The stationary version of the infinitely many neutral alleles diffusion model, as studied by Ethier and Kurtz, is constructed and analyzed using the theory of Dirichlet forms. We prove that the ‘boundary' of the state space is hit by the process iff θ < 1, where θ is the mutation parameter.

The ergodic properties of birth and death processes are studied. We obtain some explicit estimates for the rate of convergence by the methods of theory of differential equations.

If Xi, i = 1, ···, n are independent exponential random variables with parameters λ1, · ··, λ n, and if Yi, i = 1, ···, n are independent exponential random variables with common parameter equal to (λ1 + · ·· + λ n)/n, then there is a monotone coupling of the order statistics X(1), · ··, X(n) and Y(1), · ··, Y(n); that is, it is possible to construct on a common probability space random variables X′i, Y′i, i = 1, ···, n, such that for each i, Y′(i)≦X′(i) a.s., where the law of the X′i (respectively, the Y′i) is the same as the law of the Xi (respectively, the Yi.) This result is due to Proschan and Sethuraman, and independently to Ball. We shall here prove an extension to a more general class of distributions for which the failure rate function r(x) is decreasing, and xr(x) is increasing. This very strong order relation allows comparison of properties of epidemic processes where rates of infection are not uniform with the corresponding properties for the homogeneous case. We further prove that for a sequence Zi, i = 1, ···, n of independent random variables whose failure rates at any time add to 1, the order statistics are stochastically larger than the order statistics of a sample of n independent exponential random variables of mean n, but that the strong monotone coupling referred to above is impossible in general.

Let {Xk} be an i.i.d. sequence taking values in ℝ2 with the radial and spherical components independent and the radial component having a distribution with slowly varying tail. The number of extreme points in the convex hull of {X1, · ··, Xn} is shown to have a limiting distribution which is obtained explicitly. Precise information about the mean and variance of the limit distribution is obtained.

For a given transition rate, i.e., a Q-matrix Q = (qij) on a countable state space, the uniqueness of the Q-semigroup P(t) = (Pij(t)), the recurrence and the positive recurrence of the corresponding Markov chain are three fundamental and classical problems, treated in many textbooks. As an addition, this paper introduces some practical results motivated from the study of a type of interacting particle systems, reaction diffusion processes. The main results are theorems (1.11), (1.17) and (1.18). Their proofs are quite straightforward.

The heaps process (also known as a Tsetlin library) provides a model for a self-regulating filing system. Items are requested from time to time according to their popularity and returned to the top of the heap after use. The size-biased permutation of a collection of popularities is a particular random permutation of those popularities, which arises naturally in a number of applications and is of independent interest. For a slightly non-standard formulation of the heaps process we prove that it converges to the size-biased permutation of its initial distribution. This leads to a number of new characterizations of the property of invariance under size-biased permutation, notably what might be described as invariance under ‘partial size-biasing' of any order. Finally we consider in detail the heaps process with Poisson–Dirichlet initial distribution, exhibiting the tractable nature of its equilibrium distribution and explicitly calculating a number of quantities of interest.

In this paper we consider the Markov chain formed by the operation of the move-to-front scheme. We show that the eigenvalues of the transition probability matrix are of the form pi, pi + pj, ···, where pi is the probability of selecting the ith item and N is the number of items; further, that the multiplicity of the eigenvalues of the form Σpi where the summation is over m items is equal to the number of permutations of N – m objects, ordered in some way, such that no object is in its natural position. Finally, we show that the Markov chain is lumpable – many times over.

We find the first-hit distributions of symmetric randomized random walks on ℝ with exponential lifetime using prediction formulas for Gaussian stationary autoregressive processes, associated with the random walks.

A multivariate reward process defined on a semi-Markov process is studied. Transform results for the distributions of the multivariate reward and related processes are derived through the method of supplementary variables and the Markov renewal equations. These transform results enable the asymptotic behavior to be analyzed. A class of first-passage time distributions of the multivariate reward processes is also investigated.

We investigate a generalized variability ordering and its weaker versions among non-negative random variables (lifetimes of components). Our results include a necessary and sufficient condition which justifies the generalized variability interpretation of this dominance relation between life distributions, relationships to some weakly aging classes in reliability theory, closure properties and inequalities for the mean life of series and parallel systems under such ordering.

In this study we examine repairable systems with random lifetime. Upon failure, a maintenance action, specifying the degree of repair, is taken by a controller. The objective is to determine an age-dependent maintenance strategy which minimizes the total expected discounted cost over an infinite planning horizon. Using several properties of the optimal policy which are derived in this study, we propose analytical and numerical methods for determining the optimal maintenance strategy. In order to obtain a better insight regarding the structure and nature of the optimal policy and to illustrate computational procedures, a numerical example is analysed. The proposed maintenance model outlines a new research channel in the area of reliability with interesting theoretical issues and a wide range of potential applications in various fields such as product design, inventory systems for spare parts, and management of maintenance crews.

We obtain the equilibrium solutions of an R-out-of-N system subject to random breakdown. There are M spares and a single repairman, who instals good spares into the system when breakdowns occur and also repairs the failed items. Installation has pre-emptive repeat priority over repairs. Arbitrary distributions of installation and repair times are allowed. Equilibrium availability and downtime distributions are obtained from the equilibrium state distribution.

Product-form equilibrium distributions in networks of queues in which customers move singly have been known since 1957, when Jackson derived some surprising independence results. A product-form equilibrium distribution has also recently been shown to be valid for certain queueing networks with batch arrivals, batch services and even correlated routing.

This paper derives the joint equilibrium distribution of states immediately before and after a batch of customers is released into the network. The results are valid for either discrete- or continuous-time queueing networks: previously obtained results can be seen as marginal distributions within a more general framework. A generalisation of the classical ‘arrival theorem' for continuous-time networks is given, which compares the equilibrium distribution as seen by arrivals to the time-averaged equilibrium distribution.

We study sojourn times in M/G/1 multiple vacation systems and multiqueue cyclic-service (polling) systems with instantaneous Bernoulli feedback. Three service disciplines, exhaustive, gated, and 1-limited, are considered for both M/G/1 vacation and polling systems. The Laplace-Stieltjes transforms of the sojourn time distributions in the three vacation systems are derived. For polling systems, we provide explicit expressions for the mean sojourn times in symmetric cases. Furthermore a pseudo-conservation law with respect to the mean sojourn times is derived for a polling system with a mixture of the three service disciplines.

The attained waiting time of customers in service of the G/G/1 queue is compared for various work-conserving service disciplines. It is proved that the attained waiting time distribution is minimized (maximized) in convex order when the discipline is FCFS (PR-LCFS). We apply the result to characterize finiteness of moments of the attained waiting time in the GI/GI/1 queue with an arbitrary work-conserving service discipline. In this discussion, some interesting relationships are obtained for a PR-LCFS queue.

This paper deals with the stationary distribution of the virtual waiting time, i.e. the time between the arrival and the beginning of service of a customer in a single-server queue that operates as follows. If the server is busy at an arrival time, the customer is rejected. This customer attempts service again after some random delay and continues to do so until the first time at which the server is idle. At this time, the customer is served and leaves the system after service completion. Interarrival times and delays are assumed to be two independent sequences of i.i.d. exponentially distributed random variables. Service times are also i.i.d., generally distributed, and independent of the previous sequences.

Semi-Pareto processes, of which Pareto processes form a proper sub-class, are discussed here. A semi-Pareto process has semi-Pareto inputs. Asymptotic properties of the maximum and minimum of the first n observations are examined as well as the geometric maximum and geometric minimum. A characterization of the semi-Pareto distribution is given. A canonical representation of a special class of Pareto process is also given.

We consider a monotone case of the non-zero-sum stopping game with discrete time parameter which is called the Dynkin problem. Marner (1987) has investigated a stopping game with general monotone reward structures, but his monotonicity is too strong to apply our problem. We establish that there exists an explicit equilibrium point in our monotone case. We also give a simple example applicable to a duopolistic exit game.

A new class of life distributions, namely new better than used in convex ordering (NBUC), and its dual, new worse than used in convex ordering (NWUC), are introduced. Their relations to other classes of life distributions, closure properties under three reliability operations, and heritage properties under shock model and Laplace-Stieltjes transform are discussed.