.2, whereas in

a business city it may be as high as three or more. Another useful information is to know

how much of the day’s total traffic is carried during the busy hour. This is measured in

terms of day-to-day 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.

The traffic load on a given network may be on the local switching unit, inter office

trunk lines or other common sub systems. For analytical treatment all the common

subsystems of a telecommunication network are termed as servers or link or trunk. 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

Ao = Period for which server is occupied / total period of observation.

Generally the period of observation is taken as one hour. Ao is obviously

dimensionless. It is called erlang (E) to honour the Danish telephone engineer

A.K.Erlang, who did pioneering work in traffic engineering. A server is said to have 1

erlang of traffic if it is occupied for the entire period of observation. Traffic intensity may

also be specified over a number of servers.

Erlang measure indicates the average number of servers occupied and is useful

in deriving the average number of calls put through during the period of observation.

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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. We may note

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

Whenever we use the terminology “subscriber traffic” or “trunk office”, we mean

the traffic intensity contributed by a subscriber or the traffic intensity on a trunk. As

mentioned above, the 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

We can then express the load offered to the network in terms of these parameter

as

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.

In the above discussions 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

sometime exceeds the network capacity. There are two ways in which this overload

traffic may be handled. The overload traffic may be rejected without being serviced or

held in a queue until the network facilities become available. In the first case, the calls

are lost and in the second case the calls are delayed. Correspondingly, two types of

systems called loss systems and delay systems are encountered. Conventional

automatic telephone exchanges behave like loss systems. Under overload traffic

conditions a user call is blocked and is not serviced unless the user makes a retry. On

the other hand, operator oriented manual exchanges can be considered as delay

systems. A good operator registers the user request and establishes connection as

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soon as network facilities become available without the user having to make another

request. In data networks, circuit switched networks behave as loss systems whereas

store and forward (S&F) message or packet networks behave as delay systems. For

example, in S&F network if the queue buffers become full, then further requests have to

be rejected.

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 delays may be important

under different circumstances. The traffic models used for studying loss systems are

known as blocking or congestion models and the ones used for studying delay systems

are called queuing models.

4.2 Grade of Service and Blocking probability

4.2.1 Grade of Service

For non-blocking service of an exchange, it is necessary to provide as many lines

as there are subscribers. But it is not economical. So, some calls have to be rejected

and retried when the lines are being used by other subscribers. The grade of service

refers to the proportion of unsuccessful calls relative to the total number of calls. GOS is

defined as the ratio of lost traffic to offered traffic.

GOS = yHourCallsOfferedBus

yHourCallsBlockedBus

GOS = A

AoA

Where Ao = carried traffic

A = offered traffic

A-Ao = loss traffic

The smaller the value of grade of service, the better is the service. The

recommended GOS is 0.002, i.e. 2 call per 1000 offered may lost. In a system, with

equal no. of servers and subscribers, GOS is equal to zero.

GOS is applied to a terminal to terminal connection. But usually a switching

centre is broken into following components

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(a) an internal call (subscriber to switching office)

(b) an outgoing call to the trunk network (switching office to trunk)

(c) the trunk network (trunk to trunk)

(d) a terminating call (switching office to subscriber).

The GOS calculated for each component is called component GOS. The overall

GOS is in fact approximately the sum of the component grade of service.

There are two possibilities of call blocking. They are (a) Lost system and (b)

Waiting system. In lost system, a suitable GOS is a percentage of calls which are lost

because no equipment is available at the instant of call request. In waiting system, a

GOS objective could be either the percentage of calls which are delayed or the

percentage which are delayed more than a certain length of time.

Example : During a busy hour, 1400 calls were offered to a group of trunks and 14calls

were lost. The average call duration has 3 minutes. Find (a) Traffic offered (b) Traffic

carried (c) GOS and (d) The total duration of period of congestion.

Given data: n = 1400

h = 3

T = 60, lost calls = 14

Sol.

(a) Traffic offered A = 60

31400= 70E

(b) Traffic carried A0 = 60

31386= 69.3E

(c) GOS = A

AoA

where A – A0 = 70 – 69.3 = 0.7 E (lost traffic)

GOS = 3.69

7.0= 0.01

(d) Total duration = 0.01 × 3600 = 36 seconds.

4.2.2 Blocking Probability

The value of the blocking probability is one aspect of the telephone company’s

grade of service. The basic difference between GOS and blocking probability is that

GOS is a measure from subscriber point of view whereas the blocking probability is a

measure from the network or switching point of view. Based on the number of rejected

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calls, GOS is calculated, whereas by observing the busy servers in the switching

system, blocking probability will be calculated. The blocking probabilities can be

evaluated by using various techniques. Lee graphs and Jacobaeus methods are

popular and accurate methods . The blocking probability B is defined as the probability

that all the servers in a system are busy.

Congestion theory deals with the probability that the offered traffic load exceeds

some value. Thus, during congestion, no new calls can be accepted. There are two

ways of specifying congestion. They are time congestion and call congestion. Time

congestion is the percentage of time that all servers in a group are busy. The call or

demand congestion is the proportion of calls arising that do not find a free server. In

general GOS is called call congestion or loss probability and the blocking probability is

called time congestion.

If the number of sources is equal to the number of servers, the time congestion is

finite, but the call congestion is zero. When the number of sources is large, the

probability of a new call arising is independent of the number already in progress and

therefore the call congestion is equal to time congestion.

4.3 Modeling Switching System

To analyze the statistical characteristics of a switching system, traffic flow and

service time, it is necessary to have a mathematical model of the traffic offered to

telecommunication systems. The model is a mathematical expression of physical

quantity to represents the behavior of the quantity under consideration. Also the model

provides an analytical solution to a teletraffic problem. As the switching system may be

represented in different ways, different models are possible. Depending on the

particular system and particular circumstance, a suitable model can be selected.

In practice, the facilities of the switching systems are shared by many users. This

arrangement may introduce the possibility of call setup inability due to lack of available

facilities. Also in data transfer, a system has to buffer message while waiting for

transmission. Here size of the buffer depends on traffic flow. As serving the number of

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subscribers subject to fluctuation (due to random generation of subscriber calls,

variations in holding time, location of the exchange, limitation in servers etc), modeling

of traffic is studied using the concepts and methods of the theory of probability.

If a subscriber finds no available server for his call attempt, he will wait in a line

(queue) or leave immediately. This phenomenon may be regarded as a queuing

system. The mathematical description of the queuing system characteristic is called a

queuing model.

Once a mathematical model is obtained, various analytical and computational

tools can be used for analysis and synthesis purposes.

4.3.1 Elements of Probability

The traffic generated by a subscriber is random in telecommunication switching

system. The behavior of the network for the random request for service is random

process. In this section, some elemental ideas of probability theory and probability

distributions are discussed.

Probability: Probability can be defined as the relative frequency of occurrence of

a random event. Each event has a probability defined as the ratio of the number of

times it occurs to the total number of trials. Thus the probability of the occurrence of an

event A for N trials is

P (A) = Lim N N

NA

The probability of occurrence of an event P is a positive number and that 0 ≤ P ≤

1. If an event is not possible P = 0, while if an event is certain, P = 1.

Mutually Exclusive events: Two possible outcomes of an experiment are

defined as being mutually exclusive if the occurrence of one outcome precludes the

occurrence of the other. If the events are A and B with probabilities P(A) and P(B), then

the probability of occurrence of either A or B is written as

P (A or B) = P(A) + P(B)

For more than two mutually exclusive outcomes, say A1, A2……An

P (A1 or A2 or ….. AL) =

L

j

AjP1

)(

For example, in tossing a coin, if head occurs, occurrence of tail cannot take

place.

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Conditional and Joint probability. Suppose that we contemplate two

experiments A and B with outcomes A1, A2 ... and B1, B2 ... The probability of outcome

Bk, given that Aj is known to have occurred is called conditional probability given as

P (Bk | Aj) = )(

),(

AjP

BkAjP

Similarly, the probability of outcome Aj, given that Bk is known to have occurred

is given as

P (Aj |Bk) = )(

),(

BkP

BkAjP

Where P (Aj, Bk) is called joint probability that is the joint occurrence of Aj and

Bk.

P (Aj, Bk) = P(Bk|Aj) P(Aj) = P(Aj, Bk) P(Bk)

P(Aj | Bk) = )|()(

)(AjBkP

BkP

AjP

This result is known as Bayes’ theorem.

If the outcome of Bk does not depend at all on which outcome Aj accompanies it,

we say that the outcomes Aj and Bk are independent. When outcomes are independent,

the probability of a joint occurrence of particular outcomes is the product of the

probabilities of the individual independent outcomes. It is given as

P(Aj, Bk) = P(Aj) P(Bk)

This result may be extended to any arbitrary number of outcomes. Thus

P(Aj, Bk, Ci, ...) = P(Aj) P(Bk) P(Ci)

Random variables and Random process. Subscribers generate calls in

random manner. The call generation by the subscribers and therefore the behavior of

the network or the switching system is described as a random process. It is also

referred as stochastic process. In random process, one or more quantities vary with

time in such a way that the instantaneous values of the quantities are not determinable

but are predictable with certain probability. The quantities are called random variables.

In telecommunication system, telephone traffic is referred as random process

and the number of simultaneous active subscribers and simultaneous busy servers are

assumed as random variables. The variation of traffic over a period of time (30 minutes

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or 60 minutes) is a typical random process. The random process may be discrete or

continuous. In telecommunication, the variable representing the number of

simultaneous calls is discrete. Thus, in our modeling we use discrete state stochastic

processes.

Definition of statistical terms:

Mean: E(x) =

N

i N

xi

1

where x = a random variable

N = number of trials

xi = the outcome of the individual trials.

Mean is also referred as the expectation or average of x and represented by .

Variance: The variance of x is

Var (x) = N

xExiN

i

1

2))((= 21

2

)()(

xEN

xiN

i

The variance of a process is a measure of how the individual outcomes differ

from the mean.

Standard deviation: The standard deviation of a random variable x is given by

)var()( xx

The ratio of standard deviation to the mean of a random variable x is given by

)(

)()(

xE

xx

4.3.2 Discrete Probability Distributions

While the mean and variance tell a great deal about a random variable, they do not tell

us everything. The most complete information is given by the distribution of the random

variable. The distribution is the probability associated with each possible outcome.

Mean and variance. The values of P(x1), P(x2), ..., P(xn) for a discrete random

process X = x1, x2, ..., x may be plotted as shown in Fig.

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The value of x corresponding to the centre of gravity of the above diagram is

called the average or mean or expectation E(x)

E(x) =

1

)(j

jxjP

P(xj) must be non-negative and must sum to 1. The mean is also known as the

first moment of the random variable. k , the kth moment of the random variable is

defined as

)()( xjPxjk k

The variance Var(X) or 2 is a measure of the dispersion or spread of the

histogram. It is defined as the mean squared deviation of x from the mean.

Var (x) =

1

22 )()(j

xjPxj

= 2

1

2))((

j

xjxjP

The variance is generally referred as second central moment. The kth central

moment is defined as

Ck = k

j

xjxjP ))((1

The standard deviation, if defined as the square root of the variance.

)var(xx

If X and Y are independent random variable, the mean and variance of sum or

difference is given by

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E( X + Y) = E(X) + E(Y)

Var (X + Y) = Var (X) + Var (Y)

E(X – Y) = E(X) – E(Y)

Var (X – Y) = Var (X) + Var (Y)

Bernouilli or binomial distribution: The distribution of x repetitions of an event (say

head of toss) with two possible outcomes is called a binomial distribution and the

numbers above are called binomial coefficients. Consider the series of trials (n) satisfies

the following conditions:

(a) Each trial can have two possible outcomes.

(b) The outcome of each trial is an independent random event.

(c) Statistical equilibrium

The number of ways choosing an x things out of n trial is given as

)!(!

!),(

xnx

nxnC

x

n

C(n, x) is called binomial coefficient.

The probability of one particular combination of x success and n – x failures is

px (1 – p)n–x. Given two disjoint events with probabilities p and (1 – p), the probability of

first occurring x times and second occurring (n – x) times in n trial, the most general

form of binomial distribution is

P(n,x,p) = C(n,x) Px (1-P)n-x

The mean is np

The Variance is )1(2 pnp

4.3.3 Continuous Probability DistributionsThe random process may be discrete or continuous. For continuous function, the

histogram becomes a continuous. As probability P(x) tends to zero, the probability

density function p(x) is plotted against x. The probability density p(x) is defined by

p(x) = P(a ≤ x ≤ b) = b

a

dxxp )(

and b

a

dxxp )( = 1

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For continuous distribution,

Mean,

dxxp )(

Variance, 2

dxxpx 22 )(

For independent random variable X and Y,

mean, E(X ± Y) = E(X) + E(Y)

Var (X ± Y) = Var (X) + Var (Y)

Negative exponential distribution. The probability that the interval between event T,

will exceed time t is given by

p (T ≥ t) = Tte /

where T is the mean interval between events.

Gaussian or normal distribution. A random variable, x is said to be normally

distributed if its density function has the form

n(22 2/)(

22

1)():,

xexpx

where = mean

= standard deviation.

The equation is plotted in figure.

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The central limit theorem states that the probability density function of the sum of

a large number of independent variables tends towards n( , : x) as ‘n’ increases.

Substituting, t = (x – )/ in equation , and neglecting

P(t) = 2/2

2

1 te

This equation is called standard normal distribution.

4.3.4 Statistical Parameters

The random process may be discrete or continuous. Similarly the time index of

random variables can be discrete or continuous. Thus, there are four different types of

process namely (a) continuous time continuous state (b) continuous time discrete state

(c) discrete time continuous state and (d) discrete time discrete state. In

telecommunication switching system, our interest is discrete random process and

therefore for modeling a switching system, we use discrete state stochastic process. A

discrete state stochastic process is often called a chain.

A statistical properties of a random process may be obtained in two ways :

(i) Observing the behavior of the system to be modeled over a period of time

repeatedly. The data obtained is called a single sample. The average determined by

measurements on a single sample function at successive times will yield a time

average.

(ii) Simultaneous measurements of the output of a large number of statistically

identical random sources. Such a collection of sources is called an ensemble and the

individual noise waveform is called the sample function. The statistical average made

at some fixed time t = t1 on all the sample functions of the ensample is the ensemble

average.

The above two ways are analogous to obtaining the statistics from tossing a die

repeatedly (large number) or tossing one time the large number of dice.

In general, time average and ensemble average are not the same due to various

reasons. When the statistical characteristics of the sample functions do not change with

time, the random process is described as being stationary. The random process which

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have identical time and ensemble average are known as ergodic processes. An

ergodic process is stationary, but a stationary process is not necessarily ergodic.

Telephone traffic is non stationary. But the traffic obtained during busy hour may

be considered as stationary (which is important for modeling) as modeling non-

stationary is difficult.

4.4 Loss Systems

The service of incoming calls depends on the number of lines. If number of lines

equal to the number of subscribers, there is no question of traffic analysis. But it is not

only uneconomical but not possible also. So, if the incoming call finds all available lines

busy, the call is said to be blocked. The blocked calls can be handled in two ways.

The type of system by which a blocked call is simply refused and is lost is called

loss system. Most notably, traditional analog telephone systems simply block calls

from entering the system, if no line available. Modern telephone networks can

statistically multiplex calls or even packetize for lower blocking at the cost of delay. In

the case of data networks, if dedicated buffer and lines are not available, they block

calls from entering the system.

In the second type of system, a blocked call remains in the system and waits for

a free line. This type of system is known as delay system. In this section loss system is

described.

These two types differs in network, way of obtaining solution for the problem and

GOS. For loss system, the GOS is probability of blocking. For delay system, GOS is the

probability of waiting.

Erlang determined the GOS of loss systems having N trunks, with offered traffic

A, with the following assumptions. (a) Pure chance traffic (b) Statistical equilibrium (c)

Full availablity and (d) Calls which encounter congestion are lost. The first two are

explained in previous section. A system with a collection of lines is said to be a fully-

accessible system, if all the lines are equally accessible to all in arriving calls. For

example, the trunk lines for inter office calls are fully accessible lines. The lost call

assumption implies that any attempted call which encounters congestion is immediately

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cleared from the system. In such a case, the user may try again and it may cause more

traffic during busy hour.

The Erlang loss system may be defined by the following specifications.

1. The arrival process of calls is assumed to be Poisson with a rate of calls per

hour.

2. The holding times are assumed to be mutually independent and identically

distributed random variables following an exponential distribution with 1/

seconds.

4. Calls are served in the order of arrival.

There are three models of loss systems. They are :

1. Lost calls cleared (LCC)

2. Lost calls returned (LCR)

3. Lost calls held (LCH)

All the three models are described in this section.

4.4.1 Lost Calls Cleared (LCC) System

The LCC model assumes that, the subscriber who does not avail the service,

hangs up the call, and tries later. The next attempt is assumed as a new call. Hence,

the call is said to be cleared. This also referred as blocked calls lost assumption. The

first person to account fully and accurately for the effect of cleared calls in the

calculation of blocking probabilities was A.K. Erlang in 1917.

Consider the Erlang loss system with N fully accessible lines and exponential

holding times. The Erlang loss system can be modeled by birth and death process with

birth and death rate as follows.

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The probability P(0) is determined by the normalization condition

The probability distribution is called the truncated Poisson distribution or Erlang’s

loss distribution. In particular when k = N, the probability of loss is given by

This result is variously referred to as Erlang’s formula of the first kind, the Erlangs-B

formula or Erlangs loss formula.

The above Equation specifies the probability of blocking for a system with

random arrivals from an infinite source and arbitrary holding time distributions. The

Erlang B formula gives the time congestion of the system and relates the probability of

blocking to the offered traffic and the number of trunk lines.

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In design problems, it is necessary to find the number of trunk lines needed for a

given offered traffic and a specified grade of service. The offered load generated by a

Poisson input process with a rate calls per hour may be defined as

where t = average number of increasing calls at fixed time internal

h = average holding time

The average number of occupied or busy trunks is defined as the carried load

Thus, the carried load is the position of the offered load that is not lost from the system.

The carried load per line is known as the trunk occupancy.

The trunk occupancy is a measure of the degree of utilization of a group of lines and

is sometimes called the utilization factor.

In designing a telephone system, it is necessary to ensure that the system will operate

satisfactorily under the moderate overload condition.

4.4.2 Lost Calls Returned (LCR) System

In LCC system, it is assumed that unserviceable requests leave the system and

never return. This assumption is appropriate where traffic overflow occurs and the other

routes are in other calls service. If the repeated calls not exist, LCC system is used. But

in many cases, blocked calls return to the system in the form of retries. Some examples

are subscriber concentrator systems, corporate tie lines and PBX trunks, calls to busy

telephone numbers and access to WATS lines. Including the retried calls, the offered

traffic now comprises two components viz., new traffic and retries traffic. The model

used for this analysis is known as lost calls returned (LCR) model. The following

assumptions are made to analyze the CLR model.

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1. All blocked calls return to the system and eventually get serviced, even if multiple

retries are required.

2. Time between call blocking and regeneration is random statistically independent of

each other. This assumption avoids complications arising when retries are correlated to

each other and tend to cause recurring traffic peaks at a particular waiting time interval.

3. Time between call blocking and retry is somewhat longer than average holding time

of a connection. If retries are immediate, congestion may occur or the network operation

becomes delay system.

Consider a system with first attempt call arrival ratio of (say 100). If a

percentage B (say 8%) of the calls blocked, B times retries (i.e. 8 calls retries). Of

these retries, however a percentage B will be blocked again.

Hence by infinite series, total arrival rate is given as

Where B is the blocking probability from a lost calls cleared (LCC) analysis.

The effect of returning traffic is insignificant when operating at low blocking

probabilities. At high blocking probabilities, it is necessary to incorporate the effects of

the returning traffic into analysis.

4.4.3 Lost Calls Held (LCH) System

In a lost calls held system, blocked calls are held by the system and serviced

when the necessary facilities become available. The total time spend by a call is the

sum of waiting time and the service time. Each arrival requires service for a continuous

period of time and terminates its request independently of its being serviced or not. If

number of calls blocked, a portion of it is lost until a server becomes free to service a

call. An example of LCH system is the time assigned speech interpolation (TASI)

system.

LCH systems generally arise in real time applications in which the sources are

continuously in need of service, whether or not the facilities are available. Normally,

telephone network does not operate in a lost call held manner. The LCH analysis

produces a conservative design that helps account for retries and day to day variations

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in the busy horn calling intensities. A TASI system concentrates some number of voice

sources onto a smaller number

of transmission channels. A source receives service only when it is active. If a source

becomes active when all channels are busy, it is blocked and speech clipping occurs.

Each speech segment starts and stops independently of whether it is served or not.

Digital circuit multiplication (DCM) systems in contrast with original TASI, can delay

speech for a small amount of time, when necessary to minimize the clipping.

LCH are easily analyzed to determine the probability of the total number of calls

in the system at any one time. The number of active calls in the system at any time is

identical to the number of active sources in a system capable of carrying all traffic as it

arises. Thus the distribution of the number in the system is the Poisson distribution. The

Poisson distribution given as

The probability that k sources requesting service are being blocked is simply the

probability that k + N sources are active when N is the number of servers.

4.5 DELAY SYSTEMS

The delay system places the call or message arrivals in a queue if it finds all N

servers (or lines) occupied. This system delays non-serviceable requests until the

necessary facilities become available. These systems are variously referred to as delay

system, waiting-call systems and queuing systems. The delay systems are analyzed

using queuing theory which is sometimes known as waiting line theory. This delay

system has wide applications outside the telecommunications. Some of the more

common applications are data processing, supermarket checkout counters, aircraft

landings, inventory control and various forms of services.

Consider that there are k calls (in service and waiting) in the system and N lines

to serve the calls. If k ≤ N, k lines are occupied and no calls are waiting. If k > N, all N

lines are occupied and k – N calls waiting. Hence a delay operation allows for greater

utilization of servers than does a loss system. Even though arrivals to the system are

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random, the servers see a somewhat regular arrival pattern. A queuing model for the

Erlang delay system is shown in Fig.

The basic purpose of the investigation of delay system is to determine the

probability distribution of waiting times. From this, the average waiting time W as

random variable can be easily determined. The waiting times are dependent on the

following factors:

1. Number of sources

2. Number of servers

3. Intensity and probabilistic nature of the offered traffic

4. Distribution of service times

5. Service discipline of the queue.

In a delay system, there may be a finite number of sources in a physical sense

but an infinite number of sources in an operational sense because each source may

have an arbitrary number of requests outstanding. If the offered traffic intensity is less

than the servers, no statistical limit exists on the arrival of calls in a short period of time.

In practice, only finite queue can be realized. There are two service time distributions.

They are constant service times and exponential service times. With constant service

times, the service time is deterministic and with exponential, it is random. The service

discipline of the que involves two important factors.

1. Waiting calls are selected on of first-come, first served (FCFS) or first-

in-first-out (FIFO) service.

2. The second aspect of the service discipline is the length of the queue.

Under heavy loads, blocking occurs. The blocking probability or delay

probability in the system is based on the queue size in comparison with

number of effective sources.

We can model the Erlang delay system by the birth and death process with the

following birth and death rates respectively.

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And

Under equilibrium conditions, the state probability distribution P(k) can be

obtained by substituting these birth rates into the following equation

Under normalized condition,

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We know

Now, the probability of waiting (the probability of finding all lines occupied) is

equal to

P(W > 0) = C(N, A)

where B = Blocking probability for a LCC system

N = Number of servers

A = Offered load (Erlangs)

The above two equations are referred as Erlang’s second formula, Erlang’s

delay formula or Erlang’s C formula.

For single server systems (N = 1), the probability of delay reduces to , which is

simply the output utilization or traffic carried by the server. Thus the probability of delay

for a single server system is also.

The distribution of waiting times for random arrivals, random service times, and a FIFO

service discipline is

Where P(> 0) = probability of delay

h = average service time of negative exponential service time distribution.

By integrating equation, the average waiting time for all arrivals can be

determined as

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W(t)avg. is the expected delay for all arrivals. The average delay of only those

arrivals that get delayed is commonly denoted as

SUMMARY

In the introduction we have discussed the need for traffic engineering. The basic

purpose of the traffic engineering is to determine the conditions under which

adequate service is provided to subscribers while making economical use of the

resources providing the service.

Various parameters related to traffic engineering were discussed like Busy Hour,

Peak Busy Hour, Time Consistent Busy Hour, Call Completion Rate (CCR), Busy

Hour Call Attempts (BHCA), Erlang etc

In the end we have discussed the traffic design requirements, various probability

distributions, loss systems and delay systems.

KEY TERMS

Traffic Load

Busy Hour (BH)

Peak Busy Hour

Time Consistent Busy Hour

Call completion rate

Busy Hour Call Attempt (BHCA)

Traffic Intensity

Erlang (E)

Centum Call Second (CCS)

Call Minute (CM)

Call Second (CS)

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Call arrival rate

Call holding or service time

Offered load

Grade of Service (GOS)

Blocking Probability

Loss system

Delay system

Blocking Model

Birth Death process

Markov Process

Random or stochastic process

Lost call cleared (LCC)

Lost call returned (LCR)

Lost call held (LCH)

SELF TEST

1. Call completion rate is defined as

(a) ratio of no. of successful call to no. of rejected call

(b) ratio of no. of rejected call to no. of call attempts

(c) ratio of no. of successful call to no. of call attempts

(d) None

2. Busy hour calling rate is defined as

(a) no. of calls originated during a day

(b) no. of calls originated during busy hour

(c) no. of call attempted during busy hour

(d) ratio of no. of call originated during busy hour to total no. of calls in a

day

3. Traffic intensity is

(a) ratio of period for which server occupied to total period of observation

(b) ratio of no. of successful call to total call attempts

(c) no. of calls originated during busy hour

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(d) none

4. Which of the following relation is correct?

(a) 1 E = 3600 CCS=36 CS

(b) 1 E = 36 CCS = 3600 CS

(c) 1 E = 36 CM

(d) 1 E = 3600CM

5. What type of model is used to study the loss system?

(a) blocking model

(b) queuing model

(c) either a or b

(d) neither a nor b

6. What type of model is used to study the delay system?

(a) blocking model

(b) queuing model

(c) either a or b

(d) neither a nor b

7. For a good telecommunication system, GOS value should be

(a) Greater

(b) Smaller

(c) Infinite

(d) Negative

8. Which one of the following parameter is used to measure the performance of the

loss system from network point of view?

(a) Call termination rate

(b) GOS

(c) Call arrival rate

(d) Blocking probability

9. Which one of the following parameter is used to measure the performance of the

loss system from subscriber point of view?

(a) Call termination rate

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(b) GOS

(c) Call arrival rate

(d) Blocking probability

10.What is called as steady state of loss system?

(a) all the subscribers are busy

(b) all the servers are busy

(c) either a or b

(d) neither a nor b

11. Over 20 minutes observation, 40 subscribers initiate calls. Total duration of all

call is 80 minutes. Calculate load offered to the network.

(a) 0.1 E

(b) 2 E

(c) 4 E

(d) 8 E

12.Which one of the following statement is correct?

(a) Markov process and B-D process are used to model loss system

(b) B-D process is same as Markov process except that state is transited

only to adjacent states

(c) When all the servers are busy then B-D process become stationary

process

(d) All are correct

13.A pure birth process is also known as

(a) renewal process

(b) poisson process

(c) B-D process

(d) Markov process

14.A pure death process is used to analyze

(a) incoming traffic

(b) service time

(c) delay system

(d) call arrival rate

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15.The lost call delay system becomes unstable

(a) if offered traffic is less than no. of servers

(b) if offered traffic is greater than no. of servers

(c) if offered traffic is greater than lost traffic

(d) if offered traffic is less than lost traffic

Review Questions

PART A

1. Define calling rate and holding time.

2. What is traffic pattern?

3. Write short notes on (a) Erlangs and (b) CCS

Answers for Self Test

1. ratio of no. of successful call to no. of call attempts

2. no. of calls originated during busy hour

3. ratio of period for which server occupied to total period of observation

4. 1 E = 36 CCS = 3600 CS

5. blocking model

6. queuing model

7. Smaller

8. Blocking probability

9. GOS

10. all the servers are busy

11. 4 E

12. All are correct

13. renewal process

14. service time

15. if offered traffic is greater than no. of servers

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4. What is Erlang loss system? Name three models of the loss system.

5. Give the expression of Erlang’s-B formula.

6. Compare LCR and LCH system.

7. Draw the queuing model of delay systems.

8. What is the purpose of traffic engineering?

9. Define traffic intensity.

10.Draw the queuing model of delay systems.

11.Define arrival rate and service rate.

12.What is meant by Erlangs?

13.What are blocking model and congestion model?

14.How do you model a switching system for traffic analysis?

15.Compare LCR and LCH system.

16.What is the condition for a stable queue in the delay system?

17.Define blocking probability and grade of service.

PART B

1. Explain the two types of congestion.

2. Comment on modeling of traffic.

3. Give the expression of Erlang’s-C formula.

4. Comment on combined loss and delay system.

5. Derive an expression for the state probability P(k) using B-D process.

6. Derive an expression to obtain the Erlang’s formula for the first kind of loss

system.

7. A group of 10 trunks is offered 5E of traffic, find (a) GOS (b) the probability that

only one trunk is busy (c) the probability that only one trunk is free and (d) the

probability that at least one trunk is free.

8. Derive an expression to obtain the Erlang’s second formula of delay system.