A class of non-negative alternating regenerative processes is considered, where the process stays at zero random time (waiting period), then it jumps to a random positive level and hits zero after some random period (life period), depending on the evolution of the process. It is assumed that the waiting time and the lifetime belong to the domain of attraction of stable laws with parameters in the interval (½,1]. An integral representation for the distribution functions of the regenerative process is obtained, using the spent time distributions of the corresponding alternating renewal process. Given the asymptotic behaviour of the process in the regenerative cycle, different types of limiting distributions are proved, applying some new results for the corresponding renewal process and two limit theorems for the convergence in distribution.

The aim of this paper is to study genealogical processes in a geographically structured population with weak migration. The coalescence time for sampled genes from different colonies diverges to infinity as the migration rates among colonies are close to zero. We investigate the moment generating functions of the coalescence time, the number of segregating sites and the number of allele types in sampled genes when there is low migration. Employing a perturbation method, we obtain a system of recurrence relations for the approximate solutions of these moment generating functions and solve them in some cases.

We consider a repairable system with a finite state space which evolves in time according to a Markov process as long as it is working. We assume that this system is getting worse and worse while running: if the up-states are ranked according to their degree of increasing degradation, this is expressed by the fact that the Markov process is assumed to be monotone with respect to the reversed hazard rate and to have an upper triangular generator. We study this kind of process and apply the results to derive some properties of the stationary availability of the system. Namely, we show that, if the duration of the repair is independent of its completeness degree, then the more complete the repair, the higher the stationary availability, where the completeness degree of the repair is measured with the reversed hazard rate ordering.

This paper presents an algorithmic analysis of the busy period for the M/M/c queueing system. By setting the busy period equal to the time interval during which at least one server is busy, we develop a first step analysis which gives the Laplace-Stieltjes transform of the busy period as the solution of a finite system of linear equations. This approach is useful in obtaining a suitable recursive procedure for computing the moments of the length of a busy period and the number of customers served during it. The maximum entropy formalism is then used to analyse what is the influence of a given set of moments on the distribution of the busy period and to estimate the true busy period distribution. Our study supplements a recent work of Daley and Servi (1998) and other studies where the busy period of a multiserver queue follows a different definition, i.e., a busy period is the time interval during which all servers are engaged.

Some new results about the NBU(2) class of life distributions are obtained. Firstly, it is proved that the decreasing with time of the increasing concave ordering of the excess lifetime in a renewal process leads to the NBU(2) property of the interarrival times. Secondly, the NBU(2) class of life distributions is proved to be closed under the formation of series systems. Finally, it is also shown that the NBU(2) class is closed under convolution operation.

We consider a real-valued random walk which drifts to -∞ and is such that the step distribution is heavy tailed, say, subexponential. We investigate the asymptotic tail behaviour of the distribution of the upwards first passage times. As an application, we obtain the exact rate of convergence for the ruin probability in finite time. Our result supplements similar theorems in risk theory.

An annihilating process is an interacting particle system in which the only interaction is that a particle may kill a neighbouring particle. Since there is no birth and no movement, once a particle has no neighbours its site remains occupied for ever. It is shown that with initial configuration ℤ the distribution of particles at all times is a renewal process and that the probability that a site remains occupied for all time tends to 1/e. Time-dependent behaviour is also calculated for the tree 𝕋r .

This paper studies the geometric convergence rate of a discrete renewal sequence to its limit. A general convergence rate is first derived from the hazard rates of the renewal lifetimes. This result is used to extract a good convergence rate when the lifetimes are ordered in the sense of new better than used or increasing hazard rate. A bound for the best possible geometric convergence rate is derived for lifetimes having a finite support. Examples demonstrating the utility and sharpness of the results are presented. Several of the examples study convergence rates for Markov chains.

In this paper, we study a maintenance model with general repair and two types of replacement: failure and preventive replacement. When the system fails a decision is made whether to replace or repair it. The repair degree that affects the virtual age of the system is assumed to be a random function of the repair-cost and the virtual age at failure time. The system can be preventively replaced at any time before failure. The objective is to find the repair/replacement policy minimizing the long-run expected average cost per unit time. It is shown that a generalized repair-cost-limit policy is optimal and the preventive replacement time depends on the virtual age of the system and on the length of the operating time since the last repair. Computational procedures for finding the optimal repair-cost limit and the optimal average cost are developed. This model includes many well-known models as special cases and the approach provides a unified treatment of a wide class of maintenance models.

The present paper provides simple inequalities for the number of lost customers during a busy period of a GI/M/1/n queueing system.

The paper studies a single-server two-queue priority system with changeover times and switching threshold. The server serves queue 1 exhaustively and does not remain at an empty queue if the other one is non-empty. It immediately switches from queue 2 to queue 1 when the length of the latter reaches some level M. Whenever service is changed from one queue to the other a changeover time is required. Arrivals are Poisson, service times and changeover times are independent and exponentially distributed. Using an analytic method we obtain the steady-state joint probability generating function of the lengths of the two queues. By means of this probability generating function some performance measures of the system such as mean length of queue and mean delay can be calculated.

In first-passage percolation models, the passage time T(0,L) from the origin to a point L is expected to exhibit deviations of order |L|χ from its mean, while minimizing paths are expected to exhibit fluctuations of order |L|ξ away from the straight line segment . Here, for Euclidean models in dimension d, we establish the lower bounds ξ ≥ 1/(d+1) and χ ≥(1-(d-1)ξ)/2. Combining this latter bound with the known upper bound ξ ≤ 3/4 yields that χ ≥ 1/8 for d=2.

In this paper, we investigate how fast the stationary distribution π (K) of an embedded Markov chain (time-stationary distribution q (K) of the GI/M/1/K queue converges to the stationary distribution π of the embedded Markov chain (time-stationary distribution q ) of the GI/M/1 queue as K tends to infinity. Simonot (1997) proved certain equalities. We obtain sharper results than these by finding limit values limK→∞σ-K ||π(K) - π|| and limK→∞σ-K ||q (K) - q || explicitly.

It is common practice to approximate the cell loss probability (CLP) of cells entering a finite buffer by the overflow probability (OVFL) of a corresponding infinite buffer queue, since the CLP is typically harder to estimate. We obtain exact asymptotic results for CLP and OVFL for time-slotted queues where block arrivals in different time slots are i.i.d. and one cell is served per time slot. In this case the ratio of CLP to OVFL is asymptotically (1-ρ)/ρ, where ρ is the use or, equivalently, the mean arrival rate per time slot. Analogous asymptotic results are obtained for continuous time M/G/1 queues. In this case the ratio of CLP to OVFL is asymptotically 1-ρ.

This paper presents the large-buffer asymptotics for a multiplexer which serves N types of heterogeneous sessions which have long-tailed session lengths. Specifically, the model considered is that sessions of type i ∈ {1,…,N} arrive as a Poisson process with rate λi . Each type of session (independently) remains active for a random duration, say τi , where P(τi > x) ~ αi x -(1 + β i) for positive numbers αi and βi . While active, a session transmits at a rate r i . Under the assumption that the average load ρ = ∑N i=1 r i λi E[τi ] < C, where C denotes the server capacity, we show that both the tail distribution of the stationary buffer content and the loss asymptotics in finite buffers of size z behave approximately as z -κ J 0 , where κJ0 depends not only on the βi but also on the transmission rates r i ; it is the ratio of βi to r i which determines κJ0 . When specialized to the homogeneous case, i.e., when r i =r and βi = β for all i, the result coincides with results reported in the literature which have been shown under more restrictive hypotheses. Finally, it is a simple observation that light-tailed sessions only have the effect of reducing the available capacity for long-tailed sessions, but do not contribute otherwise to the definition of κJ0 .

For continuous-time Markov chains with semigroups P, P' taking values in a partially ordered set, such that P ≤ stP', we show the existence of an order-preserving Markovian coupling and give a way to construct it. From our proof, we also obtain the conditions of Brandt and Last for stochastic domination in terms of the associated intensity matrices. Our result is applied to get necessary and sufficient conditions for the existence of Markovian couplings between two Jackson networks.

A new burn-in procedure for a repairable component is proposed. During a burn-in period, the failed component is only minimally repaired rather than being completely repaired. This procedure is shown to be economical and efficient when the minimal repair method is applicable during a burn-in process. The properties of the optimal burn-in time b ∗ and block replacement policy T ∗ are also given.

Although the M/D/1/N queueing model is well solved from a computational point of view, there is no known analytical expression of the queue length distribution. In this paper, we derive closed-form formulae for the distribution of the number of customers in the system in the finite-capacity M/D/1 queue. We also give an explicit solution for the mean queue length and the average waiting time.

We consider a fluid queue fed by the superposition of n homogeneous on-off sources with generally distributed on and off periods. The buffer space B and link rate C are scaled by n, so that we get nb and nc, respectively. Then we let n grow large. In this regime, the overflow probability decays exponentially in the number of sources n. We specifically examine the scenario where b is also large. We obtain explicit asymptotics for the case where the on periods have a subexponential distribution, e.g., Pareto, Lognormal, or Weibull.

The results show a sharp dichotomy in the qualitative behavior, depending on the shape of the function v(t) := - logP(A* > t) for large t, A* representing the residual on period. If v(.) is regularly varying of index 0 (e.g., Pareto, Lognormal), then, during the path to overflow, the input rate will only slightly exceed the link rate. Consequently, the buffer will fill ‘slowly’, and the typical time to overflow will be ‘more than linear’ in the buffer size. In contrast, if v(.) is regularly varying of index strictly between 0 and 1 (e.g., Weibull), then the input rate will significantly exceed the link rate, and the time to overflow is roughly proportional to the buffer size.

In both cases there is a substantial fraction of the sources that remain in the on state during the entire path to overflow, while the others contribute at their mean rates. These observations lead to approximations for the overflow probability. The approximations may be extended to the case of heterogeneous sources. The results provide further insight into the so-called reduced-load approximation.

New bounds are found for the reliability of consecutive-k-out-of-n:F systems with equal component failure probabilities. The expressions involved are simple, thus allowing a direct use in the derivation of theoretical properties.

These bounds can also be employed in numerical computations when the value of n or k is so large that the exact calculation of the reliability is not achievable. Comparisons show that the approximation errors exhibited by these new formulas are lower than those of other widely used bounds.