This paper is devoted to a study of the integral of the workload process of the single server queue, in particular during one busy period. Firstly, we find asymptotics of the area 𝒜 swept under the workload process W(t) during the busy period when the service time distribution has a regularly varying tail. We also investigate the case of a light-tailed service time distribution. Secondly, we consider the problem of obtaining an explicit expression for the distribution of 𝒜. In the general GI/G/1 case, we use a sequential approximation to find the Laplace—Stieltjes transform of 𝒜. In the M/M/1 case, this transform is obtained explicitly in terms of Whittaker functions. Thirdly, we consider moments of 𝒜 in the GI/G/1 queue. Finally, we show asymptotic normality of .

An object is hidden in one of two boxes and occasionally moves between the boxes in accordance with some specified continuous-time Markov process. The objective is to find the object with a minimal expected cost. In this paper it is assumed that search efforts are unlimited. In addition to the search costs, the ‘real time' until the object is found is also taken into account in the cost structure. Our main results are that the optimal policy may consist of five regions and that the controls applied should be of the extreme 0 or ∞ type. The resulting expected cost compares favorably with that of the expected cost with bounded controls studied previously in the search literature.

This work is concerned with a class of mean-field models given by a switching diffusion with a continuous-state-dependent switching process. Focusing on asymptotic properties, the regularity or nonexplosiveness, Feller continuity, and strong Feller continuity are established by means of introducing certain auxiliary processes and by making use of the truncations. Based on these results, exponential ergodicity is obtained under the Foster–Lyapunov drift conditions. By virtue of the coupling methods, the strong ergodicity or uniform ergodicity in the sense of convergence in the variation norm is established for the mean-field model with a Markovian switching process. Besides this, several examples are presented for demonstration and illustration.

We consider a generalized telegraph process which follows an alternating renewal process and is subject to random jumps. More specifically, consider a particle at the origin of the real line at time t=0. Then it goes along two alternating velocities with opposite directions, and performs a random jump toward the alternating direction at each velocity reversal. We develop the distribution of the location of the particle at an arbitrary fixed time t, and study this distribution under the assumption of exponentially distributed alternating random times. The cases of jumps having exponential distributions with constant rates and with linearly increasing rates are treated in detail.

The performance and effectiveness of an age replacement policy can be assessed by its mean time to failure (MTTF) function. We develop shock model theory in different scenarios for classes of life distributions based on the MTTF function where the probabilities $\bar{P}_k$ of surviving the first k shocks are assumed to have discrete DMTTF, IMTTF and IDMTTF properties. The cumulative damage model of A-Hameed and Proschan [1] is studied in this context and analogous results are established. Weak convergence and moment convergence issues within the IDMTTF class of life distributions are explored. The preservation of the IDMTTF property under some basic reliability operations is also investigated. Finally we show that the intersection of IDMRL and IDMTTF classes contains the BFR family and establish results outlining the positions of various non-monotonic ageing classes in the hierarchy.

Let (A1A2, · ··, An) be a set of n events on a probability space. Let be the sum of the probabilities of all intersections of r events, and Mn the number of events in the set which occur. The classical Bonferroni inequalities provide upper and lower bounds for the probabilities P(Mn = m), and equal to partial sums of series of the form which give the exact probabilities. These inequalities have recently been extended by J. Galambos to give sharper bounds.

Here we present straightforward proofs of the Bonferroni inequalities, using indicator functions, and show how they lead naturally to new simple proofs of the Galambos inequalities.

Let X1, X2, · ·· be i.i.d. and Yk = max{X1, X2, · ··, Xk}. Then Xj is a record value of the sequence if Yj > Yj-1). Denoting record value times by Ln, n = 1, 2, ···, we prove that the independence of XLj − XLi and XLi with Lj > characterizes the exponential distribution.

We study the scaling limit of random fields which are solutions of a non-linear partial differential equation, known as the Burgers equation, under stochastic initial conditions. These are assumed to be of a non-local shot noise type and driven by a Cox process. Previous work by Bulinskii and Molchanov (1991), Surgailis and Woyczynski (1993a), and Funaki et al. (1994) concentrated on the case of local shot noise data which permitted use of techniques from the theory of random fields with finite range dependence. Those are not available for the non-local case being considered in this paper.

Burgers' equation is known to describe various physical phenomena such as non-linear and shock waves, distribution of self-gravitating matter in the universe, and other flow satisfying conservation laws (see e.g. Woyczynski (1993)).

Under certain conditions it is shown that for jobs with increasing completion rates the expected flow time is minimised by a non-pre-emptive service discipline.

For truncated birth-and-death processes with two absorbing or two reflecting boundaries, necessary and sufficient conditions on the transition rates are given such that the transition probabilities satisfy a suitable spatial symmetry relation. This allows one to obtain simple expressions for first-passage-time densities and for certain avoiding transition probabilities. An application to an M/M/1 queueing system with two finite sequential queueing rooms of equal sizes is finally provided.

Let G = (V, E) be a graph with N vertices and A a set of N real numbers. Then a one-to-one mapping ϕ: V → A is called a numbering of G. The elements of A will always be assumed ordered a1 ≦ a2 ≦ … ≦ aN. In [3] and [7] it was shown how to construct numberings of the n-cube, in fact all numberings, which minimize Σe ∈ E Δe where ∆e = |ϕ(v) – ϕ(w)| and e is the edge between vertices v and w. A variant problem of considerable interest is to do the same for Σe ∈ E (Δe)2. It is conjectured that when A = {1,2, …,2n} the natural numbering is the unique minimizer of Σe ∈ E (Δe)2 for every n, but this has so far resisted all efforts. Theorem 1 in the following is the result of attempts to get weaker results, namely to thin out the ranks of those numberings which could possibly minimize Σe ∈ E (Δe)2. The second problem posed and solved in this paper is a generalization of the results in [3], where the n-cube becomes the n-torus.

Estimation of the offspring distribution from a single realization of a supercritical Galton-Watson process is studied. It is shown that, based on population totals, a parameter of the offspring distribution cannot be estimated unless it is determined by the mean, variance, lattice size and lattice offset of the offspring distribution.

In the storage model proposed by Kendall [6], the input is assumed to be a stochastic process with stationary independent increments, and the release is continuous and is at a unit rate. The storage process arising from this model is known to reach a steady state if the expected net input per unit time is negative. In this paper we consider the situation where this expected net input is nonnegative and derive limiting distributions for the wet period, total dry period during a given time interval and the dam content.

Let where {Zn, ℱn} is a non-negative submartingale satisfying Ei(Zn log Zn) →∞. It is shown that When {Zn} is a simple critical Galton–Watson process and is slowly varying at∞, conditions are given ensuring that This gives an alternative proof of a result recently established by Kammerle and Schuh.

We consider adaptive importance sampling for a Markov chain with scoring. It is shown that convergence to the zero-variance importance sampling chain for the mean total score occurs exponentially fast under general conditions. These results extend previous work in Kollman (1993) and in Kollman et al. (1999) for finite state spaces.

A series of bounds for the availability and unavailability in a fixed time interval, I, for a system of maintained, interdependent components are given in Natvig (1980) in the traditional binary case, and in Funnemark and Natvig (1985) in the multistate case. For the special case of independent components the only assumption needed to arrive at these bounds is that the marginal performance process of each component is associated in I. When these processes are Markovian and binary, a sufficient condition for this to hold is given by Esary and Proschan (1970). In the present paper we generalize this condition to the multistate case, and give an equivalent and much more convenient condition in terms of the transition intensities.

For a K-stage cyclic queueing network with N customers and general service times, we provide bounds on the nth departure time from each stage. Furthermore, we analyze the asymptotic tail behavior of cycle times and waiting times given that at least one service-time distribution is subexponential.

Gibbs fields are constructed and studied which correspond to systems of real-valued spins (e.g. systems of interacting anharmonic oscillators) indexed by the vertices of unbounded degree graphs of a certain type, for which the Gaussian Gibbs fields need not be existing. In these graphs, the vertex degree growth is controlled by a summability requirement formulated with the help of a generalized Randić index. In particular, it is proven that the Gibbs fields obey uniform integrability estimates, which are then used in the study of the topological properties of the set of Gibbs fields. In the second part, a class of graphs is introduced in which the mentioned summability is obtained by assuming that the vertices of large degree are located at large distances from each other. This is a stronger version of the metric property employed in Bassalygo and Dobrushin (1986).

This paper aims to show how certain known martingales for epidemic models may be derived using general techniques from the theory of stochastic integration, and hence to extend the allowable infection and removal rate functions of the model as far as possible. Denoting by x, y the numbers of susceptible and infective individuals in the population, then we assume that new infections occur at rate βxyxy and infectives are removed at rate γxyy, where the ratio βxy / γxy can be written in the form q(x+y) / xp(x) for appropriate functions p,q. Under this condition, we find equations giving the distribution of the number of susceptibles remaining in the population at appropriately defined stopping times. Using results on Abel–Gontcharoff pseudopolynomials we also derive an expression for the expectation of any function of the number of susceptibles at these times, as well as considering certain integrals over the course of the epidemic. Finally, some simple examples are given to illustrate our results.

We consider some important systems in reliability theory situated in a random environment, where shocks occur and cause component failure in a specific way. We study some appropriate coefficients, which play an important role in the reduction of our systems to a linear combination of parallel subsystems.