In this paper, we consider a single-server queue with stationary input, where each job joining the queue has an associated deadline. The deadline is a time constraint on job sojourn time and may be finite or infinite. If the job does not complete service before its deadline expires, it abandons the queue and the partial service it may have received up to that point is wasted. When the queue operates under a first-come-first served discipline, we establish conditions under which the actual workload process—that is, the work the server eventually processes—is unstable, weakly stable, and strongly stable. An interesting phenomenon observed is that in a nontrivial portion of the parameter space, the queue is weakly stable, but not strongly stable. We also indicate how our results apply to other nonidling service disciplines. We finally extend the results for a single node to acyclic (feed-forward) networks of queues with either per-queue or network-wide deadlines.

We consider polling systems with mixtures of exhaustive and gated service in which the server visits the queues periodically according to a general polling table. We derive exact expressions for the steady-state delay incurred at each of the queues under standard heavy-traffic scalings. The expressions require the solution of a set of only M—N linear equations, where M is the length of the polling table and N is the number of queues, but are otherwise explicit. The equations can even be expressed in closed form for several routeing schemes commonly used in practice, such as the star and elevator visit order, in a general parameter setting. The results reveal a number of asymptotic properties of the behavior of polling systems. In addition, the results lead to simple and fast approximations for the distributions and the moments of the delay in stable polling systems with periodic server routeing. Numerical results demonstrate that the approximations are highly accurate for medium and heavily loaded systems.

This paper is a first study of two-person zero-sum games for denumerable continuous-time Markov chains determined by given transition rates, with an average payoff criterion. The transition rates are allowed to be unbounded, and the payoff rates may have neither upper nor lower bounds. In the spirit of the ‘drift and monotonicity’ conditions for continuous-time Markov processes, we give conditions on the controlled system's primitive data under which the existence of the value of the game and a pair of strong optimal stationary strategies is ensured by using the Shapley equations. Also, we present a ‘martingale characterization’ of a pair of strong optimal stationary strategies. Our results are illustrated with a birth-and-death game.

Consider a sequence of outcomes from Markov dependent two-state (success-failure) trials. In this paper, the exact distributions are derived for three longest-run statistics: the longest failure run, longest success run, and the maximum of the two. The method of finite Markov chain imbedding is used to obtain these exact distributions, and their bounds and large deviation approximation are also studied. Numerical comparisons among the exact distributions, bounds, and approximations are provided to illustrate the theoretical results. With some modifications, we show that the results can be easily extended to Markov dependent multistate trials.

We consider the following ordering for stochastic processes as introduced by Irle and Gani (2001). A process (Yt)t is said to be slower in level crossing than a process (Zt)t if it takes (Yt)t stochastically longer than (Zt)t to exceed any given level. In Irle and Gani (2001), this ordering was investigated for Markov chains in discrete time. Here these results are carried over to semi-Markov processes with particular attention to birth-and-death processes and also to Wiener processes.

We show how good multivariate Poisson mixtures can be approximated by multivariate Poisson distributions and related finite signed measures. Upper bounds for the total variation distance with applications to risk theory and generalized negative multinomial distributions are given. Furthermore, it turns out that the ideas used in this paper also lead to improvements in the Poisson approximation of generalized multinomial distributions.

In this paper, we propose a customer-arrival-based insurance risk model, in which customers' potential claims are described as independent and identically distributed heavy-tailed random variables and premiums are the same for each policy. We obtain some precise large deviation results for the prospective-loss process under a mild assumption on the random index (in our case, the customer-arrival process), which is much weaker than that in the literature.

The aim of this paper is to study the distribution of colours, {Xn}, in a generalized Pólya urn model with L colours, an urn function and a random environment. In this setting, the number of actions to be taken can be greater than L, and the total number of balls added in each step can be random. The process {Xn} is expressed as a stochastic recurrent equation that fits a Robbins—Monro scheme. Since this process evolves in the (L—1)-simplex, the stability of the solutions of the ordinary differential equation associated with the Robbins—Monro scheme can be studied by means of differential algebraic equations. This approach provides a method of obtaining strong laws for the process {Xn}.

The aim of this paper is to investigate discrete approximations of the exponential functional of Brownian motion (which plays an important role in Asian options of financial mathematics) with the help of simple, symmetric random walks. In some applications the discrete model could be even more natural than the continuous one. The properties of the discrete exponential functional are rather different from the continuous one: typically its distribution is singular with respect to Lebesgue measure, all of its positive integer moments are finite and they characterize the distribution. On the other hand, using suitable random walk approximations to Brownian motion, the resulting discrete exponential functionals converge almost surely to the exponential functional of Brownian motion; hence their limit distribution is the same as in the continuous case, namely that of the reciprocal of a gamma random variable, and so is absolutely continuous with respect to Lebesgue measure. In this way, we also give a new and elementary proof of an earlier result by Dufresne and Yor.

Using random walk theory, we first establish explicitly the exact distribution of the maximal partial sum of a sequence of independent and identically distributed random variables. This result allows us to obtain a new approximation of the distribution of the local score of one sequence. This approximation improves the one given by Karlin et al., which can be deduced from this new formula. We obtain a more accurate asymptotic expression with additional terms. Examples of application are given.

We study linear jump parameter systems of differential and difference equations whose coefficients depend on the state of a semi-Markov process. We derive systems of equations for the first two moments of the random solutions of these jump parameter systems, and illustrate how moment equations can be used in examining their asymptotic stability.

We study the uniform ergodicity of Markov processes (Zn, n ≥ 1) of order 2 with a general state space (Z, 𝒵). Markov processes of order higher than 1 were defined in the literature long ago, but scarcely treated in detail. We take as the basis for our considerations the natural transition probability Q of such a process. A Markov process of order 2 is transformed into one of order 1 by combining two consecutive variables Z2n–1 and Z2n into one variable Yn with values in the Cartesian product space (Z × Z, 𝒵 ⊗ 𝒵). Thus, a Markov process (Yn, n ≥ 1) of order 1 with transition probability R is generated. Uniform ergodicity for the process (Zn, n ≥ 1) is defined in terms of the same property for (Yn, n ≥ 1). We give some conditions on the transition probability Q which transfer to R and thus ensure the uniform ergodicity of (Zn, n ≥ 1). We apply the general results to study the uniform ergodicity of Markov processes of order 2 which arise in some nonlinear time series models and as sequences of smoothed values in sequential smoothing procedures of Markovian observations. As for the time series models, Markovian noise sequences are covered.

Block, Savits and Singh previously presented a preservation result for bathtub-shaped functions. In this paper, we strengthen and extend their results to a class of functions which allow for more than one change of monotonicity. Various applications are also considered.

Populations of specific components are often heterogeneous and consist of a small number of different sub-populations. For example there are often two groups: defective components with shorter lifetimes and standard components with longer lifetimes. Another heterogeneous population results when components produced by two different manufacturing lines are combined. In either case a mixture results. The resulting population can be described using the statistical concept of a mixture. It is a well-known result that a mixture of distributions with decreasing failure rates has a decreasing failure rate. However, little is known about the monotonicity of a mixture when the various subpopulations have failure rates which are not necessarily decreasing. In this paper we study and attempt to determine the shape as well as the overall behavior of the failure rate of a mixture from two subpopulations each of which has increasing linear failure rate.

We prove a result concerning the joint distribution of alleles at linked loci on a chromosome drawn from the population at stationarity. For a neutral locus, the allele is a draw from the stationary distribution of the mutation process. Furthermore, this allele is independent of the alleles at different loci on any chromosomes in the population.

Consider the classical coupon-collector's problem in which items of m distinct types arrive in sequence. An arriving item is installed in system i ≥ 1 if i is the smallest index such that system i does not contain an item of the arrival's type. We study the expected number of items in system j at the moment when system 1 first contains an item of each type

We consider the policy in a finite dam in which the input of water is formed by a compound Poisson process and the rate of water release is changed instantaneously from a to M and from M to a (M > a) at the moments when the level of water exceeds λ and downcrosses τ (λ > τ) respectively. After assigning costs to the changes of release rate, a reward to each unit of output, and a cost related to the level of water in the reservoir, we determine the long-run average cost per unit time.