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- 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)
30
Teletrafic Theory
Telecommunication networks, like roads. are saidcarry ‘traffic’, consisting not of vehicles but of
to
telephone callsor 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, K. Erlang. Erlang defined a number of parameters and
A.
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
overatelecommunicationsnetwork,butthiscouldcoveranynumber 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
- 530 TELETRAFFIC
in conversation and also the time prior to conversation when the call being set-up;
is
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,
of
but the holding time is particularly influential to the carrying capacity and congestion
telephone and other circuit-switchednetworks. As far as the science of teletraffic is
concerned, the traffic volume is normally defined the third definition above. In other
by
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 whatsize 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
wouldrepresenta 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 measureof the totalcall
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-
exchangetraffic (that passingacross an exchangefromincoming 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 routewas 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
MEASUREMENT
(ERLANG) 531
S
S circuitroute
L
5
Exchange Exchange
Circuit 1
number
2
3
L = Circuit in use
5
1 2 3 L S 6 7 8 9 10Tlme
Figure 30.1 Circuit usage on a route between exchanges
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 resultantfromthecarried calls, and it is the value of traffic intensityactually
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 theoryleadsto a set of tables andgraphs which enabletherequired
network circuit numbers to be related to the ofered trujic in Erlangs (i.e. the demand).
30.3 PRACTICALTRAFFIC INTENSITY (ERLANG) MEASUREMENT
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 circuitholding 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,
- 532 TELETRAFFIC THEORY
Table 30.1 Samples of number of circuits in use
Elapsed time 1 1
; 2; 3; g 5; 6f 7f 8; 9;
Circuits in use 3 5 5 3 4 5 4 2 4 4
2$min, . . . ,9$min, we could assume that this was representative of the respective time
periods, 0-1 min, 1-2min,. . . , etc. Thus, because after $ minute has elapsed, we find
that three circuits are in use; we assume that three circuits will be in use for the whole
period of the first minute. By this method we obtain 10 ‘snapshots’ of the number of
circuitsinuse,corresponding tothe 10 individualoneminuteperiodswhich we
sampled. The results are as shown in Table 30.1.
Averaging the sample values will give us an estimate of the traffic intensity over the
whole period. This value comes out anestimated 3.9 Erlangs. This is calculated as the
at
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
samplingthecircuitusageratherthanusingexactmeasurement.Theerrorcan 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 calledscan rate).
3.5
The imreased sampling rate Table 30.2 estimates the traffic correctly as Erlangs
of
(70 X 0.5/10). The exactness of the estimate on this occassion is fortuitous. Nonethe-
less, the principle is well illustrated that tooslow 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 exact.They do
are
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
time
Elapsed 1
4 3 14 ‘3 24 23 3; 39 4 +
in
Circuits use 3 3 4 4 5 5 3 3 4 4
- 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 itsgreatest.
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 distributioninFigure 30.2 is atypical 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
0000 0800 1600 2Loo
Traffic ( b l residential
area
Traffic (C) U.K-Austrolio (limited by
tlmezonedifference 1
Time of day
Figure 30.2 Some typical daily traffic distributions
- 534 TELETRAFFIC THEORY
hour. The third exampleis of an international route,where there is often a single peak,
constrained in duration by the time zone difference between the countries.
Multiple busy hours necessitate the calculation of separate busy-hour traffic values for
each individual busy hour,and design of the network to meet each one. It is not enough
to use only one of the sets of busy-hour traffic values. In Figure 30.2 for example, the
overall effect of distributions (a) and (b),when combined, is a two-humped pattern with
overall busy hours in morning and afternoonof approximately equal magnitude. How-
ever, if we were to design the whole network on the basis of its morning and afternoon
busy hour traffic values alone, we would not provide enough circuits at the residential
exchange to meet its evening busyhour. Normal practiceis therefore to dimension each
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 networksis 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 besignificantly higher than thehour’s average traffic, it is
sometimes convenientto redefine the busy hour asbeing 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 oftraffic which has a short duration peak between 15 and 30 minutes.
of
In the example of Figure 30.3, we were to use a 60 minute busy hour measurement
if
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 useof much shorter busy hourperiods and attempts to
pinpointpeaks of traffic whichlastonlya few minutesarenot really important;
customers who suffer congestion at 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-
of
work design and dimensioning can then be based on our forecasts what the values of
these parameters will be in the future (forecasting methods are covered in Chapter 31).
Actual I
peak
*- I
Number of clrcuits
in use
Short-term
fluctuations
Time
Figure 30.3 A short duration busy hour
- MULA THE TRAFFIC INTENSITY 535
30.5 THE FORMULA FORTRAFFIC INTENSITY
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 =the 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 calls multiplied by
of
the average holding time, then
where h =average call holding time, and therefore
It is interesting to calculatethe call arrivalrate, in particularthenumber 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
=hxc/T
= ch/T = A
In other words, the number of calls expected to be generated during theaverage 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
30.6 THE TRAFFIC-CARRYINGCAPACITY OF A SINGLE CIRCUIT
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 fromthe left-hand end until finds a free circuit.
it
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 on) until the first free circuit is found. Finally, at the end
so
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 callis 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
Circuitnumber
New calls start at this end scon
and ocross the
circuits turn
in until1
finding o free
circuit
Figure 30.4 Scanning for a freecircuit
T
ldle Idle I dle
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,- Repeatcycle
-p -- Repeat cycle ,Repeat
- -
L
-, -
cycle
T Ime
Nextcall
arrival
Figure 30.6 Average activity cycle on circuit number 1
Our imaginary cycle is as follows.
After the arrival of the call, we expect circuit number one to be busy for a period
first
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
A + 1 - h ( A + 1)
hx- - ~
A A
We also know that
circuit number one busy during each
is cycle for a periodof duration h.
of
Therefore the average proportionthe time for which circuit number one busy is given
is
by
A -
A
occupancyofcircuitnumber 1 =h X ~ -
h(l+A)-l+A
This value is the so-called circuit occupancy of circuit number one. It is numerically
equal to theaverage number of calls in progress on circuit number one, and is therefore
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 proportionof 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
A
B(b1ocking ratio - for one circuit) = -
l+A
where A = offered traffic.
- 538 TELETRAFFIC
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,halfofthecallsfail(meetingcongestion),andonlyhalfarecarried.This
confirms our earlierconclusion thatthe circuitnumbers needed tocarry 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 manynetworkoperatingcompanies.Putin
practical terms, the carrying capacity of a single circuitinisolation 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 (equalto 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
B(1,A) = -
1 l+A
E(N, A )
or 2! 3!
= c /! ( l + A + - +A-2+ .A 3
N . . + -N
B(N, A )
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 TheErlanglost-callformula
- RCUIT-SWITCHED
DIMENSIONING 539
Circuit 5 erlangs (Total area under plot 1
occupancy
(in erlangs)
I I
Areashadedrepresents traffic
carried by the Elh clrcult
I 11 circuits
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 circuitoccupancy (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 onewe end up
with that carried by circuitnumbertwo. In asimilarmanner,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 occupancieswhen 5 Erlangs of
traffic is offered to an infinite circuit grading.
As expected,thelow-numberedcircuitscarrynearly 1 Erlangandare innear-
constant use, whereas higher numberedcircuitscarry progressively less traffic. The
traffic carried by the first eleven circuits is also shown. From the formula Figure 30.7,
of
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
Thefuture circuitrequirementsforeachroute 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 valuethe 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 organizations
and haveproduced modified
versions of the Erlang method, more advanced and complicated techniques intended
- 540 TELETRAFFIC
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 dimensioningisunwarranted.In practicethetraffic demand mayvary
greatly from one day or month to the and the practicality is such that circuits have
next
to be provided in whole numbers, often indeed in multiples of 12 or 30. The decision
say
then is whether 1 or 2, 12 or 24, 30 or 60 circuitsshouldbeprovided. It is rather
academic to decide whether23 or 24 circuits are actually necessary when least 30 will
at
be provided.
Table 30.3 illustrates a typical traffic table. The one shown hasbeen calculated from
the Erlang lost-call formula. Down the left hand column of the table the numbercir- of
cuits on a particular route are listed. Across the top the table various different grades
of
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 designgrade 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 goodpracticetherefore to ensure that a
minimum number of circuits (say five) are provided on every route.
Table 30.3 A simpleErlangtraffic table
Grade of service ( B ( N ,A ) )
1 lost call in
Number 50 100 200 1000
of circuits (0.02) (0.01) (0.005) (0.001)
~~ ~~~~~~
Erlangs Erlangs Erlangs Erlangs
1 0.020 0.010 0.005 0.001
0.15 2 0.22 0.105 0.046
0.60 3 0.35 0.19
4 1.1 0.9 0.7 0.44
1.7 5
1.9 6 2.3 1.1
7 2.9 2.5 2.2 1.6
3.6 8 2.7 3.2 2.1
4.3 9 3.3 2.6
10 5.1 4.5 4.0 3.1
11 4.6 5.8 5.2 3.6
6.6 12 5.9 5.3 4.2
- DIMENSIONING CIRCUIT-SWITCHED
NETWORKS 541
Circuit requirement
12
11
10
Curvesrepresent :
9 1 lostcallin 50 (gos 0 . 0 2 )
1 lost call in 100 (gos 0.011
1 lostcall in 200 (90s 0.005)
8
1 lost call in 1000 (gos 0.001)
7
6
Required circuit number L.7
( 5 in practice1
5
L Intended grade of service
better
than 0.01
3
Offered
traffic of 1 erlang
2
1
1 2 3 L 5 6 7
Capacity traffic
in intensity(erlongs )
Figure 30.9 Graphical representation of the Erlang formula
- 542 TELETRAFFIC
Exchange
C
Exchange Exchange
A B
Secondchoiceroute
First choiceroute
L
Figure 30.10 Overflow routing
The information held in Table 30.3 is sometimes presented graphically, and Figure30.9
illustrates this. The traffic offered in Erlangs is usually plotted along the horizontalaxis,
andthe circuit numbersupthe verticalaxis. A numberof different curvesthen
correspond to different grades of service. To determine how many circuits are required
for a given offered traffic, the offered Erlang value is read along the horizontalaxis, then
a vertical line is drawn upwards to the curve corresponding to the required grade of
service, and a horizontal line is drawn from this point to the vertical axis, where the
circuitrequirementcan be read off. Our earlierexample is repeated on the figure,
confirming that five circuits are needed to carry l Erlang at 1% grade of service.
The traffic-to-circuit relationshipshowninTable30.3andFigure 30.9, andthe
Erlang lost call formula on which they are based are only suitable for dimensioning
full-availability networks (as defined in Chapter 6). For limited availability networks,
slightly different formulae and traffic tables must be used. Furthermore, anotherslightly
different set of traffic tables is called for when dimensioningnetworks which use
overflow routing. Such tables are available from various publishers. Figure 30.10 shows
an example of overflow routing in which traffic from exchange A to exchange B first
tries to route directly, but is allowed to overjowl via exchange C if all direct circuits are
busy. Only the link A-B in this network can be dimensioned by the simple Erlang lost-
call formula (provided we specify an overflow grade of service, e.g. 10% of calls), but as
we shall find out in Chapter 32, only a slight modification to the method is necessary to
permit us to dimension all the other links.
30.8 EXAMPLE ROUTE DIMENSIONING
As an example of the use of the Erlang lost-call formula in route dimensioning, let us
calculate the number of circuits required to carry offered traffic A = 55 Erlangs at a
grade of service B = 0.01.
We shall show in Chapter 32 that an estimateof the number of circuits B required for
1% grade of service is given by the formulae
N =6 + (A/0.85) where A < 75
N = 14 + (A/0.97) where 75 < A < 400
N=29+A where 400 < A
- CALL WAITING SYSTEMS 543
so that for our case, A = 55
N(approx) = 70 circuits
So far so good, but from here on we rely on trial and error and the formula in Figure
30.7.
substituting A = 55, N = 70 we get B = 0.007
This is slightly better than we need, so let us try
A = 55, N = 69. This time B = 0.009
Again
A = 55, N = 68,then B = 0.012
68 circuits give a slightly worse grade of service than we need, so 69 circuits must be
provided!
30.9 CALL WAITING SYSTEMS
Whereas in telephone and other circuit switched networks the dimensioning is carried
outto a given grade-of-service (or call loss), this technique is not suitablefor
dimensioning all types of network. Data networks and server-based services (such as
Internet bureaux and operator switchrooms or telemarketing bureaux) are examples of
queuing (i.e. bufSer) or call-waiting systems. A queue (or bufSer) minimizes, and may
even eliminate, any loss of offered traffic. As we saw in Chapter 9, arriving packets from
the separate virtual connections can be queued (buflered) in order until the line is free.
Similarly, a caller accessing an Internet service must wait a little longer at times of
heavy traffic demand on the network, but will be answered eventually.
In the case of operator services or call-in telemarketing bureaux, incoming callers
waitinaqueue listening to music or a recorded message untila human operator
becomes free.
In such networks the dimensioning techniques must be oriented towards ensuring
thatthe capacity of adatalink,thenumber of servers or thenumber of human
operators is sufficient to keep the queue waiting time within acceptable bounds. If there
are insufficient operators the queue and the consequent waiting time will get longer and
longer. We must also ensure that the queue capacity is long enough for all the people
waiting or the data packets being temporarily stored in the bujier.
Consider the operator switchroom or telemarketing bureaux first, because what hap-
pens there follows on from the Erlang lost-call formula already discussed. Figure 30.11
illustrates a networkin which an operator switchroom is serving customers of the
telephone network.
In all, N operators (called servers in teletraffic jargon) are working and K places
(confusingly also called servers) are available in the queue.
The switchroom manager wants to make sure that the number operator positions
of
staffed (N) at any pointtime during the daygreat enough to cope thetraffic and
of is with
so keep down the queue waiting time. Furthermore, the switchroom designer wants to
ensure that the queue capacity ( K ) is great enough to minimize the number of lost calls.
- 544 TELETRAFFIC
7 I N operators
Calls
I Operator
switchroom
Figure 30.11 Call queueing in an operator switchroom
Both NandK can be determined from adapted form the Erlanglost-cull method,
an of
using various forms of the Erlung Waiting Cull formula, as shown in Table 30.4.
The formulae of Table 30.4 are unfortunately complex and this is not the place to
describe their derivation or any of the more complex versions of them that have also
been developed. We can, however, discuss their practical use, as follows. A tolerable
delay t is set for thetime which telephone callers Figure 30.11
in will have to wait for an
operator (e.g. 15 seconds), and we then decide what percentage of calls will be answered
withinthisdelaytime(atypicaltargetwouldbe 95%). It is thenatrial-and-error
exercise, using formulae (i) and (iv) of Table 30.4, to give the number of servers ( N )
which will meet the target constraints.
The value N will always be greater than the offered traffic A . If it were not so, the
operators would be reducing the queue at a rate slower than that at which the queue
was forming. The queue and the delay could only become longer. (SubstituteN = A in
formula (ii) and you will see that the average delay is infinite).
Let us use an example to illustrate the method. We shall assume that the switchroom
handles an average of 3000 calls per hour, and that they take an average of d = 60
seconds of the operator’s time (i.e. service time) to deal with.
. . . Then the offered trufic, A (average number of calls in progress)
= 3000 X 60/3600 = 50
Guessing a value of N = 55 to meet our target that 95% calls will be answered within
15 seconds, we calculate in order. . .
A = 50 Erlangs (offered trufJic)
N = 55 (active servers (operators))
B = 0.054 from the lost-cull formula
C = 0.388 from formula (i)
t = 15 seconds
(target
answer
time)
d= 60 seconds
(average
service
time)
Probability that delay exceeds 15 seconds from formula (iv) =0.11
- CALL WAITING SYSTEMS 545
Table 30.4 The Erlang waiting call formula
Probability of delay (i.e. that a customer will have to wait to be served)
NB
=c=
N- A(l -B)
This is called the Erlang call-waiting formula
Average delay (held in queue)
Average number of waiting calls
Probability of delay exceeding t seconds
Probability of; or more waiting calls
Probability of X servers being busy
(a) No queue, X = number of operators busy
AX
P(X) = P(0) 0 IX 5 N
(b) Queue, all operators busy, J queue places taken
X= N+j
p ( N + j ) = c(1-$) ()
i 1
Notation
N = number of servers (those processing the calls, e.g. operators)
j = number of calls in queue
B = lost call probability if there were no queues derived from the Erlang lost call formula
~
= E(N, A)
A = offered traffic in Erlangs
d = average service time required by an active server to process a call
As this proportion is higher than the target of 0.05, wewill need more servers, but
probably not too many more because we are not too far adrift.
Repeatingfor N = 56, we findtheprobability of answer in 15 seconds is 0.07.
Almost, but not quite! Repeat again for N = 57, and the probability of answer taking
longer than 15 seconds is only 0.04. Thus 57 operators will be needed in our
switchroom.
- 546 TELETRAFFIC THEORY
Having determined the numberof operators ( N ) , the number of queue places needed
is found by using formula (v) of Table 30.4. The planner first decides what small prop-
ortion of calls it is acceptable to lose because there are no free places in the queue. Then
he finds the value j from the formula which satisfiesthe condition that the probability
of
lost calls (more waiting calls than queue positions) is met.
Going back to our example, let us aim to lose only1 % of calls because of inadequate
queue capacity. Then the probability o f j or more waiting calls must be less than 0.01,
where j is the designed capacity of the queue. The calculation is as follows.. .
A = 50 Erlangs offered traffic
N = 57 active operators
B = 0.039 Erlang lost call formula
C = 0.246 Erlang waiting call formula
So, from formula (v)
0.01 = C ( A / N ) j
rearranging
j = (logO.01 - log C)/(log A - log N )
j = 24.4
rounding up, 25 queue places will be needed.
Out of curiosity, why not calculate theaverage waiting time and the average number
of callers inthe queue? The average waiting timeis 2.1 seconds, and the average number
of waiting calls is 1.76.
Calculating the number of operators needed to answer calls on a company’s office
switchboard is carried out in an identical way to that used above.
Theoretically, different formulae must be used on occasions where the offered traffic
distribution does not follow Erlang’s detailed mathematical assumptions (given in the
appendix). Inappropriateuse of the formulae may lead errorsin the available network
to
capacity, but on the other hand most practitioners prefer to use simple dimensioning
methods and rely ontheirpracticalexperienceratherthanworrytoomuchabout
complex statistical theory. If interested, a number of more complex methods and their
detailed derivations are given in numerous specialist texts on queueing theory.
30.10 DIMENSIONING DATA
NETWORKS
Terminals and other devices sending information on data networks generally do so in a
packet-, frame- or cell-oriented format over a packet-, frame- or cell-switched network.
In the case of Figure 30.12(b), the data network system has been designed to flatten
the profile of the traffic by using buffers and statistical multiplexing. Such a network
demands the use of a call-waiting dimensioning technique to decide the required data
link bit rate and buffer capacity.
- DIMENSIONING DATA NETWORKS 547
Link
speed restriction
4
Data speed
demanded
( bit /s)
(a)
Figure 30.12 Peaked bursts of data demand
The time for which individual packet, frame cell must wait in the
an or buffer for the line
to become free may be a significant proportion of the total time needed for it totraverse
the network as a whole. Thus the waiting time in the buffer contributes significantly to
the propagation time. The propagation time in turn may affect the apparent speed of
response of a computer reacting to the typed other commands of the computer user.
or
Most computer applications can withstand some propagation delay, though above a
given threshold value further delay may be unacceptable. Thus, for example, many
packet-switched data networks are designed to keep delays lower than 200ms.
The switch and network designers of a data network must ensure that the bitspeed of
the line is greater than theaverage offered bitrate of all the packets, framesor cells. If the
line were not fast enough, a build-up data would occur in the
of buffer at the transmitting
end, causingconsiderabledelay. Inaddition,the buffer must be largeenoughto
temporarily store all waiting data. If the buffer were not big enough, then arriving
packets will be lost at times when it is full and ‘overflows’.
The formulae of Table 30.4 may be adapted for data network dimensioning by sub-
situting N = 1. This corresponds to a single connection between points. It is equivalent
to a single data link between terminals or packet-switched data exchanges. The link will
have a given bit speed capacity, and the data flowing over the link could be thought of
as being measured in Erlangs. For example, an average bit rate of 7500 bit/s being
carried on a 14 400 bit/s link is equivalent to 7500/14 400 or 0.52 Erlangs.
Table 30.5 The Erlang call-waiting formula applied to data networks
_____ ~~ ~
(i) probability of delay = C = A =average line loading(typicallymonitored in %)
(ii)
average
delay= D =D P
(1/A - 1)L
A*
(iii) averagenumber of waitingpackets or frames = -
l-A
(iv) probabilityofpacketdelay(due to outputbuffering)exceeding t seconds
= Aexp(-(l - A)tL/p)
(v) probability o f j or more
waitingpackets = A’+’
where p =data packet, frame or cell size in bits
L = datalink line speed in bit/s
- 548 TELETRAFFIC THEORY
Setting N = 1 into the Erlang lost-call formula to calculate B, and then substituting
this value into the formulae of Table 30.4, and by renaming some of the parameters to
some more appropriate to data communication, we obtain the formulae of Table 30.5.
It is interesting to substitute afew values into the formulaeof Table 30.5 to give prac-
tical insight into the operation of data networks. First, let us consider the maximum
acceptable line loading for a typical X.25-based packet switching network based upon
9600 bit/s trunk lines. To do so, we consider formula (iv), substituting L = 9600, a
packet size p of 128 bytes = 1024 bits and an ‘acceptable’ buffering delay of 200 ms,
together with a ‘maximum acceptable’ buffering delayof 500 ms. Substituting different
values of A , the line loading, we obtain the probabilities of the packet delays exceeding
the 200ms and 500ms thresholds, as shown the graphs of Figures 30.13 and 30.14
in
Note how the probability of the buffering delay exceeding the 500ms ‘maximum
acceptable’ increases rapidly above line loadings of about 40%. At 40% line loading,
this probability is only around 2% (Figure30.14). The conclusion is that we should not
expect to run a 9600 bit/s packet-switched data network much above about 40-5070
line loading, if we do not want to experience ‘unacceptable’ delays.
Whataboutthe effect ofincreasing the linespeed while keepingthepacket size
constant you might ask? What we increase the linespeed fourfoldto 28 800bit/s? This
if
is a speed widely available nowadays using analogue modems. The result is shown in
Figure 30.15, where we again plot the probability of exceeding the 500 ms ‘maximum
acceptable’ buffering delay limit. Now we are able to operate the line at around 70%
loading without exceeding the 500 ms delay limit more than 1YOof the time.
Finally, let us also consider the practical problem of how large the waiting packet
buffer mustbeforourtwo examples.Here we decide on a ‘maximumacceptable’
proportion of packets which may be lost or corrupted (due to buffer overflow) of say
0.01%. We use formula (v) to calculate the required buffer size and derive the graph
shown in Figure 30.16.
0.7000
t
g 0.6oOo
U#
p 0.5000
U
I
2
- 0.4000
d
B 0.3000
.-
-
5
5 0.2000
m
I
n
g 0.1000
10% 20% 30% 40% 50% 60% 70% 80% 9%
0
line loading
Figure 30.13 Probability of buffering delay exceeding 200ms (for linespeed = 9600 bit/s, packet
size 128 bytes). Total packet propagation delay
around 300 ms
nguon tai.lieu . vn
