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INVARIANCE AND SECONDARY STOCHASTIC PROCESSES C.E.M. Pearce, B.Sc., M.Sc., Ph.D., University of Adelaide, Australia.

Sunnary

Pain and Taktfcs have shown that when renewal traffic is presented to a finite group of trunks with negative exponential holding times, the consequent overflow stream is itself of renewal type. Such renewal character is destroyed by pooling, except in the case when all the component streams involved are Poisson.

It is shown that pooling of independent renewal streams results in a more general class of traffic stream which preserves its char­ acter if made to overflow from a finite set of trunks or to undergo pooling with further independent streams. We give a general discuss­ ion of this class of traffic, with some indication as to practical calculation procedures for quantities of interest. A rigorous justification of the formulae presented is omitted.

1. Introduction

Suppose that renewal traffic is offered to a finite group of trunks with negative exponential holding times. The transient and steady-state distributions of the resultant trunk occupancies have been studied by Palm [31], TakScs [38, 39] and Cohen [4]. See also Descloux [5], who weakens slightly the assumption on holding times. Palm [31] and TakScs [38] have shown that the overflow traffic in this situation is also a renewal stream, rendering facile an analysis of the trunk occupancy when the overflow is offered to a second (finite or infinite) group. Earlier studies (Kosten [16], Riordan [40], Brockmeyer [2]) of the special case of Poisson offered traffic analyse the second choice group via the joint occupancy of the two groups, and obtain quite complicated formulae. (For a study of the joint distribution for a general renewal offered traffic see Le Gall [20].) General techniques also exist such as combinatorial methods working with the joint distributions of the epochs of events of the arrival process and Fortet's stochastic integrals approach [7,8,9]. With these the traffic carried by finite or infinite groups with a Proceedings of the 14th Annual Conference of the Operational Research Society of New Zealand, May 1978, University of Canterbury, Christchurch, N.Z. - Vol. I - Theory -20-

general holding time distribution has been investigated for completely general arrival processes (independent of occupancies). See Le Gall [18, 19] and more generally in [22], Finch [6 ] and the author [32],

While the expressions derived can lead to series for the calcul­ ation of quantities of practical interest, results obtained in this way often do not uncover the simple forms (and consequent practical calculation procedures) that frequently exist in particular cases. This is also a disadvantage of the powerful complex variable tools of Pollaczek [36, 37]. Pollaczek develops integral equations describing the case of a finite group of trunks with identically and independently distributed holding times following a general distribution and a renew­ al stream of offered traffic. The information is substantially locked in the integral equations, to which solutions have been found only in certain special cases [36]. With a restriction to "pseudo-Poisson" offered traffic Le Gall [23] has been able to obtain some simple formulae relating to models with general holding times.

Several studies exist of trunk occupancies arising from offered traffic consisting of the aggregate of two or more independent pooled renewal streams. Bech [1] derives matrix equations characterising the solution for a particular example and the more general case is consid­ ered by Le Gall [21]. Padgett and Tsokos [30] show that the random variable giving the number of trunks occupied satisfies a Fortet integral equation possessing a unique solution under certain assumpt­ ions. A detailed treatment of the case of two streams, one general, one Poisson, is given by Kuczura [17].

A crucial feature with the above is the assumption of mutual independence between the pooled traffic streams. Except in the trivial case when all the component traffics are Poisson, the pooling of renewal streams results in a non-renewal stream, and the components are no long­ er independent after an overflow. Only one exact study is known to the author in the area of problems involving the overflow stream resulting from pooled renewal offered traffic. This is a model of Neal [27] in which Poisson traffic is divided and offered to separate trunk groups and the (correlated) overflow traffics then recombined. The loss of independence and the simple renewal character of the traffic is a severe setback to exact analysis, and practical work with probabilistic models almost invariably proceeds via Wilkinson's Equivalent Random approximation [40] and its extensions (see Mina [25]).

As noted earlier the convenient renewal property of the stream is left invariant when offered traffic is forced to overflow from a finite trunk group, only the precise functional form of the inter-arrival time distribution function of calls changing. In this paper we shall ob­ serve that pooled independent renewal streams constitute an example of a more general class of traffic with certain invariance properties on overflowing. These include a muted renewal property which is not lost when pooled renewal traffics overflow and so lose their mutual indep­ endence. Once pooling has occurred, such independence or the lack of it is subsequently irrelevant. Further pooling with independent streams of the same class may also occur without loss of any crucial - 21-

property. The renewal property is utilised to derive equations suff­ icient to determine overflow and carricd traffics. Procedures are discussed for the solution of these equations.

The methods considered are thus in principle suitable for the piecemeal analysis of compound ful1 -availabi1 ity loss systems which do not allow later feedback from a lower choice route to a higher choice route. Our aim in this paper is to give a descriptive account of a general model, with some indication as to practical calculation pro­ cedures for quantities of interest. A number of questions of con­ vergence, ergodicity etc. arise in connection with the model delineated which must, for reasons of space, be considered elsewhere. However, in a practical context, the existence of the relevant invariant meas­ ures or convergence of series for physical quantities, for instance, are not normally in question. It is therefore hoped that, despite its omissions, the paper may seem relatively complete.

2. General Semi-Markov Stream

Let (Z(t), t >. 0} be a general semi-Markov process with under­ lying Markov chain D defined on a probability space (S, B, v). Denote by P(t, x, E), x e S, E e B, t ^ 0, the probability of a transition into a state of the set E in time less than or equal to t, given that the process has initially just entered state x. If the state transit­ ion epochs t are taken as the advent instants of calls in a traffic stream, we sRall refer to the traffic as a general semi-Markov stream, and label the call at time t by the state x e S of the process at time t +0. (Cf. the conditioning sequence (z , n >, 0} of Loynes [24], who shows that in some situations the steady-state waiting time dis­ tribution in a single-server queue with non-recurrent input may be calculated.)

We assume that the underlying Markov chain is v-recurrent with invariant probability measure \ defined on the sets E e B. (See Orey [29] for a discussion of these concepts.) Physically A(E) is just the steady-state probability that an arriving call has a lable in the set E. It is given as the (unique) non-negative solution to the equations

X(E) = A(dx)P(~, x, E) E e B

X(S) = 1.

Various special cases of the general semi-Markov stream correspond to classes of stream of interest in telephony and queueing theory. In the following we shall write x = t _tnl» n 2 0 . (a) Renewal Stream n n This corresponds to the case when S consists of a single point. (b) Semi-Markov Input This arrises when S consists of a finite set of points, each corr­ esponding to a state of the Markov chain underlying the semi-Markov input. The infinite and finite trunk groups (loss system) with semi- Markov input have been studied by Franken [10] and the infinite trunk -22-

group by Neuts and Chen [28]. (c) Generalised Moving Average Input P t = I g (U ), n 6m n+p-m m=o r where the g are random functions and {U , n >, 0} an identically and independently distributed sequence of random variables.

The points of S can be taken as p+1 -vectors, with x = (un, un+j, ..., un+p)> where, as in (e) below, lower casensymbols denote particular values realised by the random variables denoted by the corresponding upper case symbols. This model was introduced ex­ plicitly into queueing theory by Loynes [24] in the case of determin­ istic functions gm> and an earlier instance is given by Winsten [41]. Special cases occur for an infinite trunk group in Finch [6 ] and Pearce [33]. The possibility of random functions g has been found useful in road traffic theory [35].

(d) The Singly-Connected Dependent Stream

t = U + V , n ^ 0 , n n n ’ ’ (Jarovickii [12, 13, 14]) where {U , n * 0} is an identically and independently distributed sequence and Vn depends on Un This is a special case of (c) with p = 1 . (e) EARMA (1,1) Sequence (Jacobs and Lewis [11]). This is a particular case of

Tn = 0 En + f( V l ) With An = p V l +g(En}’ where {E ; n 5 0} is an identically and independently distributed sequence of random variables and f and g are random functions. We can take x = (e , a ^). (?) Poo¥ed renewal Streams with k Independent Components We can take x = (x ., x •••> x rj. where x , m = 1, ..., k, is the backward de¥ay at t +0nto the last event in stream m. Thus with probability 1, each x e S will possess exactly one vanishing component. Let F be the distribution function for inter-event times in stream m. For y = (y^, ..., y^) and a set E e B of the form

E = {z e S| z. = 0} (1 ) we have the explicit representation

1-F (y +u) P(t,y,E) = / tn -- — ---- } I (y +u,..,y ,+u.O.y. +u,..,y.+u) 0 mjfj 1-F (y ) 1 J + 1 k J mv/m^ dF.(y,+u)

1 - F.(y.) 3 J where I is the indicator function - 23-

!/•-•* 1 if x c E !E(x) 3 0 otherwise .

For sets E c B which are not unions of sets of the fora (1) we have P(t» y. E) a 0 . The measure P is determined for disjoint unions by o -additivity in E.

3. Pooling of General Semi-Markov Streams

An important invariance property of general semi-Markov streams is that two or more such streams, if independent, pool to give another. In fact, provided the precise inter-connection of the underlying general semi-Markov processes is known, the result may be extended to correlated streams, although we shall not pursue this possibility here.

It suffices to consider two streams with underlying Markov chains with, say, state spaces (S., B., v.)» i = 1.2. Denote by P.(t, x., E.), x. e S., E. c B., the Corresponding transition prob­ abilities. 1Let us associate an event in the pooled stream at time t with the state (t., t_, x., x2), where t^ is the backward delay at time t+0 to the last 1 -stream event and x. e S. the state in the underlying chain of the i-stream associated with that event.

One-step transitions from (t^f t_, x,, x2) taking place in time t or lessss will be to sets of states of the form

(a) {(0 , t2+u, x ^ , x2); 0 < u $ t, { x ^ } = Ej e Bj)

(b) {(tj+v, 0 , Xj. x(2)); 0 < v { t, (x(2)> = E2 c B2).

Define S as the collection of all points included in sets of the forms (a) and (b), and let B be the minimal o -field generated by sets of forms (a) and (b). Then with an obvious probability measure v, the Markov chain defined on (S, B, v) in a natural way act as a Markov chain underlying the pooled stream. If the associated semi-Markov process has transition probability R(t, x, E) for a transition x -*■ E in time t or less, x e S, E e B, we have

...... t 1 'P2 (t2+U' S2> duP(V U* R(t, x, E) = f ------V . ------V E) 1-P2 (t2, x2, S2) l-PCtj, Xj, s p

for E of the form (a), with a similar formula with the roles of streams 1 and 2 interchanged for a set (b) and a natural completion for a general E e B (we note that the intersection of an (a) and a (b) set is empty.

4. The Overflow Stream

Suppose a general semi-Markov stream with one-step transition probabilities P(t, x, E), x e S, E e B, is offered to a single trunk. Here as elsewhere we take holding times to be negative exponential with parameter y. We are interested in characterising the resultant over­ -24-

flow traffic.

Imagine a call specified by state x e S arrives at time t and produces an overflow. The event in the overflow stream may again be labelled by x. Holding times at the trunk lack memory and the stoch­ astic development of the arriving stream is completely specified by x. It follows that the stochastic development of the overflow after an event at time t is also completely specified by x. The time for a first transition, say from x into E c B, in the overflow stream can be seen to be a random function of x and E alone. Hence the overflow stream is also given by a general semi-Markov process, with under­ lying Markov chain on the same state space (S, B, v). This basic invariance property extends the Palm-TakScs result.

Further, the stochastic evolution of the offered and overflow streams after time t will be the same whether or not the call at time t was obliged ?o overflow. Hence the one-step transition prob­ abilities Q(t, x, E) for the overflow stream satisfy

Q(t,x,E)=/Vuu du P(u, x, E)+ / P(t-u, x, dy)[l-e_lj(t‘u)]duQ(u,y,E) (2) If Q and P possess time-derivatives q(t, x, E), p(t, x, E)then (2) may be written as q(t,x,E) = p(t,x,E)e“pt * f f} p(t-u,x,dy)[l-e~p(t"u)]q(u,y,E)du (3). The overflow of the traffic from a finite group of L trunks can be regarded as arising from a sequence of L successive one-trunk over­ flows. Thus the arguments of this section can be extended to this case by induction. Hence the basic invariance obtains for arbitrary overflows. If (2) can be solved to give Q in terms of P, then iterat­ ion of this procedure will give the one-step transitions prescribing the overflow stream arising from a finite trunk group.

The solution of (2) (or (3)) can frequently be carried out with some facility. Consider the space T of real-valued functions f(t,x,E), teR,xeS,EeB, for which the norm

I I f I I = [sup ff f2 (t, x,dy)dt]ly^2 x is defined, and suppose p e T. If p denotes the metric given by

P(f,g) = II f-g !I , f, g e T, then it can be seen that (T, p) is a complete metric space. Define a mapping A on T by

Af=p(t,x,E)e p(t-u,x,dy)(1-e w^t'u^)f(u,y,E)du, f e T.

Then for f,g e T, we have

Af-Ag = //* p(t-u,x,dy) [1-e u-* ] [f (u,y,E)-g(u,y,E) ]du.

Let us now take norms on both sides of this equation and employ - 25-

Schwarz's inequality on the right hand side. In a variety of situat­ ions the presence of the damping factor l-exp(-p(t-u)) can cause the right hand side to be less than k||f-g|| for some constant k, 0 < k< 1, so that

p(Af,Ag) = ||Af-Agj| < k||f-g||= kp(f,g), f,g e T.

(It may sometimes be apposite to choose some other metric.)

In this event A is a contraction mapping, and so by the contraction mapping theorem (cf. Kolmogorov and Fomin [15]) there exists a unique element q e T such that Aq * q. Equation (3) will then determine the physical q uniquely. The contraction mapping theorem also states that for an arbitrary q e T, q is given as the limit of the sequence {qn ; n z 0) defined by q . = A qn, n i 0, so that we have a con­ structive procedure for the determination of q. This algorithm is efficient in the sense that the convergence of the solution is geo­ metrically fast, that is,

P(q„. q) ^ k" p(qQ ,q) / (1-k).

5. Trunk Occupancy

(a) Infinite Trunk Group Suppose that a general semi-Markov stream characterised by the one-step transition probabilities P(t,x,E), x e S, E e B, is offered to an infinite trunk group. We assume without further discussion that the semi-Markov stream is such as to induce well-defined proper stationary distributions (n (E); n i 0}, {q (E); n >. 0). Here *n(E) represents the steady-state probability that an arriving call is characterised by state x e E and finds n calls occupying trunks and a (E) the steady-state probability that at an arbitrary instant of time n calls are present and the last arriving call was characterised by state x e E.

The trunk occupancy state (n, x) at any instant will belong either to the set C given by n < j, x e E or to its complement. By the generalised principle of statistical equilibrium of Morris and Wolman [26], the mean transition rates into and out of C must balance in the steady state. This leads to a rather complicated relation between the n - and q -probabilities which we shall not pursue here.

For the determination of the probabilities "(E), consider the transitions of the system between two consecutive^arrival instants in the steady-state. We have

nn(E) = I ff V m - 1(du)(nr n)e",,tn(l-e"yt)" dtP(t’U,E)’ n * 0, E e S. m=o

These equations may be reduced as in the case of a renewal input ([34]). If we define 00 hn (E) = 1 0 V E)* n ^ 0, E e B (4) j=n -26-

our equations may be recast as

hn (E) = //[hn l (dx)+hn (dx)] e‘nutdtP(t,x,E). (5)

In the renewal case the h are constants and h =1, so that (5) gives a trivial recurrence relation for the h . In our present general case, h (E) is, from (4), the probability A^E) of the Markov chain under­ lying the offered traffic. If A(E) is known, (5) is available for the successive determination of the h . The presence of the exponential term in (5) for n 5 1 may give rise to the possibility of solving for h^ (when hR ^ is known) through the contraction mapping theorem.

The inversion of (4) gives CO IT (E) = Z (-l)n_j (") h (E), j * 0. n=j J

(b) Finite Trunk Group Let us now look at the case of a general semi-Markov stream offered to a finite group of L trunks. For 0 < j s L, define w.(E) as before and set L H (E) = I (J) ir.(E), 0 s n < L, E e B. (6) n j=n n J

Proceeding as in the previous section we find that

Hn (E)=//[Hn l (dx)+Hn(dx)]e'nptdtP(t,x,E)-//irL(dx)e-niJt(n5;i)dtP(t,x,E)

E e B, 0 * n $ L (7)

(cf. [34]). As before Hq (E) = A(E), while

ttl(E) = Hl(E) = Hl(S) Al(E), (8) where A denotes the probability invariant measure for the overflow stream From the L trunks. Thus ^(E) is of the form t ,(E) = dAL(E). If we replace d by unity, equation (7) is then available for a downward recursive determination of the H . The constant d is determined as the scale factor necessary ?o make H (S) so found equal to unity. 0

Inversion of (6) provides L ir.(E) = Z (-l)n-;,(") H (E). n=j

(c) Engineering Calculations In practical work a full knowledge of the distributions {q.}, {it.} is not necessarily required, as one is normally interested^only in^such quantities as the mean and variance of offered or carried traffic and the probability of overflow. These quantities are much easier to determine than the full distributions. Thus we note that - 27-

for a finite trunk group the mean and variance of-the imbedded chain distribution are simply Hj(S) and 2H_(S)+H,(S)-ll. (S). Quantities with a set E other than S in the argument are sometimes also of direct practical interest and not merely variables needed incidentally in calculations. Thus the set E may be chosen to represent just those states for which the call is of some specific type (cf. equation (1) in which the choice of E corresponds to a j-stream call). Such a choice has direct bearing on problems in which we wish to keep track of each component in a pooled stream.

6. References

[1] Bech, N.I., "Metode til bcregning af spaerring i altemativ trunking-og gradingsystemes", Teleteknik, Vol. 5, No. 5, 1954 [2] Brockmeyer, E., "Det simple overFTowproblem i telefontra fict- eorien", Teleteknik, Vol. 5, No. 4, pp. 361-374, 1954 [3] Burke, P.J., "The Overflow Distribution for Constant Holding Time", Bell System Technical Journal, Vol. 50, pp. 3195-3210, 1971 [4] Cohen, J.W., "The Full Availability Group of Trunks with an Arbitrary Distribution of the Inter-arrival Times and the Negative Exponential Holding Time Distribution", Simon Stevin Natuurkundig Tijdschrift, Vol. 26, pp. 169-181, 1957 [5] Descloux, A., "On Markovian Servers with Recurrent Input", Sixth International Teletraffic Congress, Munich, pp. 331/1-6 1970 [6] Finch, P.D., "The Co-incidence Problem in Telephone Traffic with Non-recurrent Arrival Process", Journal of the Australian Mathematical Society, Vol. 3, pp. 237-240, 1963 [7] Forget, R., "Random Distributions with an Application to Telephone Engineering", Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Vol. 2, pp. 81-88, 1956 [8] Fortet, R., and Canceill, B., "Probability de perte en select­ ion conjugSe", First International Teletraffic Congress, Copen­ hagen, 1955-Teleteknik, Vol. 1, No. 1, pp. 41-55, 1957 [9] Fortet, R., "Applications of Characteristic Functionals in Traffic Theory", Fifth International Teletraffic Congress, New York, p. 287, 1967 [10] Franken, P., "Erlangsche Formeln fur Semimarkowschen Eingang", Elektron In format ionsverarbe it Kybemetik, Vol. 4, pp. 197-204, 1968. [11] Jacobs, P.A., and Lewis, P.A.W., "A Mixed Autoregressive-moving Average Exponential Sequence and Point Process (EARMA 1,1)", Journal of Applied Probability, Vol. 9, pp. 87-104, 1977 [12] Jarovickii, M.V., "On the Outgoing Flow of a Unilinear Service System with Losses", (in Ukrainian), Dopovidi Akad. Nauk. Ukrain. R.S.R., pp. 1251-4, 1961 -28-

Jarovickii, M.V., "Vectorial Simply-connected Flows", Dopovidi Akad. Nauk. Ukrain. R.S.R., pp. 715-9, 1962 Jarovickii, M.V., "On Some Properties of Simply-connected Dependent Streams", (in Russian), Ukrain. Mat. Z.,Vol. 14, pp. 170-9, 1962 Kolmogorov, A.N., and Fomin, S.V., ELEMENTS OF THE THEORY OF FUNCTIONS AND FUNCTIONAL ANALYSIS, Graylock Press, 1957 Kosten, L., "Sur la probability d'encombrement dans les multi­ plages gradu£s", Annales des P.T.T., pp. 1002-1019, 1937 Kuczura, A., "Loss Systems with Mixed Renewal and Poisson Inputs" Seventh International Teletraffic Congress, Stockholm, pp. 412/ 1-5, 1973 Le Gall, P., "Methods de calcul de 11encombrement dans les systfemes t616phoniques automatiques & marquage", Ann. T61£com., Vol. 12, No. 11, pp. 374-386, 1957 ~ ~ Le Gall, P., "Les calculs d’organes dans les centraux t61£phon- iques modemes", Collection Techniques et Scientifiques du C.N. E.T., £dit. de la revue d'optique, pp. 1-76, 1959 Le Gall, P., "Le trafic de d^bordenment", Annals des Tdl£communic- ations, Vol. 16, Nos. 9-10, pp. 226-238, 1961 Le Gall, P., "Le trafic de d€bordement", Third International Teletraffic Congress, Paris, pp. 26/1-23 and 3 appendices, 1961 Le Gall, P., "Random Processes Applied to Traffic Theory and Engineering", Seventh International Teletraffic Congress, Stock­ holm, pp. 221/1-26, 1973 Le Gall, P., "General Telecommunications Traffic without Delay", Eighth International Teletraffic Congress, Melbourne, pp. 125/ 1-8, supplement 1-7, 1976 Loynes, R.M., "Stationary Waiting-time Distributions for Single­ server Queues", Annals of Mathematical Statistics, Vol. 33, pp. 1323-1369, 1962 Mina, R.R., "A Solution to the Problem of Smooth Traffic", Fourth International Teletraffic Congress, London, pp.76/1-16 and ~ addendum 1-4, 1964 Morris, R., and Wolman, E., "A Note on Statistical Equilibrium", Operations Research, Vol. 9, pp. 751-3, 1961 Neal, S., "Combining Correlated Streams of Non-random Traffic", Bell System Technical Journal, Vol. 50, pp. 2015-2037, 1971 Neuts, M.F. and Chen, S.-Z., "The Infinite Server Queue with Semi-Markovian Arrivals and Negative Exponential Services", Journal of Applied Probability, Vol. 9, pp. 178-184, 1972 Orey, S., LIMIT THEOREMS FOR MARKOV CHAIN TRANSITION PROBABILIT­ IES, Van Nostrand Reinhold, 1971 Padgett, W.J., and Tsokos, C.P., "On a Stochastic Integral Equation of the Volterra Type in Telephone Traffic Theory", Journal of Applied Probability, Vol. 8, pp. 269-275, 1971 Palm, C., "Intensitatsschwangkungen in Femsprechvehrkehr", Ericsson Technics, Vol. 44, pp. 1-189, 1943 Pearce, C.E.M., "A Queueing System with Non-recurrent Input and Batch Servicing", Journal of Applied Probability, Vol. 2, pp. 442-448, 1965 Pearce, C.E.M., "On a Many Server Queue with Non-recurrent Input and Negative Exponential Servers", Journal of the Australian - 29-

Mathematical Society, Vol. 8, pp. 706-715, 1968 [34] Pearce, C.E.M. and Potter, R.M., "Some Formulae Old and New for Overflow Traffic in Telephony", Australian Telecommunication Research, Vol. 11, pp. 92-97, 1977 [35] Pearce, C.E.M., "Superimposed Alternating Renewal Streams and Delay Models Involving Multi-lane Major Road Traffic at an Uncontrolled Intersection", Proceedings of the Seventh Intcr- National Symposium on Transportation and Traffic Flow, Kyoto, pp. 217-245, 1977 [36] Pollaczek, F., "Problfemes de calcul des probability relatifs & des systfemes t61€phoniques sans possibility d'attente", Annales de l'Institut H. Poincare, Vol. 12, No. 2,pp. 57-96, 1951 [37] Pollaczek, F., "Generalisation de la thfiorie probabiliste des systfemes teiephoniques sans dispositif d'attente", C.R. Acad. Sci. Paris, Vol. 236, No. 15, pp. 1469-70, 1953 [38] Takics, L., "On the Generalisation of Erlang's Formula", Acta. Math. Acad. Sci. Hung., Vol. 7, pp. 419-433, 1956 [39] TakScs, L., "On the Limiting Distribution of the Number of Coin­ cidences Concerning Telephone Traffic", Annals of Mathematical Statistics, Vol. 30, pp. 134-141, 1959 [40] Wilkinson, R.I., "Theories for Toll Traffic Engineering in the U.S.A.", Bell System Technical Journal, Vol. 35, pp. 421-514, (appendix by J. Riordan), 1956 [41] Winsten, C.B., "Geometric Distributions in the Theory of Queues", Journal of the Royal Statistical Society, Series B, Vol. 21, pp. 1-35, 1959