We show that the solution of a quasi-renewal equation with an exponential distribution of the renewals converges at infinity and we compute explicitly the limit, hence generalizing the classical renewal theorem. We apply this result to a stochastic model of DNA replication introduced by Cowan and Chiu (1994).

We consider a network with N infinite-buffer queues with service rates λ, and global task arrival rate Nν. Each task is allocated L queues among N with uniform probability and joins the least loaded one, ties being resolved uniformly. We prove Q-chaoticity on path space for chaotic initial conditions and in equilibrium: any fixed finite subnetwork behaves in the limit N goes to infinity as an i.i.d. system of queues of law Q. The law Q is characterized as the unique solution for a non-linear martingale problem; if the initial conditions are q-chaotic, then Q 0 = q, and in equilibrium Q 0 = q ρ is the globally attractive stable point of the Kolmogorov equation corresponding to the martingale problem. This result is equivalent to a law of large numbers on path space with limit Q, and implies a functional law of large numbers with limit (Q t )t≥0. The significant improvement in buffer utilization, due to the resource pooling coming from the choices, is precisely quantified at the limit.

In various stochastic models the random equation of implicit renewal theory appears where the real random variable S and the stochastic process Ψ with index space and state space R are independent. By use of stochastic approximation the distribution function of S is recursively estimated on the basis of independent or ergodic copies of Ψ. Under integrability assumptions almost sure L 1-convergence is proved. The choice of gains in the recursion is discussed. Applications are given to insurance mathematics (perpetuities) and queueing theory (stationary waiting and queueing times).

In this paper we consider an irreducible random walk in Z + defined by X(m+1) = max(0, X(m) + A(m+1)) with E{A} < 0 and for an s ≥ 0 where a + = max(0,a). Let π be the stationary distribution of X. We show that one can find probability distributions πn supported by {0,n} such that ||πn - π||1 ≤ Cn -s , where the constant C is computable in terms of the moments of A, and also that ||πn - π||1 = o(n -s ). Moreover, this upper bound reveals exact for s ≥ 1, in the sense that, for any positive ε, we can find a random walk fulfilling the above assumptions and for which the relation ||πn - π||1 = o(n -s-ε) does not hold. This result is used to derive the exact convergence rate of the time stationary distribution of an M/GI/1/n queueing system to the time stationary distribution of the corresponding M/GI/1 queueing system when n tends to infinity.

Minimal repairs have been given considerable attention in the reliability literature. Instead of replacing a failed system by a new one, such a minimal repair restores the system to the state it had just before failure. But the state just before failure depends on the information which is available about the system. Different information levels are possible. This paper gives a general definition characterizing point processes which describe time points of minimal repairs with respect to a certain information level. Some examples demonstrate the wide range of applications.

We consider a fluid model similar to that of Kella and Whitt [32], but with a buffer having finite capacity K. The connections between the infinite buffer fluid model and the G/G/1 queue established by Kella and Whitt are extended to the finite buffer case: it is shown that the stationary distribution of the buffer content is related to the stationary distribution of the finite dam. We also derive a number of new results for the latter model. In particular, an asymptotic expansion for the loss fraction is given for the case of subexponential service times. The stationary buffer content distribution of the fluid model is also related to that of the corresponding model with infinite buffer size, by showing that the two corresponding probability measures are proportional on [0,K) if the silence periods are exponentially distributed. These results are applied to obtain large buffer asymptotics for the loss fraction and the mean buffer content when the fluid queue is fed by N On-Off sources with subexponential on-periods. The asymptotic results show a significant influence of heavy-tailed input characteristics on the performance of the fluid queue.

We consider a fluid queue fed by a superposition of a finite number of On/Off sources, the distribution of the On period being subexponential for some of them and exponential for the others. We provide general lower and upper bounds for the tail of the stationary buffer content distribution in terms of the so-called minimal subsets of sources. We then show that this tail decays at exponential or subexponential speed according as a certain parameter is smaller or larger than the ouput rate. If we replace the subexponential tails by regularly varying tails, the upper bound and the lower bound are sharp in that they differ only by a multiplicative factor.

By applying a perturbation method to the structured coalescent process in population genetics theory, we obtain the approximate solutions of the moment generating functions for the total coalescence time, the number of segregating sites among sampled DNA sequences and the number of allele types in a sample in the case of strong migration.

In 1991 Perkins [7] showed that the normalized critical binary branching process is a time inhomogeneous Fleming-Viot process. In the present paper we extend this result to jump-type branching processes and we show that the normalized jump-type branching processes are in a new class of probability measure-valued processes which will be called ‘jump-type Fleming-Viot processes’. Furthermore we also show that by using these processes it is possible to introduce another new class of measure-valued processes which are obtained by the combination of jump-type branching processes and Fleming-Viot processes.

The use of bias optimality to distinguish among gain optimal policies was recently studied by Haviv and Puterman [1] and extended in Lewis et al. [2]. In [1], upon arrival to an M/M/1 queue, customers offer the gatekeeper a reward R. If accepted, the gatekeeper immediately receives the reward, but is charged a holding cost, c(s), depending on the number of customers in the system. The gatekeeper, whose objective is to ‘maximize’ rewards, must decide whether to admit the customer. If the customer is accepted, the customer joins the queue and awaits service. Haviv and Puterman [1] showed there can be only two Markovian, stationary, deterministic gain optimal policies and that only the policy which uses the larger control limit is bias optimal. This showed the usefulness of bias optimality to distinguish between gain optimal policies. In the same paper, they conjectured that if the gatekeeper receives the reward upon completion of a job instead of upon entry, the bias optimal policy will be the lower control limit. This note confirms that conjecture.

In this paper we extend the notion of quasi-reversibility and apply it to the study of queueing networks with instantaneous movements and signals. The signals treated here are considerably more general than those in the existing literature. The approach not only provides a unified view for queueing networks with tractable stationary distributions, it also enables us to find several new classes of product form queueing networks, including networks with positive and negative signals that instantly add or remove customers from a sequence of nodes, networks with batch arrivals, batch services and assembly-transfer features, and models with concurrent batch additions and batch deletions along a fixed or a random route of the network.

We prove a new heavy traffic limit result for a simple queueing network under a ‘join the shorter queue’ policy, with the amount of traffic which has a routeing choice tending to zero as heavy traffic is approached. In this limit, the system considered does not exhibit state space collapse as in previous work by Foschini and Salz, and Reiman, but there is nevertheless some resource pooling gain over a policy of random routeing.

The estimation of critical values is one of the most interesting problems in the study of interacting particle systems. The bounds obtained analytically are not usually very tight and, therefore, computer simulation has been proved to be very useful in the estimation of these values. In this paper we present a new method for the estimation of critical values in any interacting particle system with an absorbing state. The method, based on the asymptotic behaviour of the absorption time of the process, is very easy to implement and provides good estimates. It can also be applied to processes different from particle systems.

Let G be an infinite, locally finite, connected graph with bounded degree. We show that G supports phase transition in all or none of the following five models: bond percolation, site percolation, the Ising model, the Widom-Rowlinson model and the beach model. Some, but not all, of these implications hold without the bounded degree assumption. We finally give two examples of (random) unbounded degree graphs in which phase transition in all five models can be established: supercritical Galton-Watson trees, and Poisson-Voronoi tessellations of ℝd for d ≥ 2.

In this note, we derive an inequality for the renewal process. Then, using this inequality, together with an identity in terms of the renewal process for the tails of random sums, we prove that a class of random sums is always new worse than used (NWU). Thus, the well-known NWU property of geometric sums is extended to the class of random sums. This class is illustrated by some examples, including geometric sums, mixed geometric sums, certain mixed Poisson distributions and certain negative binomial sums.

A major design challenge of Asynchronous Transfer Mode (ATM) networks is to efficiently provide the quality of service (QOS) specified by users with different demands. We classify sources so that sources in one class join the same buffer and have the same requirement for the ATM cell loss ratio. It is important to search for the service discipline that minimizes the accumulated cell loss under the constraint that the cell loss ratios of the sources are proportional to their QOS requirements. In this paper we consider a model that has N finite buffers and a single server. Buffer i, of size B i , is assigned a positive number w i . The server serves from one of the non-empty buffers whose indices are equal to argmin w i (B i -Q i ), where Q i is the queue length of buffer i. This scheduling policy is called the smallest weighted available buffer policy (SWAB). We show that in a completely symmetric setting, the SWAB policy minimizes the discounted expected loss of cells under some technical conditions. For asymmetric models, we show that the accumulated loss of cells of the SWAB service discipline is asymptotically optimal under heavy traffic conditions in the diffusion limit. Finally, we obtain the expression of w i so that the cell loss ratios of the sources in the diffusion limit are proportional to their QOS requirements.

Let L n be the number of losses during a busy period of an M/GI/1/n queueing system. We develop a coupling between L n and L n+1 and use the resulting relationship to provide a simple proof that when the mean service time equals the mean interarrival time, EL n = 1 for all n. We also show that L n is increasing in the convex sense when the mean service time equals the mean interarrival time, and it is increasing in the increasing convex sense when the mean service time is less than the mean interarrival time.

For systems subject to inspections at Poisson random times, we present an analytic method which gives upper and lower bounds for the reliability. We also study its asymptotic behaviour and derive the asymptotic failure rate.

The convergence to equilibrium of the renormalized M/M/N/N queue is analysed. Upper bounds on the distance to equilibrium are obtained and the cut-off property for two regimes of this queue is proved. Simple probabilistic methods, such as coupling techniques and martingales, are used to obtain these results.

In this paper, we give necessary and sufficient conditions to ensure the validity of confidence intervals, based on the central limit theorem, in simulations of highly reliable Markovian systems. We resort to simulations because of the frequently huge state space in practical systems. So far the literature has focused on the property of bounded relative error. In this paper we focus on ‘bounded normal approximation’ which asserts that the approximation of the normal law, suggested by the central limit theorem, does not deteriorate as the reliability of the system increases. Here we see that the set of systems with bounded normal approximation is (strictly) included in the set of systems with bounded relative error.