We apply the splitting method to three well-known counting problems, namely 3-SAT, random graphs with prescribed degrees, and binary contingency tables. We present an enhanced version of the splitting method based on the capture-recapture technique, and show by experiments the superiority of this technique for SAT problems in terms of variance of the associated estimators, and speed of the algorithms.

We construct matrix-valued covariance functions and in ℝ2 and ℝ3, starting from an arbitrary scalar-valued variogram. It is shown that sufficiently smooth random vector fields (RVFs) with these covariance functions have divergence-free and curl-free sample paths, respectively. Conversely, essentially all models with such sample paths can be obtained via our construction. Extensions to space-time RVFs are possible. RVFs with divergence-free and curl-free sample paths can be utilised in meteorological applications e.g. for modelling and interpolating wind fields.

Heavy traffic limit theorems are established for a class of single server queueing models including those with heavy-tailed or long-range dependent arrivals and time-varying service rates. The models are motivated by wireless queueing systems for which there is an increasing evidence of the presence of heavy-tailed or long-range dependent arrivals, and where the service rates vary with the changes in the wireless medium. The main focus of the paper is to obtain the different possible limit processes that can arise depending on the relationship between scalings for both the arrival and departure processes. The limit processes obtained here are driven by either Brownian motion (when the contribution from the departure process dominates the limit) or the limits of properly scaled arrivals (when the contribution from the arrival process dominates the limit), typical examples being stable Lévy motion or fractional Brownian motion. In particular, for the case where arrival process is given by the infinite source Poisson process, this relationship, which determines the type of the limiting queue-length process, is a simple condition involving the heavy tail exponent, arrival rate and channel variation parameter in the wireless medium model. To establish these limit results, two approaches are studied. In one approach, when the limit is driven by Brownian motion, the perturbed test function method is extended to incorporate reflection. In contrast, the second approach allows for non-Markovian and/or non-Gaussian driving processes in the limit. Both approaches involve averaging in the drift term arising from random service rates at the departures. In the second approach, this averaging is carried out directly and pathwise, thus sidestepping the assumption of driving Brownian motion used in the perturbed test function method.

We use semi-regenerative, fluctuations methods, and sequential games to examine a queueing system with server performing a secondary maintenance work. When the buffer becomes empty, the server leaves the system to service packets of jobs at a maintenance facility where his time there is restricted by T (a random time). He returns to the system when his sojourn time expires and if the last packet is fully processed. If the buffer contents is below N, the server departs from the system again to continue with maintenance; this time rendering just one packet at a time until the buffer is replenished. We obtain compact and explicit functionals for the queueing process in equilibrium for both discrete and continuous time parameter processes.

Consider two insurance companies (or two branches of the same company) that receive premiums at different rates and then split the amount they pay in fixed proportions for each claim (for simplicity we assume that they are equal). We model the occurrence of claims according to a Poisson process. The ruin is achieved when the corresponding two-dimensional risk process first leaves the positive quadrant. We will consider two scenarios of the controlled process: refraction and impulse control. In the first case the dividends are payed out when the two-dimensional risk process exits the fixed region. In the second scenario, whenever the process hits the horizontal line, it is reduced by paying dividends to some fixed point in the positive quadrant where it waits for the next claim to arrive. In both models we calculate the discounted cumulative dividend payments until the ruin. This article is the first attempt to understand the effect of dependencies of two portfolios on the joint optimal strategy of paying dividends. For example in case of proportional reinsurance one can observe the interesting phenomenon that choice of the optimal barrier depends on the initial reserves. This is in contrast with the one-dimensional Cramér-Lundberg model where the optimal choice of the barrier is uniform for all initial reserves.

We characterize the value function of maximizing the total discounted utility of dividend payments for a compound Poisson insurance risk model when strictly positive transaction costs are included, leading to an impulse control problem. We illustrate that well known simple strategies can be optimal in the case of exponential claim amounts. Finally we develop a numerical procedure to deal with general claim amount distributions.

We consider a spatial stochastic model for telecommunication networks, the stochastic subscriber line model, and we investigate the distribution of the typical shortest path length between network components. Therefore, we derive a representation formula for the probability density of this distribution which is based on functionals of the so-called typical serving zone. Using this formula, we construct an estimator for the density of the typical shortest path length and we analyze the statistical properties of this estimator. Moreover, we introduce new simulation algorithms for the typical serving zone which are used in a numerical study in order to estimate the density and moments of the typical shortest path length for different specific models.

The article deals with the queueing system of GI/G/1 type with individual service and batch arrival of customers. The method of integral equations on a half-axis [0, ∞) is applied to obtain new results for the actual waiting time w n of nth arriving customer. The transient and steady state as n tends to infinity are considered. Some simplifications and numerical results for M/M/1, M/E 2/1, and M 2/M/1 queueing systems are derived as well.

The purpose of this article is to study the bivariate distributions, proposed by Hassan and Lii[ 19 ], in modeling marked point processes, from the reliability point of view. The reliability functions, viz. the survival function, the failure rate, and the mean residual life function, are derived and their monotonicity is discussed for the marginals as well as the conditional models. The monotonicity of the failure rate, in the case of series system, is also investigated. Finally, Clayton's association measure is obtained in terms of the hazard gradient and its value, in the case of our model, is derived.

This paper studies perturbations of the single server queue with breakdowns, where the perturbations consist of introducing dependent breakdowns. Problems that are modeled by queueing models with breakdowns and repairs are often very complicated, and they are resolved only through approximations. Therefore, it is very important to justify these approximations and to estimate the resultant error. For this purpose, we use the strong stability method to approximate the characteristics of the M/G/1 queue with dependent breakdowns by those of the M/G/1 queue with classical (constant) breakdowns. This latter queue is simpler and more exploitable in practice. Thus, we prove the stability conditions and next obtain stability inequalities with exactly computing of the constants. These results give with precision the error, on the queue size stationary distribution, due to the approximation. For this, we elaborate from the obtained theoretical results an algorithm allowing one to verify the approximation conditions and to provide the made numerical error. The accuracy of the approach is evaluated by comparison with true distance values.