The steady-state distribution of the inventory position for a continuous review (s, S) inventory system is derived. The demand for items in inventory is dependent on an external environment. During an interval of time in which the environment is in a fixed state, the demand follows a discrete-valued compound Poisson process. The parameters of the compound Poisson process depend completely on the state of the environment. The environmental process is modeled as a continuous-time Markov process.

In this paper the L2-convergence of a superadditive bisexual Galton–Watson branching process is studied. Necessary and sufficient conditions for the convergence of the suitably normed process are given. In the final section, a result about one of the most important bisexual models is proved.

In this paper, we present an iterative procedure to calculate explicitly the Laplace transform of the distribution of the maximum for a Lévy process with positive jumps of phase type. We derive error estimates showing that this iteration converges geometrically fast. Subsequently, we determine the Laplace transform of the law of the upcrossing ladder process and give an explicit pathwise construction of this process.

We consider weak and strong survival for branching random walks on multigraphs with bounded degree. We prove that, at the strong critical value, the process dies out locally almost surely. We relate the weak critical value to a geometric parameter of the multigraph. For a large class of multigraphs (which enlarges the class of quasi-transitive or regular graphs), we prove that, at the weak critical value, the process dies out globally almost surely. Moreover, for the same class, we prove that the existence of a pure weak phase is equivalent to nonamenability. The results are extended to branching random walks on weighted graphs.

We consider an M/G/1 retrial queue, where the service time distribution has a finite exponential moment. We show that the tail of the queue size distribution is asymptotically given by a geometric function multiplied by a power function. The result is obtained by investigating analytic properties of probability generating functions for the queue size and the server state.

We introduce a stochastic process that describes a finite-velocity damped motion on the real line. Differently from the telegraph process, the random times between consecutive velocity changes have exponential distribution with linearly increasing parameters. We obtain the probability law of the motion, which admits a logistic stationary limit in a special case. Various results on the distributions of the maximum of the process and of the first passage time through a constant boundary are also given.

The paper deals with a problem which arises in the Monte Carlo optimization of steady state or ergodic systems which can be modelled by Markov chains. The transition probability depends on a parameter, and one wishes to find the parameter value at which some performance function is minimum. The only available data are obtained from either simulation or actual operating information. For such a problem one needs good statistical estimates of the derivatives. Conditions are given for the existence of the derivative of the stationary measure with respect to the parameter, in the sense that the derivative is a signed measure, and is the limit of the natural approximating sequence. Some properties and a useful characterization of the derivative are obtained. It is also shown that, under appropriate conditions, the derivative of the n-step transition function converges to the derivative of the stationary measure as n tends to ∞. This latter result is of particular importance whether one is simply estimating or is actually optimizing via some sequential Monte Carlo procedure, since the basic observations are always taken over a finite time interval.

Whittle has given a sufficient condition for a certain class of Markov decision processes to have an optimal strategy of a particularly simple form. We show this condition to be necessary in a certain sense and give some examples of its application.

We consider scheduling problems with m machines in parallel and n jobs. The machines are subject to breakdown and repair. Jobs have exponentially distributed processing times and possibly random release dates. For cost functions that only depend on the set of uncompleted jobs at time t we provide necessary and sufficient conditions for the LEPT rule to minimize the expected cost at all t within the class of preemptive policies. This encompasses results that are known for makespan, and provides new results for the work remaining at time t. An application is that if the cµ rule has the same priority assignment as the LEPT rule then it minimizes the expected weighted number of jobs in the system for all t. Given appropriate conditions, we also show that the cµ rule minimizes the expected value of other objective functions, such as weighted sum of job completion times, weighted number of late jobs, or weighted sum of job tardinesses, when jobs have a common random due date.

We consider several classical notions of partial orderings among life distributions which have been used to describe ageing properties and tail domination. We show that if a distribution G dominates another distribution F in one of these partial orderings introduced here, and if two moments of G agree with those of F, including the moment that describes this partial ordering, then G = F. This leads to a characterization of the exponential distribution among HNBUE and HNWUE life distribution classes, and thus extends the results of Basu and Bhattacharjee (1984) and rectifies an error in that paper.

This paper presents an algorithmic analysis of the busy period for the M/M/c queueing system. By setting the busy period equal to the time interval during which at least one server is busy, we develop a first step analysis which gives the Laplace-Stieltjes transform of the busy period as the solution of a finite system of linear equations. This approach is useful in obtaining a suitable recursive procedure for computing the moments of the length of a busy period and the number of customers served during it. The maximum entropy formalism is then used to analyse what is the influence of a given set of moments on the distribution of the busy period and to estimate the true busy period distribution. Our study supplements a recent work of Daley and Servi (1998) and other studies where the busy period of a multiserver queue follows a different definition, i.e., a busy period is the time interval during which all servers are engaged.

Let X1, X2, · ··, Xn be independent identically distributed random points with common density f(x), taking values in a bounded region (p ≧ 1). We obtain the limit distribution, as n → ∞, for the maximum value of the suitably ‘weighted' (according to f(x)) rth-nearest-neighbour distances of Χ1, · ··, Χ n (r ≧ 1 fixed) provided that f(x) is bounded from below by a positive constant and a weak continuity condition holds. This is achieved by refining an argument used by the author (Henze (1981)) to derive the limit distribution in the special case r = 1. Edge-effects are eliminated by defining, for each Xi, the distance to the boundary of G to be the ‘rth-nearest-neighbour distance' if it is smaller than the distance to the rth nearest neighbour among the remaining points. Applications to a multivariate test of goodness of fit are given.

We consider a vector, discrete, time sequence x(n), n = 0, ± 1 ···, of p components Xj(n), j = 1, ··· p. For the most part we shall assume x(n) to be strictly stationary and with finite variances. Thus if μ is the mean vector we shall have

This paper is concerned with the problem of controlling a simple immigration–birth process, which represents a pest population, by the introduction of catastrophes which, when they occur, reduce the population size to zero. The optimality criterion is that of minimising the long-term average cost per unit time of the process. Firstly, an optimal policy is found within a restricted class of stationary policies, which introduce catastrophes if and only if the population size is greater than or equal to some critical value x. The optimality of this policy within the wider class of all stationary policies is then verified by applying the general results of Bather (1976).

We consider a fork-join network with arrival and service times forming a stationary and ergodic process. The usual stability condition, namely that the input rate is strictly less than all the service rates, is proved to be valid in this general case. Finally we extend the result to the case where there is random routing.

A parallel server system with n identical servers is considered. The service time distribution has a finite mean 1 / μ, but otherwise is arbitrary. Arriving customers are routed to one of the servers immediately upon arrival. The join-idle-queue routeing algorithm is studied, under which an arriving customer is sent to an idle server, if such is available, and to a randomly uniformly chosen server, otherwise. We consider the asymptotic regime where n → ∞ and the customer input flow rate is λn. Under the condition λ / μ < ½, we prove that, as n → ∞, the sequence of (appropriately scaled) stationary distributions concentrates at the natural equilibrium point, with the fraction of occupied servers being constant at λ / μ. In particular, this implies that the steady-state probability of an arriving customer waiting for service vanishes.

Methods using gambling teams and martingales are developed and applied to find formulae for the expected value and the generating function of the waiting time to observation of an element of a finite collection of patterns in a sequence generated by a two-state Markov chain of first, or higher, order.

For general risk processes, we introduce and study the expected time-integrated negative part of the process on a fixed time interval. Differentiation theorems are stated and proved. They make it possible to derive the expected value of this risk measure, and to link it with the average total time below 0, studied by Dos Reis, and the probability of ruin. We carry out differentiation of other functionals of one-dimensional and multidimensional risk processes with respect to the initial reserve level. Applications to ruin theory, and to the determination of the optimal allocation of the global initial reserve that minimizes one of these risk measures, illustrate the variety of fields of application and the benefits deriving from an efficient and effective use of such tools.

Exact expressions are derived for the transition probabilities of the birth-death process with parameters and which serves as a queueing model where potential customers are discouraged by queue length.

This paper investigates the effect of migration between two colonies each of which undergoes a simple birth and death process. Expressions are obtained for the first two moments and approximate solutions are developed for the probability generating function of the colony sizes.