We consider an absorbing semi-Markov chain for which each time absorption occurs there is a resetting of the chain according to some initial (replacement) distribution. The new process is a semi-Markov replacement chain and we study its properties in terms of those of the imbedded Markov replacement chain. A time-dependent version of the model is also defined and analysed asymptotically for two types of environmental behaviour, i.e. either convergent or cyclic. The results contribute to the control theory of semi-Markov chains and extend in a natural manner a wide variety of applied probability models. An application to the modelling of populations with semi-Markovian replacements is also presented.

Many disordered random systems in applications can be described by N randomly coupled Ito stochastic differential equations in :

where is a sequence of independent copies of the one-dimensional Brownian motion W and ( is a sequence of independent copies of the ℝp-valued random vector ξ. We show that under suitable conditions on the functions b, σ, K and Φ the dynamical behaviour of this system in the N → (limit can be described by the non-linear stochastic differential equation

where P(t, dx dy) is the joint probability law of ξ and X(t).

Consider the total service time of a job on an unreliable server under preemptive-repeat-different and preemptive-resume service disciplines. With identical initial conditions, for both cases, we notice that the distributions of the total service time under these two disciplines coincide, when the original service time (without interruptions due to server failures) is exponential and independent of the server reliability. We show that this fact under varying server reliability is a characterization of the exponential distribution. Further we show, under the same initial conditions, that the coincidence of the mean values also leads to the same characterization.

In this paper a particular loss network consisting of two links with C1 and C2 circuits, respectively, and two fixed routes, is investigated. A call on route 1 uses a circuit from both links, and a call on route 2 uses a circuit from only the second link. Calls requesting routes 1 and 2 arrive as independent Poisson streams. A call requesting route 1 is blocked and lost if there are no free circuits on either link, and a call requesting route 2 is blocked and lost if there is no free circuit on the second link. Otherwise the call is connected and holds a circuit from each link on its route for the holding period of the call.

The case in which the capacities C1, and C2, and the traffic intensities v1, and v2, all become large of O(N) where N » 1, but with their ratios fixed, is considered. The loss probabilities L1 and L2 for calls requesting routes 1 and 2, respectively, are investigated. The asymptotic behavior of L1 and L2 as N→ ∞ is determined with the help of double contour integral representations and saddlepoint approximations. The results differ in various regions of the parameter space (C1, C2, v1, v2). In some of these results the loss probabilities are given in terms of the Erlang loss function, with appropriate arguments, to within an exponentially small relative error. The results provide new information when the loss probabilities are exponentially small in N. This situation is of practical interest, e.g. in cellular systems, and in asynchronous transfer mode networks, where very small loss probabilities are desired.

The accuracy of the Erlang fixed-point approximations to the loss probabilities is also investigated. In particular, it is shown that the fixed-point approximation E2 to L2 is inaccurate in a certain region of the parameter space, since L2 « E2 there. On the other hand, in some regions of the parameter space the fixed-point approximations to both L1 and L2 are accurate to within an exponentially small relative error.

A generalization of the notion of a stationary Markov chain in more than one dimension is proposed, and is found to be a special class of homogeneous Markov random fields. Stationary Markov chains in many dimensions are shown to possess a maximum entropy property, analogous to the corresponding property for Markov chains in one dimension. In addition, a representation of Markov chains in many dimensions is provided, together with a method for their generation that converges to their stationary distribution.

Obtaining good estimates for the distribution function of random variables like (‘perpetuity’) and (‘aggregate claim amount’), where the (Yi), (Zi) are independent i.i.d. sequences and (N(t)) is a general point process, is a key question in insurance mathematics. In this paper, we show how suitably chosen metrics provide a theoretical justification for bootstrap estimation in these cases. In the perpetuity case, we also give a detailed discussion of how the method works in practice.

This paper shows how to calculate solutions to Poisson's equation for the waiting time sequence of the recurrent M/G/l queue. The solutions are used to construct martingales that permit us to study additive functionals associated with the waiting time sequence. These martingales provide asymptotic expressions, for the mean of additive functionals, that reflect dependence on the initial state of the process. In addition, we show how to explicitly calculate the scaling constants that appear in the central limit theorems for additive functionals of the waiting time sequence.

We present structural properties of optimal policies for the problem of scheduling a single server in a forest network of N queues (without arrivals) subject to switching penalties. In addition to linear holding costs, we impose either lump sum switching costs or batch set-up delays which are incurred at each instant the server processes a job in a queue different from the previous one. We use reward rate notions to unearth conditions on the holding costs and service distributions for which an exhaustive policy is optimal. For the case of two nodes connected probabilistically in tandem, we explicitly define an optimal policy under similar conditions.

The paper deals with large trunk line systems of the type appearing in telephone networks. There are many nodes or input sources, each pair of which is connected by a trunk line containing many individual circuits. Traffic arriving at either end of a trunk line wishes to communicate to the node at the other end. If the direct route is full, a rerouting might be attempted via an alternative route containing several trunks and connecting the same endpoints. The basic questions concern whether to reroute, and if so how to choose the alternative path. If the network is ‘large’ and fully connected, then the overflow traffic which is offered for rerouting to any trunk comes from many other trunks in the network with no one dominating. In this case one expects that some sort of averaging method can be used to approximate the rerouting requests and hence simplify the analysis. Essentially, the overflow traffic that a trunk offers the network for rerouting is in some average sense similar to the overflow traffic offered to that trunk. Indeed, a formalization of this idea involves the widely used (but generally heuristic) ‘fixed point' approximation method. One sets up the fixed point equations for appropriate rerouting strategies and then solves them to obtain an approximation to the system loss. In this paper we work in the heavy traffic regime, where the external offered traffic to any trunk is close to the service capacity of that trunk. It is shown that, as the number of links and circuits within each link go to infinity and for a variety of rerouting strategies, the system can be represented by an averaged limit. This limit is a reflected diffusion of the McKean–Vlasov (propagation of chaos) type, where the driving terms depend on the mean values of the solution of the equation. The averages occur due to the symmetry of the network and the averaging effects of the many interactions. This provides a partial justification for the fixed point method. The concrete dynamical systems flavor of the approach and the representations of the limit processes provide a useful way of visualizing the system and promise to be useful for the development of numerical methods and further analysis.

Burn-in is a widely used method to improve quality of products after they have been produced. For a repairable component there are two common types of repair, complete repair and minimal repair. Preventive maintenance policies such as age replacement and block replacement are often employed in field operation. The present paper takes burn-in, maintenance and repair into consideration at the same time and considers related cost structures. The properties of the corresponding optimal burn-in times and optimal maintenance policies are discussed.

Let Φ = {Φ n} be an aperiodic, positive recurrent Markov chain on a general state space, π its invariant probability measure and f ≧ 1. We consider the rate of (uniform) convergence of Ex[g(Φ n)] to the stationary limit π (g) for |g| ≦ f: specifically, we find conditions under which

as n →∞, for suitable subgeometric rate functions r. We give sufficient conditions for this convergence to hold in terms of

(i) the existence of suitably regular sets, i.e. sets on which (f, r)-modulated hitting time moments are bounded, and

(ii) the existence of (f, r)-modulated drift conditions (Foster–Lyapunov conditions).

The results are illustrated for random walks and for more general state space models.

We consider the standard slotted ALOHA system with a finite number of buffered users. Stability analysis of such a system was initiated by Tsybakov and Mikhailov (1979). Since then several bounds on the stability region have been established; however, the exact stability region is known only for the symmetric system and two-user ALOHA. This paper proves necessary and sufficient conditions for stability of the ALOHA system. We accomplish this by means of a novel technique based on three simple observations: applying mathematical induction to a smaller copy of the system, isolating a single queue for which Loynes' stability criteria is adopted, and finally using stochastic dominance to verify the required stationarity assumptions in the Loynes criterion. We also point out that our technique can be used to assess stability regions for other multidimensional systems. We illustrate it by deriving stability regions for buffered systems with conflict resolution algorithms (see also Georgiadis and Szpankowski (1992) for similar approach applied to stability of token-passing rings).

In this paper we study the extremal properties of several scheduling policies in an in-forest network consisting of multiserver queues. Each customer has a due date, and we assume that service times at the different queues form mutually independent sequences of independent and identically distributed random variables independent of the arrival times and due dates. Furthermore, the network is assumed to consist of a mixture of nodes, some of which permit only non-preemptive service policies whereas the others permit preemptive resume policies. In the case of non-preemptive queues, service times may be generally distributed if there is only one server; otherwise the service times are required to be increasing in likelihood ratio (ILR). In the case of preemptive queues, service times are restricted to exponential distributions. Using stochastic majorizations and partial orders on permutations, we establish that, within the class of work conserving service policies, the stochastically smallest due date (SSDD) and the stochastically largest due date (SLDD) policies minimize and maximize, respectively, the vector of the customer latenesses of the first n customers in the sense of the Schur-convex order and some weaker orders, provided the due dates are comparable in some stochastic sense. It then follows that the first come-first served (FCFS) and the last come-first served (LCFS) policies minimize and maximize, respectively, the vector of the response times of the first n customers in the sense of the Schur-convex order. We also show that the FCFS and LCFS policies minimize and maximize, respectively, the vector of customer end-to-end delays in the sense of the strong stochastic order. Extensions to the class of non-idling policies and to the stationary regime are also given.

Existence and finiteness of the sample-mean limit of sojourn times of jobs in a queueing system are investigated. The queueing system operates under rather general multiprocessor disciplines allowing job classes and priorities. The input stream of jobs consisting of job classes and interarrival and processing times is stationary and ergodic and may contain batch arrivals. Existence of the sample-mean limit is proved by means of the superadditive ergodic theorem, and its finiteness is controlled by uniform mixing of the input stream.

Learning from Matheron's representation (1973), and using the increment vector (PIV) methodology introduced by Cressie (1988) and developed by Chen and Anderson (1994), this paper presents a theory for the representation and decomposition of integrated stationary time series and gives some applications.

We consider a problem in which a single server must serve a stream of customers whose arrivals are distributed over a finite-size convex space. Under the assumption that the server has full information on the customer location, obvious service policies are the FCFS and the greedy (serve-the-closest-customer) approaches. These algorithms are, however, either inefficient (FCFS) or ‘unfair' (greedy).

We propose and study two alternative algorithms, the gated-greedy policy and the gated-scan policy, which are more ‘fair' than the pure greedy method. We show that the stability conditions of the gated-greedy are p < 1 (where p is the expected rate at which work arrives at the system), implying that the method is at least as efficient (in terms of system stability) as any other discipline, in particular the greedy one. For the gated-scan policy we show that for any p < 1 one can design a stable gated-scan policy; however, for any fixed gated-scan policy there exists p < 1 for which the policy is unstable. We evaluate the performance of the gated-scan policy, and present bounds for the performance of the gated-greedy policy.

These results are derived for systems in which the arrivals occur on a two-dimensional space (a square) but they are not limited to this configuration; rather they hold for more complex N-dimensional spaces, in particular for serving customers in (three-dimensional) convex space and serving customers on a line.

Consider the GI/G/1 queueing system with traffic intensity 1 and let wk and lk denote the actual waiting time of the kth unit and the number of units present in the system at the kth arrival including the kth unit, respectively. Furthermore let τ denote the number of units served during the first busy period and μ the intensity of the service. It is shown that as k →∞, where a is some known constant, , , and are independent, is a Brownian meander and is a Wiener process. A similar result is also given for the difference of virtual waiting time and queue length processes. These results are also extended to a wider class of queueing systems than GI/G/1 queues and a scheme of series of queues.

Using the electric network approach, we give closed-form formulas for the expected hitting times in the Ehrenfest urn model.

Consider GI/G/1 processor sharing queues with traffic intensity tending to 1. Using the theory of random measures and the theory of branching processes we investigate the limiting behaviour of the queue length, sojourn time and random measures describing attained and residual processing times of customers present.

The mean busy period of a Markov-modulated queue or fluid model is computed by an extension of the time-reversal argument connecting the steady-state distribution and the maximum of a related Markov additive process.