The asymptotic moments of the concentration of a pollutant subject to rainout and dry removal are given. Both the lengths of the wet and dry periods and the wet and dry removal rates are allowed to be random, although the removal rate is assumed to be fixed within any one wet or dry period. The results hold in a more general setting than removal processes, so its is hoped that they will be applicable to other problems.

Let {ε t, t = 1, 2, ···, n} be a sequence of mutually independent standard normal random variables. Let Xn(λ) and Yn(λ) be respectively the real and imaginary parts of exp iλ t, and let . It is shown that as n tends to∞, the distribution functions of the normalized maxima of the processes {Xn(λ)}, (Yn(λ)}, {In(λ)} over the interval λ∈ [0,π] each converge to the extremal distribution function exp [–e–x], —∞ < x <∞.

It is also shown that these results can be extended to the case where {ε t} is a stationary Gaussian sequence with a moving-average representation.

Queueing networks continue to be of both theoretical interest and applied interest. One of the operations performed on a point process, representing customer flows in the network, thins a given process into two (or more) subprocesses each of which may thereafter be the arrival process to subsequent servers. A fair amount is known about the properties of each subprocess.

A new class of reliability point-process models for dependent components is introduced. The dependence is expressed through a regression, following a form suggested by Cox (1972) for survival data analysis involving the current life-length of the components. After formulating the current-life process as a Markov process with stationary transitions and stating some general results on asymptotic behavior, we describe the stationary distributions in some bivariate examples. Finally, we discuss statistical inference for the new models, exhibiting and justifying full- and partial-likelihood methods for their analysis.

The paper deals with some properties of the (Gaussian) likelihood function for multivariable ARMA models. Its behaviour at the boundary of the parameter space is described; its continuity properties as well as the question of the existence of a maximum are discussed. We have not been able to show in general the existence of the maximum over the usual parameter spaces. However, the maximum always exists over a suitably enlarged parameter space (given that the data are non-degenerate), which includes parameters corresponding to processes with discrete spectral components.

Let K be a finite graph with vertex set V = {x0, x1, …, xσ–1} and automorphism group G. It is assumed that G acts transitively on V. We can imagine that the vertices of K represent σ cities and a traveler visits the cities in a series of random flights. The traveler starts at a given city and in each flight, independently of the past journey, chooses a city at random as the destination. Denote by vn (n = 1, 2, …) the location of the traveler at the end of the nth flight, and by v0 the initial location. It is assumed that the transition probabilities P{vn = xj | vn–1 = xi}, xi ϵ V, xj ϵ V, do not depend on n and are invariant under the action of G on V. The main result of this paper consists in determining p(n), the probability that the traveler returns to the initial position at the end of the nth flight.

Using operator methods, we prove a general decomposition theorem for Jackson networks. For its transient joint queue-length distribution, we can stochastically bound it above by a network that decouples into smaller independent Jackson networks.

In this paper we study some distribution properties of the system failure time in general shock models associated with correlated renewal sequences (Xn, Yn) . Two models, depending on whether the magnitude of the nth shock Xn is correlated to the length Yn of the interval since the last shock, or to the length of the subsequent interval to the next shock, are considered. Sufficient conditions under which the system failure time is completely monotone, new better than used, new better than used in expectation, and harmonic new better than used in expectation are given for these two models.

Many queues and related stochastic models, and in particular those that have a matrix-geometric stationary probability vector, have steady-state queue-length densities that are asymptotically geometric. The graph of the asymptotic rate η of these densities as a function of the traffic intensity ρ is the caudal characteristic curve. This is an informative graph from which a number of qualitative inferences about the behavior of the queue may be drawn.

In a closed cycle of exponential queues where the first and the last nodes are multiserver queues while the other nodes are single-server queues, the cycle-time distribution has a simple product form. The same result holds for passage-time distributions on overtake-free paths in Gordon–Newell networks. In brief, we prove Burke's theorem on passage times in closed networks.

A body receives a sequence of rotations through a fixed angle about an axis whose direction is arbitrary. The probability distribution governing the resulting orientation of the body is determined. The problem is generalized to the case where the axis of each individual rotation makes the same angle, Θ, with an axis fixed in the body but is otherwise random. The resulting distribution is shown, in the case , to reduce to the Roberts-Ursell distribution for random walk on a sphere. Some diffusion limits are examined.

We consider a production-inventory problem in which the production rate can be continuously controlled in order to cope with random fluctuations in the demand. The demand process for a single product is a compound Poisson process. Excess demand is backlogged. Two production rates are available and the inventory level is continuously controlled by a switch-over rule characterized by two critical numbers. In accordance with common practice, we consider service measures such as the average number of stockouts per unit time and the fraction of demand to be met directly from stock on hand. The purpose of the paper is to derive practically useful approximations for the switch-over levels of the control rule such that a pre-specified value of the service level is achieved.

We consider a single-server queueing system in which a customer gives up whenever his waiting time is larger than a random threshold, his patience time. In the case of a GI/GI/1 queue with i.i.d. patience times, we establish the extensions of the classical GI/GI/1 formulae concerning the stability condition and the relation between actual and virtual waiting-time distribution functions. We also prove that these last two distribution functions coincide in the case of a Poisson input process and determine their common law.

We consider a k-component system where the failure rates of the components interact, but where interaction depends only on the current ages of the components. The stationary behaviour of such systems has been studied by, for example, Franken and Streller (1980). Our main aim is to study the transient behaviour, and in particular to show that under easily verifiable conditions the transient behaviour tends to a stationary situation.

We consider an inventory system to which arrival of items stored is a renewal process, and the demand is a Poisson process. Items stored have finite and fixed lifetimes. The blood-bank model inspired this study. Three models are studied. In the first one, we assume that each demand is for one unit and unsatisfied demands leave the system immediately. Using results on this model one is able to study a model in which arrival of items is Poisson but demands are for several units, and a model in which demands are willing to wait. We compute ergodic limits for the lost demands and the lost items processes and the limiting distribution of the number of items stored. The main tool in this analysis is an analogy to M/G/1 queueing systems with impatient customers.

We give here concrete formulas relating the transition generatrix functions of any random walk on a finite group to the irreducible representations of this group. Some examples of such explicit calculations for the permutation groups A4, S4, and A5 are included.

We consider queueing systems where the stationary state probabilities are insensitive with respect to the distribution of certain basic random variables such as service requirements, interarrival times, repair times, etc. The conditional expected sojourn times are stated as Radon–Nikodym densities of the stationary distribution at jump points of the queueing system. The conditions are the given values of such basic random variables for which the insensitivity is valid. We use stationary point processes as our main tool. This means that dependences between certain basic random variables are permitted. Conditional expected real service times, conditional mean response times in closed queueing networks, and similar conditional expected values, are dealt with as special cases.

In Part I (Hunter) a study of feedback queueing models was initiated. For such models the queue-length process embedded at all transition points was formulated as a Markov renewal process (MRP). This led to the observation that the queue-length processes embedded at any of the ‘arrival', ‘departure', ‘feedback', ‘input', ‘output' or ‘external' transition epochs are also MRP. Part I concentrated on the properties of the embedded discrete-time Markov chains. In this part we examine the semi-Markov processes associated with each of these embedded MRP and derive expressions for the stationary distributions associated with their irreducible subspaces. The special cases of birth-death queues with instantaneous state-dependent feedback, M/M/1/N and M/M/1 queues with instantaneous Bernoulli feedback are considered in detail. The results obtained complement those derived in Part II (Hunter) for birth-death queues without feedback.

A state-space model is presented for a queueing system where two classes of customer compete in discrete-time for the service attention of a single server with infinite buffer capacity. The arrivals are modelled by an independent identically distributed random sequence of a general type while the service completions are generated by independent Bernoulli streams; the allocation of service attention is governed by feedback policies which are based on past decisions and buffer content histories. The cost of operation per unit time is a linear function of the queue sizes. Under the model assumptions, a fixed prioritization scheme, known as the μc -rule, is shown to be optimal when the expected long-run average criterion and the expected discounted criterion, over both finite and infinite horizons, are used. This static prioritization of the two classes of customers is done solely on the basis of service and cost parameters. The analysis is based on the dynamic programming methodology for Markov decision processes and takes advantage of the sample-path properties of the adopted state-space model.

Let with Z(0) = 0 be a random process under investigation and N be a point process associated with Z. Both Z and N are defined on the same probability space. Let with R0 = 0 denote the consecutive positions of points of N on the half-line . In this paper we present sufficient conditions under which (Z, R) is a new better than used (NBU) process and give several examples of NBU processes satisfying these conditions. In particular we consider the processes in which N is a renewal and a general point process. The NBU property of some semi-Markov processes is also presented.