Stationary Waiting-Time Distributions for Single-Server Queues

Abstract

In a single-server first-come-first-served queue the waiting-times of successive customers are related by the equation \begin{equation*}\tag{1}w_{n + 1} = \lbrack w_n + U_n\rbrack^+\end{equation*} where \begin{equation*}\tag{2}U_n = S_n - T_n.\end{equation*} Here $S_n$ and $T_n$ are the service-time of the $n$th customer, and the time between the arrivals of the $n$th and the $(n + 1)$th customers, respectively. In the particular case of mutually independent identically distributed $U_n$ the basic investigation of this equation was carried out by Lindley [1], who found a simple necessary and sufficient condition for the existence of a stationary waiting-time distribution and derived this distribution in certain special cases. The theory was developed by Smith [2], who under fairly weak conditions gave a systematic treatment of the Wiener-Hopf equation obtained by Lindley. In the less restricted case when $U_n$ is a strictly stationary process it has been shown by the author (Loynes [3]) that under a simple condition the existence of a unique stationary distribution is again ensured: it is the purpose of this paper to show how (when this condition is satisfied) the stationary distribution may sometimes be found, and to obtain some qualitative results. The theory will be developed in Sections 3 and 4, and some examples discussed in Section 5. On the assumption that $U_n$ is a metrically transitive sequence the condition just referred to is \begin{equation*}\tag{3}E\lbrack U_n\rbrack < 0,\end{equation*} and we shall therefore suppose this satisfied throughout the paper. The assumption that $U_n$ is metrically transitive does not affect most of the arguments in this paper, but it will be made for convenience. For the existence of a unique stationary distribution (3) is both necessary and sufficient, but there is sometimes more than one stationary distribution when the inequality in (3) is replaced by equality. Such situations have been ignored here for several reasons: they are not very common, some of the arguments either break down or need adaptation, and in any case the problems are not then as difficult, for it follows from equation (7) of [3] that the function $\psi$ occurring in (5) below is then identically equal to one, so that the only unknown is $\phi$. There is never a stationary distribution when the inequality in (3) is reversed. Throughout the entire investigation we shall suppose the queue has a structure which satisfies the following condition (H). This is actually no restriction at all when the queue is in the stationary state, since the variables $z_n$ occurring in it may be taken as $(W_{n + 1}, T_{n + 1})$, where $\{W_n\}$ is the stationary waiting-time sequence. However, all the results will refer to the distribution of the waiting-time conditional on $z_n$, so that it will always be tacitly assumed that the distribution of $z_n$ is known in order to allow deductions about the unconditional distribution of the waiting-time. (H): There exists a sequence $\{z_n\}$ (defined on the same probability space as $w_n$) of random vectors--i.e., in finite-dimensional Euclidean space--with the following properties: (i) $\{z_n, S_n, T_n\}$ is a strictly stationary process. (ii) $S_n, T_n$, and $w_n$ are conditionally independent given $z_{n - 1}, z_n$. (iii) $w_n$ and $z_n$ are conditionally independent given $z_{n - 1}$. The assumption that (H) is satisfied allows us to write down the equation satisfied by the stationary (conditional) waiting-time distribution function. We cannot, however, in such a general situation actually find the solution to the equation, and to discover appropriate further restrictions which may make this possible we may consider further the case treated by Lindley and Smith. The general solution given by Smith was analytically complex and apparently only obtainable from the Wiener-Hopf technique or the equally specialized method due to Spitzer [4]. If, however, either the service-times or the inter-arrival times had a distribution with a rational characteristic function the solutions were much simpler in form, and in at least a few cases had been obtained previously by other methods. With this in mind we impose similar conditions here on the conditional characteristic functions; on that of the interarrival time in Section 3, and on that of the service-time in Section 4. The lack of success in treating the problem without these restrictions is probably no great loss in practice, in view of the complex nature of the Wiener-Hopf solution for the simple case. The results obtained under the conditions imposed so far are in the nature of aids to the solution of the problem, rather than the solution itself. A simple situation for which the required distribution may then be obtained straightforwardly occurs when the following "finite-matrix" condition is satisfied: (HF): Condition (H) holds, and $z_n$ has (with probability one) only a finite number $k$ of different possible values. An interesting result obtained by Smith showed that, roughly speaking, the characteristic functions of the waiting-time and the service-time had the same number of poles, and a somewhat similar result related the distribution of the inter-arrival times to that of the waiting-time. Under (HF) it will be found that similar results are also true. Smith in fact gave explicit formulae of a simple type for the two special cases, but we shall content ourselves here with pointing out how corresponding formulae could be obtained, since it appears that they are no longer simple when $k$ has a value other than one. As we have already remarked, all simple queues are included in (H) when in the stationary state. Even under the condition that the distribution of $z_n$ be known, a large class of queues satisfies (H), such as that in which $S_n$ and $T_n$ are independent stationary Markov processes. An example of a queue of considerable interest which apparently does not admit a specification of this type is that in which customers are due at regular intervals but are late independently with a lateness distribution which extends to $\infty$ (if the lateness distribution and service time distribution are both negative exponential, then (H) is satisfied, and a brief discussion of this example will be found in Section 5). It will be seen that by taking $z_n = 1$ the classical case with independence is included under (HF); two other types of queue satisfying (HF) are of some interest, and are discussed in Section 5--one being a simple queue whose input is a mixture of two streams of customers, and the other a queue whose input consists of customers who have already been served in a first queue. In addition to any other queues which may fall into this class, it is presumably true, that any queue may be approximated by those of this "soluble" class, although there are of course very great difficulties in deciding how such an approximation should be carried out. Finally we observe that since the content of a semi-infinite dam in discrete time is described by the same equation (1), the results obtained here may be applied in that context also, and in particular the finite-matrix case gives a class of dams with serially dependent inputs for which explicit solutions can be found.