A Markov chain model for a battle between two opposing forces is formulated, which is a stochastic version of one studied by F. W. Lanchester. Solutions of the backward equations for the final state yield martingales and stopping identities, but a more powerful technique is a time-reversal analogue of a known method for studying urn models. A general version of a remarkable result of Williams and McIlroy (1998) is proved.

We consider one-dimensional processes in which particles annihilate their neighbours, grow until they meet their neighbours or are deposited onto surfaces. All of the models considered have the property that they are connected to exponential series often by an inclusion-exclusion argument.

In this paper, we introduce a bisexual Galton-Watson branching process with mating function dependent on the population size in each generation. Necessary and sufficient conditions for the process to become extinct with probability 1 are investigated for two possible conditions on the sequence of mating functions. Some results for the probability generating functions associated with the process are also given.

Often in the study of reliability and its applications, the goal is to maximize or minimize certain reliability characteristics or some cost functions. For example, burn-in is a procedure used to improve the quality of products before they are used in the field. A natural question which arises is how long the burn-in procedure should last in order to maximize the mean residual life or the conditional survival probability. In the literature, an upper bound for the optimal burn-in time is obtained by assuming that the underlying distribution of the products has a bathtub-shaped failure rate function; however, no lower bound is available. A similar question arises in studying replacement policy, warranty policy, and inspection models. This article gives a lower bound for the optimal burn-in time, and lower and upper bounds for the optimal replacement and warranty policies, under the same bathtub-shape assumption.

We consider, in discrete time, a single machine system that operates for a period of time represented by a general distribution. This machine is subject to failures during operations and the occurrence of these failures depends on how many times the machine has previously failed. Some failures are repairable and the repair times may or may not depend on the number of times the machine was previously repaired. Repair times also have a general distribution. The operating times of the machine depend on how many times it has failed and was subjected to repairs. Secondly, when the machine experiences a nonrepairable failure, it is replaced by another machine. The replacement machine may be new or a refurbished one. After the Nth failure, the machine is automatically replaced with a new one. We present a detailed analysis of special cases of this system, and we obtain the stationary distribution of the system and the optimal time for replacing the machine with a new one.

In this paper, we consider the compound Poisson process that is perturbed by diffusion (CPD). We derive formulae for the Laplace transform, expectation and variance of total duration of negative surplus for the CPD and also present some examples of the CPD with an exponential individual claim amount distribution and a mixture exponential individual claim amount distribution.

Recently Bassan and Spizzichino (1999) have given some new concepts of multivariate ageing for exchangeable random variables, such as a special type of bivariate IFR, by comparing distributions of residual lifetimes of dependent components of different ages. In the same vein, we further study some properties of these concepts of IFR in the bivariate case. Then we introduce certain concepts of bivariate DMRL ageing and we develop a treatment that parallels those developed for bivariate IFR. For both the IFR and DMRL concepts, we analyse a weak and a strong version, and discuss some of the differences between them.

We deal with compound geometric sums of independent positive random variables and study the moment problem for the distributions of such sums (the Stieltjes moment problem). We find conditions under which the distributions are uniquely determined by their moments. We also treat related topics, including the Hamburger moment problem involving random variables on the whole real line. Some conjectures are outlined.

Testing in order to produce software of high reliability is an area of major concern in software engineering. In an effort to find efficient methods of testing, the comparison of partition and random sampling testing methods has received considerable attention in the literature. A standard criterion for comparisons between random and partition testing, based on their expected efficacy in program debugging, is the probability of detecting at least one failure causing input in the program's domain. However, the goal in software testing is usually to find as many faults as possible in a reasonable period of time, and therefore stochastic comparisons of the number of faults obtained in partition and random testing may provide more valuable information on which testing procedures to use. We establish various conditions which guarantee that the number of faults discovered in partition testing is stochastically greater than the number discovered in random testing (using a fixed total sample size) for many of the well-established stochastic orders (including the usual stochastic order, the hazard rate order, the likelihood ratio order, and the variability order). The results established also allow us to obtain both upper and lower bounds with these stochastic orders for the sum of n independent Bernoulli random trials (with varying probability of success) in terms of the binomial distribution with parameters n and p.

In this paper, we consider a Hilbert-space-valued autoregressive stochastic sequence (Xn) with several regimes. We suppose that the underlying process (In) which drives the evolution of (Xn) is stationary. Under some dependence assumptions on (In), we prove the existence of a unique stationary solution, and with a symmetric compact autocorrelation operator, we can state a law of large numbers with rates and the consistency of the covariance estimator. An overall hypothesis states that the regimes where the autocorrelation operator's norm is greater than 1 should be rarely visited.

We study the closure properties of the family ℒ(α) of classes of life distributions introduced by Lin (1998) under general compounding. We define a discrete analogue of this family and study some properties.

We consider an infinite server resequencing queue, where arrivals are generated by jumps of a semi-Markov process and service times depend on the jumps of this process. The stationary distribution of the sojourn time, conditioned on the state of the semi-Markov process, is obtained both for the case of hyperexponential service times and for the case of a Markovian arrival process. For the general model, an accurate approximation is derived based on a discretisation of interarrival and service times.

We consider a fluid queue with downward jumps, where the fluid flow rate and the downward jumps are controlled by a background Markov chain with a finite state space. We show that the stationary distribution of a buffer content has a matrix exponential form, and identify the exponent matrix. We derive these results using time-reversed arguments and the background state distribution at the hitting time concerning the corresponding fluid flow with upward jumps. This distribution was recently studied for a fluid queue with upward jumps under a stability condition. We give an alternative proof for this result using the rate conservation law. This proof not only simplifies the proof, but also explains an underlying Markov structure and enables us to study more complex cases such that the fluid flow has jumps subject to a nondecreasing Lévy process, a Brownian component, and countably many background states.

We consider a discrete-time queueing system with the discrete autoregressive process of order 1 (DAR(1)) as an input process and obtain the actual waiting time distribution and the virtual waiting time distribution. As shown in the analysis, our approach provides a natural numerical algorithm to compute the waiting time distributions, based on the theory of the GI/G/1 queue, and consequently we can easily investigate the effect of the parameters of the DAR(1) on the waiting time distributions. We also derive a simple approximation of the asymptotic decay rate of the tail probabilities for the virtual waiting time in the heavy traffic case.

Consider a ·/G/k finite-buffer queue with a stationary ergodic arrival process and delayed customer feedback, where customers after service may repeatedly return to the back of the queue after an independent general feedback delay whose distribution has a continuous density function. We use coupling methods to show that, under some mild conditions, the feedback flow of customers returning to the back of the queue converges to a Poisson process as the feedback delay distribution is scaled up. This allows for easy waiting-time approximations in the setting of Poisson arrivals, and also gives a new coupling proof of a classic highway traffic result of Breiman (1963). We also consider the case of nonindependent feedback delays.

In this paper we find conditions under which the epoch times of two nonhomogeneous Poisson processes are ordered in the multivariate dispersive order. Some consequences and examples of this result are given. These results extend a recent result of Brown and Shanthikumar (1998).

We consider a variant of the house-selling problem in which the seller has n objects for sale and the buyers come sequentially one by one. The objective of this paper is to find the seller's optimal strategy for choosing his selling price so as to maximize the average reward per selling attempt until the nth house is sold.

We generalize Banach's matchbox problem: demands of random size are made on one of two containers, both initially with content t, where the container is selected at random in the successive steps. Let Zt be the content of the other container at the moment when the selected container is found to be insufficient. We obtain the asymptotic distribution of Zt as t → ∞ under quite general conditions. The case of exponentially distributed demands is considered in more detail.

The benefit obtained by a selfish robot by cheating in a real multirobotic system can be represented by the random variable Xn,q: the number of cheating interactions needed before all the members in a cooperative team of robots, playing a TIT FOR TAT strategy, recognize the selfish robot. Stability of cooperation depends on the ratio between the benefit obtained by selfish and cooperative robots. In this paper, we establish the probability model for Xn,q. If the values of the parameters n and q are known, then this model allows us to make predictions about the stability of cooperation. Moreover, if these parameters are modifiable, it is possible to tune them to guarantee the viability of cooperation.