Keilson and Servi introduced in [5] a variation of a GI/G/1 queue with vacation, in which at the end of a service time, either the server is not idle, and he starts serving the first customer in the queue with probability p, or goes on vacation with probability 1 – p, or he is idle, and he takes a vacation. At the end of a vacation, either customers are present, and the server starts serving the first customer, or he is idle, and he takes a vacation. The case p = 1, called the GI/G/1/V queue, was studied analytically by Gelenbe and Iasnogorodski [3] (see also [4]) and then by Doshi [1] and Fricker [2] who obtained stochastic decomposition results on the waiting-time of the nth customer extending the law decomposition result of [3]. Keilson and Servi [5] give a more complete analytic method of treating both the GI/G/1/V model and the Bernoulli vacation model: instead of the waiting time, they use a bivariate process at the service and vacation initiation epochs and the waiting-time distribution is computed as a conditional distribution of the above. In this note the law decomposition result is obtained from a reduction to the GI/G/1/V model with a modified service-time distribution just using the waiting time, with simple path arguments so that by [1] and [2] stochastic decomposition results are valid, which extend the result of [5].

We study cyclic polling models with exhaustive service at each queue under a variety of non-FCFS (first-come-first-served) local service orders, namely last-come-first-served with and without preemption, random-order-of-service, processor sharing, the multi-class priority scheduling with and without preemption, shortest-job-first, and the shortest remaining processing time policy. For each of these policies, we first express the waiting-time distributions in terms of intervisit-time distributions. Next, we use these expressions to derive the asymptotic waiting-time distributions under heavy-traffic assumptions, i.e. when the system tends to saturate. The results show that in all cases the asymptotic waiting-time distribution at queue i is fully characterized and of the form Γ Θi, with Γ and Θi independent, and where Γ is gamma distributed with known parameters (and the same for all scheduling policies). We derive the distribution of the random variable Θi which explicitly expresses the impact of the local service order on the asymptotic waiting-time distribution. The results provide new fundamental insight into the impact of the local scheduling policy on the performance of a general class of polling models. The asymptotic results suggest simple closed-form approximations for the complete waiting-time distributions for stable systems with arbitrary load values.

Many regenerative arguments in stochastic processes use random times which are akin to stopping times, but which are determined by the future as well as the past behaviour of the process of interest. Such arguments based on ‘conditioning on the future’ are usually developed in an ad-hoc way in the context of the application under consideration, thereby obscuring the underlying structure. In this paper we give a simple, unified, and more general treatment of such conditioning theory. We further give a number of novel applications to various particle system models, in particular to various flavours of contact processes and to infinite-bin models. We give a number of new results for existing and new models. We further make connections with the theory of Harris ergodicity.

Consider randomly scattered radio transceivers in ℝd, each of which can transmit signals to all transceivers in a given randomly chosen region about itself. If a signal is retransmitted by every transceiver that receives it, under what circumstances will a signal propagate to a large distance from its starting point? Put more formally, place points {xi} in ℝd according to a Poisson process with intensity 1. Then, independently for each xi, choose a bounded region Axi from some fixed distribution and let be the random directed graph with vertex set whenever xj ∈ xi + Axi. We show that, for any will almost surely have an infinite directed path, provided the expected number of transceivers that can receive a signal directly from xi is at least 1 + η, and the regions xi + Axi do not overlap too much (in a sense that we shall make precise). One example where these conditions hold, and so gives rise to percolation, is in ℝd, with each Axi a ball of volume 1 + η centred at xi, where η → 0 as d → ∞. Another example is in two dimensions, where the Axi are sectors of angle ε γ and area 1 + η, uniformly randomly oriented within a fixed angle (1 + ε)θ. In this case we can let η → 0 as ε → 0 and still obtain percolation. The result is already known for the annulus, i.e. that the critical area tends to 1 as the ratio of the radii tends to 1, while it is known to be false for the square (l∞) annulus. Our results show that it does however hold for the randomly oriented square annulus.

We study the ruin probability where the claim sizes are modeled by a stationary ergodic symmetric α-stable process. We exploit the flow representation of such processes, and we consider the processes generated by conservative flows. We focus on two classes of conservative α-stable processes (one discrete-time and one continuous-time), and give results for the order of magnitude of the ruin probability as the initial capital goes to infinity. We also prove a solidarity property for null-recurrent Markov chains as an auxiliary result, which might be of independent interest.

Let X1, X2,… be real-valued random variables. For u>0, define the time of ruin T = T(u) by T = inf{n: X1+⋯+Xn>u} or T=∞ if X1+⋯+Xn≤u for every n = 1,2,…. We are interested in the ruin probabilities of general processes {Xn} for large u. In the presence of heavy tails, one often finds power estimates. Our objective is to specify the associated powers and provide the crude estimate P(T≤xu)≈u−R(x) for large u, for a given x∈ℝ. The rate R(x) will be described by means of tails of partial sums and maxima of {Xn}. We also extend our results to the case of the infinite time horizon.

We propose and study a number of layer methods for Navier‒Stokes equations (NSEs) with spatial periodic boundary conditions. The methods are constructed using probabilistic representations of solutions to NSEs and exploiting ideas of the weak sense numerical integration of stochastic differential equations. Despite their probabilistic nature, the layer methods are nevertheless deterministic.

This paper deals with the thrown string problem posed by Synge [12]. The models studied are those of Willenborg [14] and Kingman [7]. The convergence results obtained are complementary to those of Kingman, although they are derived by quite different methods involving the use of martingales.

In their paper [2] Linnik and Zinger proved a theorem which has an important role in both theoretical and applied statistics. For the proof of their theorem the authors used a condition which greatly narrows down its applicability. In this paper we prove the theorem of Linnik and Zinger without this condition, and also give other extensions of it.

This paper gives the exact evaluation of the shape density on the shape space Σ(S2, 3) for a labelled random spherical triangle whose vertices are i.i.d.-uniform in a ‘cap' of S2 bounded by a ‘small' circle of angular radius ρ0.

In this paper we introduce and solve a generalization of the classic average cost Brownian control problem in which a system manager dynamically controls the drift rate of a diffusion process X. At each instant, the system manager chooses the drift rate from a pair {u, v} of available rates and can invoke instantaneous controls either to keep X from falling or to keep it from rising. The objective is to minimize the long-run average cost consisting of holding or delay costs, processing costs, costs for invoking instantaneous controls, and fixed costs for changing the drift rate. We provide necessary and sufficient conditions on the cost parameters to ensure the problem admits a finite optimal solution. When it does, a simple control band policy specifying economic buffer sizes (α, Ω) and up to two switching points is optimal. The controller should invoke instantaneous controls to keep X in the interval (α, Ω). A policy with no switching points relies on a single drift rate exclusively. When there is no cost to change the drift rate, a policy with a single switching point s indicates that the controller should change to the slower drift rate when X exceeds s and use the faster drift rate otherwise. When there is a cost to change the drift rate, a policy with two switching points s < S indicates that the controller should maintain the faster drift rate until X exceeds S and maintain the slower drift rate until X falls below s.

We consider de Finetti’s problem for spectrally one-sided Lévy risk models with control strategies that are absolutely continuous with respect to the Lebesgue measure. Furthermore, we consider the version with a constraint on the time of ruin. To characterize the solution to the aforementioned models, we first solve the optimal dividend problem with a terminal value at ruin and show the optimality of threshold strategies. Next, we introduce the dual Lagrangian problem and show that the complementary slackness conditions are satisfied, characterizing the optimal Lagrange multiplier. Finally, we illustrate our findings with a series of numerical examples.

In this paper, we solve exit problems for a one-sided Markov additive process (MAP) which is exponentially killed with a bivariate killing intensity $\omega(\cdot,\cdot)$ dependent on the present level of the process and the current state of the environment. Moreover, we analyze the respective resolvents. All identities are expressed in terms of new generalizations of classical scale matrices for MAPs. We also remark on a number of applications of the obtained identities to (controlled) insurance risk processes. In particular, we show that our results can be applied to the Omega model, where bankruptcy takes place at rate $\omega(\cdot,\cdot)$ when the surplus process becomes negative. Finally, we consider Markov-modulated Brownian motion (MMBM) as a special case and present analytical and numerical results for a particular choice of piecewise intensity function $\omega(\cdot,\cdot)$ .

The ‘scanning process' Z(t), t ∈ ℝk of the title is a Gaussian random field obtained by associating with Z(t) the value of a set-indexed Brownian motion on the translate t + A0 of some ‘scanning set' A0. We study the basic properties of the random field Z relating, for example, its continuity and other sample path properties to the geometrical properties of A0. We ask if the set A0 determines the scanning process, and investigate when, and how, it is possible to recover the structure of A0 from realisations of the sample paths of the random field Z.

Recently, systematic sampling on the circle and the sphere has been studied by Gual-Arnau and Cruz-Orive (2000) from a design-based point of view. In this note, it is shown that their mathematical model for the covariogram is, in a model-based statistical setting, a special case of the p-order shape model suggested by Hobolth, Pedersen and Jensen (2000) and Hobolth, Kent and Dryden (2002) for planar objects without landmarks. Benefits of this observation include an alternative variance estimator, applicable in the original problem of systematic sampling. In a wider perspective, the paper contributes to the discussion concerning design-based versus model-based stereology.

This paper studies the GI/G/1 queueing system in which no customer can stay longer than a fixed interval D. This is also a model for the dam with finite capacity, instantaneous water supply and constant release rule. Using analytical method together with the property that the queueing process ‘starts anew’ probabilistically whenever an arriving customer initiates a busy period, we obtain various transient and stationary results for the system.

Generalizing the classical Banach matchbox problem, we consider the process of removing two types of ‘items’ from a ‘pile’ with selection probabilities for the type of the next item to be removed depending on the current numbers of remaining items, and thus changing sequentially. Under various conditions on the probability pn1,n2 that the next removal will take away an item of type I, given that n1 and n2 are the current numbers of items of the two types, we derive asymptotic formulas (as the initial pile size tends to infinity) for the probability that the items of type I are completely removed first and for the number of items left. In some special cases we also obtain explicit results.