267

Congestion Probabilities in a Circuit-Switched Integrated Services Network Zbigniew Dziong Institute of Telecommunications, Warsaw University of Technologv, Nowowiejska 15/19, 00-655 Warsaw; Poland

James W. Roberts Centre National d'Etudes des T~l&ommunications, 38-40, rue du G~n&al Leclerc, 92131 lssy les Moulineaux, France

Received 10 January 1987 Revised 3 May 1987

We derive recurrence relations enabling the direct calculation of the joint distribution of the number of busy channels in a circuit-switched network whose offered traffic streams require a variable number of channels on each link. Approximate congestion probability calculation methods are developed based on the estimation of marginal link occupancy distributions taking account of link traffic correlation. These approximations are tested on a number of simple two-link examples.

Keywords: Integrated Services Digital Network, End-to-End Blocking, Multi-Rate Circuit Switching, Joint Occupancy Distribu- tion, Link Traffic Correlation.

I. Introduction

We consider the problem of evaluating congestion probabilities in circuit-switched telecommunication networks when the number of channels seized by a call on each link of its path is a variable determined by

Zbigniev Dziong was born in Poland in 1952. He received his M.Sc. and Ph.D. degrees from the Warsaw University of Technology, Poland in 1974 and 1980, respectively, both in Electrical Engineering. He is currently an Assistant Professor at the Warsaw University of Technology where he lectures on digital switching systems and modelling and performance evaluation of telecommunications systems. His current research interests include modelling and optimisation of communication networks, teletraffic theory and system design methodology based on the CCITT defined System Description Language. He has taken sabbatical leave at the Centre National d'Etudes des T~16communications, France (six months in 1985/1986) and at the Department of Communication Systems, Lund Institute of Technology, Sweden (three months in 1987).

James W. Roberts received the B.Sc. degree in Mathematics from the University of Surrey, United Kingdom n 1970. He worked from 1971 to 1975 in the Teletraffic Division of the British Post Office before moving to Zrance to join the company SOCOTEL. He has been with the Centre National d'Etudes des T616communica- ions since 1978 where he has worked on different aspects of teletraffic and performance evaluation. His :urrent interests include the performance evaluation problems arising in multiservice networks with )articular emphasis on the integration of high bit rate services.

North-Holland Performance Evaluation 7 (1987) 267-284

0166-5316/87/$3.50 © 1987, Elsevier Science Publishers B.V. (North-Holland)

268 Z Dziong, J. W. Roberts / Congestion probabilities m circuit-switched ISN

the call's bit rate requirement. The considered networks operate with loss (blocked calls disappear from the system) and use fixed routine (one and only one path per traffic stream). Our prime motivation is to develop tools for evaluating the performance of a future integrated services network offering circuit switched bearer services at multiples of 64 Kb i t / s (up to 2 Mbit /s) , although the studied models may be applied in any communication or service system where multiple resources of different types are shared by a number of heterogeneous user classes.

In general, modern telecommunications networks do not employ fixed routing but offer each traffic stream a variety of alternative paths defined by their routing strategy. However, the calculation of blocking probabilities in such networks is very difficult and practical procedures are often based on an approximate decomposition into a series of fixed routing problems. In [4], in particular, it is shown how end-to-end blocking evaluation can be broken down into a sequence of individual path blocking calculations. This approach is preferred to the more usual link-by-link decomposition when the correlation between link occupancy states is high due to a large proportion of common traffic. In the case of integrated services networks with multiple channel requirements for high bit rate calls, the amount of common traffic might be rather more significant on certain paths. This has led us to develop evaluation methods which, like the Equivalent Path Method in [4], take account of link traffic correlation.

Two recent papers [9,14] have addressed the evaluation of blocking probabilities in fixed routing circuit-switched networks. (This is, in fact, a classical teletraffic problem and has already been addressed by many authors; references [4,9,14] contain an extensive bibliography and indicate some related applications.) The papers notably establish the validity of certain bounds and asymptotic results frequently used in approximations based on a link-by-link decomposition. In the present work we adopt a different approach based on a generalisation of the recurrence relations satisfied by the occupancy distribution of a single link offered multiple bit rate traffic [12].

These single link recurrence relations were derived for constant rate Poisson traffic by Kaufman [8] and Roberts [11] and were generalised to the so-called Bernoulli-Poisson-Pascal state-dependent arrival processes by Delbrouck [3]. We further generalise the last result to obtain a set of relations satisfied by the joint occupancy distribution of all network links. Although this distribution has considerably fewer dimensions than the product form distribution of the number of calls present in each traffic stream, the state space remains too large for direct application in blocking calculations. We therefore study approxi- mations in which each link's marginal occupancy distribution is computed from modified recurrence relations taking account of inter-link traffic correlation.

After introducing basic assumptions and notation in Section 2, we derive the general recurrence relation in Section 3. We also express the mean, variance and covariance of the offered multiple bit rate traffic and suggest a first approximation based on the multi-variate normal distribution with the same moments. In Section 4 we propose methods for estimating traffic stream time congestion using approximate link marginal occupancy distributions. We also generalise the reduced load approximation studied in [9,14] to the present case of multiple bit rate traffic. The different approximations are tested and compared in Section 5 on a number of elementary two-link network examples. We draw preliminary conclusions in Section 6.

2. Assumptions and notation

We describe the network quite generally as a set of trunk groups or links numbered from 1 to R. Trunk group r has L r channels. Routing in this network is defined by the identities of the trunk groups used by calls of a given traffic stream, these trunk groups constituting the network path. We assume fixed routing with calls lost if their path is blocked. Call set up is instantaneous.

The network is offered traffic from M independent streams. Traffic in the network at time t is described by the vector process

x ( t ) = { K , ( / ) , / q ( t ) . . . . .

where Ki(t) is the number of calls of stream i present in the system at time t.

Z. Dziong, J. W. Roberts / Congestion probabilities in circuit-switched ISN 269

Traffic stream i is characterised by the following: • call arrivals form a Poisson process with intensity ~ ( k ) when K~(t) = k; we consider, in particular,

arrival intensities of the form

X , ( k ) = a i + B~k; this type of process has been termed BPP [2] since, depending on the values of the parameters, the distribution of the number of busy trunks on an infinite capacity link would be Bernoulli (fl, < 0, ai/fli a negative integer), Poisson (fli = 0), or Pascal (fli > 0);

• call holding times are independent and identically distributed with mean 1//~i (note that we do not need to assume an exponential distribution);

• each call requires 6jr channels on link r, r -- 1, 2 . . . . . R, throughout its holding time; we shall use the following notation:

A = ( 6 i r } an M X R m a t r i x ,

6~.= ( 6 n, 6i2 . . . . . 6in ) the ith row of A,

6-r = {61r, 82 . . . . . . 6 M r } T the r t h c o l u m n of A

(for the considered circuit-switched network we would normally have

= f d i if l i n k r i s i n a s t r e a m i p a t h , Sir 0 if not,

where d i is the channel requirement of stream i calls; however, the developed formulae do not require this restriction and it is clear that the described model could be applicable to other service systems with multiple resource types and customer classes).

We define link occupancy random variables Nr(t ) by M

Nr(t ) = ~_~ g i ( t ) r i r . i = 1

The vector N ( t ) = (Nl( t ) , N2(t) . . . . . Nn(t )} is thus given by

N(t) =X(t)a. The set of all possible occupancy states is denoted '/':

n ~ V P ~ O<~nr<~L r f o r r = l , 2 . . . . . R.

The occupancy states may be partitioned according to whether or not there remains sufficient idle capacity to carry a new call of stream i. Define 12~ to be the set of states with sufficient capacity:

n ~ [ 2 i ~ O ~ n r ~ L r - 6 i r f o r r = l , 2 . . . . . R,

and let ~,, the complement of I2 i, be the set of stream i blocking states. Calls of stream i arriving when N ( t ) ~ ~, are lost.

With appropriate conditions on the arrival intensities, the above assumptions imply that the process K( t ) has a stationary probabili ty distribution. Let this distribution be

P ( k ) = lim P r ( K , ( t ) = k ~ , i = l , 2 . . . . . M } . l~OO

The corresponding limiting distribution of N ( t ) is

Q ( n ) = lim P r ( N r ( t ) = n r, r = 1 , 2 . . . . . R } . t ~ 00

In the stationary limit we shall denote the random variables by K, and N r. The marginal distribution of N r will be written Qr(n):

Qr( n ) = E Q ( n ) . ( , :nr=n)

270 Z. Dziong, J. W. Roberts / Congestion probabilities in circuit-switched ISN

3. The joint occupancy distribution

We give exact and approximate relations for evaluating the joint occupancy distribution Q(n).

3.1. A recurrence relation for Q(n)

With the assumptions of Section 2, the stationary distribution P(k) has the product form:

P ( k ) = P(O) 1-[ X~(k - 1) / l~ 'ki! (1) i = 1

for all states k such that M Y'.ki3ir<~L r f o r r = l , 2 . . . . ,R .

i = l

This expression constitutes a slight generalisation of the result proved in [1]: we additionally allow state-dependent arrival rates X,(k) and variable capacity requirements 3,r. These modifications do not change the substance of the demonstration which is therefore omitted here. The validity of equation (1) can also be established by a trivial extension to multiple resources of the product form and insensitivity results proved in [81.

For BPP arrivals (X,(k) = a t + fl~k), the occupancy distribution satisfies the following set of recurrence relations:

nrQ(n) = ( M , ~ o a ~ 3 i r @ ~ ( n ) i f n E ~ a n d n s > O f ° r s ° m e s ' = a #~- fl~ (2)

i f n ~ g',

for r = 1, 2 . . . . . R, with

( ( ~ - fli)O(n - 8~.) + fli~)i(n - - ¢~i.) if n -- 6~.~ ~/', # ~ O i ( n ) = 0 if n - - 8 i . ~ ~ , (3)

for i = 1 , 2 . . . . ,M. These relations together with the normalising condition

Y'. Q(n) = 1 (4) n E ~

completely determine Q(n). Forgetting, for the moment, practical computing problems of memory requirements and precision, these relations can be used to progressively calculate all the Q(n) in terms of some arbitrarily chosen value for Q(0). The latter will be given its correct value on renormalising the Q(n) at the end of the recursive calculations.

To derive equations (2) and (3), write

( ) O~ i IJrQ(B) = E ~=lki3ir P ( k ) = ~ ~ ~irOi(n),

{k:kA=n) i i = 1

where

o,(.) E k,e(k)- E +(k'-l)B' Oli (k:kA=n} O~i {k:ka=n} ~i

using the notation k - e i = ( k 1 . . . . . k i - ] . . . . . k M }. z

Relabelling the summation indices k" - k~ - 1 and kj = kj for j =~ i, we have

- + k" e ( k ' ) , O/i ( k ' : k'A=n-Si. ) I£i

P ( k - e i ) ,

Z. Dziong~ J. W. Roberts / Congestion probabilities in circuit-switched I S N 271

finally giving (3):

O,(n) I ~ ' - fl~ Q ( n - 8~.) + fl@6),(n - 8, . ) . ~i l'ti

The functions Oi(n ) may be eliminated in (2) and (3) to give another form to the recurrence relation:

nrQr(n ) = • --~8~r~7~ Q(n--JSi.) (5) i = 1

for r = 1, 2 . . . . . R and n ~ '/', where the summat ion over j extends f rom 1 to the highest value such that nr --jSir >1 0 for r = 1, 2 . . . . . R. Recurrence (5) generalises the form given initially by Delbrouck [3] for R = I .

3.2. Traffic moments

It is cus tomary to describe the telephone traffic offered to an isolated trunk group by the moments of the number of busy trunks if the group were of infinite dimension. By extension, we can describe the traffic in the present system by the moments of the N r and their correlations, assuming a network of infinite capacity. Specifically, we shall use the mean, variance and covariances of the N r.

Let E ( K i } = m i and Var{Ki} = v~ and denote by M r, V~, and Qs the mean and variance of N r and the covariance of N r and Ns, respectively.

Since N r = E K~8ir,we immediately have M

M r = Y] m i ~ i r for r = 1, 2 . . . . . R. (6) i = 1

To calculate second moments we need

E ( N r N s } = E ( ~ - ' K i S i r ' Y ' K j ~ j , } = y" E(Kf}~ir~is + ~_, mimjSir~js i = 1 j = l

j4: i

M M M

= E Oi~ir~is ~- £ m i~ir ° £ m j ~ j s - i = 1 i = 1 j = l

Since, by definition,

V ~ = E { N r Z } - E ( N r } 2 and C r s = E ( N r N s } - E ( N ~ } . E ( N s } ,

we find M

V r = £ O,~ 2 for r = 1, 2 . . . . . R , (7) i = 1

M

Crs = E ViSirSis for r = 1, 2 . . . . . R and s = 1, 2 . . . . . R. (8) i = 1

For BPP arrival processes, the individual s tream means and variances are

OL i Oli~£ i m i U i (9) B,' B,) 2

3.3. A multivariate normal approximation

To avoid calculating Q(n) recursively we might a t tempt fitting a s tandard distr ibution with the same first moments. An obvious candidate is the multivariate normal distr ibution with density (cf. [10, p. 272])

1 e x p ( - ½ ( x - M ) U - l ( x - M ) x} (10) f ( x ) = (2~r)R/EIUI1/2

272 Z. Dzion~ J. W. Roberts / Congestion probabilities in circuit-switched ISN

where M = ( M 1, M 2 . . . . . M R ) is the vector of expectations and

{V~ if r = s , U = (Urs } with Urs = Cr s if r ~ s ,

is the dispersion matrix. Note that the single variate normal distribution has previously been suggested as an approximation for

traffic occupancy distributions (see, for example, [13] and works referred to therein). To approximate the occupancy distribution we must appropriately truncate and discretise this density.

We might, for example, assume

] /[ fL'+ll2"'" s_L~;2112f(x) dx1 ...d.X,R] Q(n) = [ L"", - i/2f''+'/2"'" ",,R + l/zfnR+l/2f(x) dxl " " d x " l / L " - a l 2

(11)

(in most cases approximating the numerator by f ( n ) and the denominator by the integral from -o o ) . This approximation is not purely empirical. From (2) we can write

( i ~i ) (~iLli~ir n , - E ,~_ R 3,, Q(n) = ~., (#~,O,(n) - fl, O,(n) - (#.t,- n , )Q(n ) ) . 13,) 2 i

Using (3) and (9) this becomes

(nr-~imi~ir)Q(n)=Evi~ir(~)i(R)-~)i(n'J¢-~i'))'i (12)

Now, by (3),

# ~ , ( O , ( n ) - @ , ( n + S i . ) ) - B , ( O , ( n - 8 , . ) - O , ( n ) ) = ( # . t , - f l i ) ( Q ( n - 8 , . ) - Q ( n ) ) . (13) If 3it is small with respect to nr, for r = 1, 2 . . . . . R, we should have

e , ( n ) - O, (n + ¢ 3 = O, (n - ¢ . ) - e i ( n ) .

If this approximation holds or if r , is small compared to /~ , then, from (13),

(g i (n ) - @,(n + 8,.) = Q ( n - 8,.) - Q ( n ) .

If we now put this expression into (12) and replace Q ( n - 8~.) by the first terms of its Taylor expansion

Q ( n - 8,.) = Q ( n ) - Y',3,, 3Q(n------!) + . . . s ~ns

(assuming n real), we obtain a family of partial differential equations:

R 3 Q ( n ) for r = 1, 2 . . . . . R. (14) (n r -- M , ) Q ( n ) = - E u, , 3n , s~l

The solution of (14), normalised over all real vectors n, is precisely (10). We thus expect approximation (11) to be accurate when the gradients of Q are relatively constant in a neighbourhood of width 8~., for i = 1, 2 . . . . . M. Practically, we would require the 3ir to be small compared to the mean traffic offered and network capacities.

While attractive in principle, this approximation proves difficult to apply in practice due to the absence of efficient routines for integrating the multivariate normal density. However, we shall use the following interesting properties in developing other approximations in Section 4.

Let (Xa, X2, . . . , X R ) be a set of random variables with density given by equation (10). We then have [10, pp. 276-280]: • the marginal distribution of any subset of { Xa, X 2 . . . . . X R ) is also multivariate normal with the same

means, variances and covariances as in the initial distribution; • the conditional distribution of a subset of ( X1, X 2 . . . . . X R } given the values of the remaining variables is

also multivariate normal with known expectation vector and dispersion matrix.

Z. Dziong 3. 14d Roberts / Congestion probabilities in circuit-switched I S N 273

In particular, the distribution of X, given X r is univariate normal with mean and variance, respectively:

E ( X s I Xr = x } = M s - ( C r s / g r ) ( M r - x ) , (15)

Var{ X, l Xr= x } = V s - C2/Vr. (16)

4. Congestion probabilities

We express congestion probabilities in terms of the joint occupancy distribution and derive approxima- tions based on estimates of the marginal distributions.

4.1. Time congestion and call congestion

The time congestion of stream i calls (i.e., the probability that the network is in a blocking state for stream i calls at an arbitrary instant) is

E,= ~, O ( n ) = ~, P (k ) . (17) ne~, {k:kAe~,}

Call congestion (i.e., the probability that a stream i call arrival is blocked) may be expressed by

{k: k a ~ , } {k: ka~q,}

For BPP arrivals, this expression simplifies and we can write

B, = Y', e ' ( k ) , {k : k Z ~ }

where P'(k) is the stationary distribution of traffic streams in the same network where, however, stream i has a modified intensity

X" (ki) = (a, + [3i) + flik,.

In other words, stream i call congestion in the considered network is the same as stream i time congestion in the same network with modified traffic parameters. In what follows we shall therefore devote our attention to evaluating time congestion.

4.2. Decomposition of the blocking probability

Rather than performing the summation in (17) it is common practice in network performance evaluation to estimate E i by assuming that the various trunk group occupancies N r are statistically independent, so that

R

E i = l - I-I Pr{Nr<~Lr-3ir}. (18) r= l

To determine the marginal probabilities in (18) it is often further assumed that these are 'functionally independent' in the sense that their offered traffic is not affected by congestion on other links. Improved approximations are obtained by reducing the arrival intensities of a given stream uniformly by a factor equal to the probability of blocking on the other links of that stream's path (cf. Section 4.5 below).

Numerical evaluation of specific examples shows that the assumption of functional independence is, in most cases, a greater source of error than the assumption of statistical independence.

Fig. I illustrates the effects of both assumptions on the time congestion estimates in a simple symmetric two-link network with constant rate Poisson single channel per call offered traffic. Each link has 30

274

Time

congestion

. 016

. 012

• 008

Z Dziong Z W. Roberts / Congestion probabilities in circuit-switched ISN

Link caoacitv 30

Mean traffic 20

Traffic mix all streams single channel, Poisson.

~ r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Exact

. . . . . . . . . . Statistical independence

Statistical and functional independenc

I I I

0 % 50 % common traffic 100 %

percentage

Fig. 1. Effect of independence assumptions.

channels. There are three offered streams whose intensity is set to give a mean link offered traffic of 20 Erlangs:

- stream 1 calls require one channel on link 1; - stream 2 calls require one channel on link 2; - stream 3 calls require one channel on each link.

The curves in the figure give stream 3 time congestion as a function of the proportion of end-to-end traffic. The horizontal dashed line corresponds to time congestion probabilities assuming complete statistical and functional independence; they are calculated using Erlang's formula:

E l = 30! / i__~.,1 ~-.i ( = E2) ' E 3 = 1 - (1 - El ) 2.

where A ( = 20 Erlangs) is the link offered traffic. The solid line gives the value of E 3 calculated exactly using (17) with state probabilities given by the

formulae in Section 3.1. The same relations are used to obtain E l , which value is then used to estimate E 3

assuming statistical independence: E 3 = 1 - (1 - E1)2; the latter estimate is plotted in Fig. 1 as the dotted line.

The difference between the exact value of E 3 and the latter estimate is small except when end-to-end traffic accounts for more than about 80% of the total. Statistical dependence only dominates functional dependence at an end-to-end traffic proportion greater than 90%. On the other hand, the total indepen- dence assumption gives relatively poor results for more than a very low proport ion of end-to-end traffic.

In view of these results, in the next section we aim at deriving approximations for the marginal distributions which take functional dependence into account.

Z. Dziong, J. W. Roberts / Congestion probabilities in circuit-switched ISN 275

4.3. Approximate marginal distributions

The marginal distribution of the random variable Nr is

Qr( n ) = E Q(n) . (n:n~,nr=n)

By (5), by summing over all arguments except n r, we have

nrQr(nr) = E - ~j i=1 I~i gr Pr{Ns<~Ls-JSgs,• sCr[Nr=nr -Jr i r }Qr (nr - j rg~) . (19)

We now proceed to approximate the conditional probabilities in (19). First we assume the Ns, for s 4: r, to be mutually independent so that

Pr(Ns<~ls, s ~ r l N r =n} = I-I Vr{Us<~l~lNr=n}. (20) s~r

We then approximate the terms in the product by the normal distribution with mean and variance given by (15) and (16) respectively and renormalised so that

Pr{Ns<Ls lNr=n} = 1 .

In other words, we assume

where q)(t) is the standard normal integral

Thus, (19) with (20) and (21) gives us a first approximate recurrence relation for the marginal distribution:

nrQr(nr)-~" E ~. gr~., ~ Sirj(nr)Qr(nr-J~ir), (22) i=1 ~xi j

where

Relation (22) is somewhat complicated. Simpler approximate recurrence relations are established below. By definition,

(~i ) ~ i "i OLi-- ' i ~irOri(nr)' nrQr(nr) = E kiSirP(k) E {k : kS.r=nr)

where

{~ri(nr) ]'Li- fli E k iP(k) . O~i {k:kc$.r=nr }

276 Z. Dziong, J. W. Roberts / Congestion probabilities in circuit-switched I S N

The state probabilities P (k) , given by (1), satisfy the local balance equations

k~ l~ iP(k ) = ~ ( k i - 1 ) P ( k - e+) (24)

for states k such that kA ~< L. Now consider link r as an isolated resource and assume that, instead of (24), we have approximately

k dz ,P( k ) = X~( k~ - 1)~ri( n r - - 8ir ) P ( k - e, )

for states k such that k ¢ $ . r = nr, where ~ ( n ) is the probabili ty that a stream i call is not blocked on other links, given N~ = n.

In fact, we only need to assume

~., k i t ~ i P ( k ) - - l ; , i ( n ~ - t S i r ) ~_, X i ( k i - 1 ) P ( k - e i ) . {k: kS.r=rlr} { / : ]¢~.r=nr }

Writing Xi(k~ - 1) = a~ + ]3~(k, - 1), we then deduce the approximate relation

IZ iOri ( n r ) = ~ri ( n r -- ~ir ) ( ( t~ i - l~i ) Qr ( n r - ~ir ) h- t~i~ri ( rl r -- 6Jr))"

Using the normal approximation for ~ri we arrive at our second recurrence relation:

with

O~ i n~Qr(nr ) = ~_, ~ i r O r i ( n r )

i t '~ i - -[J i (25)

<Vs- cj / 1/ In a two-finkpath, stream i time congestion may be written as

E i = E i l + E i 2 - E 1 2 .

L ~ - M ~ + C r ~ ( M r - n ~ ) / V ~ ) - - - - ~ ~ - . ( 2 9 )

Cr,/Vr)

(30)

t~,6)ri(nr) = S i r l (n r ) ( ( l~ i - f l , ) Q r ( n r - ~,r) "~- ~ i ~ r i ( n r - ~ir)) , (26)

where S[l(nr) ( = ~ r i ( n r - - $ i r ) ) is given by (23) with j = 1. This relation is computationally more efficient than (22) and requires fewer evaluations of the normal

integral. Both are, of course, equivalent for constant intensity Poisson traffics (fli = 0 for i = 1, 2 . . . . . M).

4. 4. S t ream blocking probabilities

Knowing (approximately) the occupancy distributions of each link r, we can calculate its contribution to the time congestion of stream i,

t r

E r = Y'= Or ( n~ ) . (27) nr= L r - S i r + l

To calculate the stream time congestion, we could assume statistical independence as in (18),

R

E , = I - I-I (1 - e,r). r = l

A more accurate result would be obtained by estimating the joint time congestion of links in the stream i path, taking account of their traffic correlation. In particular, we can express the joint stream i time congestion of links r and s as

Lr

E, rs = E Q r ( n r ) ~ r S ( n r ) , (28) n • = L r - 8it + 1

where ~r~(nr) is the conditional probability of congestion on link s, given that link r has n~ busy channels. Again using the normal distribution to approximate ~[s, we take:

Z. Dzion~ J. W. Roberts / Congestion probabilities in circuit-switched 1SN 277

4.5. Reduced load approx imat ions

When link traffic correlations are small (case of large networks with many streams, each individually of low intensity), the stream congestion probabilities can accurately be estimated using a reduced load approximation [9,14]. To calculate the marginal distribution on a given link, arrival rates of offered streams are uniformly reduced by a factor equal to the probability that a new call of that stream would not be blocked on the other links of its path. This probability is simply the product of the stream i time congestion on each of these links. In the present context, this procedure may be expressed as follows.

The marginal link r occupancy distribution satisfies the recurrence relations (deduced from (2) and (3) with R = 1)

p

ai (31) n r e r ( n r ) = E iz ̀ _ f l f ~ir~)ri(nr) '

with

#rOri( n~ ) = (l~, - f l f ) Q r ( nr -- ~ir ) "JV ~ / Or i ( nr - ~ir ), (32)

where we have modified arrival rates

~" (k~) = a; + B / k , = X , ( k , ) 1-I (1 - E / ) , sq:r

and E/ is the link s stream i time congestion (27). This procedure is clearly iterative. It has been shown in [9] that the above relations are satisfied by a

unique set of E f (if arrival rates are constant, i.e., /~ i = 0 for i = 1, 2 . . . . . M) but there is no gaurantee of convergence (see [14, p. 1816]).

We can also apply a reduced load approximation to improve the marginal distributions of Section 4.3 above. We simply propose to reduce the parameters of the traffic offered to link s in the normal approximation (23) to account for blocking on links other than r and s. We calculate Ms, V~, and Crs from (6), (7), and (8) with modified stream arrival rates

X;(k,) = Xi(ki) I-I (1 - E i t ) , (33) t~r , s

for i = 1, 2 . . . . . M. This calculation again requires iteration until convergence to a set of Ef satisfying (27).

5. Numerical results

To give some idea of the accuracy and range of application of the formulae derived above we have performed a number of numerical experiments. These only constitute a preliminary test of the developed models and their application to real network performance evaluation (notably, the calculation of end-to-end blocking) requires further work.

5.1. S tud ied sys tem

We consider the simplest nontrivial network consisting of two links in series offered traffic streams which may be either particular to one link or common to both (see Fig. 2). We further restrict our attention to symmetric systems where the two links have the same capacity and identical offered traffic. We investigate the effect of link traffic correlation on the accuracy of studied time congestion approxima- tions by varying the proportion of common traffic while keeping the mean link traffic constant.

Considered traffic streams require either 1, 6, or 30 channels per call and total traffic is expressed in 'Erlang channels' - - the mean number of busy channels. The different approaches are compared through their estimates of the time congestion of common streams.

2 7 8 Z. Dziong, J. IV,, Roberts / Congestion probabilities in circuit-switched 1SN

L channels L channels

Oi IO

i nd iv idus1 traffic

w

common traffic

Fig. 2. Studied two-link network.

5.2. Time congestion estimates

In previous sections we have derived a number of exact and approximate methods for calculating time congestion. We compare below the results obtained with the following four methods:

EX :the exact occupancy distribution given by recurrence relations (2) and (3) with time congestion calculated from (17),

AM1 :approximate marginal distribution given by (22) with common stream time congestion given by (30),

AM2 :approximate marginal distribution given by (25) and (26) and time congestion by (30),

RL :reduced load approximation where marginals are computed iteratively using (31) and (32) and common stream time congestion is estimated assuming statistical independence.

These methods have been compared in a number of two-link networks. A representative sample of results is presented in the next sections.

5.3. Effect of link traffic correlation

In the example of Fig. 3, two 150 channel links are offered a mixture of single and 6 channel traffic streams. Arrival processes are Pascal with parameters a i and fli chosen so that peakedness (i.e., the ratio vs/m~) is equal to 2 and the mean offered traffic on each link is 110 Erlang channels. Single channel and 6 channel calls each contribute half the traffic offered.

Fig. 3 plots time congestion of (a) six channel and (b) single channel common streams as a function of the proportion of common traffic. We note that AM1 and AM2 both give results close to the exact curve while RL can significantly overestimate time congestion for high proportions of common traffic.

In the example of Fig. 4, common traffic is composed of half 30 channel calls and half single channel calls while individual traffic is entirely composed of single channel calls. Thus, unlike the previous example, as the proportion of common traffic increases, so does the variance of offered link traffic. The offered streams are all Poisson so that approximations AM1 and AM2 coincide. Link capacity is of 300 channels and mean traffic offered is fixed at 220 Erlang channels.

Fig. 4 gives the variation of time congestion of 30 and single channel streams as the proportion of common traffic increases from 0 to 100%. The AM approximations again give very good estimates while RL performs rather worse than in the previous example due to the higher correlation between link traffics. The oscillations in time congestion are due to the changing traffic mix. This phenomenon is explained in [6] and more recently in [7]: single channel time congestion local maxima occur when the difference between link capacity and mean single channel traffic is an integral multiple of the capacity required for one multi-channel call.

5. 4. Effect of stream peakedness

In Fig. 5 we illustrate how the accuracy of time congestion estimates depends on the peakedness of offered streams. (Note that we are only testing the accuracy of our approximations; we do not test the

Z. Dziong, J. IV.. Roberts / Congestion probabilities in circuit-switched I S N 279

Link capacitv

Mean traffic

1 5 0

1 1 0

E 6

.06

.05

04

T r a f f i c mix for common and individual traffic,

- 50 % single channel, peakedness 2

- 50 % six c h a n n e l , peakedness 2.

(a) 6 channel stream

E 1

• oog --

i)07 --

0 0 5

K e y

EX

. . . . . . . . . . . . . . . . . . . . . A M 2

. . . . . . . . . . . . . . . AM1

R L

[ b ) 1 c h a n n e l s t r e a m

\\ 0 % 50 % 100 %

Common traffic percentage

Fig. 3. Time congestion vs, common traffic percentage--example 1.

280 Z. Dzion~ J. W. Roberts / Congestion probabilities in circuit-switched ISN

E30 _

. 1 5

. 1 0

• 0 5

E 1 - -

. 0 0 4 _

• 002

Link capacity : 300

Mean traffic : 220

Traffic mix : common traffic,

- 50 % single channel, Poisson

- 50 % t h i r t y channel, Poisson

individual traffic,

- 100 % single channel, Poisson

(a) BO channel stream

1 1 /

i / /

/ /

/

I I I

j /

[ b ] 1 channel s t r e a m

/

I [ [

0 % 50 % I00 %

common traffic percentage

Fig. 4. Time congestion vs. common traffic percentage--example 2.

Z. Dziong, J.W. Roberts / Congestion probabilities in circuit-switched ISN 281

Link cBpacity 150

Meen trBffic 100

T r a f f i c mix Common e n d individual treffic,

- 50 % single channel,

- 50 % six channel.

Relative error

0.2

0 , 1 _

0 - -

- 0 1 - -

0 . 2

Common traffic ~ercentage - 50 %

6 channel @treBm

f f l

J

J /

K ~ ' :

A M 1

AM2

RL

I I I ]

0.5 1.0 1.5 2.0 2.5 3. S t r e a m p e a k e d n e s s

Fig. 5. Relative error of 6 channel time congestion vs. stream peakedness.

appropriateness of BPP arrival processes for representing the peakedness of traffic streams in a real network, which is another quest ion--cf . [2].) The studied configuration has two 150 channel links offered a half and half mix of single channel and 6 channel streams. The c o m m o n traffic proportion is fixed at 50% and mean link traffic is 100 Erlang channels.

Fig. 5 gives the relative error- - (E(approx) - E(exact))/E(exact)--of time congestion estimates for the 6 channel c o m m o n stream as a function of offered stream peakedness. Results for the single channel stream are practically identical. AM1 and AM2 remain very accurate for the studied range while the RL approximation error increases with stream peakedness.

5.5. Effect of time congestion level

For the same configuration as in Section 5.4 we study, in Fig. 6, the effect on relative error of the time congestion level when stream peakedness is fixed at 1.5. In this example, however, c o m m o n traffic counts for 75% of offered link traffic. The figure shows the variation of the relative error of 6 channel c o m m o n stream time congestion estimates as this quantity increases from 1 0 - 4 to 1. Time congestion is varied by changing the overall offered load while keeping the same proportional traffic mix.

The most important observation to be made here is that AM methods lose accuracy as time congestion increases beyond 30%. This is presumably due to the inappropriateness of the truncated normal distribu- tion for estimating conditional probabilities in (23) when congestion is high. The RL approach is to be preferred in this congestion range.

282 Z. Dz ion~ J. W. Roberts / Congestion probabilities in circuit-switched 1 S N

L i n k cepacitv : 150

T r a f f i c mix : Common and individual traffic,

- 50 % sin~le channel, peakeoness 1.5

- 50 % six channel, peakedness 1.5

Common traffic percentaQe - 75 %

R e l a t i v e

e r r o r

0 . 2

0 1

- 0 . 1

- 0 . 2

B channel stream

f

. . . . . . . . . . . . Li-ll i i " \ /

Key :

AMI

. . . . . . . . . . . . . . . . . . AM2

RL

I I I I I I I I I I I I I I I I I I I I I l I I I I 1

@-~ 1 0 -~ 1 0 -~ 1 0-' I I I 1 1 1 1

Time congestion

Fig. 6. Rela t ive e r r o r o f 6 c h a n n e l t ime c o n g e s t i o n vs. t ime c o n g e s t i o n level.

T a b l e 1

T ime conges t i on for d i f fe ren t t ra f f ic mixes.

L ink capac i ty : L r = 150

Traf f ic : mean , M r = 110; va r i ance , V r = 770

C o m m o n S t r e a m p a r a m e t e r s fo r c o m m o n a n d ind iv idua l t ra f f ic

t ra f f ic

pe r cen t a d l = l , v l / m 1 = 2 d 1 = 1 , v l / m I = 1 d 1 = 5,

d 2 = 6, v z / m 2 = 2 d 2 = 13, v 2 / m 2 = 1 d 2 = 9,

v l / m 1 = 1

v2 / /m2 = 1

0.5 % E 1 0 .0096 0.0101 0 .0113

E 6 0 .0636 0 .0669 0 .0700

25% E~ 0 .0089 0 .0093 0.0105

E 6 0 .0584 0 .0617 0 .0647

50% E 1 0 .0080 0 .0084 0.0095

E 6 0 .0527 0 .0557 0.0585

75% E 1 0 .0069 0 .0073 0 .0083

E 6 0 .0459 0 .0486 0 .0512

99.5% E 1 0 .0049 0 .0052 0 .0058

E 6 0.0331 0 .0348 0.0365

a I t m a y be verif ied tha t , in these con f igu ra t i ons , the c o m m o n t ra f f ic p e r c e n t a g e is equa l to the l ink t ra f f ic co r r e l a t i on coef f ic ien t ( Cl:/(Vy2)w2).

Z. Dziong, J. 14I. Roberts / Congestion probabilities in circuit-switched ISN 283

5.6. Effect of third and higher traffic moments

For a single resource it has been observed that we obtain approximately the same occupancy distribution for different traffic mixes having the same overall mean and variance [12]. We have investigated whether this result extends to the present case of a two-link system if we also conserve the link traffic covariance. Typical results obtained for a particular configuration are given in Table 1.

The first column of time congestion probabilities in this table reproduces some results already presented graphically in Fig. 2. The system configuration is thus that described in Section 5.3 and we evaluate the time congestion of single channel and 6 channel streams for five common traffic proportions: 0.5%, 25%, 50%, 75%, and 99.5%. The next two columns give the same congestion probabilities for two different traffic mixes producing the same mean, variance, and covariance. The second column corresponds to a mix of two Poisson traffic streams requiring respectively 1 and 13 channels per call for both common and individual traffic while column 3 traffic is composed of 5 channel and 9 channel Poisson streams. In all three cases, time congestions are calculated exactly.

These results show that the observation of approximate insensitivity to higher moments for a single resource (cf. results for a common traffic proportion of 0.5% and 99.5%) is equally valid for two links with correlated traffic. The difference in time congestion can, however, be significant (around 17% difference between columns 1 and 3 for single channel congestion) compared to the differences between exact and approximate methods noted above. We nevertheless feel that there might be a basis here for an efficient approximation for end-to-end blocking calculations. Replacing peaky traffic streams by Poisson traffic streams with a greater channel requirement is akin to the Hayward approximation which has already proved its worth in link-by-link blocking calculations [5].

6. Conclusions

For a circuit-switched telecommunications network with fixed routing we have obtained recurrence relations satisfied by the number of busy channels in each link. The recurrence relations are valid for multiple bit rate traffic streams having Bernoulli-Poisson-Pascal arrival rates. They enable the exact calculation of time congestion probabilities for small networks. For more realistically sized networks, however, the number of terms in the joint distribution is too large for direct application.

We have therefore developed approximate methods which take account of the correlation between link occupancies due to common traffic streams. These methods are based on the computation of marginal link occupancy distributions calculated from one-dimensional recurrence relations in which, roughly speaking, stream arrival rates are reduced by the probability a call would be blocked on the other links of its path. The latter depend on the link occupancy and are estimated here using a truncated normal distribution with appropriately chosen mean and variance.

The proposed approximations have been tested on a number of simple two-link examples. They give very good accuracy except when the time congestion is greater than 30%. They generally perform better than an iterative reduced load approximation based on link-by-link decomposition, especially when the proportion of common traffic is high. We do recognize, however, that the presented examples exaggerate the weakness of the reduced load approximation and that a high correlation between link traffic is unlikely in practice. Furthermore, we only account for pairwise dependence between link traffics and the effect of higher order dependence cannot be seen in the considered two-link configurations.

Further work is necessary to extend the presented methods to develop efficient procedures for calculating end-to-end blocking probabilities in integrated services networks. A possible approach could be the substitution of offered peaky traffic streams by Poisson traffic with greater bit rate requirements (cf. Hayward's approximation) since preliminary numerical results suggest that the occupancy distribution can reasonably well be estimated on fitting just the first and second moments (mean, variance, and covariance) of offered traffic.

284 Z. Dziong J. W. Roberts / Congestion probabilities in circuit-switched ISN

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