Mạng và viễn thông P30

530 TELETRAFFIC THEORY in conversation and also the time prior to conversation when the call is being set-up; holding time is the total time for which the network is in use 0 the total

30 Teletrafic Theory

Telecommunication networks, like roads are said to carry ‘traffic’, consisting not of vehicles but of telephone calls or data messages The more traffic there is, the more circuits and exchanges must be provided On a road network the more cars and lorries, the more roads and roundabouts are needed In any kind of network, if traffic exceeds the design capacity then there will be pockets of congestion On the road this means traffic jams; on the telephone the frustrated caller receives

frequent ‘busy tones’; in a data network unacceptably long ‘response times’ are experienced

Short of providing an infinite number of lines, it is impossible to know in advance precisely

how much equipment to build into a telecommunications network to meet demand without con- gestion However, there is a tool for ‘dimensioning’ network links and exchanges It is the rather

complex statistical science of ‘teletraffic theory’ (sometimes called ‘teletraffic engineering’) This is the subject of this chapter

We begin with the teletraffic dimensioning method used for circuit-switched networks, first

published in 1917 by a Danish scientist, A K Erlang Erlang defined a number of parameters and

developed a set of formulae, which together give a framework of rules for planners to design

and monitor the performamce of telephone, telex and circuit-switched data networks The latter

part of the chapter deals with the dimensioning of data networks, reviewing the techniques in

such a way as to offer the reader a practical method of network design

30.1 TELECOMMUNICATIONS TRAFFIC

Trajic is the term to describe the amount of telephone calls or data messages conveyed

over a telecommunications network, but this could cover any number of different

scientific definitions Possible definitions of trajic include

0 the total number of calls or messages

0 the total conversation time (i.e the number of calls multiplied by the conversation

time on each)

0 the total circuit holding time This is the number of calls multiplied by the holding

time on each; the holding time includes the time peridod during which the parties are

529

Networks and Telecommunications: Design and Operation, Second Edition.

Martin P Clark Copyright © 1991, 1997 John Wiley & Sons Ltd ISBNs: 0-471-97346-7 (Hardback); 0-470-84158-3 (Electronic)

530 TELETRAFFIC THEORY

in conversation and also the time prior to conversation when the call is being set-up;

holding time is the total time for which the network is in use

0 the total number of data characters conveyed

All the above statistics impact on the performance of teleconmmunication networks,

but the holding time is particularly influential to the carrying capacity and congestion of

telephone and other circuit-switched networks As far as the science of teletraffic is

concerned, the traffic volume is normally defined by the third definition above In other

words, the total trafic volume is equal to the total network holding time The traffic

volume, however, cannot be directly used in the determination of exactly how many

circuits on a trunk or what size of exchanges will need to be provided What we need is

some idea of the maximum usage of the network at any one time For this reason, it is

normal to measure the traffic intensity of circuit-switched networks

30.2 TRAFFIC INTENSITY (CIRCUIT-SWITCHED NETWORKS)

The trafic intensity of a circuit-switched network is defined to be the average number of

calls simultaneously in progress during a particular period of time It is measured in

units of Erlangs Thus an average of one call in progress during a particular period

would represent a trafic intensity of one Erlang The traffic intensity on any route

between two exchanges can also be quoted in Erlangs It is measured by first summing

the total holding time of all the circuits within the route and then dividing this by the

time period T , over which the measurement was made

In some countries, including the United States, trafic intensity is measured not in

Erlangs but in units called CCS (hundred call seconds) CCS is a measure of the total call

holding time during the network or route busy hour The two units, CCS and Erlang are

very simply related because

1 Erlang = 3600 call seconds = 36 CCS

The definition of trafic intensity is not restricted to traffic between exchanges Cross-

exchange traffic (that passing across an exchange from incoming ports to outgoing

ports) can also be measured and quoted in Erlangs

As an example, the route between exchanges A and B in Figure 30.1 consists of five

circuits The chart in Figure 30.1 also shows the periods of usage of each of these

circuits during a 10 minute period

It will be seen from the chart in Figure 30.1 that the total number of minutes of cir-

cuit usage during the 10 minute period was 35 minutes This is the sum of the individual

circuit holding times, respectively: 6min, 7.5 min, 7.5 min, 8 min, 6min Dividing the

total of 35 minutes by the length of the monitoring period (10 minutes) we may deduce

that the trafJic intensity on the route was 3.5 Erlangs In other words, an average of 3.5

circuits were in use throughout the whole of the period

However, usually the challenge facing a telecommunications network planner is how

many circuits to provide to carry a given volume of traffic The general principle is that

more circuits will be needed on a route than the numerical value of the route traffic

intensity measured in Erlangs It would be easy to succumb to the mistaken belief that a

PRACTICAL TRAFFIC INTENSITY (ERLANG) MEASUREMENT 531

trajic intensity of 3.5 Erlangs (3.5 simultaneous calls) could be carried on four circuits

Figure 30.1 clearly illustrates an example in which this is not the case All five circuits

were simultaneously in use twice, between times 2 and 3, and between times 5.5 and 6

Here we must digress briefly to examine the concepts of ofered and carried trafic,

though their names may give them away Offered trafic is a theoretical concept It is a

measure of the unsuppressed trajic intensity that would be transported on a particular

route if all the customers’ calls were connected without congestion Carried trajic is

that resultant from the carried calls, and it is the value of traffic intensity actually

measured For a network without congestion the carried traflc is equal to the offered

trafic However, if there is congestion in the network, then the offered traffic will be

higher than that carried, the difference being the calls which cannot be connected

Teletraffic theory leads to a set of tables and graphs which enable the required

network circuit numbers to be related to the ofered trujic in Erlangs (i.e the demand)

The trafic intensity on a given route may be measured using one of two main methods,

either by sampling or by an absolute measurement The latter method was used in the

example of Figure 30 I Historically, however, electro-mechanical exchanges did not

lend themselves to easy calculation of the absolute value of the total holding time

Instead, it was common practice to use a sampling technique We can obtain an esti-

mate of the total circuit holding time by looking at the instantaneous state of the

circuits at a number of sample points in time If we take ten samples, after imin, limin,

Averaging the sample values will give us an estimate of the traffic intensity over the whole period This value comes out at an estimated 3.9 Erlangs This is calculated as the sum of the ten sample values (39), divided by the overall duration (10) Compare this with the actual value of 3.5 Erlangs The error arises from the fact that we are only sampling the circuit usage rather than using exact measurement The error can be reduced by increasing the frequency of samples Table 30.2 gives the values calculated

at a $ minute (as opposed to a l-minute) sampling rate (sometimes also called scan rate)

The imreased sampling rate of Table 30.2 estimates the traffic correctly as 3.5 Erlangs (70 X 0.5/10) The exactness of the estimate on this occassion is fortuitous Nonethe- less, the principle is well illustrated that too slow a sampling rate will produce unreliable traffic intensity estimates and that the estimate is improved in accuracy by a higher sampling rate A good guide is that the sample period length should be no more than

about one-third of the average call holding time In the example of Figure 30.1, the average call holding time is 35 min/l7 calls = 2.06 min, so a sample should be taken at least every 0.7min to derive a reliable estimate of the traffic intensity The minimum monitoring period should be about three times the average call holding time This helps

us to avoid unrepresentative peaks or troughs in call intensity

Nowadays, stored program controlled ( S P C ) (i.e computer-controlled) exchanges

have made the absolute measurement of call holding times relatively easy So today

many exchanges produce the measurements of traffic intensity which are exact They do

so by calculating the value using data records storing the start and end time of each individual call or connection

Table 30.2 ; minute scan rate

THE BUSY HOUR 533

30.4 THE BUSY HOUR

In practice, telecommunications networks are found to have a discernible busy hour

This is the given period during the day when the trufic intensity is at its greatest

Traditionally measurements of traffic intensity have been made over a full 60-minute

period at the busy hour of day, to calculate the busy hour trufic (a shortened term for the busy hour trufic intensity) By taking samples over a few representative days, future busy hour trufJic can usually be predicted well enough to work out what the equipment

quantities and route sizes of the network should be The dimensioning is made by using

the predicted busy hour traffic as an input to the Erlang formula (presented later in this chapter) When that formula has pronounced the scale of circuits and equipment that are going to be needed at the busy hour, we can feel secure at less hectic times of day

In more recent times it has been found that exchanges are developing more than one busy hour; maybe as many as three, including morning, afternoon and evening busy hours The morning and afternoon busy hours are usually the result of business traffic The evening one results from residential, international traffic and nowadays also from

Internet access traffic (dial-up connections from home PC-users to Internet servers and

bureaux Some examples of daily traffic distribution are given in Figure 30.2

The first distribution in Figure 30.2 is a typical business-serving exchange, with morning and afternoon busy hours The second example shows the traffic in a residen-

tial exchange, where morning and afternoon busy hours are less than the evening busy

Traffic I ( a ) business area

Traffic ( b l residential area

Traffic ( C ) U.K-Austrolio (limited by

tlme zone difference 1

Time of day Figure 30.2 Some typical daily traffic distributions

exchange matrix according to its own exchange busy hour, and each route according to its individual route busy hour

Another point to consider when dimensioning networks is that a route or exchange is unlikely to be equally busy throughout the entire 60 minute busy hour period To meet the peak demand, which might be significantly higher than the hour’s average traffic, it is sometimes convenient to redefine the busy hour as being of less than 60 minute duration

It may seem paradoxical to have a busy hour of less than 60 minutes, but Figure 30.3 illustrates a case of traffic which has a short duration peak of between 15 and 30 minutes

In the example of Figure 30.3, if we were to use a 60 minute busy hour measurement period, we would estimate a busy-hour traffic of value Q as shown in the diagram This would be a gross underestimate of the actual traffic peak, and would guarantee that the busiest period would be heavily congested By redefining the busy hour to be of only

30min duration we obtain an estimate P which is much nearer the actual traffic peak Surprisingly, however, the use of much shorter busy hour periods and attempts to pinpoint peaks of traffic which last only a few minutes are not really important; customers who suffer congestion a t this time are more than likely to get through on a repeat call attempt within a few minutes anyway!

To sum up, it is usual to monitor exchange busy hour and route busy hour traffic

values as a gauge of current customer usage and network capacity needs Future net- work design and dimensioning can then be based on our forecasts of what the values of these parameters will be in the future (forecasting methods are covered in Chapter 31)

THE FORMULA FOR TRAFFIC INTENSITY 535

Our next step is to develop the formula for traffic intensity, as a basis for subsequent discussion of the Erlang method of network dimensioning Recapping in mathematical terms, the traffic intensity is given by the expression

Trafic intensity= the sum of circuit holding times

(carried traffic) the duration of the monitoring period Now let

A = th e traffic intensity in Erlangs

T = the duration of the monitoring period

h; = the holding time of the ith individual call

c = the total number of calls in the period of mathematical summation Then, from above

Now, because the sum of the holding times is equal to the number of calls multiplied by the average holding time, then

where h =average call holding time, and therefore

It is interesting to calculate the call arrival rate, in particular the number of calls expected to arrive during the average holding time Let N be this number of calls, then

N = no of call arrivals during a period equal to the average holding time

= h X call arrival rate per unit of time

= h x c / T

= c h / T = A

In other words, the number of calls expected to be generated during the average holding time of a call is equal to the traffic intensity A This is perhaps a surprising result, but one which sometimes proves extremely valuable

536 TELETRAFFIC THEORY

In this section we discuss the traffic-carrying capacity of a single circuit This leads on

to a mathematical derivation of the Erlang formula, which is the formal method to calculate the traffic-carrying capacity of a circuit group of any size

For our explanation let us assume we have provided an infinite number of circuits, laid out in a line or grading, as shown in Figure 30.4 The infinite number provides

enough circuits to carry any value of traffic intensity Now let us further assume that each new call scans across the circuits from the left-hand end until it finds a free circuit Then let us try to determine how much traffic each of the individual circuits carries First, let us consider circuit number one Figure 30.5 shows a timeplot of the typical activity we might expect on this circuit, either busy carrying a call, or idle awaiting for another call to arrive The timeplot of Figure 30.5 starts with the arrival of the first call

This causes the circuit to become busy for the duration of the call While the circuit is busy a number of other calls will arrive, which circuit number one will be incapable of carrying These other calls will scan across towards a higher-numbered circuit (circuit number two, then three, and so on) until the first free circuit is found Finally, at the end

of the call on circuit number one, the circuit will be returned to the idle state This state will prevail until the next new call arrives

Let us try to determine the proportion of time for which circuit number one is busy For this purpose, let us assume each call is of a duration equal to the average call hold- ing time h This is not mathematically rigorous but it makes for simpler explanation

Let us also invent an imaginary cycle of activity on the circuit

Circuit outlets

Circuit number

New calls start at this end and scon ocross the circuits in turn until1 finding o free circuit

Figure 30.4 Scanning for a free circuit

T ldle I dle Idle Arrival time

of the first call

Figure 30.5 Activity pattern of circuit number l

THE TRAFFIC-CARRYING CAPACITY OF A SINGLE CIRCUIT 537

Cycle r,- Repeat cycle -L- Repeat cycle ,Repeat cycle

T Ime

Next call arrival Figure 30.6 Average activity cycle on circuit number 1

Our imaginary cycle is as follows

After the arrival of the first call, we expect circuit number one to be busy for a period

of time equal to h As we learned in the last section we can expect a total of A calls to arrive during the average holding time, where A is the offered traffic ( A - 1 ) of these

calls (i.e all but the first) will scan over circuit number one to find a free circuit among the higher numbered circuits At the end of the first call circuit number one will be

released, and an idle period will follow until the next call arrives (i.e the ( A + 1)th) We can imagine this cycle repeating itself over and over again

As we can see from our imaginary ‘average’ cycle, shown in Figure 30.6, the total

number of calls arriving during the cycle is A + 1 The total duration of the cycle is therefore

We also know that circuit number one is busy during each cycle for a period of duration h

Therefore the average proportion of the time for which circuit number one is busy is given

the intensity of the traffic carried on circuit number one, and is measured in Erlangs

accordingly Thus if one Erlang were offered to the grading ( A = l), then the first circuit would carry half an Erlang The remaining half Erlang is carried by other circuits

Taking one last step, if we assume that new calls arrive at random instants of time,

then the proportion of calls rejected by circuit number one is equal to the proportion of time during which the circuit is busy, i.e A / ( 1 + A ) In our simple case, if only one

circuit were available, then A / ( 1 + A ) proportion of calls cannot be carried This is

called the blocking ratio B, and is usually written

538 TELETRAFFIC THEORY

Though not proven above in a mathematically rigorous fashion, the above result is

the foundation of the Erlang method of circuit group dimensioning Before going on,

however, it is worth studying some of the implications of the formula a little more

deeply for two cases

First, take the case of one Erlang of offered traffic to a single circuit Substituting in

our formula A = 1, we conclude that the blocking value is 1/(1 + 1) = 1/2 In other

words, half of the calls fail (meeting congestion), and only half are carried This

confirms our earlier conclusion that the circuit numbers needed to carry a given

intensity of traffic are greater than the numerical value of that traffic

With traffic intensity of A = 0.01, then the proportion of blocked calls would have

been only O.Ol/l.Ol (=O.Ol), or about one call in one-hundred blocked This is the

proportion of lost calls targetted by many network operating companies Put in

practical terms, the carrying capacity of a single circuit in isolation is around only

0.01 Erlangs

Next let us consider a very large traffic intensity offered to our single circuit In this

case most of the traffic is blocked (if A = 99, then the formula states that 99% blocking

is incurred, i.e is not carried by our particular circuit) However, the corollary is that

the traffic carried by the circuit (equal to the proportion of the time for which the circuit

is busy) is 0.99 Erlangs In other words the circuit is in use almost without let-up This is

what we expect, because as soon as the circuit is released by one caller, a new call is

offered almost immediately

In the appendix the full Erlang lost cull formula is derived using a more rigorous

mathematical derivation to gain an insight into the traffic-carrying characteristics of all

the other circuits in Figure 30.4 For the time being, however, Figure 30.7 simply states

the formula

To confirm the result from our previous analysis, let us substitute N = 1 into the

formula of Figure 30.5 As before, we obtain a proportion of lost calls for a single

circuit (offered traffic A ) of

A/(1 + A ) A

E ( N , A ) B(N, A )

or = c / ( l + A + - + - + N ! A 2 2! A 3 3! + - N

where

E(N, A ) = proportion of lost calls, and probability of blocking

A = offered traffic intensity

N = available number or circuits

N ! = factorial N

Figure 30.7 The Erlang lost-call formula

DIMENSIONING CIRCUIT-SWITCHED NETWORKS 539

Circuit occupancy (in erlangs) I 5 I

erlangs (Total area under plot 1

carried by the Elh clrcult Area shaded represents traffic

1 2 3 L 5 6 7 8 9 Circuit number

( i n order of selection) Figure 30.8 Circuit occupancies

This of course is also equal to the circuit occupancy (the traffic carried by it) The advantage of our new formula is that we may now calculate the occupancy of all the other circuits of Figure 30.4 By calculating the lost traffic from two circuits we can derive the carried traffic Subtracting the traffic carried on circuit number one we end up with that carried by circuit number two In a similar manner, the traffic carrying contributions of the other circuits can be calculated Eventually we are able to plot the graph of Figure 30.8, which shows the individual circuit occupancies when 5 Erlangs of

traffic is offered to an infinite circuit grading

As expected, the low-numbered circuits carry nearly 1 Erlang and are in near- constant use, whereas higher numbered circuits carry progressively less traffic The traffic carried by the first eleven circuits is also shown From the formula of Figure 30.7, this is the number of circuits needed to guarantee less than l % proportion of calls lost Thus the right-hand shaded area in Figure 30.10 represents the small proportion of lost calls if 11 circuits are provided The ability to calculate individual circuit occupancies is crucial to grading design (see Chapter 6)

30.7 DIMENSIONING CIRCUIT-SWITCHED NETWORKS

The future circuit requirements for each route of a circuit-switched network (i.e telephone, telex, circuit switched data) may be determined from the Erlang lost call

formula We do so by substituting the predicted ofleered traffic intensity A , and using trial-and-error values of N to determine the value which gives a slightly better

performance than the target blocking or grade of service B A commonly used grade of

service for interchange traffic routes is 0.01 or 1 % blocking

It is not an easy task by direct calculation to determine the value of N (circuits

required), and for this reason it is usual to use either a suitably programmed computer

or a set of trafJic tables

In recent years, numerous authors and organizations have produced modified versions of the Erlang method, more advanced and complicated techniques intended

540 TELETRAFFIC THEORY

to predict accurately the traffic-carrying capacity of various sized circuit groups for

different grades ofservice All have their place but in practice it comes down to finding

the most appropriate method by trying several for the best fit for given circumstances

In my own experience, the extra effort required by the more refined and complicated

methods of dimensioning is unwarranted In practice the traffic demand may vary

greatly from one day or month to the next and the practicality is such that circuits have

to be provided in whole numbers, often indeed in multiples of say 12 or 30 The decision

then is whether 1 or 2, 12 or 24, 30 or 60 circuits should be provided It is rather

academic to decide whether 23 or 24 circuits are actually necessary when at least 30 will

be provided

Table 30.3 illustrates a typical traffic table The one shown has been calculated from

the Erlang lost-call formula Down the left hand column of the table the number of cir-

cuits on a particular route are listed Across the top of the table various different grades

of service are shown In the middle of the table, the values represent the maximum

offered Erlang capacity corresponding to the route size and grade of service chosen

Thus a route of four circuits, working to a design grade of service of 0.01, has a

maximum offered traffic capacity of 0.9 Erlangs

We can also use Table 30.3 to determine how many circuits are required to provide

a 0.01 grade of service, given an offered traffic of 1 Erlang In this case the answer

is five circuits The maximum carrying capacity of five circuits at 1% grade of service

is 1.4Erlangs, slightly greater than needed, but the capacity of four circuits is only

0.9 Erlangs

The problem with traffic routes of only a few circuits is that only a small increase in

traffic is needed to cause congestion It is good practice therefore to ensure that a

minimum number of circuits (say five) are provided on every route

Table 30.3 A simple Erlang traffic table

DIMENSIONING CIRCUIT-SWITCHED NETWORKS 541

1 lost call in 50 (gos 0 0 2 )

1 lost call in 100 (gos 0.011

1 lost call in 200 (90s 0.005)

1 lost call in 1000 (gos 0.001)

Required circuit number L.7

Capacity in traffic intensity (erlongs )

Figure 30.9 Graphical representation of the Erlang formula