A basic problem in the mathematical theory of traffic flow is to obtain the expected delay to a vehicle at a fixed-cycle traffic light controlling an intersection. The factors affecting this expected delay are the cycle time, the fraction of time the traffic light is effectively red, the number of vehicles which can pass through when the signal is green, and the nature of the (random) arrival process for the vehicles. Once a solution is known, it is possible to fix the cycle time and the effective red period in such a way that the expected delay is minimized.

We find the distribution of delay to minor road vehicles waiting to merge or cross a single stream of major road traffic. The decision to cross is taken on the basis of a gap-acceptance function. The model turns out to be a simple queueing problem in which a customer finding an empty queue has a different service time distribution from queueing customers. The key to this representation is given in Section 3. Some numerical results in Section 6 indicate that in most circumstances a simple model will give adequate results.

A perturbation formalism is presented which shows how the stationary distribution and fundamental matrix of a Markov chain containing a single irreducible set of states change as the transition probabilities vary. Expressions are given for the partial derivatives of the stationary distribution and fundamental matrix with respect to the transition probabilities. Semi-group properties of the generators of transformations from one Markov chain to another are investigated. It is shown that a perturbation formalism exists in the multiple subchain case if and only if the change in the transition probabilities does not alter the number of, or intermix the various subchains. The formalism is presented when this condition is satisfied.

In this paper, we shall consider a storage model in which a bin of finite capacity is fed by specified r(r > 0) or zero input depending on the content of the bin and the operation of the input machines (i.e., whether or not they are working on the bin). These machines are allowed to fill the bin until its level reaches maximum and they are then used to produce some secondary products not relevant to the storage problem. When the content of the bin decreases to a given minimum level, these machines are again set to feed the bin. The whole process is then repeated. It is often desirable in practice to keep the minimum level as low as possible, to attain stability in the process, or for maintenance purposes. When the machines are subject to breakdowns, it becomes necessary to consider the reliability of the operational system.

In a previous paper [4] the author studied the stochastic process {wn, n = 1,2, …}, recursively defined by with K a positive constant, τ1, τ2, … σ1, σ2, …, independent, nonnegative stochastic variables. τ1,τ2…, are identically distributed, and σ1,σ2,…, are also identically distributed variables. For this process the generating function of the Laplace-Stieltjes transforms of the joint distribution of Wn, σ2 + … + σn and τ1 + … + τn−1 has been obtained. Closely related to the process {wn, n = 1, 2,…} is the process {un, n = 1, 2,…} with {un = K + [wn + τn − K]−, n = 1,2,…; these are dual processes.

In the present paper we study the stationary distributions of the processes {wn, n= 1,2, …} and {un, n = 1,2, …}, and the distributions ot the entrance times and return times of the events “wn, n = 0” and “un = K” for some n, for discrete as well as for continuous time. For these events various taboo probabilities are also investigated. The mathematical descri ption of the processes {wn, n = 1,2, …} and {un, n= 1,2, …} gives all the necessary information about the time-dependent behaviour for the general dam model with finite capacity K, since the process {wn, n= 1,2, …} is the basic process for such dam models. In Sections 5, 6 and 7 the general theory is applied to the models M/G/1 and G/M/1. Complete explicit solutions are obtained for these models.

The present theory also leads to new and important results for the queueing system or dam model G/G/1 with infinite capacity. For instance the joint distribution of the busy period (or wet period) and of the supremum of the dam content dunng this period is obtained.

In this paper we discuss a counter system whose output is a stochastic point process such that the time intervals between pairs of successive events form a first order Markov chain. Such processes may be regarded as next, in order of complexity, in a hierarchy of stochastic point processes, to “renewal” processes, which latter have been studied extensively. The main virtue of the particular system which is studied here is that virtually all its important statistical properties can be obtained in closed form and that it is physically realizable as an electronic device. As such it forms the basis for a laboratory generator whose output may be used for experimental work involving processes of this kind.

Such statistical properties as the one and two-dimensional probability densities for the time intervals are considered in both the stationary and nonstationary state and also discussed are corresponding properties of the successive numbers arising in the stores of the counter system. In particular it is shown that the degree of coupling between successive time intervals may be adjusted in practice without altering the one dimensional probability density for the interval lengths. It is pointed out that operation of the counter system may also be regarded as a problem in queueing theory involving one server alternately serving two queues. A generalization of the counter system, whose inputs are normally a pair of statistically independent Poisson processes, to the case where one of the inputs is a renewal process is considered and leads to some interesting functional equations.

Let S denote a subset of the states of a finite continuous-time Markov chain and let Y(a) denote the time that elapses until a weighted sum of a time units have been spent in S. Formulae are derived for the generating functions of Y(a) and of Y(a + b) – Y(b).

Consider an interval of the real line (0, x), x > 0; and place in it a random subinterval S(x) defined by the random variables Xx and Yx, the position of the center of S(x) and the length of S(x). The set (0, x)– S(x) consists of two intervals of length δ and η. Let a > 0 be a fixed constant. If δ ≦ a, then a random interval S(δ) defined by Xδ, Yδ is placed in the interval of length δ. If δ < a, the placement of the second interval is not made. The same is done for the interval of length η. Continue to place non-intersecting random subintervals in (0, x), and require that the lengths of all the random subintervals be ≦ a. The process terminates after a finite number of steps when all the segments of (0, x) uncovered by random subintervals are of length < a. At this stage, we say that (0, x) is saturated. Define N(a, x) as the number of random subintervals that have been placed when the process terminates. We are interested in the asymptotic behavior of the moments of N(a, x), for large x.

A general proposition is proved stating that the exponential ergodicity of a stationary Markov chain is preserved for derived Markov chains as defined by Cohen [2], [3]. An application to a certain type of continuous time Markov chains is included.

A service unit I, with Poisson input and general service times is in series with a unit II, with negative-exponential service times. The intermediate waitingroom can accomodate at most k persons and a customer cannot leave unit I when the waitingroom is full.

The paper shows that this system of queues can be studied in terms of an imbedded semi-Markov process. Equations for the time dependent distributions are given, but the main emphasis of the paper is on the equilibrium conditions and on asymptotic results.

We follow the terminology of Khintchine's monograph [4].

Let us consider a random stream of calls entering a service system which consists of n lines. The service durations are assumed to be identically distributed random variables which are independent both of each other and of the course of the incoming stream.

Suppose that the arrival rate λ(t) of customers to a service facility increases with time at a nearly constant rate, dλ(t)/dt = a, so as to pass through the saturation condition, λ(t) = μ = service capacity, at some time which we label as t = 0. The stochastic properties of the queue are investigated here through use of the diffusion approximation (Fokker-Planck equation). It is shown that there is a characteristic time Tproportional to α–2/3 such that if , then the queue distribution stays close to the prevailing equilibrium distribution associated with the λ(t) and μ, evaluated at time t. For |t| = O(T), however, the mean queue length is much less than the equilibrium mean, and is measured in units of some characteristic length L which is proportional to α–1/3. For , the queue is approximately normally distributed with a mean of the order L larger than that predicted by deterministic queueing models. Numerical estimates are given for the mean and variance of the distribution for all t. The queue distributions are also evaluated in non-dimensional units.

Suppose that the state variables x = (x1,…,xn)′ where the dot refers to derivatives with respect to time t, and u ∊ U is a vector of controls. The object is to transfer x to x1 by choosing the controls so that the functional takes on its minimum value J(x) called the Bellman function (although we shall define it in a different way). The Dynamic Programming Principle leads to the maximisation with respect to u of and equality is obtained upon maximisation.

Customers arrive in a recurrent process and get served by one of the s (≧1) servers wit han exponential service time distribution. The equilibrium behavior of the queue length process has been studied by earlier authors. In this paper the transient behavior of this process is investigated.

Let n be a positive integer. Denote by Mn the convex, closed, bounded, and n-dimensional set of all n-tuples (c1, c2,···, cn) such that for some probability measure a on the Borel subsets of the unit interval I = [0,1].

In the storage model proposed by Kendall [6], the input is assumed to be a stochastic process with stationary independent increments, and the release is continuous and is at a unit rate. The storage process arising from this model is known to reach a steady state if the expected net input per unit time is negative. In this paper we consider the situation where this expected net input is nonnegative and derive limiting distributions for the wet period, total dry period during a given time interval and the dam content.

Consider n independent vector valued point processes. Superposition is defined component by component as a natural extension of the definition for the one-dimensional case. Under proper conditions as n → ∞, it is shown that the superposed process is a many-dimensional Poisson process with independent components. The results are applied to the superposition of Markov renewal processes.

A queueing system is considered where each item has a property associated with it, and where the service time interposed between two items depends on the properties of both of these items. The steady state of a single-channel queue of this type, with Poisson input, is investigated. It is shown how the probability generating function of the number of items waiting can be found. Easily applied approximations are given for the mean number of items waiting and for the average waiting time.

In this note, we consider a queueing system under any discipline which does not affect the distribution of the number of customers in the queue at any time. We shall show that the variance of the waiting time is a maximum when the queue discipline is “last come, first served”. This result complements that of Kingman [1] who showed that, under the same assumptions, the mean waiting time is independent of the queue discipline and the variance of the waiting time is a minimum when the customers are served in the order of their arrival.

This paper applies the invariance principle to the problem of testing a distribution on a compact homogeneous space for uniformity. The notion of using a reduction by invariance in such a situation is due to Ajne[1], who considers tests invariant under rotation on a circle. In his paper, he derives the distribution of the maximal invariant and gives the general form of the most powerful invariant test for uniformity on the circle.