In this paper we investigate the properties of stationary point processes motivated by the following traffic model. Suppose there is a dichotomy of slow and fast points (cars) on a road with limited overtaking. It is assumed that fast points are delayed behind (or are clustered at) a slow point in accordance with the principles of a GI/G/s queue, the order of service being irrelevant. Thus each slow point represents a service station, with the input into each station consisting of a fixed (but random) displacement of the output of the previous queueing station. It is found that tractable results for stationary point processes occur for the cases M/M/s (s = 1, 2, ···, ∞) and M/G/∞. In particular, it is found that for these cases the steady state point processes are compound Poisson and that for the M/M/1 case the successive headways form a two state Markov renewal process. In addition it is shown that the input, output, and queue size processes in a steady state M/G/∞ queue are independent at any fixed time; this is a result I have been unable to find in the literature.

A realization of a doubly stochastic Poisson field is assumed to be observed in a Borel set S ⊂ Rk. The best linear estimate of the realization of the intensity field (at an arbitrary but fixed point x ∈ S) which generated the observation is obtained.

Suppose there are k kinds of balls identical except for their colours. The number of balls of the ith colour is ri (i = 1, 2, ···, k) and Put down the R balls in a sequence (a line), then an acceptable sequence is one which does not have two adjacent balls of the same colour. The problem is to generate a random sample of the set of acceptable sequences S(r1, r2, ···, rk).

In a recent paper Rényi (1964) has obtained the following characterization of the Poisson process.

A basic process of simple queueing theory is S(t), an integer valued stochastic process which represents the “number of clients present, including the one in service” at epoch t. The queueing context physically limits S(t) to non-negative values, but if this impenetrable barrier is removed thereby permitting S(t) to assume negative values as well, we have a “randomized random walk” in which S(t) represents the position at time t of a mythical particle which moves on the x-axis according to the rules of the walk. The nomenclature “randomized random walk” is due to Feller (1966a). S(t) changes by unit positive or negative amounts, and our basic assumption is that the time intervals τ/σ separating successive positive/negative steps are i.i.d. with continuous d.f.'s. A(t)/B(t) (A(0)/B(0) = 0) possessing p.d.f.'s. a(t)/b(t): τ and σ are further assumed to be statistically independent. When A(t) = 1 – e–λt and B(t) = 1 – e–μt, we have a generalization of the M/M/1 queueing process which we shall denote by M/M. The walk generated by A(t) = 1 – e–λt and B(t) arbitrary may similarly be denoted by M/G. For G/M we clearly mean A(t) arbitrary and B(t) = 1 – e–μt. M/M has received extensive treatment elsewhere (cf. Feller (1966a), (1966b); Takács (1967); Gibson (1968); Conolly (1971)), and will not be considered here. Our interest centers on certain aspects of M/G and G/M, but since each is the dual of the other (loosely, G/M is M/G “upside down”) it is sufficient to restrict attention to one of them, as convenient; pointing out, where necessary, the interpretation in terms of the other.

Let {Xn} be a random walk between reflecting barriers at 0 and a > 0 with jumps {Zn}. By we mean the random walk between absorbing barriers at — a and 0+ with the same jumps {Zn}. It has been known for some time that when {Zn} is a sequence of mutually independent and identically distributed random variables, and 0 ≦x <a, we have for all n:

In problems of traffic theory, we are frequently concerned with the queues of vehicles which form at an intersection. A standard model is the vehicle queue whose length is increased during unit time intervals [t, t + 1) by non-negative integer inputs which form a sequence of i.i.d. or Markovian random variables; see Gani (1970) and Lehoczky ((1969), (1971)). The output from the queue at the end of each unit of time is one vehicle if the queue is non-empty, and no vehicle if there are none waiting.

Consider first a Markov chain with two ergodic states E1 and E2, and discrete time parameter set {0, 1, 2, ···, n}. Define the random variables Z0, Z1, Z2, ···, Znby then the conditional probabilities for k = 1,2,···, n, are independent of k. Thus the matrix of transition probabilities is

A theorem is proved which, in essence, says the following. If, for any queueing system, (i) the arrival process is stationary, (ii) the queue discipline is first-in-first-out (FIFO), and (iii) the waiting time of each customer is statistically independent of the number of arrivals during any time interval after his arrival, then the stationary random queue size has the same distribution as the number of customers who arrive during a random time interval distributed as the stationary waiting time.

A certain single server queueing system with negative exponential service with mean rate nμ, when the system contains n customers, and Poisson arrivals, is formally equivalent to the infinite capacity system M/M/∞. This equivalence is exploited to yield in a very simple manner results for the single server system which were previously obtained by difficult analysis (see Hadidi (1969)).

The Laplace-Stieltjes transform of the distribution of the busy period for the M/G/1 system with infinite waiting room can be obtained by using an argument from branching theory. In the present paper it is shown that by applying this argument it is rather easy to derive the expression for the joint distribution of the busy period and the maximum number of customers present simultaneously during this busy period for the M/G/1 system with infinite waiting room as well as the expression for the distribution of the busy period for the M/G/1 system with finite waiting room.

Dr. J. Alan Steppe has kindly pointed out two serious flaws, and some others less serious, in the proof of our paper [2]. However these can all be corrected, so that the result remains valid.

Diffusion and birth-death processes have basic theoretical and practical importance for statistics. Insight into the structure of transition distributions and passage time distributions for such processes has been given in recent years by Feller. Karlin, Kemperman, D. G. Kendall, Reuter and many others. An elementary account of this work and partial bibliography has been given elsewhere ([9], [10)]. Certain key passage time densities and sojourn time densities for such processes have a simple property of log-concavity or log-convexity and associated unimodality. Such properties provide information on the character of distributions unavailable from the spectral representations, Laplace transforms and series of convolutions at hand. These properties may also have value for purposes of estimation and optimization.

Kingman (1962) studied the effect of queue discipline on the mean and variance of the waiting time. He made no assumptions regarding the stochastic nature of the input and the service distributions, except that the input and service processes are independent of each other. When the following two conditions hold:

1. (a) no server sits idle while there are customers waiting to be served;

2. (b) the busy period is finite with probability one (i.e., the queue empties infinitely often with probability one);

The game of pure strategy, sometimes called Goofspiel or Gops (see [2] and [3]), is played by two players, using a normal deck of cards, as follows. The 13 clubs are first taken out of the deck and of the remaining 39 cards the 13 hearts are given to Player I, the 13 diamonds to Player II, and the 13 spades are placed face down in the center. The spades are shuffled and one is turned face up. At this point, the two players choose one of their cards and then simultaneously discard it. The one who discards the higher card (ace being low, king high) wins from the other an amount equal to the value of the upturned spade (ace = 1, king = 13). If both players discard the same card, then neither wins. The three cards are then thrown away, a new spade upturned and the game continues. After 13 plays, there are no remaining cards and the game ends.

In a short article McKean [2] inaugurated the study of a class of time-inhomogeneous Markov processes which arise from the reaction of a single particle (the “tagged” particle) with an evolving environment of other particles. Other more recent references on these processes are [1], [4] and [5], in which some existence and uniqueness theorems for the solution of the resulting non-linear equation are obtained. We confine attention here to a very particular type of these processes which arises when two interacting particles simply coalesce, and the mass is taken as the process variable.

In [4], the limiting behaviour of a stochastic system with two types of input was investigated by reducing the problem to the solution of an integral equation. In this note we use the same approach to study the equilibrium waiting time problem for the general single server queue with preemptive service interruptions. (For a comprehensive account of the existing literature on queues with service interruptions we refer to [2] and [3].)

An important problem, arising in connection with the estimation of mathematical expectation of a homogeneous random field X(x1, ···, xn) in Rn by means of the arithmetic mean of observed values, is to determine the number of observations for which the variance of the estimate attains its minimum. Vilenkin [2] has shown, that in the case of a stationary random process X(x) such a finite number exists, provided that the covariance function satisfies certain conditions.