- 19- 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.