We consider a multitype branching random walk with independent Poisson random fields of each type of particle initially. The existence of limiting random fields as the generation number, is studied, when the intensity of the initial field is renormalized in such a way that the mean measures converge. Spatial laws of large numbers and central limit theorems are given for the limiting random field, when it is non-trivial.

Consider the shot noise process X(t):= Σih(t – τi), , where h is a bounded positive non-increasing function supported on a finite interval, and the are the points of a renewal process η on [0, ). In this paper, the extremal properties of {X(t)} are studied. It is shown that these properties can be investigated in a natural way through a discrete-time process which records the states of {X(t)} at the points of η. The important special case where η is Poisson is treated in detail, and a domain-of-attraction result for the compound Poisson distribution is obtained as a by-product.

A sequence Xn, 1 of independent and identically distributed random vectors is considered. Under a condition of regular variation, the number of vertices of the convex hull of {X1, …, Xn} converges in distribution to the number of vertices of the convex hull of a certain Poisson point process. In this paper, it is proved without sharpening the conditions that the expectation of this number also converges; expressions are found for its limit, generalizing results of Davis et al. (1987). We also present some results concerning other quantities of interest, such as area and perimeter of the convex hull and the probability that a given point belongs to the convex hull.

This expository paper deals with many problems concerning bounded objects arranged randomly in space. The objects are of rather general shapes and sizes, whilst the random mechanisms for positioning and orienting them are also fairly general. There are no restrictions on the dependence between shapes, sizes, orientations and positions of objects. The only substantive assumption is that the objects are distributed in a ‘statistically uniform' way throughout the whole of the space. We focus on the statistical properties of features seen in an observation window, itself of general size and shape.

In this paper we identify the shape space Σ(S2, k) for k labelled points on the sphere S2 that gives a mathematical model applicable to data sets whose elements are, or can be represented by, configurations of labelled sequences of points on S2 and for which the fundamental properties of interest are the shapes of these configurations, and we examine the geometric structures on the space, especially the riemannian structure on Σ(S2, 3). In a companion paper (pp. 581–594) we investigate the statistical properties of such shapes when the k points are generated by a random mechanism.

This paper gives the exact evaluation of the shape density on the shape space Σ(S2, 3) for a labelled random spherical triangle whose vertices are i.i.d.-uniform in a ‘cap' of S2 bounded by a ‘small' circle of angular radius ρ0.

We present conditions under which a point process of certain jump times of a Markov process is a Poisson process. The central idea is that if the Markov process is stationary and the compensator of the point process in reverse time has a constant intensity a, then the point process is Poisson with rate a. A known example is that the output flow from an M/M/1 queueing system is Poisson. We present similar Poisson characterizations of more general marked point process functionals of a Markov process. These results yield easy-to-use criteria for a collection of such processes to be multivariate Poisson, compound Poisson, or marked Poisson with a specified dependence or independence. We discuss several applications for queueing systems with batch arrivals and services and for networks of queues. We also indicate how our results extend to functionals of non-Markovian processes.

Many authors have discussed the equivalence of partial balance and insensitivity in in stochastic processes. When speeds are introduced into a stochastic process there arises a difficulty in proving the necessity of partial balance for insensitivity. Previous authors have overcome this difficulty by assuming that a process has the property of instantaneous attention. This property enforces the requirement that no lifetime can be created in a state in which that lifetime has zero speed.

In this paper it is shown that for processes with a finite state space it is unnecessary to make this assumption provided the notion of partial balance is slightly changed. Thus we give a criterion, analogous to partial balance, which is necessary and sufficient for insensitivity even in processes which do not possess the property of instantaneous attention. When a process does have instantaneous attention this criterion is equivalent to partial balance.

A simple queueing system, known as the fork-join queue, is considered with basic performance measure defined as the delay between the fork and join dates. Simple lower and upper bounds are derived for some of the statistics of this quantity. They are obtained, in both transient and steady-state regimes, by stochastically comparing the original system to other queueing systems with a structure simpler than the original system, yet with identical stability characteristics. In steady-state, under renewal assumptions, the computation reduces to standard GI/GI/1 calculations and the bounds constitute a first sizing-up of system performance. These bounds can also be used to show that for homogeneous fork-join queue system under assumptions, the moments of the system response time grow logarithmically in the number of parallel processors provided the service time distribution has rational Laplace–Stieltjes transform. The bounding arguments combine ideas from the theory of stochastic ordering with the notion of associated random variables, and are of independent interest to study various other queueing systems with synchronization constraints. The paper is an abridged version of a more complete report on the matter [6].

Implied costs in loss networks are measures of the rate of change of an objective function with respect to the parameters of the network. This paper considers these costs and the costs predicted by the Erlang fixed-point approximation. We derive exact expressions for the implied costs and consider the asymptotic accuracy of the approximation. We show that the approximation is asymptotically valid in some cases but is not valid in one important limiting regime. We also show that a linearity approximation for the implied costs is asymptotically correct when taken over suitable subsets of links.

We consider a single server first-come-first-served queue with a stationary and ergodic input. The service rate is a general function of the workload in the queue. We provide the necessary and sufficient conditions for the stability of the system and the asymptotic convergence of the workload process to a finite stationary process at large times. Then, we consider acyclic networks of queues in which the service rate of any queue is a function of the workloads of this and of all the preceding queues. The stability problem is again studied. The results are then extended to analogous systems with periodic inputs.

It is known that if an irreducible and aperiodic Markov chain satisfies a ‘drift' condition in terms of a non-negative measurable function g(x), it is geometrically ergodic. See, e.g. Nummelin (1984), p. 90. We extend the analysis to show that the distance between the nth-step transition probability and the invariant probability measure is bounded above by ρ n(a + bg(x)) for some constants a, b> 0 and ρ < 1. The result is then applied to obtain convergence rates to the invariant probability measures for an autoregressive process and a random walk on a half line.

By use of a method of dissection, a formula is derived for the variance in the number of lattice points covered by a rectangle of arbitrary size, lying on a square lattice of unit spacing and inclined to the lattice at an angle of which the tangent is a rational fraction.

Power series ΣzXn are studied, where {Xn} is a strictly increasing integer-valued stochastic process.

A simple proof is given for the best strategy when the game of pure strategy is played.

Suppose each of the two devices is subjected to shocks occurring randomly as events in a Poisson process with constant intensity λ. Let Pk denote the probability that the first device will survive the first k shocks and let denote such a probability for the second device. Let and denote the survival functions of the first and the second device respectively. In this note we show that some partial orderings, namely likelihood ratio ordering, failure rate ordering, stochastic ordering, variable ordering and mean residual-life ordering between the shock survival probabilities and are preserved by the corresponding survival functions and .

The IFR property of the stochastic process governing a one-component system supported by an inactive standby and a repair facility when the lifetime of one component and the repair time of the other component are dependent, is established. We solve the problem of selecting repair rates to maximize the steady-state availability for given component failure rates when a lower bound for the MTBF and upper bounds for the steady-state expected number of repairs of the components per unit time and expected number of failures of the system per unit time are given.

In this work we investigate under what circumstances the equilibrium distribution of the numbers of groups of various sizes in a certain M/G/k group-arrival group-departure loss system can be obtained in a closed product form.