This paper is concerned with nearest-neighbour systems on the coloured lattice (unordered state space). It extends the paper of Strauss (1975) on clustering in the two-colour case. Comparison is made with the more general methods of Besag (1974). Some tests are developed, and illustrated with an example.

Several new limit theorems for one- and two-sample statistics of the Kolmogorov-Smirnov type are obtained. These generalize results of Takács, Vincze, and others. The proofs combine combinatorial and weak convergence techniques.

We consider the problem of product pricing when the firm's market share is changing stochastically according to a birth and death process. The current market share together with the price prevailing determine the current rate of profit made as well as the birth and death rates. The optimal pricing policy must balance the immediate advantage of setting a high price in terms of increased current profit against the disadvantage in terms of a possible erosion of the future market share. We formulate a continuous-time Markov decision model and analyse it using a recent technique developed by Lippman [6] for optimization of exponential queueing systems. The optimal pricing policy is characterized as having a sort of monotonicity property. We also analyse the dependence of the optimal policy on the problem parameters and indicate further extensions of the model.

An account is given of the autocorrelation theory of seasonal moving average models, using a vector representation defined by the author. It is shown that the autocorrelation structure of a seasonal moving average is reducible to that of a certain non-seasonal moving average, independently of the value of the seasonal periodicity if sufficiently large. It is pointed out that existing methods for the Cramér–Wold factorization discussed in Godolphin (1976) can easily be adapted to contend with the seasonal case.

In this paper we shall solve the optimal policy for the Markovian replacement model in which the state of a machine is not observable. We shall consider both discounted and average costs and discuss two examples.

A dynamic programming approach for the investigation of learning systems is taken. Making use of one-stage decision models and dynamic programs, respectively, two learning models are formulated and the existence of optimal strategies for learning in the respective models is proven.

We consider the variation in size of raw silk in the fixed size raw silk reeling process, and the relations between occurrences of feeding ends and some properties of cocoons from the stochastic viewpoint. The results obtained by stochastic analysis appear reasonable for the reeling experiment using three groups of cocoons.

During the course of an R and D project, it is often meaningful and possible to evaluate its status, so that this information may be used for making better financing decisions over time. The project status changes stochastically due to the internal (technological) and the external (market) uncertainties, the former being partially controlled by expenditure of resources. In addition to the resource expenditure strategy, the manager must also decide when to terminate the project. Once the project is terminated, a terminal return is collected, whose value depends on the final project status. It is shown that the project should be terminated if the current status is either too low or too high to make further expenditure worthwhile. Otherwise, for an intermediate (promising) status of the project, an aggressive investment strategy is shown to be optimal. Thus, the model unifies the problems of optimally undertaking, financing and terminating an R and D project in face of various uncertainties.

The D.A.I. theorem of Gittins and Jones has proved a powerful tool in solving sequential statistical problems. A generalisation of this theorem is presented. This generalisation enables us to solve certain stochastic scheduling problems where the items or jobs to be scheduled have random times to completion, the random times having distributions dependent upon parameters to which prior distributions are allocated. Such problems are of interest in many areas where scheduling is important.

For independent random variables X and Y, if the conditional distribution of X given X + Y satisfies certain conditions, then the joint distribution of X and Y is a member of a certain one-parameter exponential family. Extensions for n independent random variables are given. A characterization for independent random variables involving order statistics is also given.

For fixed accuracy functions, let Gnm be a non-discrete duel in which the amounts of ammunition available to the two players are n and m, respectively. Let vnm be the value of Gnm and let the support of the optimal pure strategies for the players be (anm, 1). The asymptotic relationships among the quantities n, m, vnm and anm are investigated.

Two types of processes occurring in queues with stationary inputs are considered. They are called ‘time processes’ and ‘customer processes’. Sufficient conditions for the convergence of sample averages and the existence of limiting distributions for each type of processes are given. The results generalize those of Loynes (1962). The results are applied to three queueing processes and the Little's formula L = λW is obtained under rather general conditions.

The asymptotic behaviour of the Ek/G/1 queue with finite waiting room is studied. Using a combination of the supplementary variable and phase techniques, queue length and waiting time distributions are obtained. Also the idle period distribution and mean length of the idle and busy periods are found.

The optimality of the one step look-ahead stopping rule is shown to hold under conditions different from those discussed by Chow, Robbins and Seigmund [5]. These results are corollaries of the following theorem: Let {Xn, n = 0, 1, …}; X0 = x be a discrete-time homogeneous Markov process with state space (E, ℬ). For any ℬ-measurable function g and α in (0, 1], define Aαg(x) = αExg(X1) – g(x) to be the infinitesimal generator of g. If τ is any stopping time satisfying the conditions: Ex[αNg(XN)I(τ > N)]→0 as as N → ∞, then Applications of the results are considered.

The distribution of the average noise power from a Poisson stream of vehicles is, properly normalised, shown to converge to a normal distribution although the corresponding stationary process is deterministic. The speed of convergence is estimated. Finally the asymptotic efficiency of a sampling procedure is discussed.

Consider a single-server queue with service times distributed as a general random variable S and with non-stationary Poisson input. It is assumed that the arrival rate function λ (·) is periodic with average value λ and that ρ = λE(S) < 1. Both weak and strong limit theorems are proved for the waiting-time process W = {W1, W2, · ··} and the server load (or virtual waiting-time process) Z = {Z(t), t ≧ 0}. The asymptotic distributions associated with Z and W are shown to be related in various ways. In particular, we extend to the case of periodic Poisson input a well-known formula (due to Takács) relating the limiting virtual and actual waiting-time distributions of a GI/G/1 queue.

Several authors have noted that simple models for the evolution of a reproducing and spatially distributed population have no limiting distribution, although a Poisson process in statistical equilibrium has sometimes been implicitly assumed. It is shown that, even when a mechanism for restricting population density is postulated, a Poisson process is usually impossible to achieve, essentially because of an assumption of independent displacements. When this assumption is abandoned, a Poisson process is possible, at least for some highly idealised models.

This paper considers an infinite dam fed by a discrete input Xt during the time interval [t, t + 1), t = 0, 1, 2, ···. At time t – 0 there is an output Yt = min(Zt–1, + Xt–1, r) from the dam leaving behind the amount Zt = max(0, Zt–1, + Xt–1, r). The probability Pr(Zt = i), i = 0, 1, 2, ··· is discussed under the strict assumption that r > 1 and the given initial condition that Z0 = u, u = 1, 2, ···. The generating function technique has been used throughout the paper.

It is shown that certain estimators of the matrix of offspring means and the vector of stationary means in a subcritical multitype branching process with immigration are strongly consistent and obey the central limit theorem under fairly natural conditions. The results give some evidence of the robustness of the analogous autoregressive time series model in the case of large samples.

In this paper we consider a finite dam with unit release. Ratio identities are derived for the various random walks involved, and from them are derived the p.g.f. of the time of first emptiness with and without overflow and the complete time-dependent distribution of the dam content.