Motivated by a statistical application, we consider continuum percolation in two or more dimensions, restricted to a large finite box, when above the critical point. We derive surface order large deviation estimates for the volume of the largest cluster and for its intersection with the boundary of the box. We also give some natural extensions to known, analogous results on lattice percolation.

Let N be a stationary Markov-modulated marked point process on ℝ with intensity β∗ and consider a real-valued functional ψ(N). In this paper we study expansions of the form Eψ(N) = a0 + β∗a1 + ·· ·+ (β∗)nan + o((β∗)n) for β∗→ 0. Formulas for the coefficients ai are derived in terms of factorial moment measures of N. We compute a1 and a2 for the probability of ruin φ u with initial capital u for the risk process in the Markov-modulated environment; a0 = 0. Moreover, we give a sufficient condition for ϕu to be an analytic function of β∗. We allow the premium rate function p(x) to depend on the actual risk reserve.

We consider a heavy traffic regime in queueing systems with identical service. These systems belong to the class of multi-phase systems with dependent service times in different service nodes. We study the limit behaviour of the waiting time vector in heavy traffic. Both transient behaviour and the stationary regime are considered. Our analysis is based on the conception of ‘approximated functionals', which appeared to be fruitful in weak convergence theory of stochastic processes.

An exponential inequality is generalized to one involving the tail of a decreasing failure rate distribution. The results are then applied in various situations, notably when the exponential inequality does not apply.

Using sample path analysis we show that under the same load the mean delay in queue in the M/G/2 system is smaller than that in the corresponding M/G/1 system, when the service time has either the DMRL or NBU property and the service discipline is FCFS. The proof technique uses a new device that equalizes the work in a two server system with that in a single sterver system. Other interesting quantities such as the average difference in work between the two servers in the GI/G/2 system and an exact alternate derivation of the mean delay in the M/M/2 system from sample path analysis are presented. For the same load, we also show that the mean delay in the M/G/C system with general service time distribution is smaller than that in the M/G/1 system when the traffic intensity is less than 1/c.

The comparison of optimal strategies in a simple stochastic replacement model for two types of machines with identical cost characteristics when one of them is more reliable than the other is conducted. It is proven that the scheduled replacement period for the more reliable type is always less than for the less reliable one. An example is presented when even the expected period of use of a more reliable machine is less than the expected period for the less reliable one. Some related problems are briefly discussed.

The use of Mineka coupling is extended to a case with a continuous state space: an efficient coupling of random walks S and S' in can be made such that S' — S is virtually a one-dimensional simple random walk. This insight settles a zero-two law of ergodicity. One more proof of Blackwell's renewal theorem is also presented.

This paper presents an analysis of the single-server discrete-time finite-buffer queue with general interarrival and geometric service time, GI/Geom/1/N. Using the supplementary variable technique, and considering the remaining interarrival time as a supplementary variable, two variations of this model, namely the late arrival system with delayed access (LAS-DA) and early arrival system (EAS), have been examined. For both cases, steady-state distributions for outside observers as well as at random and prearrival epochs have been obtained. The waiting time analysis has also been carried out. Results for the Geom/G/1/N queue with LAS-DA have been obtained from the GI/Geom/1/N queue with EAS. We also give various performance measures. An algorithm for computing state probabilities is given in an appendix.

We obtain a necessary and sufficient condition for the interaction between a service system and an external environment under which the stationary joint distribution of the set of busy channels and the state of the external environment is given by a product-form formula.

We investigate the asymptotic behaviour of the distribution of the number of comparisons needed by a quicksort-style selection algorithm that finds the lth smallest in a set of n numbers. Letting n tend to infinity and considering the values l = 1, ···,n simultaneously we obtain a limiting stochastic process. This process admits various interpretations: it arises in connection with a representation of real numbers induced by nested random partitions and also in connection with expected path lengths of a random walk in a random environment on a binary tree.

Conditions that have been widely used to establish stochastic ordering of random vectors include those given by Kamae et al. (1977), Barlow and Proschan (1975) and Langberg (1988). In this note we offer conditions that are weaker than those above, thus providing potential application to a wider class of problems.

This paper considers fluid queuing models of Markov-modulated traffic that, due to large differences in the time-scales of events, possess structural characteristics that yield a nearly completely decomposable (NCD) state-space. Extension of domain decomposition and aggregation techniques that apply to approximating the eigensystem of Markov chains permits the approximate subdivision of the full system to a number of small, independent subsystems (decomposition phase), plus an ‘aggregative' system featuring a state-space that distinguishes only one index per subsystem (aggregation phase). Perturbation analysis reveals that the error incurred by the approximation is of an order of magnitude equal to the weak coupling of the NCD Markov chain.

The study in this paper is then extended to the structure of NCD fluid models describing source superposition (multiplexing). It is shown that efficient spectral factorization techniques that arise from the Kronecker sum form of the global matrices can be applied through and combined with the decomposition and aggregation procedures. All structural characteristics and system parameters are expressible in terms of the individual sources multiplexed together, rendering the construction of the global system unnecessary.

Finally, besides providing efficient computational algorithms, the work in this paper can be recast as a conceptual framework for the better understanding of queueing systems under the presence of events happening in widely differing time-scales.

Conditions for a correlated cumulative shock model under which the system failure time is HNBUE are given. It is shown that the proof of a theorem given by Sumita and Shanthikumar (1985) relative to this property is not correct and a correct proof of the theorem is given.

The cumulative distribution of the finite sum of the binary sequence of order k is studied and some of its applications discussed. Certain properties of this sequence are investigated and uniformly superior bounds for the cumulative distribution under minimal information on the ‘success' probabilities are derived.

First, some basic concepts from the theory of point processes are recalled and expanded. Then some notions of stochastic comparisons, which compare whole processes, are introduced. The use of these notions is illustrated by stochastically comparing renewal and related processes. Finally, applications of the different notions of stochastic ordering of point processes to many replacement policies are given.

A set-valued analog of the elementary renewal theorem for Minkowski sums of random closed sets is considered. The corresponding renewal function is defined as where are Minkowski (element-wise) sums of i.i.d. random compact convex sets. In this paper we determine the limit of H(tK)/t as t tends to infinity. For K containing the origin as an interior point, where hK(u) is the support function of K and is the set of all unit vectors u with EhA(u) > 0. Other set-valued generalizations of the renewal function are also suggested.

In this paper we consider a family of product-form loss models, including loss networks (or circuit-switched communication networks) and a class of resource-sharing models. There can be multiple classes of requests for multiple resources. Requests arrive according to independent Poisson processes. The requests can be for multiple units in each resource (the multi-rate case, e.g. several circuits on a trunk). There can be upper-limit and guaranteed-minimum sharing policies as well as the standard complete-sharing policy. If all the requirements of a request cannot be met upon arrival, then the request is blocked and lost. We develop an algorithm for computing the (exact) steady-state blocking probability of each class and other steady state descriptions in these loss models. The algorithm is based on numerically inverting generating functions of the normalization constants. In a previous paper we introduced this approach to product-form models and developed a full algorithm for a class of closed queueing networks. The inversion algorithm promises to be even more useful for loss models than for closed queueing networks because fewer alternative algorithms are available for loss models. Indeed, for many loss models with sharing policies other than traditional complete sharing, our algorithm is the first effective algorithm. Unlike some recursive algorithms, our algorithm has a low storage requirement. To treat the loss models here, we derive the generating functions of the normalization constants and develop a new scaling algorithm especially tailored to the loss models. In general, the computational complexity grows exponentially in the number of resources, but the computation can often be reduced dramatically by exploiting conditional decomposition based on special structure and by appropriately truncating large finite sums. We illustrate our numerical inversion algorithm by applying it to several examples. To validate our algorithm on small models, we also develop a direct algorithm. The direct algorithm itself is of interest, because it tends to be more efficient when the number of resources is large, but the number of request classes is small. Furthermore, it also allows a form of conditional decomposition based on special structure.

Consider a single-server queue with zero buffer. The arrival process is a three-level Markov modulated Poisson process with an arbitrary transition matrix. The time the system remains at level i (i = 1, 2, 3) is exponentially distributed with rate cα i. The arrival rate at level i is λ i and the service time is exponentially distributed with rate μ i. In this paper we first derive an explicit formula for the loss probability and then prove that it is decreasing in the parameter c. This proves a conjecture of Ross and Rolski's for a single-server queue with zero buffer.

In this paper, we develop mathematical machinery for verifying that a broad class of general state space Markov chains reacts smoothly to certain types of perturbations in the underlying transition structure. Our main result provides conditions under which the stationary probability measure of an ergodic Harris-recurrent Markov chain is differentiable in a certain strong sense. The approach is based on likelihood ratio ‘change-of-measure' arguments, and leads directly to a ‘likelihood ratio gradient estimator' that can be computed numerically.

We consider a family of M(t)/M(t)/1/1 loss systems with arrival and service intensities (λt (c), μt (c)) = (λct, μct), where (λt, μt) are governed by an irreducible Markov process with infinitesimal generator Q = (qij)m × m such that (λt, μt) = (λi, μi) when the Markov process is in state i. Based on matrix analysis we show that the blocking probability is decreasing in c in the interval [0, c∗], where c∗ = 1/maxi Σj≠iqij/(λi + μi). Two special cases are studied for which the result can be extended to all c. These results support Ross's conjecture that a more regular arrival (and service) process leads to a smaller blocking probability.