Little [8] gave conditions sufficient to obtain the formula L = λW, where L is the time average queue size, W is the expected wait in queue, and λ is the arrival rate of customers. This formula is applicable to a very general class of queuing models and is one of the most useful results in queuing theory.

The present theory of finite reservoirs is not rich in general theorems even when of the simple Moran type, with unit draft and stationary discrete independent-sequence inflows. For the corresponding systems with unbounded capacity however there are two classes of results which have been known for some time. One of these classes is concerned with the time-dependent solution, where the theory provides a functional equation for the generating function of the time to first emptiness (Kendall (1957)), and the other with the asymptotic stationary distribution of reservoir contents, for which an explicit formula for the generating function is available (Moran (1959)).

Queueing models have been used to describe the slowing or stopping of a stream of vehicles due to a traffic light, a merging disturbance, or an accident. The buildup of stopped vehicles, and their eventual dispersal after the delaying factor has been removed, has been likened to the busy period of a queue, e.g., Jewell (1964). Some studies have dealt with the total delay to all drivers involved in such a queue, Daley and Jacobs (1969), Gaver (1969); or the combination of this total delay and the distance of the last vehicle from the originating incident, McNeil (1969), Shaw (1970).

In this paper we derive the time pattern of optimal planned expenditures for R and D projects in which the total effort required to complete the research satisfactorily is not known. Theorem 1 summarizes our results under alternative assumptions regarding the behavior of the function governing the supposed probability of completing the R and D satisfactorily with incremental effort. In addition, an easily employed necessary condition for the R and D project to be worthwhile is provided in Theorem 2.

Consider the real-valued stochastic process {S(t), 0 ≦ t < ∞} which assumes values in an arbitrary space X. For a given subset T ⊂ X we define which represents the length in time of a visit to state T. We shall restrict ourselves to processes such that τT is a random variable having a differentiable distribution function which is independent of the time t0 at which the visit to state T begins.

We shall consider a single-server queue with r priority classes of customers and a preemptive-resume discipline. In this system customers are served in order of their priority while customers of the same priority are served in order of their arrival. Higher priority customers, immediately upon arrival, replace lower priority customers at the server, while customers displaced in this way return to the server before any other customers of the same priority receive service. When a displaced customer returns to the server, his remaining service time is the uncompleted portion of his original service time (cf. Jaiswal (1968)).

This paper describes two one-for-one ordering (S − 1, S) inventory models in which the time required for order replenishment is state-dependent. The demand is assumed to follow a compound Poisson distribution, and that portion of the leadtime corresponding to the actual filling of orders is assumed to depend on the number of outstanding orders. Since the orders placed are assumed to go into a single-server queue, queuing results are used to obtain the expected inventory cost as a function of S in order to obtain an optimal value of S.

In mathematical models of phenomena occurring in the general areas of the engineering, biological, and physical sciences, random or stochastic equations appear frequently. In this paper we shall formulate a problem in telephone traffic theory which leads to a stochastic integral equation which is a special case of the Volterra type of the form where:

1. (i) ω∊Ω, where Ω is the supporting set of the probability measure space (Ω,B,P);

2. (ii) x(t; ω) is the unknown random variable for t ∊ R+, where R+ = [0, ∞);

3. (iii) y(t; ω) is the stochastic free term or free random variable for t ∊ R+;

4. (iv) k(t, τ; ω) is the stochastic kernel, defined for 0 ≦ τ ≦ t < ∞; and

5. (v) f(t, x) is a scalar function defined for t ∊ R+ and x ∊ R, where R is the real line.

Let Λ = R1 × {1, 2, ···, N} denote N copies of the real line and ξ(t) = (X(t), α(t))be a right-continuous Markov process taking values in A having transition function of the form P(t, (x, α), A × {β}) = Fαβ(t, A – x). Fukushima and Hitsuda [2] have found the most general such transition function; the (matrix) logarithm of its characteristic function is decomposed into a Lévy-Khintchine integral on the diagonal and multiples of characteristic functions off the diagonal.

The first emptiness time of an infinite reservoir with unit release and discrete input forming a stationary. Markov chain is investigated. The exact distribution of the first emptiness time is derived without the use of moment generating functions. The first and second moments of this distribution are given explicity. The close relationship between the process with stationary independent input and Markov chain input is emphasized.

The first moment of the area beneath the sample path up to first emptiness is computed. This area is often used as a measure of total delay in traffic flow theory.

Bounds are obtained for the difference between the expected number in the queue found by an arrival and the time average expected number in the queue for the stationary GI/G/m queue. The lower bound is completely general but the upper bound requires that the class of inter-arrival distributions be restricted. When the upper bound applies, these quantities differ by at most one customer. Analogous results are obtained for the difference between the arrival average and time average number in the system for the GI/G/1 queue. An upper bound is also determined for the kth factorial moment of the number found in the queue by an arrival in terms of the kth. moment about the origin of the wait in the queue. Inequalities on the mean virtual wait are found in terms of the mean actual wait which show that under the same restrictions, these two measures of congestion differ by no more than half the mean inter-arrival time for the GI/G/1 queue.

Avi-Itzhak, Maxwell and Miller (1965) studied a queueing model with a single server serving two service units with alternating priority. Their model explored the possibility of having the alternating priority model treated in this paper with a single server serving alternately between two service units in tandem.

Here we study the distribution of busy period, virtual waiting time and queue length and their limiting behavior.

A weighted Markov chain technique is used to find the generating functions for several restricted one-dimensional random walks. Examples with simple restrictions concern the number of penetrations of a penetrable barrier. Examples with compound restrictions include the number of full crossings of the origin. A typical asymptotic evaluation is carried out.

This is an extension of the paper [1] by Bather and Chernoff, where approximations are obtained to the solution of a sequential control problem involving the decision of when to use fuel to alter course during a space-ship's approach to a target. It is desired to come as close as possible to the target. If fuel is used too early, it may be used incorrectly due to imprecise information about the location of the target; however if the fuel is used too late it may be ineffective in reducing the miss distance.

This paper develops expressions for the derivatives with respect to a parameter μ of the stable probabilities of a class of Markov chains whose transition matrices are of the form Q + μW. These expressions lead to iterative schemes for calculation which in turn suggest gradient algorithms for finding locally optimal chains.

This paper is concerned with the spectrum of the intervals between responses in the renewal inhibited Poisson process, and continues the author's earlier work on this type of process ((1970a), (1970b), (1971)). A generating function for all the pairwise joint distributions of the synchronous intervals following an average response is obtained and leads directly to the associated serial correlations. It is shown that these correlations are equivalent to those predicted on different assumptions by the general stationary point theory. The results are then used to obtain the interval spectrum, and to exhibit a relationship between the sum of the serial correlations and the variance-time function. Explicit results for the spectrum of the renewal inhibited Poisson process are given for gamma inhibitory distributions, and the qualitative behavior is determined. Possible further developments are briefly discussed.

In recent years many applications of probability theory have involved such concepts as records, inter-record times and extreme order statistics. The results have generally been proved by diverse methods. In the present work a unifying structure is presented, which makes possible the simplification and extension of some of these results. The approach taken is to place all relevant processes on the same sample space. The underlying sample space is a homogeneous two-dimensional Poisson process.

The distance from Gaussianity of the shot noise process is considered, where ti are the random times of a Poisson process with average density λ(t). With F(x) the distribution function of x(t) and G(x) that of a normal process with the same mean and variance as x(t) it is shown that where If the process x(t) is stationary with λ(t) =λ and h(t, τ) = h(t – τ) and the function h(t) is bandlimited by ωc, then the above yields

Because there are no boundary conditions, extra properties are required in order to identify the correct potential cost function. A solution of the Dynamic Programming equation for one-dimensional processes leads to an optimal solution within a wide class of alternatives (Theorem 1), and is completely optimal if certain conditions are satisfied (Theorem 2). Necessary conditions are also given. Several examples are solved, and some extension to the multidimensional case is shown.

The aim of this paper is to investigate the existence of a random solution and the stochastic absolute stability of the differential systems (1.0)–(1.1) and (1.2)–(1.3) with random parameters. These objectives are accomplished by reducing the differential systems into a stochastic integral equation of the convolution type of the form (1.4) and utilizing a generalized version of V. M. Popov's frequency response method.