A realization of a doubly stochastic Poisson field is assumed to be observed in a Borel set S ⊂ Rk. The best linear estimate of the realization of the intensity field (at an arbitrary but fixed point x ∈ S) which generated the observation is obtained.

We explicitly calculate the dynamic allocation indices (i.e. the Gittins indices) for multi-armed Bandit processes driven by superdiffusive noise sources. This class of model generalizes former results derived by Karatzas for diffusive processes. In particular, the Gittins indices do, in this soluble class of superdiffusive models, explicitly depend on the noise state.

Let ·be a sequence of k -dimensional i.i.d. random vectors and define the first-passage times for where (cvτ)v, τ= 1,· ··,k is the covariance matrix of In this paper the weak convergence of Zn in (D[0, ∞))k is proved under the assumption (0,∞) for all v = 1, ···, k. We deduce the result from the Donsker invariance principle by means of Theorem 5.5 of Billingsley (1968). This method is also used to derive a limit theorem for the first-exit time Mn = min{Nnt for fixed t1,···, tk > 0. The second result is an extension of a theorem of Hunter (1974) whose method of proof applies only if Ρ (ξ1 [0,∞)k) = 1 and μ ν = tv for all v = 1, ···, k.

The literature on supply models for manpower planning in a hierarchy shows that an important consideration is the size of the discrepancy between the age distribution of the population and the stationary age distribution which would be reached if present policies were continued indefinitely. We establish the convergence properties of age distribution in various cases and show under what circumstances a convergent age distribution exists. In the convergent case we examine both the speed of convergence and the form of the stationary age distribution.

A discrete-time storage system with a general release rule and stationary nonnegative inflows is examined. A simple condition is found for the existence of a stationary storage and outflow for a general possibly non-monotone release function. It is also shown that in the Markov case (i.e. independent inflows) these distributions are unique under certain conditions. It is demonstrated that under these conditions the stationary behaviour in the Markov case varies continuously with parametric changes in the release rule. This result is used to prove convergence of a finite state space approximation for the Markov storage system.

We consider an interacting particle system in which each site of the d-dimensional integer lattice can be in state 0, 1, or 2. Our aim is to model the spread of disease in plant populations, so think of 0 = vacant, 1 = healthy plant, 2 = infected plant. A vacant site becomes occupied by a plant at a rate which increases linearly with the number of plants within range R, up to some saturation level, F1, above which the rate is constant. Similarly, a plant becomes infected at a rate which increases linearly with the number of infected plants within range M, up to some saturation level, F2. An infected plant dies (and the site becomes vacant) at constant rate δ. We discuss coexistence results in one and two dimensions. These results depend on the relative dispersal ranges for plants and disease.

The stochastic process resulting when pairs of events are formed from two point processes is a rich source of questions. When the two point processes have different rates, the resulting stochastic process has a mean drift towards either -∞ or +∞. However, when the two processes have equal rates, we end up with a null-recurrent Markov chain and this has interesting behavior. We study this process for both discrete and continuous times and consider special cases with applications in communications networks. One interesting result for applications is the waiting time of a packet waiting for a token, a special case of this pair-formation process. Pair formation by two independent Poisson processes of equal rates results in a point process that is asymptotically a Poisson process of the same rate.

A Bayesian approach for analyzing layered defense systems is presented. This approach incorporates the dependence of penetration probabilities on the size of attackers going into any layer. A general formula is developed for computing the predictive distribution of the number of attackers surviving any layer as well as the posterior distribution of the penetration probabilities under the a priori assumptions that: (i) the probabilities are dependent and their joint distribution is Dirichlet, and (ii) the probabilities are independent. Positive dependence of the penetration probabilities as well as the number of attackers surviving the different layers is also established.

We show that the advantage that can accrue to the server in tennis does not necessarily imply that serving first changes the probability of winning the match. We demonstrate that the outcome of tie-breaks, sets and matches can be independent of who serves first. These are corollaries of a more general invariance result that we prove for n-point win-by-2 games. Our proofs are non-algebraic and self-contained.

We consider a large class of random walks on the discrete circle Z/(n), defined in terms of a piecewise Lipschitz function, and motivated by the ‘generation gap' process of Diaconis. For such walks, we show that the time until convergence to stationarity is bounded independently of n. Our techniques involve Fourier analysis and a comparison of the random walks on Z/(n) with a random walk on the continuous circle S1.

An arbitrary number of competitors are presented with independent Poisson streams of offers consisting of independent and identically distributed random variables having the uniform distribution on [0, 1]. The players each wish to accept a single offer before a known time limit is reached and each aim to maximize the expected value of their offer. Rejected offers may not be recalled, but they are passed on to the other players according to a known transition matrix. This paper finds equilibrium points for two such games, and demonstrates a two-player game with an equilibrium point under which the player with the faster stream of offers has a lower expected reward than his opponent.

Intersymbol and cochannel interference in a communications channel can often be modelled as the sum of an infinite series of random variables with weights which decay fairly rapidly. Frequently, this yields a random variable which is singular but non-atomic. The Hausdorff dimension of the distribution is estimated and methods for calculating expectations are studied. A connection is observed between the dimension and the complexity of the calculation of an expectation.

The concept of entropy in models is discussed with particular reference to the work of P.A.P. Moran. For a vector-valued Markov chain {Xk} whose states are relative-frequency (proportion) tables corresponding to a physical mixing model of a number N of particles over n urns, the definition of entropy may be based on the usual information-theoretic concept applied to the probability distribution given by the expectation . The model is used for a brief probabilistic assessment of the relationship between Boltzmann's Η-Theorem, the Ehrenfest urn model, and Poincaré's considerations on the mixing of liquids and card shuffling, centred on the property of an ultimately uniform distribution of a single particle. It is then generalized to the situation where the total number of particles fluctuates over time, and martingale results are used to establish convergence for .

This paper studies the equilibrium behaviour of the M/G/k loss system with servers subject to breakdown. Such a system has k servers, whose customers arrive in a Poisson process. They are served if there is an idle server, otherwise they leave and do not return. Each server is subject to breakdown with probability of occurrence depending on the length of the time the server has been busy since his last repair. On a breakdown the customer waits and his service is continued just after the repair of the server. Among other things, a generalization of the Erlang B-formula is given and it is shown that the equilibrium departure process is Poisson. In fact these results are obtained for the more general case where customers may balk and service and repair rates are state dependent.

A transient cumulative process based upon a sequence of possibly infinite lifetimes is defined, and examples of such a process are described. Given a mild condition on the improper lifetime distribution and given that all lifetimes observed by time t are finite, the expected value of this transient process at t is related to the expected value of a cumulative process based upon proper lifetimes. This relationship is exploited to show that the conditional behavior of the transient process is analogous to that of a proper process and, in particular, the transient process is asymptotically normal.

For a GI/G/c queue, a full busy period is an interval that begins when an arrival finds c − 1 customers in the system, and ends when, for the first time after that, a departure leaves behind c − 1 customers in the system. We present a probabilistic proof of conditions for full busy periods to have finite moments. For queues that empty, this result may be deduced from results in the literature, but our proof is much easier. For queues that do not empty, our proof still applies, and this result is new.

A multitype polymer reaction process is considered. Its dynamics is described by means of the evolution in time of the average number of reactions (of any kind) and the mean number of unreacted chain segments (of any length), assuming that the initial chain molecule consists of n unreacted units. The asymptotic behaviour (n →∞) of the variance of the extent of reaction is also studied.

An annihilating process is an interacting particle system in which the only interaction is that a particle may kill a neighbouring particle. Since there is no birth and no movement, once a particle has no neighbours its site remains occupied for ever. It is shown that with initial configuration ℤ the distribution of particles at all times is a renewal process and that the probability that a site remains occupied for all time tends to 1/e. Time-dependent behaviour is also calculated for the tree 𝕋r.

We derive the asymptotic behavior of the total, active, and inactive branch lengths of the seed bank coalescent when the initial sample size grows to infinity. These random variables have important applications for populations evolving under some seed bank effects, such as plants and bacteria, and for some cases of structured populations like metapopulations. The proof relies on the analysis of the tree at a stopping time corresponding to the first time a deactivated lineage is reactivated. We also give conditional sampling formulas for the random partition, and we study the system at the time of the first reactivation of a lineage. All these results provide a good picture of the different regimes and behaviors of the block-counting process of the seed bank coalescent.

The first part of the paper gives a multitude of essentially different representations of a stationary stochastic process. The second part gives a sufficient condition for the sum of two oscillatory processes to be again oscillatory.