s3e trunk arrangements

1S3 TRUNK ARRANGEMENTS

Radio School

Core Unit Radio Systems and TechnologyR C U R

DetectorModulator

Channel coder

Speech coder

Channel decoder

Speech decoder

S3 Trunk arrangements

S3

S3 Trunk ArrangementsIndex

busy hour

collision

DSI

Engset distribution

erlang

Erlang B

Erlang C

idle marking

Offered traffic, o

loss system, Erlang B

multiple access

packet transmission

paging channel

Poisson distribution

Poisson model

queuing system

queuing time

radio exchange

radio trunk system

single access

trunk traffic, t

trunk utilization


S3 Trunk arrangements

ContentsPage

1. Introduction 32. Queuing models 73. Signalling in radio switching systems 134. Summary 155. References 17

Appendix 18

1. IntroductionA user of a communications network generally needs service only a smallproportion of the total time. A telephone subscriber typically uses the phone for1-3 minutes during the busy hour (the clock hour during which most trafficoccurs).

Furthermore, during a telephone call, each party will be active on average forless than half of the time. This uneven loading due to bursty traffic results inpoor utilization of the transmission channels. The situation can be even moreextreme in respect of interactive data traffic between data terminals and centralcomputers (Fig. 1.1).

Fig. 1.1.

t1 t2 t3t

ta tb tc10 kb/s

T

t1t

T Terminal (T) busy onlysmall proportion of the timeTele-

networkTerminal

Level 1: Trunk system

Level 2: Speech interpolation. Packet.T is not active all the time during a call

Traffic (intensity): t1+t2+t3 0.01 erlangT

Activity factor::ta+tb+tc t1

0.4 speech0.01 interactive data traffic


Despite uneven traffic loads, good utilization of the transmission channels ispossible through the trunking of the traffic generated by a large number ofsubscribers whose service requests are largely random and uncorrelated. Thegreater the number of subscribers sharing a transmission resource, the greaterwill be the trunking gain. An asymptotic limit is that the capacity of a networkneed not be higher than the average traffic load during the busy hour. Thismeans that fluctuations in traffic flow over time are completely balanced out ina perfect trunking system.

In a practical situation, however, the available peak transmission capacity mustexceed the mean traffic intensity. Besides the actual number of subscriberssharing the trunking resource, the difference between the top capacity and themean traffic intensity is also determined by the level of inconvenience thatsubscribers are prepared to tolerate. Because the capacity of the transmissionchannel (trunk group) is lower than the number of subscribers sharing the trunkgroup, there is a chance that the available capacity will be insufficient. Thus,excess traffic must either be blocked (loss system) or placed in a queue to waitfor free capacity (delay system).A general picture of these relationships is shown in Figs. 1.2 and 1.3. Thefigures apply to a loss system based on the Erlang B formula. The traffic (inten-sity) is measured in erlangs. One erlang corresponds to 100% traffic load onone line. One subscriber who during the busy hour has two calls, for instance,each having a duration of 1.5 min, generates A = 3/60 = 0.05 erlang. If Nterminals are connected, then the total offered traffic A

o= N.A erlang. The

most important measure of the performance of a loss system is the relationshipbetween the offered traffic

o per trunk (

o= A

o/n) and the corresponding

blocking or loss probability, pb, in other words, the probability of a call beingblocked owing to there being no available trunk. Because of the blocking, aproportion, pb, of the offered traffic will be lost, which means that the carriedtraffic, t, per trunk will be:

t = (1 - pb)oAs shown in Fig. 1.2, a substantial trunking gain is obtained if a number ofsubscribers share a trunk group comprising several lines, instead of a smallernumber of subscribers having access to just one common line.

Fig. 1.2

n=1

n=10

n=36

10% 68% 85% 1

10

100

50

Telenetwork

n

11036

6400

2,052

31519

3150684

t

N radio terminals use a common trunk group comprising n traffic channels.Traffic per terminal is 3 min (0.05 erlang) during the busy hour.Permitted blocking, pb, = 10%

6 (9)4557

(9)If N>>1, the system can carry60 . o . n minutes of call time per busy hourSystem capacity: terminal traffic (call minutes/busy hour)

60 . o . n

Signallingchannel

Radio switchingn traffic channels

Idle trafficchannels

Radio channels

N terminals

Loss system withlost calls cleared

pbblocking, %

Permitted blocking Trunkutilization

Call minutesper hour

Per channel Total

Number of terminalsin network

TotalPer channel

Offered traffic/channel, o = (1 pb)t

0.5

Improved spectrum efficiency from trunking gain

or t n erlangs carried traffic

terminals


If one radio channel is shared between a relatively large number, N, of termi-nals, the permitted blocking will determine the trunk utilization (t = pb). If 10%blocking is allowed, then there will be a 90% probability that none of the otherterminals (N - 1) will be using the channel at the arbitrary moment at which thenth terminal want to be connected. Thus, the offered traffic from (N - 1) termi-nals will be:

A erlangtpbpbpb0 0 1 1 0 11= = = =

,

(n=1)

If N >> 1, we can disregard the difference between N and (N-1). In other words,the number of terminals that can share the channel is given by the equation:A

o= 0.11 N.A. However, at low values of N, N must be replaced by (N - 1)

in the above equation. In this case (n = 1 in figure 1.2) 10% blocking is obtainedif (N - 1) terminals generate 6.6 min of traffic per busy hour, i.e. (N - 1) = 2,since each terminal generates three minutes. Thus, one channel can only servethree terminals.For a trunk group consisting of more (n) traffic channels, the offered traffic perchannel,

o, will be substantially higher at 10% blocking. For n = 10 we get

o

= 0.75 (see the chart for Erlang B in Fig. 2.2 which gives t = 0.68). Themaximum number of terminals that can be connected is determined by the totaloffered traffic: (0.75 x 10 x 60) = 450 call minutes per busy hour. The trunkgroup can therefore serve (450/3) = 150 terminals. Similarly, for a trunk groupof 36 radio channels, 10% blocking gives

o= 0.95, which means that the

number of terminals it can serve is (0.95 x 36 x 60/3) = 684.The Erlang B model applies to a lost call cleared situation; in other words, oncea call has been blocked, the subscriber will no longer wish to establish thedesired connection. A more plausible case is that the subscriber whose call hasbeen blocked will try again in the hope that all trunks will no longer be busy. Ascan be seen from Fig. 1.3, retrials increase the blocking probability (see alsosection 2.2).

Fig. 1.3.

o1

1 2 3 4 5 6 7 8

Example:Capacity with lost calls returningHow many subscribers can be accommodatedper radio channel with 20% blocking withouttrunking (n = 1) and with a trunk group of four?The total holding time per terminal during busy hour is 2 min.

Blockin

g = 2

0%pb

Blockin

g = 5

%pb

Lost calls cleared

Lost calls returning

Lost calls clearedLost calls returning

Offered traffic = tsmN ts = Average holding time m = Average number of calls per busy hour per subscriber N = Number of subscribers (terminals) (N>>1)

n = Number of trunks

Solution:At n= 1 the capacity will be 0.16 erlang offered traffic;that is, 60 x 0.16 = 9.6 10 call minutes per hour allowing5 subscribers per channel.At n = 4, the capacity will be 0.63 erlang per channel, that is 60 x 0.63 = 38 call minutesper hour per radio channel.Each channel can accommodate 19 subscribers, i.e. a trunk group of four channels canaccomodate 76 subscribers.

0.5

Offered traffic (o erlang) per trunk in a loss system


The effective sharing of transmission channels is particularly important in radiosystems where the available traffic capacity is limited not only by cost but alsoby frequency shortage. The acute frequency shortage in mobile radio has alsoforced higher utilization of the channels than what is usual in the public tel-ephone network. Subscribers often experience a relatively high level of block-ing or considerable delays.

The transmission procedures described above are of the single-access type. Thismeans that the shared transmission channel connects two nodes (exchanges) towhich the subscribers put their service requests and feed the information to besent. The node therefore has full knowledge of the demands for service andavailable trunks instant by instant, and can pack the calls densely. This achievesoptimum utilization of transmission capacity through trunking of the calls.

Single access is often used in radio networks, such as in multiplexed point-to-point communications and for outgoing traffic from radio base stations toconnected terminals. However, sharing of radio channels is also often based onmultiple access (MA). In MA, there is no common traffic-controlling node thathas total knowledge of the instantaneous traffic demand and thus able to provideoptimum sharing of the transmission resource between the subscribers. Conse-quently, multiple-access systems do not utilize the transmission channels aswell as single-access systems do.

This module deals with traffic models based on single-access. The usual ar-rangement consists of a loss system, which means that when all the channels arebusy, additional incoming calls are blocked (and not returned). In traffic theory,this corresponds to the Erlang B model, which describes the relationship be-tween blocking probability, the size of the trunk group and the total offeredtraffic.

Another possibility is to have a queuing or delay system, whereby, when all thechannels are busy, additional demands for service are placed in a queue andserved when trunks are released. This corresponds to the Erlang C model. Aswell as the delay probability, the Erlang C model also gives the average delay(waiting) time as a function of the size of the trunk group and the total trafficload.


Total offered traffic = Ao = N tm = tm (=N )

Ao = 2

3 50 =

Aa = 5 erlang

The following is true during the busy hour: Offered traffic per subscriber: Call arrival rate Average holding time, t m

Traffic intensity per subscriber =

Normalized to number of trunks, o Carried traffic per trunk, t Total carried traffic, At = n t With delay system: A o = At = A o = t = With loss system: A t = Ao (1pb) t = o (1pb) pb =

Ao At : (probability of blocking)

in the same unit

AoExample for loss system: = 2 calls per hourtm = 3 minN = 50 60

t 8

Nsubscriberlines

n trunks

(Offered traffic, Ao or ) (Trunk traffic, )a aAn=

nP

ab

=

=

=

885

8%

Atn

tm erlang

= 5(1 0.08) = 0.575(from Fig 2.2)

2. Queuing models2.1. Overview

In this chapter we shall be looking at the overload characteristics of an ex-change or concentrator that combines the traffic from a relatively large number(N) of subscribers to a smaller number (n) of trunks (see Fig. 2.1). An importantdifference between a loss system (Erlang B) and a delay system (Erlang C) isthat a delay system (with infinite capacity) must accomodate all offered traffic(At = Ao), whereas a loss system rejects the excess traffic. In the latter case, weneed to distinguish between trunk utilization (trunking efficiency) t = At/n and

o= A

o/n, that is, the offered traffic load normalized to the number of trunks.

Fig. 2.1

Both the Erlang B and Erlang C models are based on the assumption that thecall arrivals are completely random, having no mutual correlation. The numberof call attempts per unit time, , is not time dependent. In the correspondingmathematical model, the interarrival times have a negative exponential distribu-tion, and the number of arrivals in a given period has Poisson distribution(see the Appendix). A further assumption is that the number of terminals islarge (N ).

Another important traffic model is the Poisson model which is used in the USAand discussed in the Appendix. The Poisson model gives somewhat higherblocking than Erlang B, which roughly takes into account the retrials, which aredisregarded in the Erlang B model.

0pb


1

0.5

0.3

0.03

0.2

0.02

0.1

0.01

0.05

0.2 0.5 0.8 1

pb = A

1. Lost calls cleared2. Call arrivals have Poisson distribution3. Number of connected subscribers, N, = N =

At = Ao (1-pb)

Example:Permitted blocking is 3%.What traffic load can be offered to a trunkgroup witha) n=4?b) n=16? What will be the correspondingtrunk utilization, t?

Solution:a)

b)

n=1

n=2

n=4

n=8

n=16

n=30

n=50

n=10

0

Loss system Erlang B Blocking probability

n trunks

Ao

Trunk utilization:

t =At

=

Ao Ab =

Ao (1pb) = o (1pb)n n n

Ab

bAo

Aono=

= 0.67 Ao = 16 0.67 = 10.7t = 0.67 0.97 = 0.65

= 0.31 Ao = 4 0.31 = 1.24t = 0.31 0.97 = 0.30

2.2. Erlang B loss systemThe Erlang B traffic model is based on the assumption that the calls blockedduring traffic overload will be cleared from the system, i.e. no calls are re-turned. It is also assumed that incoming calls have a totally random time distri-bution and that N = . This case is described in Fig. 2.2 (see also Table 1).

Fig. 2.2

Retrials resulting from blocking means that the traffic load consists of both newtraffic and return traffic. Mathematically this can be treated as an increase in theoffered traffic, provided that the return traffic is random with an average delaylonger than the average holding time. (If the interval between re-tries is tooshort, the probability of blocking will increase as the overload situation will nothave been resolved. If several subscribers whose calls have been blocked tryagain immediately, there will be an acute traffic peak, with a renewed risk ofsystem overload.)

Since several retries may be necessary, the following expression gives the totalnumber of call attempts per unit time:

pr is the probability of a blocked subscriber wanting to retry a call. This pro-

duces a corresponding increase in the offered traffic:

Return of lost calls increases the probability of blocking compared with thebasic Erlang B model.

= + pr pb + pr pb( )2 + = 1 pr pb

Ao =Ao

1 prpb


Table 1. Erlang B loss systemAttempts have Poisson distribution. Number of terminals, N, = . No retry.Blocking probability = pb. Number of trunk lines = n.

Offered subscriber traffic = Ao

Normalized = o =

A on

Carried traffic = At Trunk utilization t =t = (1pb)o

pb = 1% pb = 2% pb = 5% pb = 10%

n Ao

o

t Ao o t Ao o t Ao o t n

1 0.01 0.010 0.010 0.02 0.020 0.020 0.05 0.050 0.050 0.11 0.110 0.100 1 2 0.15 0.075 0.076 0.22 0.110 0.109 0.38 0.190 0.181 0.60 0.300 0.268 2 3 0.46 0.153 0.150 0.60 0.200 0.197 0.90 0.300 0.285 1.27 0.423 0.381 3 4 0.87 0.218 0.215 1.09 0.273 0.268 1.52 0.380 0.362 2.04 0.501 0.460 4 5 1.36 0.272 0.270 1.66 0.332 0.325 2.22 0.444 0.422 2.88 0.576 0.519 5

6 1.91 0.318 0.315 2.28 0.380 0.372 2.96 0.493 0.469 3.76 0.627 0.564 6 7 2.50 0.357 0.354 2.94 0.420 0.411 3.74 0.534 0.507 4.67 0.667 0.600 7 8 3.13 0.391 0.387 3.63 0.454 0.444 4.54 0.568 0.539 5.60 0.700 0.630 8 9 3.78 0.420 0. 416 4.34 0.482 0.473 5.37 0.597 0.567 6.55 0.728 0.655 910 4.46 0.446 0.442 5.08 0.508 0.498 6.22 0.622 0.591 7.51 0.751 0.676 10

12 5.88 0.490 0.485 6.62 0.552 0.540 7.95 0.663 0.629 9.47 0.789 0.711 1214 7.35 0.525 0.520 8.20 0.586 0.574 9.73 0.695 0.660 11.47 0.819 0.737 1416 8.88 0.555 0.549 9.83 0.614 0.682 11.54 0.721 0.685 13.50 0.844 0.759 1618 10.44 0.580 0.574 11.49 0.638 0.626 13.39 0.744 0.706 15.55 0.864 0.777 1820 12.03 0.602 0.596 13.18 0.659 0.646 15.25 0.763 0.724 17.62 0.881 0.793 20

22 13.65 0.620 0.614 14.89 0.677 0.663 17.13 0.779 0. 740 19.69 0.895 0.806 2224 15.30 0.638 0.631 16.63 0.693 0.679 19.03 0.793 0.753 21.79 0.908 0.817 2426 16.96 0.652 0.646 18.38 0.707 0.693 20.94 0.805 0.765 23.68 0.911 0.82 2628 18.64 0.666 0.659 20. 15 0.720 0.705 22.87 0.817 0.776 25.99 0.928 0.835 2830 20.34 0.678 0.671 21.93 0.731 0.716 24.60 0.820 0.785 28.11 0.937 0.843 30

32 22.05 0.682 0.682 23.73 0.742 0.727 26.74 0.836 0.794 30.23 0.945 0.850 3234 23.77 0.699 0.692 25.53 0.751 0.736 28.70 0.844 0.802 32.37 0.952 0.857 3436 25.51 0.709 0.701 27.34 0.759 0.744 30.66 0.852 0.809 34.51 0.959 0.863 3638 27.25 0.717 0.710 29.17 0.768 0.752 32.62 0.858 0.815 36.65 0.964 0.868 3840 29.01 0.725 0.718 31.09 0.777 0.759 34.60 0.865 0.822 38.79 0.970 0.873 40

Erlang B assumes that N = . Cases with low values of N and n can be analysedusing Engset distribution, which is described in published tables (e.g. seereference 2). One example given in Reference 2 applies to the case with offeredtraffic, A

o= 12 erlang and n = 12. For Erlang B (N = ), pb 1 % is obtained.

For N = 200, the Engset model gives blocking of approx. 0.8% and, for N = 50,blocking of about 0.4%.

A related case is that where N and different subscribers generate differenttraffic loads. This is discussed in references 4 and 5. The probability of blockinghere is lower for subscribers generating a high traffic load than for those gener-ating little traffic.

A tn

10

S3 TRUNK ARRANGEMENTS

pd

n=1

n=2

n=4

n=8n=16

n=30

1

0.5

0.3

0.2

0.1

0.05

0.03

0.2 0.5 0.8 1

=

=

1 (1pb)

An

Queuing system (Erlang C). Delay probabilityQueuing system

pb Probability of delay, pd, =

Infinite queuing capacity

Unlimited patience in waiting subscribers

Poisson distribution of incoming calls

Exponential distribution of service times

Number of subscriber lines, N,

Delayed traffic Ad

AdA

At

At =Ao = A

Ad

n trunks

Ao

(pbfrom Erlang B)

2.3. Erlang C delay systemIn the Erlang B model, the distribution of holding times was not important.Only the average holding time, t

m, was used in the model. As regards queuing

systems, however, the distribution of the delays for calls held in a queue and theaverage delay are influenced by the distribution of the holding times. The aver-age delay will be shorter, for instance, if the service times are constant. In nor-mal voice traffic, the distribution of service times, to a reasonable level of accu-racy, is exponential. This corresponds to a constant probability of a given callbeing terminated within a short period (the probability is not dependent on howlong the call has been in progress, and the average remaining duration is notdependent on how long the call has been in progress).

The assumption of exponentially distributed service times greatly simplifies themodel and this case is therefore generally assumed to apply. The Erlang Cmodel for a delay system is based on this assumption and also that the queuingcapacity is unlimited. In addition the assumptions stated earlier (random incom-ing calls and N = ) are introduced. Another condition is that the queuing callsare cleared in order. These assumptions together result in an exponential distri-bution of queuing delays. Thus, if the average delay is known, the distributionof delays can easily be calculated.

The parameters of greatest interest to a delay system are the blocking probability(delay probability, pd), for calls placed in a queue, and the average queuing de-lay. This can either be given in respect of all calls ( W ) or only for those de-layed ( Wd). The first of these values will be pd times the second (see Figs. 2.3,2.4 and Table 2). The table shows the average delay time, Wd , for delayed callsfor an average call duration, t

m, of 100 seconds. The corresponding values of Wd

for other values of tm

are obtained through proportioning. (For instance, iftm

= 200 s, the values of Wd given in the table must be doubled.)

Fig. 2.3

The average delay is proportional to the average service time and inverselyproportional to the number of trunks (n). If acceptable absolute delays aredetermined by the overriding system requirements, better trunk utilization willbe obtained for short service times and large trunk groups. Trunk utilizationhere will approach 100%.

11


Fig. 2.4

Table 2. Erlang C delay systemPoisson distributed arrivals. Number of terminals, N, = .Infinite queue capacity.Exponential distribution of service times.Probability of queuing = pd Number of trunks = nTraffic = A Trunk utilization, = AWd: = Average call delay in seconds (for calls in queu) if average service timeis 100 s.

= 2% = 5% = 10% = 20%

n A Wd A Wd A Wd A Wd n

1 0.02 0.020 102 0.05 0.050 105 0.10 0.10 111 0.20 0.200 125 1 2 0.21 0.105 56 0.34 0.171 60 0.50 0.25 61 0.74 0.370 79 2 3 0.55 0.185 40.9 0.79 0.262 45.2 1.04 0.347 51 1.39 0.464 62 3 4 0.99 0.249 33.3 1.32 0.330 37.3 1.65 0.413 42.6 2.10 0.525 53 4 5 1.50 0.299 28.6 1.91 0.381 32.3 2.31 0.463 37.2 2.85 0.569 46.4 5 6 2.05 0.341 25.3 2.53 0.422 28.8 3.01 0.501 33.4 3.62 0.603 42.0 6 7 2.63 0.376 22.9 3.19 0.455 26.2 3.73 0.532 30.5 4.41 0.629 38.6 7 8 3.25 0.406 21.0 3.87 0.484 24.2 4.46 0.558 28.3 5.21 0.651 35.9 8 9 3.88 0.432 19.6 4.57 0.508 22.6 5.22 0.580 26.5 6.03 0.670 33.6 910 4.54 0.454 18.3 5.29 0.529 21.2 5.99 0.599 24.9 6.85 0.685 31.8 1012 5.90 0.492 16.4 6.76 0.563 19.1 7.6 0.630 22.5 8.53 0.711 28.8 1214 7.31 0.522 15.0 8.27 0.591 17.5 9.2 0.654 20.7 10.23 0.731 26.6 1416 8.77 0.548 13.8 9.82 0.614 16.2 10.8 0.674 19.2 11.96 0.747 24.7 1618 10.25 0.570 12.9 11.40 0.633 15.2 12.4 0.691 18.0 13.70 0.761 23.2 1820 11.77 0.588 12.1 13.00 0.650 14.3 14.1 0.706 17.0 15.45 0.773 22.0 2022 13.30 0.605 11.5 14.62 0.664 13.6 15.8 0.719 16.1 17.22 9.783 20.9 2224 14.86 0.619 10.9 16.25 0.677 12.9 17.5 0.730 15.4 19.00 0.792 20.0 2426 16.44 0.632 10.4 17.91 0.689 12.4 19.2 0.739 14.8 20.79 0.800 19.2 2628 18.03 0.644 10.0 19.57 0.699 11.9 21.0 0.748 14.2 22.58 0.806 18.4 2830 19.64 0.655 9.7 21.25 0.708 11.4 22.7 0.756 13.7 24.38 0.813 17.8 3032 21.26 0.644 9.3 22.93 0.717 11.0 24.4 0.763 13.2 26.19 0.818 17.2 3234 22.89 0.673 9.0 24.63 0.724 10.7 26.2 0.770 12.8 28.01 0.824 16.7 3436 24.53 0.681 8.7 26.34 0.732 10.4 27.9 0.776 12.4 29.83 0.829 16.2 3638 26.18 0.689 8.5 28.05 0.738 10.1 29.7 0.782 12.1 31.65 0.833 15.8 3840 27.84 0.696 8.2 29.77 0.744 9.8 31.5 0.787 11.7 33.48 0.837 15.3 40

pd pd pd pd

n

10

3

1

0.3

0.10.2 0.5 0.7 0.9

P [Wd Wd*] = exp (WW d*) = exp (Wd n (1) )d mExample:

= 0.8 n=4What is the probability of a call being delayed?If it is, what is the probability that the delay exceeds 30 s?

Solution: a and b pd = 0,45a. Wd = 3 = 3.75 min P[Wd 30s] = exp ( 0.5 ) = = exp (0.133) = 88%b. W d = 0.5 = 0.625 min P[Wd 30s] = exp( 0.5 ) = = exp (0.8) = 45%

3.75 4.0.2

Delays in queuing systems(Erlang C)

d

W =.

:

4.0.2

n=1

n=2 n=4

n=8

n=16

n = 1p = W tm(1)

(1) tm

*

(ts service time has exponential distribution)If constant t s

When calls are serviced in order, the delay distribution is exponential

tm = Average service time = Offered traffic per trunk pd = Probability of delay (as per Fig. 2.3)

Average delay for all subscribers, W = p d . Wd

Average delay for queuing subscribers, W d

=

n(1) tm

Mean service time tm = a) 3 min and b) 30 s

0.625

Special case: n=1

Wt

Pnm

d=

( )1 pd

d=

2(1) tsW =

12


A considerable complication for a heavily loaded queuing system is its vulner-ability to overloading if the offered traffic during certain periods is substan-tially higher than the design value (e.g. mean traffic load during the busy hour).When the traffic intensity reaches a value corresponding to 100% trunk utiliza-tion, the system collapses because of rapidly increasing delay times and numberof calls in the queue. When designing a system, we may therefore be forced tointroduce a safety margin in respect of the permissible trunk utilization. A lostcalled clear system, such as Erlang B, is less sensitive to overloading, owing tothe greatly increased level of blocking when trunk usage approaches 100%; inother words, an increasing proportion of the offered traffic is blocked. Even athigh values of

o, t will never attain a value of 1.

On a radio trunking system for dispatch applications, call durations aregenerally much shorter than on a mobile telephone network. Consequently, anyinconvenience to users will be fairly small, even if the average queuing delay isas long or even longer than the average call duration. Delay systems are there-fore often preferable to loss systems. However, the Erlang C model has limitedapplication, since the effective number of users is mainly determined by therelatively small number of dispatchers and, in addition, the number of trunklines due to practical limitations seldom exceeds eight.

The relationships are so complex that we usually have to resort to simulation. Agood summary of this, complete with bibliography, is given in the CCIR GreenBook, Vol. VIII (Ref. 7). The report draws the conclusion that in certain condi-tions the maximum trunk utilization is achieved with a trunk group as small asfive. This is because the system must be able to handle a reasonable peak inincoming traffic. In large trunking systems, trunk utilization is usually so highthat unacceptable delays will result if a moderate safety margin is added to thedesign value of offered traffic load.

In packet transmission using single-access, a large number of packets sharea single wideband transmission channel. The simplest case is where all thepackets are of the same length and with random arrivals. This case has manysimilarities to Erlang C with n=1, but one essential distinction is that the aver-age delay here is halved. (As mentioned above, the average delay is lower withconstant service times than with varying times. The probability of delay, on theother hand, is not affected.) In this case the average normalized delay is

w

tp=

1

tp is the packet length and the channel utilization.

13


3. Signalling in radio switching systemsAn additional complication to a radio trunking system (radio switching system)is the transmission requirement to enable terminals to make requests for assign-ment of traffic channels and to enable base stations to page the terminals. Thereis no corresponding requirement when the terminals are connected by wire tothe exchange, since these lines are available all the time for signalling to andfrom idle subscribers.

The easiest way to allocate traffic channels in a radio switching system is, forthe base station to transmit a radio signal with a special identifier on the idlechannel to be used for the next call from a terminal (see Fig. 3.1). This outwardchannel is used for paging of terminals and the return channel for call requests.

Fig. 3.1

All idle terminals scan the group of radio channels that have been allocated tothe base station and look for the allocated paging channel. Once the pagingchannel has been found, the search ceases and the terminals monitor this chan-nel continuously. When the next call shall be set up, the call is assigned to thepaging channel, and the idle marking is cleared. Provided that all the availableradio channels are not busy, one of those still free is designated as the nextpaging channel. The idle terminals leave the previous paging channel as soon asidle marking is cleared, whereupon they start searching for the new pagingchannel.

The procedure is simple, as no signalling other than the idle marking is requiredto assign traffic channels. However, there are limitations. In large radio systemswith comprehensive signalling needs, it may therefore be necessary to introducea separate signalling and paging channel. This principle is shown in Fig. 3.2.

Channel 1 BT

tT1

T3

T5

T2

T4

T6

S1

S2

S3

M1

M2

M3

Terminal number(acknowledgement)

Kanal 1 TB

tTerminal number

B subscriber number

Channel 2 BT

tChannel 2 TB

t

Channel 3 BT

t

Risk of collision between call requests from terminals

Channel allocation through idle marking of an available channel

Terminals Base station

All idle terminals search for the tagged radio channel

= Idle tone

Acknow-ledgement

Call request from terminal userCall request from fixed network

Terminal number

Terminal number(acknowledgement)

14


Fig. 3.2

Signalling consists of data messages, which include the identity of the terminalin question. A wide variety of messages can be sent: paging, allocation of trafficchannel, acknowledgement of calls placed in queue to await traffic channelbecoming free. The different types of messages are identified by differentcodes.

A complication to both signalling arrangements shown in Figs. 3.1 and 3.2 isthat when signalling takes place (request for channel assignment) from termi-nals to the base station, several terminals may transmit nearly simultaneously.This can cause the signalling packets to collide or overlap in time, with theresult that the base station receiver can neither decode any of them nor identifywhich terminals were attempting to make a call. This signalling from the termi-nals to the base really constitute a multiple-access situation (Aloha). This isdiscussed in the packet-radio module. Protocols must be introduced so that, inthe event of collisions occurring, the risk will be minimized of further collisionsoccurring when the packets are repeated.

Queuing status (if any)

BT TBt

t

t

Request for channel allocation TB

TB

BT

t

Allocation of traffic channels using a separate signalling channel

Terminals Base stations

Traffic manage-ment

System signalling

Terminal number

Paging & traffic channel allocation B T

Code Code

Code

Code

Code

Traffic channel

(Acknowledgement)Terminal number

No traffic channel freeTerminal number

Terminal number Allocation of traffic channel

Terminal number

Traffic channel

Risk of collision between data messages from terminals to base (multiple access)

T

All idle terminals monitor the dedicated signalling channel. Assigned traffic channel communicated via data message B

15


Wvtm

1

0.5

0.2

0.1

0.05

0.03

n=1

n=16

Erlang CPoissonErlang B

0.2 0.5 1

pb , pd

o =

0.8

POISSON and ERLANG models. Overview

Aon

4. SummaryFig. 4.1 contains a chart showing the probability of blocking, pb, or the prob-ability of delay, pd, for Erlang B, Poisson and Erlang C, for two values of n. ThePoisson model clearly gives rise to a higher probability of blocking than ErlangB at given values of n and . As mentioned earlier, the probability of blockingin the Erlang B model is lower than in reality, because the effect of retries ofblocked calls has been disregarded. Since the Poisson model produces slightlyhigher blocking values than in Erlang B, it might be a better model for a lostcall returning system. The Poisson model is generally used in the USA.

Fig. 4.1

The principal parameters in the Erlang C queuing system, i.e. the probability ofdelay, pd, the normalized average delay for queuing subscribers, , and thenormalized average delay for all subscribers, , are shown In Fig. 4.2 for twovalues of n. An important characteristic is the behaviour when = 1, whereuponpd will be 1 and the delay time will be infinite. A queuing system can breakdown due to overload.

Fig. 4.2

Wtm

pd

pd

10

5

32

1

0.5

0.30.2

0.1

0.05

0.030.02

0.010.2 0.5 0.7 1

Wvtm

Wtm

Wvtm

Wtm

n = 1n = 8

An

=

Erlang C delay system: performance overview

16


The Erlang and Poisson distributions are based on certain assumptions; theseare shown in Fig. 4.3. The fundamental assumption is that arrivals are totallyrandom. This is reasonably valid for most of the time. However, situations canarise where a large number of subscribers may attempt to initiate calls simulta-neously, thus giving rise to traffic peaks that overload the system. Such situa-tions may be caused by extraneous events, e.g. a large number of subscribersexperiencing mains power cuts or the like. Traffic peaks induced by offers onTV programmes have also been known to jam the telephone network. Futurepersonal telephone systems based on microcells may have difficulty in copingwith concentrated peak loads, such as can occur immediately after a major con-ference or public event. (The major part of the coverage area of a microcell maybe a congress centre or sports arena/complex. In this case the assumption thatthere is no correlation between service request are evidently nor valid).

Fig. 4.3

Common assumptions

1. Random call arrivals No correlation between service requests Constant arrival rate (independent of time) Exponential distribution of interarrival times Poisson distribution of the number of service requests per unit time

2. Exponential distribution of holding (service) times Probability of a call being terminated during the period, t to t + t, is independent of t.

3. Infinite number of subscribers (N ) Offered traffic not dependent on the number of busy subscribers

(N )

17


5. References1. Kleinrock: Queuing systems

Volume 1: Theory, Wiley 1975Volume 2: Computer Applications, Wiley 1976

2. Bear: Principles of Telecommunication - Traffic EngineeringP. Pergrinus Ltd., 1976

3. J. Bellamy: Digital TelephoneWiley 1982

4. Davis, Mitchell: Studies of Small Trunking Systems for Mobile RadioCommunications 78, Birmingham

5. Davis, Mitchell: Traffic Handling Capacity of Trunked Land Mobile RadioSystemsIEEE/ICC 79

6. Dartois: Lost Call Cleared Systems with Unbalanced Traffic Sources6th Int. Teletraffic Congress Mnchen

7. CCIR Green Book 1986 Vol VIII-1, sid 110 - 124

8. Descloux: Delay Tables for Finite and Infinite Source SystemsMcGraw Hill, 1962

18


AppendixThe Poisson model. TASI/DSI

Mathematical analysis of Erlang B and C is complicated as the traffic flow isaffected by the overload characteristics of the system. In the loss system, lostcalls are cleared; in the delay system, calls are placed in a queue. This is asignificant complication and it would take too long to analyse these cases here.However, it is relatively easy to analyse the case whereby the traffic to behandled is not influenced by calls being definitively or temporarily lost when allthe trunks are busy. The influence of overloading is disregarded in the Poissonmodel (blocked calls are assumed to wait during the originally planned servicetime).

Another important case in which the same applies is that in TASI/DSI, dealtwith in Module S2. In this case, the first part of the speech segment is cut off ifa trunk cannot be allocated immediately. The probability of blocking is obtainedby calculating the percentage time (the probability) that the number of incomingactive lines is greater than the number of outgoing trunks.

Poisson distributionThe basic assumption in the analysis below is that arrivals are totally randomand that the traffic intensity is not dependent on time. This means that thenumber of arrivals during a given period of time will be a stochastic variabledetermined by the Poisson distribution (see Fig. A-1).

Fig. A-1

The Poisson distribution may also be regarded as an extreme case of the bino-mial distribution with N .

Poisson modelIf, to start with, we make the simplifying assumption that all calls have the sameduration (t

m), the probability of overload (blocking) can easily be calculated. At

P(tt*) = exp ( t*) t*

1= 37%

e

t*t*

1

0.5

1 2 0

0.40.2

1 2 3 4 5 6

Pk (t*)

ktl t2 tnT

tn + 1

t*

. e

t* k!

Poisson and exponential distributions Probability of event occurring during the period, t t + t = c . t P [t t*] = X (t*) = Probability that the interval between two events is t*. [l X (t*)] = Probability of no event occurring during the period t*. Probability that the time between two events is within the interval of t* t* + t*:

[lX (t*)] . c . t* = dX(t*) .

X (t*) = l e c . t* X (0) = 0 X() = 1

dt*

P (t t*) = l X (t*) = e c.

t*

dX = c . e ct*

dt* c

0 t

T

t*

t* = l

Probability of the time between two events exceeding t*

Example: T = t*

The probability of k events occurring during period T is given by Poisson distribution:

PK (T) = (T/ t*)k

Events

19


any given moment blocking will occur if more than n call requests have beenreceived during the previous time interval of length t

m.

If the average arrival rate is , the probability of there being k incoming callsduring the interval, t

m will be:

Since the total traffic intensity is A = tm, the probability of blocking, Pb, may

be written as:

This traffic model is known as Poisson distribution. The relationship betweenthe blocking probability and the incoming traffic normalized to the number oftrunk lines is shown in Fig. A-2.

Fig. A-2.

A more complicated calculation for an arbitrary statistical distribution of hold-ing times yields the same relationship if t

m represents the average holding time.

0.6

0.3

0.2

0.1

0.05

0.03

0.02

0.010.2 0.5 0.8 1

n=1n=2

n=4n=8

n=30

50100

n=16

o = Ao

n

Example for a TASI/DSI arrangement

Traffic intensity on incoming lines = 0.25 erlang/line

a) N = 8 A = 2 erlang n = 4 = 0.5

b) N = 20 A = 5 erlang n = 10 = 0.5

p b = 15%

p b = 4%

Poisson1. Poisson distribution of arrivals2. Exponential distribution of holding times3. Number of connected subscriber lines, N, =

Pb

Pk =tm( )k

k!e

tm

Pb = e A

k=n+1

Ak

k!

pb

20


Analysis of overload characteristics in TASI/DSILet us assume that a concentrator has N incoming lines and n outgoing lines.The probability that an incoming line will be active at a given instant is desig-nated p, the value of which is determined by the incoming traffic on the line. If,for example, this is 0.25 erlang, then p = 25%. (In this case the traffic from aterminal is determined by the active speech intervals - level 2 in fig. 1.1).

Blocking will occur if at least n of the other (N - 1) lines are active whilst thestudied incoming line is active. This is the same mathematical problem that weanalysed in conjunction with channel coding. There, the problem was to find theprobability that a code word comprising (N - 1) bits would contain at least nincorrect bits if bit errors occur randomly and have a probability of p. Here, theprobability that | incoming lines will be active is given by:

The probability of blocking, pb, is obtained by adding the probabilities of | = nup to | = (N - 1) incoming lines being active:

Another interesting parameter in a TASI/DSI system is the average length of thecut-off parts of the speech segment in question. If only a small part of the seg-ments is cut off, the resulting inconvenience will be relatively insignificant. Wecan estimate this if we first calculate the probability of blocking, pb, that is theinstantaneous probability that a given incoming call will be subjected to block-ing. If k incoming lines are active at a given instant (k > n), then (k - n) of thesewill be blocked. Thus, the number of blocked lines will be:

Px = (k n)k=n+1

N Pk dr Pk = Nk pk (1 p)Nk

is the probability that k incoming lines will be active at a given instant.

The probability that a given line will be blocked is obtained by dividing Px by

the average number of active lines, N p. From this it follows that the probabilityof blocking will be:

If the average duration of an active speech segment is T, the average time cutoff will be pbT. Note that this average is valid for all speech segments, eventhose that have not been affected by overloading. However, our interest is con-fined to those segments affected by overloading. For these, the average cut-offtime, Tbf, is given by the following equation:

The above function becomes relatively complex at high values of N and n.However, it is generally an acceptable approximation to use Poisson distributioninstead of binomial distribution.

where

P(l) = N1l

pl(1 p)(Nl)1

ps =l=n

N1 N1l pl(1 p)(Nl)1

pb =k=n+1

N k n( ) Nk

pk 1 p( )N k

1N p

Tbf =pb Tps

21


EN/LZT 123 1245/3 R3

Author professor Sven-Olof hrvikin cooperation with Ericsson Radio Systems ABunit ERA/T, Core Unit Radio System and TechnologyPublisher Ericsson Radio Systems ABT/Z Ragnar Lodn

Ericsson Radio Systems ABTorshamnsgatan 23, KistaS-164 80 Stockholm, SwedenTelephone: +46 8 757 00 00Fax +46 8 757 36 00 Ericsson Radio Systems AB,1997

ContentsIntroductionRadio ExchangeTrunking Example

2. Queuing models2.1. Overview2.2. Erlang B loss systemErlang B Table

2.3. Erlang C delay systemErlang C Table

3. Signalling in radio switching systems4. Summary5. ReferencesAppendix: Poisson Model

s3e trunk arrangements

Documents

traffic flow

excess traffic

average traffic load

themean traffic intensity

trunk systemlevel

traffic intensity

traffic load onone line

available capacity