When the interarrival and service time distributions of a queue are interchanged a new queue is obtained which can be considered as the dual of the original. Another dual system, a dam, can also be associated with the original queue. Events defined for the original system can be transformed into events defined for the duals and conversely, and hence, probabilities obtained for one system can be extended to the others. In this paper several duality relations are derived, with particular emphasis on results pertaining to a single busy period. Examples are given, most of which refer to the M/G/1 – G/M/l queues.

The block replacement policy, wherein items are replaced at regular intervals of time and on failure, is rather wasteful because sometimes almost new items are also removed. As an alternative a policy of replacement by new items at regular intervals of time and by used items on failure, is suggested. The consequences of this policy, called used item replacement policy, are studied for Erlangian and sub-exponential life-time distributions. The latter distribution which is the difference of two negative exponential distributions, does not seem to have received much attention in the literature so far.

A “correlated queue” is defined to be a queueing model in which the arrival pattern influences the service pattern or vice versa. A particular model of this nature is considered in this paper. It is such that the service time of a customer is directly proportional to the interval between his own arrival and that of his predecessor. The initial busy period, state and output processes are analyzed in detail. For completeness, a sketch is also given of the analysis of the waiting time process which forms the subject of another paper. The results are used in the analysis of the state and output processes.

This paper is a continuation of Daley (1969), referred to as (I), whose notation and numbering is continued here. We shall indicate various approaches to the study of the total waiting time in a busy period2 of a stable single-server queue with a Poisson arrival process at rate λ, and service times independently distributed with common distribution function (d.f.) B(·). Let X'i denote3 the total waiting time in a busy period which starts at an epoch when there are i (≧ 1) customers in the system (to be precise, the service of one customer is just starting and the remaining i − 1 customers are waiting for service). We shall find the first two moments of X'i, prove its asymptotic normality for i → ∞ when B(·) has finite second moment, and exhibit the Laplace-Stieltjes transform of X'i in M/M/1 as the ratio of two Bessel functions.

Let Xt denote the waiting time of customer t in a stationary GI/G/1 queue, with traffic intensity τ; let ρn denote the correlation between Xt and Xt+n. For a rational GI/G/1 queue, in which the distribution of the difference between arrival and service intervals has a rational characteristic function, it is shown that, for large n, ρn is asymptotically proportional to n–3/2e–βn, where β and the factor of proportionality are calculable. The asymptotic law n–3/2e–βn applies also to the approach of the waiting-time distribution to the stationary state in an initially empty rational GI/G/1 queue, and to the correlations in the queueing systems recently analysed by Cohen [1]. Its more general applicability is discussed.

Modern highways, particularly the freeways of large cities, carry a considerable volume of traffic during certain times of day. Thus if any interruption or retardation of flow occurs, a large reaction in the shape of a monumental and time-consuming traffic jam soon appears. For example, when an accident or mechanical breakdown gives rise to a severe flow restriction or stoppage, many other vehicles may be quickly halted, and remain stopped until the impediment is cleared away. In addition, the flow of traffic may be slowed considerably even after the original stoppage is removed owing to the existence of a queue. Consequently, vehicles that arrive long after the original restriction is removed experience prolonged, and sometimes seemingly inexplicable, delay. Our purpose is to develop a probability model for the situation described, with the aim of estimating the consequence of a temporary flow restriction. Various measures of (in)effectiveness are worth considering. We consider primarily the total vehicle-hours waited while the jam dissipates; the latter may roughly measure total social cost. The total number of vehicles involved in the jam is also of interest, as are other figures of merit.

A new two parameter family of life length distributions is presented which is derived from a model for fatigue. This derivation follows from considerations of renewal theory for the number of cycles needed to force a fatigue crack extension to exceed a critical value. Some closure properties of this family are given and some comparisons made with other families such as the lognormal which have been previously used in fatigue studies.

In the study of the busy period for a single server queueing system, three variables that have been investigated individually or at most in pairs are:

1. 1. The duration of the busy period.

2. 2. The number of customers served during the busy period.

3. 3. The maximum number of customers in the queue during the busy period.

The use of Erlangian distributions has been proposed for the approximation of more general types of distributions of interarrival and service times in single-server queueing systems. Any Erlangian approximation should have the same mean and variance as the distribution it approximates, but it is not obvious what effect the various possible approximants have on the behaviour of the system. A major difference between approximants is their degree of skewness and accordingly, numerical results for various approximants are obtained for (a) the mean time spent by a customer in a simple single-server system, and (b) the mean queue length in a system with bulk service. Skewness is shown to have little effect on these quantities.

The estimation problem is studied for a new two-parameter family of life length distributions which has been previously derived from a model of fatigue crack growth. Maximum likelihood estimates of both parameters are obtained and iterative computing procedures are given and examined. A simple estimate of the median life is exhibited, shown to be consistent and then compared, favorably, with the maximum likelihood estimate. More important, the asymptotic distribution of this estimate is shown to be within the same class of distributions as the observations themselves. This model, and these estimation procedures, are tried by fitting this distribution to several extensive sets of fatigue data and then some comparisons of practical significance are made.

In [1] the authors dealt with a particular queueing system in which arrivals occurred in a Poisson stream and the probability differential of a service completion was μσn when the queue contained n customers. Much of the theory could not be carried out further analytically for a general σn, which is a purely n-dependent quantity. To carry the analysis further to the extent of finding the “effective” service time and the waiting time distribution when σn is a linear function of n, (which is considered to be rather general and sufficient for practical purposes), constitutes the subject matter of this paper.

A Markovian process with rewards is considered. The interference into its development consists in instantaneous changes of state called replacements. It is shown that, for any replacement policy not depending on the future, the expected reward is a linear function of the expected replacement numbers. The application of this result is illustrated by examples.

A weighted Markov chain technique is used to determine the general distribution of the number of tie positions in a two candidate ballot. This may be interpreted as the distribution of the number of returns to the origin in an asymmetric random walk, when the particle at each step moves μ units in the positive direction and γ units in the negative direction, given the total number of positive steps and the number of negative steps. Explicit results are obtained when γ = 1, whereas the case of μ and γ ≠ 1 is solved in the form of a generating function.

Un théorème de Lewis [3] montre que les seuls facteurs indécomposables de la mesure de Lebesgue m sur S = [–π, π] sont certaines lois discrètes πn,r (r est premier et divise n), et que les décompositions en facteurs premiers étudiées dans [5] constituent toutes les décompositions possibles (en facteurs premiers).1 De ces décompositions se déduisent toutes les propriétés de divisibilité de m.2 Nous développons ce point de vue et posons des problèmes analogues pour le cas de la circonférence γ.

Some previously obtained asymptotic results for branching Poisson processes are extended and sharpened. It is shown that under rather general conditions the number of events in both the transient and the equilibrium processes, suitably normalized, have a unit normal distribution. Finally, unique initial conditions are derived for the equilibrium process.

We consider a queueing model in which arrivals occur according to a non-homogeneous Poisson process in batches of varying size, and in which a customer is served immediately upon arrival by one of an infinite number of servers.

In a previous paper (Leslie (1967)) the distribution of recurrence times for a particular set of success-failure patterns on a sequence of Bernoulli trials was investigated. We now consider the analogous events in continuous time and obtain the Laplace Transform (L.T.) of the distribution of recurrence times; numerical inversion yields the distribution functions.

Skorokhod (1961) demonstrated how the study of martingale sequences (and zero-mean random walks) can be reduced to the study of the Wiener process (without drift) at a sequence of random stopping times. We show how the study of certain submartingale sequences, including certain random walks with drift and log likelihood ratio sequences, can be reduced to the study of the Wiener process with drift at a sequence of stopping times (Theorem 4.1). Applications to absorption problems are given. Specifically, we present new derivations of a number of the basic approximations and inequalities of classical sequential analysis, and some variations on them — including an improvement on Wald's lower bound for the expected sample size function (Corollary 7.5).

There are now many examples in various fields where the behaviour of ‘particles' as exhibited by their transition from one state to another is described by a multidimensional stochastic process. The linear migration model in which particles move independently of one another through a number of states has been dealt with by Bartlett (1949). This process has been used by Siegert (1949) in considering the approach to equilibrium of non-interacting gas molecules and by Krieger and Gans (1960) and Gans (1960) to examine the distribution of a multicomponent system disturbed from its equilibrium distribution and relaxing by first-order processes to another equilibrium. The correspondence between the deterministic model based on the principle of mass action and the stochastic model has been discussed by Darvey and Staff (1966) in the context of unimolecular multicomponent chemical reactions.

This paper considers aspects of the following problem. Let F(x, θ) be a distribution function, d.f., in x for all θ and a Borel measurable function of θ. Define the mixture (Robbins (1948)), where Φ is a d.f., then it is of interest to determine conditions under which F(x) and F(x, θ) uniquely determine Φ. If there is only one Φ satisfying (1), F is said to be an identifiable mixture. Usually a consistency assumption is used whereby it is presumed that there exists at least one solution to (1).