We give formulae for different types of contact distribution functions for stationary (not necessarily Poisson) Voronoi tessellations in ℝd in terms of the Palm void probabilities of the generating point process. Moreover, using the well-known relationship between the linear contact distribution and the chord length distribution we derive a closed form expression for the mean chord length in terms of the two-point Palm distribution and the pair correlation function of the generating point process. The results obtained are specified for Voronoi tessellations generated by Poisson cluster and Gibbsian processes, respectively.

This paper establishes a link between the Wiener–Hopf factorization and the phase-type method for studying the GI/G/1 queue. Using the Wiener–Hopf factorization, infinite-matrix type results are established for the GI/G/1 queue. An iterative numerical procedure (‘Levinson&s method’) based on these results is described. This method does not always converge. For the situation where either the interarrival times or the service times are of the so-called almost phase type (APH) an alternative, probabilistic derivation of the same results is given. This alternative derivation shows that in the APH situation Levinson&s method converges, converges essentially monotonically, and converges to the correct values.

The algorithm has been coded and examples of numerical results are included.

Let be a sequence of independent and identically distributed random vectors drawn from the d-dimensional unit ball Bd and let Xn be the random polytope generated as the convex hull of a1,· ··, an. Furthermore, let Δ(Xn): = Vol (BdXn) be the volume of the part of the ball lying outside the random polytope. For uniformly distributed ai and 2 we prove that the limiting distribution of Δ(Xn)/Ε (Δ (Xn)) for n → ∞ (satisfies a 0–1 law. In particular, we show that Var for n → ∞. We provide analogous results for spherically symmetric distributions in Bd with regularly varying tail. In addition, we indicate similar results for the surface area and the number of facets of Xn.

The necessary and sufficient condition for unilateral characterization of Gaussian Markov fields and the Besag-Moran positivity condition for second-order autonormal bilateral models define the same tetrahedral domain of achievable regression parameters. A bijective function maps this domain to a different tetrahedral domain of parameters in the Pickard model. These two domains are identical to the corresponding ones in the Welberry-Carroll model. We obtain series solutions for correlation coefficients and study their limits near the boundaries of the first domain.

Queueing systems which can be formulated as Markov renewal processes with basic transitions of three types, ‘arrivals', ‘departures' and ‘feedbacks' are examined. The filtering procedure developed for Markov renewal processes by Çinlar (1969) is applied to such queueing models to show that the queue-length processes embedded at any of the ‘arrival', ‘departure', ‘feedback', ‘input', ‘output' or ‘external' transition epochs are also Markov renewal. In this part we focus attention on the derivation of stationary and limiting distributions (when they exist) for each of the embedded discrete-time processes, the embedded Markov chains. These results are applied to birth–death queues with instantaneous state-dependent feedback including the special cases of M/M/1/N and M/M/1 queues with instantaneous Bernoulli feedback.

Waiting-time distributions for M/G/1 systems with priority dependent on class, order of arrival, service length, etc., are difficult to obtain. For single-server multipurpose processors the difficulties are compounded. A certain ergodic post-arrival depletion time is shown to be a true maximum for all delay times of interest. Explicit numerical evaluation of the distribution of this time is available. A heavy-traffic distribution for this time is shown to provide a simple and useful engineering tool with good results and insensitivity to service-time distribution even at modest traffic intensity levels. The relationship to the diffusion approximation for heavy traffic is described.

Relative ageing describes how one system ages with respect to another. The ageing faster orders are used to compare the relative ageing of two systems. Here, we study ageing faster orders in the hazard and reversed hazard rates. We provide some sufficient conditions for one coherent system to dominate another with respect to ageing faster orders. Further, we investigate whether the active redundancy at the component level is more effective than that at the system level with respect to ageing faster orders, for a coherent system. Furthermore, a used coherent system and a coherent system made out of used components are compared with respect to ageing faster orders.

In this paper we generalize some of the classical results of Rényi and Sulanke (1963), (1964) in the context of spindle convexity. A planar convex disc S is spindle convex if it is the intersection of congruent closed circular discs. The intersection of finitely many congruent closed circular discs is called a disc polygon. We prove asymptotic formulae for the expectation of the number of vertices, missed area, and perimeter difference of uniform random disc polygons contained in a sufficiently smooth spindle convex disc.

In the study of bivariate processes, one often encounters expressions involving repeated combinations of bivariate continuum operations such as multiple bivariate convolutions, marginal convolutions, tail integration, partial differentiation and multiplication by bivariate polynomials. In many cases numerical computation of such results is quite tedious and laborious. In this paper, the bivariate Laguerre transform is developed which provides a systematic numerical tool for evaluating such bivariate continuum operations. The formalism is an extension of the univariate Laguerre transform developed by Keilson and Nunn (1979), Keilson et al. (1981) and Keilson and Sumita (1981), using the product orthonormal basis generated from Laguerre functions. The power of the procedure is proven through numerical exploration of bivariate processes arising from correlated cumulative shock models.

A stationary Markovian decision model is considered with general state and action spaces where the transition probabilities are weakened to be bounded transition measures (this is useful for many applications). New and improved bounds are given for the optimal value of stationary problems with a large planning horizon if either only a few steps of iteration are carried out or, in addition, a solution of the infinite-stage problem is known. Similar estimates are obtained for the quality of policies which are composed of nearly optimal decisions from the first few steps or from the infinite-stage solution.

Subcritical Markov branching processes {Zt} die out sooner or later, say at time T < ∞. We give results for the path to extinction {ZuT, 0 ≤ u ≤ 1} that include its finite dimensional distributions and the asymptotic behaviour of xu−1ZuT, as Z0=x → ∞. The limit reflects an interplay of branching and extreme value theory. Then we consider the population on the verge of extinction, as modelled by ZT-u, u > 0, and show that as Z0= x → ∞ this process converges to a Markov process {Yu}, which we describe completely. Emphasis is on continuous time processes, those in discrete time displaying a more complex behaviour, related to Martin boundary theory.

We introduce a new 1-dependent percolation model to describe and analyze the spread of an epidemic on a general directed and locally finite graph. We assign a two-dimensional random weight vector to each vertex of the graph in such a way that the weights of different vertices are independent and identically distributed, but the two entries of the vector assigned to a vertex need not be independent. The probability for an edge to be open depends on the weights of its end vertices, but, conditionally on the weights, the states of the edges are independent of each other. In an epidemiological setting, the vertices of a graph represent the individuals in a (social) network and the edges represent the connections in the network. The weights assigned to an individual denote its (random) infectivity and susceptibility, respectively. We show that one can bound the percolation probability and the expected size of the cluster of vertices that can be reached by an open path starting at a given vertex from above by the corresponding quantities for independent bond percolation with a certain density; this generalizes a result of Kuulasmaa (1982). Many models in the literature are special cases of our general model.

A stochastic ordering approach is applied with Stein's method for approximation by the equilibrium distribution of a birth-death process. The usual stochastic order and the more general s-convex orders are discussed. Attention is focused on Poisson and translated Poisson approximations of a sum of dependent Bernoulli random variables, for example, k-runs in independent and identically distributed Bernoulli trials. Other applications include approximation by polynomial birth-death distributions.

A new approach to the solution of optimal stopping problems for one-dimensional diffusions is developed. It arises by imbedding the stochastic problem in a linear programming problem over a space of measures. Optimizing over a smaller class of stopping rules provides a lower bound on the value of the original problem. Then the weak duality of a restricted form of the dual linear program provides an upper bound on the value. An explicit formula for the reward earned using a two-point hitting time stopping rule allows us to prove strong duality between these problems and, therefore, allows us to either optimize over these simpler stopping rules or to solve the restricted dual program. Each optimization problem is parameterized by the initial value of the diffusion and, thus, we are able to construct the value function by solving the family of optimization problems. This methodology requires little regularity of the terminal reward function. When the reward function is smooth, the optimal stopping locations are shown to satisfy the smooth pasting principle. The procedure is illustrated using two examples.

A sequence of random triangles is constructed by choosing successively the three vertices of one triangle at random in the interior of its predecessor. A way is found to prove that the shapes of these triangles converge, almost surely, to collinear shapes, thus closing a gap in one of the central arguments of Mannion [5]. The new approach is based on a representation of the triangle process by a sequence of products of i.i.d. random matrices. We succeed in calculating the corresponding Lyapounov exponent.

Empirical point processes of exceedances play an important role in extreme value theory, and their limiting behaviour has been extensively studied. Here, we provide explicit bounds on the accuracy of approximating an exceedance process by a compound Poisson or Poisson cluster process, in terms of a Wasserstein metric that is generally more suitable for the purpose than the total variation metric. The bounds only involve properties of the finite, empirical sequence that is under consideration, and not of any limiting process. The argument uses Bernstein blocks and Lindeberg's method of compositions.

A single server faces a renewal stream of low and another one of high priority customers, one of them being a Poisson stream. For the preemptive resume and head-of-the line disciplines the transient behaviour of the queue is studied. Various limiting distributions are also obtained.

The systems considered are single-server, though the theory has wider application to models of adaptive queueing systems. Arrival and service mechanisms are governed by state (n)-dependent mean arrival and service rates λn and µn. It is assumed that the choice of λn and µn leads to a stable regime. Formulae are sought that provide easy means of computing statistics of effectiveness of systems. A measure of traffic intensity is first defined in terms of ‘effective’ service time and inter-arrival intervals. It is shown that the latter have a renewal type connection with appropriately defined mean effective arrival and service rates λ∗ and µ∗ and that in consequence the ratio λ∗/µ∗ is the traffic intensity, equal moreover to where is the stable probability of an empty system, consistent with other systems. It is also shown that for first come, first served discipline the equivalent of Little's formula holds, where and are the mean waiting time of an arrival and mean system size at an arbitrary epoch. In addition it appears that stable regime output intervals are statistically identical with effective inter-arrival intervals. Symmetrical moment formulae of arbitrary order are derived algebraically for effective inter-arrival and service intervals, for waiting time, for busy period and for output.

We consider a single-type supercritical or near-critical size-dependent branching process {Nn}n such that the offspring mean converges to a limit m ≥ 1 with a rate of convergence of order as the population size Nn grows to ∞ and the variance may vary at the rate where −1 ≤ β < 1. The offspring mean m(N) = m + μN-α + o(N-α) depends on an unknown parameter θ0 belonging either to the asymptotic model (θ0 = m) or to the transient model (θ0 = μ). We estimate θ0 on the nonextinction set from the observations {Nh,…,Nn} by using the conditional least-squares method weighted by (where γ ∈ ℝ) in the approximate model mθ,ν̂n(·), where ν̂n is any estimation of the parameter of the nuisance part (O(N-α) if θ0 = m and o(N-α) if θ0 = μ). We study the strong consistency of the estimator of θ0 as γ varies, with either h or n - h remaining constant as n → ∞. We use either a minimum-contrast method or a Taylor approximation of the first derivative of the contrast. The main condition for obtaining strong consistency concerns the asymptotic behavior of the process. We also give the asymptotic distribution of the estimator by using a central-limit theorem for random sums and we show that the best rate of convergence is attained when γ = 1 + β.

This paper studies optimal routing and jockeying policies in a two-station parallel queueing system. It is assumed that jobs arrive to the system in a Poisson stream with rate λand are routed to one of two parallel stations. Each station has a single server and a buffer of infinite capacity. The service times are exponential with server-dependent rates, μ1 and μ2. Jockeying between stations is permitted. The jockeying cost is cij when a job in station i jockeys to station j, i ≠ j. There is no cost when a new job joins either station. The holding cost in station j is hj, h1 ≦ h2, per job per unit time. We characterize the structure of the dynamic routing and jockeying policies that minimize the expected total (holding plus jockeying) cost, for both discounted and long-run average cost criteria. We show that the optimal routing and jockeying controls are described by three monotonically non-decreasing functions. We study the properties of these control functions, their relationships, and their asymptotic behavior. We show that some well-known queueing control models, such as optimal routing to symmetric and asymmetric queues, preemptive or non-preemptive scheduling on homogeneous or heterogeneous servers, are special cases of our system.