We consider ageing properties of a general repair process. Under certain assumptions we prove that the expectation of an age at the beginning of the next cycle in this process is smaller than the initial age of the previous cycle. Using this reasoning, we show that the sequence of random ages at the start (end) of consecutive cycles is stochastically increasing and is converging to a limiting distribution. We discuss possible applications and interpretations of our results.

In this paper we present counter-intuitive examples for the multiclass queueing network, where each station may serve more than one job class with differentiated service priority and each job may require service sequentially by more than one service station. In our examples, the network performance is improved even when more jobs are admitted for service.

We introduce and study a new model: zero-automatic queues. Roughly, zero-automatic queues are characterized by a special buffering mechanism evolving like a random walk on some infinite group or monoid. The salient result is that all stable zero-automatic queues have a product form stationary distribution and a Poisson output process. When considering the two simplest and extremal cases of zero-automatic queues, we recover the simple M/M/1 queue and Gelenbe's G-queue with positive and negative customers.

In this paper telegraph processes on geodesic lines of the Poincaré half-space and Poincaré disk are introduced and the behavior of their hyperbolic distances examined. Explicit distributions of the processes are obtained and the related governing equations derived. By means of the processes on geodesic lines, planar random motions (with independent components) in the Poincaré half-space and disk are defined and their hyperbolic random distances studied. The limiting case of one-dimensional and planar motions together with their hyperbolic distances is discussed with the aim of establishing connections with the well-known stochastic representations of hyperbolic Brownian motion. Extensions of motions with finite velocity to the three-dimensional space are also hinted at, in the final section.

Assume that the surplus of an insurer follows a compound Poisson surplus process. When the surplus is below zero or the insurer is on deficit, the insurer could borrow money at a debit interest rate to pay claims. Meanwhile, the insurer will repay the debts from her premium income. The negative surplus may return to a positive level. However, when the negative surplus is below a certain critical level, the surplus is no longer able to be positive. Absolute ruin occurs at this moment. In this paper, we study absolute ruin questions by defining an expected discounted penalty function at absolute ruin. The function includes the absolute ruin probability, the Laplace transform of the time to absolute ruin, the deficit at absolute ruin, the surplus just before absolute ruin, and many other quantities related to absolute ruin. First, we derive a system of integro-differential equations satisfied by the function and obtain a defective renewal equation that links the integro-differential equations in the system. Second, we show that when the initial surplus goes to infinity, the absolute ruin probability and the classical ruin probability are asymptotically equal for heavy-tailed claims while the ratio of the absolute ruin probability to the classical ruin probability goes to a positive constant that is less than one for light-tailed claims. Finally, we give explicit expressions for the function for exponential claims.

Computer analysis of biological sequences often detects deviations from a random model. In the usual model, sequence letters are chosen independently, according to some fixed distribution over the relevant alphabet. Real biological sequences often contain simple repeats, however, which can be broadly characterized as multiple contiguous copies (usually inexact) of a specific word. This paper quantifies inexact simple repeats as local sums in a Markov additive process (MAP). The maximum of the local sums has an asymptotic distribution with two parameters (λ and k), which are given by general MAP formulas. The general MAP formulas are usually computationally intractable, but an essential simplification in the case of repeats permits λ and k to be computed from matrices whose dimension equals the size of the relevant alphabet. The simplification applies to some MAPs where the summand distributions do not depend on consecutive pairs of Markov states as usual, but on pairs with a fixed time-lag larger than one.

In the classical risk model with initial capital u, let τ(u) be the time of ruin, X+(u) be the risk reserve just before ruin, and Y+(u) be the deficit at ruin. Gerber and Shiu (1998) defined the function mδ(u) =E[e−δ τ(u)w(X+(u), Y+(u)) 1 (τ(u) < ∞)], where δ ≥ 0 can be interpreted as a force of interest and w(r,s) as a penalty function, meaning that mδ(u) is the expected discounted penalty payable at ruin. This function is known to satisfy a defective renewal equation, but easy explicit formulae for mδ(u) are only available for certain special cases for the claim size distribution. Approximations thus arise by approximating the desired mδ(u) by that associated with one of these special cases. In this paper a functional approach is taken, giving rise to first-order correction terms for the above approximations.

We consider a constant rate traffic which shares a buffer with a random cross traffic. A first come first served or priority service discipline is applied at the buffer. After service at the first buffer the constant rate traffic moves to a play-out buffer. Both buffers provide service at constant rate and infinite waiting room. We investigate logarithmic large and moderate deviation asymptotics for the tail probabilities of the steady-state queue length distribution at the play-out buffer for long-range dependent cross traffic in critical loading. We characterize the asymptotic behavior of the cross traffic which leads to a large queue length at the play-out buffer and compare it to the one for renewal cross traffic.

A network belongs to the monotone separable class if its state variables are homogeneous and monotone functions of the epochs of the arrival process. This framework contains several classical queueing network models, including generalized Jackson networks, max-plus networks, polling systems, multiserver queues, and various classes of stochastic Petri nets. We use comparison relationships between networks of this class with independent and identically distributed driving sequences and the GI/GI/1/1 queue to obtain the tail asymptotics of the stationary maximal dater under light-tailed assumptions for service times. The exponential rate of decay is given as a function of a logarithmic moment generating function. We exemplify an explicit computation of this rate for the case of queues in tandem under various stochastic assumptions.

In this paper we study reliability properties of consecutive-k-out-of-n systems with exchangeable components. For 2k ≦ n, we show that the reliability functions of these systems can be written as negative mixtures (i.e. mixtures with some negative weights) of two series (or parallel) systems. Some monotonicity and asymptotic properties for the mean residual lifetime function are obtained and some ordering properties between these systems are established. We prove that, under some assumptions, the mean residual lifetime function of the consecutive-k-out-of-n: G system (i.e. a system that functions if and only if at least k consecutive components function) is asymptotically equivalent to that of a series system with k components. When the components are independent and identically distributed, we show that consecutive-k-out-of-n systems are ordered in the likelihood ratio order and, hence, in the mean residual lifetime order, for 2k ≦ n. However, we show that this is not necessarily true when the components are dependent.

We show that there exist symmetric properties in the discrete n-cube whose threshold widths range asymptotically between 1/√n and 1/logn. These properties are built using a combination of failure sets arising in reliability theory. This combination of sets is simply called a product. Some general results on the threshold width of the product of two sets A and B in terms of the threshold locations and widths of A and B are provided.

Two cumulative damage models are considered, the inverse gamma process and a composed gamma process. They can be seen as ‘continuous’ analogues of Poisson and compound Poisson processes, respectively. For these models the first passage time distribution functions are derived. Inhomogeneous versions of these processes lead to models closely related to the Weibull failure model. All models show interesting size effects.

Let ξ, ξ1, ξ2,… be a sequence of independent and identically distributed random variables, and let Sn=ξ1+⋯+ξn and Mn=maxk≤nSk. Let τ=min{n≥1: Sn≤0}. We assume that ξ has a heavy-tailed distribution and negative, finite mean E(ξ)<0. We find the asymptotics of P{Mτ ∈ (x, x+T]} as x→∞, for a fixed, positive constant T.

A large dam model is the object of study of this paper. The parameters Llower and Lupper define its lower and upper levels, L = Lupper - Llower is large, and if the current level of water is between these bounds, the dam is assumed to be in a normal state. Passage across one or other of the levels leads to damage. Let J1 and J2 denote the damage costs of crossing the lower and, respectively, the upper levels. It is assumed that the input stream of water is described by a Poisson process, while the output stream is state dependent. Let Lt denote the dam level at time t, and let p1 = limt→∞P{Lt = Llower} and p2 = limt→∞P{Lt > Lupper} exist. The long-run average cost, J = p1J1 + p2J2, is a performance measure. The aim of the paper is to choose the parameter controlling the output stream so as to minimize J.

We give a simple and direct treatment of insensitivity in stochastic networks which is quite general and provides probabilistic insight into the phenomenon. In the case of multi-class networks, our results generalise those of Bonald and Proutière (2002), (2003).

A Euclidean first passage percolation model describing the competing growth between k different types of infection is considered. We focus on the long-time behavior of this multitype growth process and we derive multitype shape results related to its morphology.

Most devices (systems) are operated under different environmental conditions. The failure process of a system not only depends on the intrinsic characteristics of the system itself but also on the external environmental conditions under which the system is being operated. In this paper we study a stochastic failure model in a random environment and investigate the effect of the environmental factors on the failure process of the system.

The ‘square root formula’ in the Internet transmission control protocol (TCP) states that if the probability p of packet loss becomes small and there is independence between packets, then the stationary distribution of the congestion window W is such that the distribution of W√p is almost independent of p and is completely characterizable. This paper gives an elementary proof of the convergence of the stationary distributions for a much wider class of processes that includes classical TCP as well as T. Kelly's ‘scalable TCP’. This paper also gives stochastic dominance results that translate to a rate of convergence.

The distribution theory for reward functions on semi-Markov processes has been of interest since the early 1960s. The relevant asymptotic distribution theory has been satisfactorily developed. On the other hand, it has been noticed that it is difficult to find exact distribution results which lead to the effective computation of such distributions. Note that there is no satisfactory exact distribution result for rewards accumulated over deterministic time intervals [0, t], even in the special case of continuous-time Markov chains. The present paper provides neat general results which lead to explicit closed-form expressions for the relevant Laplace transforms of general reward functions on semi-Markov and Markov additive processes.

The small-world phenomenon, the principle that we are all linked by a short chain of intermediate acquaintances, has been investigated in mathematics and social sciences. It has been shown to be pervasive both in nature and in engineering systems like the World Wide Web. Work of Jon Kleinberg has shown that people, using only local information, are very effective at finding short paths in a network of social contacts. In this paper we argue that the underlying key to finding short paths is scale invariance. In order to appreciate scale invariance we suggest a continuum setting, since true scale invariance happens at all scales, something which cannot be observed in a discrete model. We introduce a random-connection model that is related to continuum percolation, and we prove the existence of a unique scale-free model, among a large class of models, that allows the construction, with high probability, of short paths between pairs of points separated by any distance.