We study a variant of Gilbert's disc model, in which discs are positioned at the points of a Poisson process in ℝ2 with radii determined by an underlying stationary and ergodic random field φ:ℝ2→[0,∞), independent of the Poisson process. This setting, in which the random field is independent of the point process, is often referred to as geostatistical marking. We examine how typical properties of interest in stochastic geometry and percolation theory, such as coverage probabilities and the existence of long-range connections, differ between Gilbert's model with radii given by some random field and Gilbert's model with radii assigned independently, but with the same marginal distribution. Among our main observations we find that complete coverage of ℝ2 does not necessarily happen simultaneously, and that the spatial dependence induced by the random field may both increase as well as decrease the critical threshold for percolation.

In this paper, we study a large multi-server loss model under the SQ(d) routeing scheme when the service time distributions are general with finite mean. Previous works have addressed the exponential service time case when the number of servers goes to infinity, giving rise to a mean field model. The fixed point of the limiting mean field equations (MFEs) was seen to be insensitive to the service time distribution in simulations, but no proof was available. While insensitivity is well known for loss systems, the models, even with state-dependent inputs, belong to the class of linear Markov models. In the context of SQ(d) routeing, the resulting model belongs to the class of nonlinear Markov processes (processes whose generator itself depends on the distribution) for which traditional arguments do not directly apply. Showing insensitivity to the general service time distributions has thus remained an open problem. Obtaining the MFEs in this case poses a challenge due to the resulting Markov description of the system being in positive orthant as opposed to a finite chain in the exponential case. In this paper, we first obtain the MFEs and then show that the MFEs have a unique fixed point that coincides with the fixed point in the exponential case, thus establishing insensitivity. The approach is via a measure-valued Markov process representation and the martingale problem to establish the mean field limit.

This paper is the third in a series with the general title ‘How to look at the five-dimensional shape space Σ4,3. Parts I and II were concerned with distributions and diffusions respectively. Here we ‘look at’ geodesics.

Let (Y, Z) denote the solution to a forward-backward stochastic differential equation (FBSDE). If one constructs a random walk $B^n$ from the underlying Brownian motion B by Skorokhod embedding, one can show $L_2$-convergence of the corresponding solutions $(Y^n,Z^n)$ to $(Y, Z).$ We estimate the rate of convergence based on smoothness properties, especially for a terminal condition function in $C^{2,\alpha}$. The proof relies on an approximative representation of $Z^n$ and uses the concept of discretized Malliavin calculus. Moreover, we use growth and smoothness properties of the partial differential equation associated to the FBSDE, as well as of the finite difference equations associated to the approximating stochastic equations. We derive these properties by probabilistic methods.

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.

The purpose of the paper is to study the asymptotic geometry of a smooth-grained Boolean model (X[t])t≥0 restricted to a bounded domain as the intensity parameter t goes to ∞. Our approach is based on investigating the asymptotic properties as t → ∞ of the random sets X[t;β], β≥0, defined as the Gibbsian modifications of X[t] with the Hamiltonian given by βtμ(·), where μ is a certain normalized measure on the setting space. We show that our model exhibits a phase transition at a certain critical value of the inverse temperature β and we prove that at higher temperatures the behaviour of X[t;β] is qualitatively very similar to that of X[t] but it becomes essentially different in the low-temperature region. From these facts we derive information about the asymptotic properties of the original process X[t]. The results obtained include large- and moderate-deviation principles. We conclude the paper with an example application of our methods to analyse the asymptotic moderate-deviation properties of convex hulls of large uniform samples on a multidimensional ball. To translate the above problem to the Boolean model setting considered we use an appropriate representation of convex sets in terms of their support functions.

A single machine is available to process a collection of jobs J, each of which evolves stochastically under processing. Jobs incur costs while awaiting the machine at a rate which is state dependent and processing must respect a set of precedence constraints Γ. Index policies are optimal in a variety of scenarios. The indices concerned are characterised as values of restart problems with the average reward criterion. This characterisation yields a range of efficient approaches to their computation. Index-based suboptimality bounds are derived for general processing policies. These bounds enable us to develop sensitivity analyses and to evaluate scheduling heuristics.

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.

A Fourier series is obtained for the variance of the number of points lying in a parallelogram thrown randomly on to a square lattice at a fixed angle.

Within an anticipative stochastic calculus framework, we study a market game with asymmetric information and feedback effects. We derive necessary and sufficient criteria for the existence of Nash equilibria and study how general welfare is affected by the level of information. In particular, we show that, under certain conditions in a competitive environment, an increased level of information may in fact lower the level of general welfare, leading to the so-called Hirshleifer effect (see Hirshleifer (1971)). Finally, we determine equilibrium prices for particular pieces of information, by extending our market game with a pre-stage, in which information is traded.

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.

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.

Partial differential equations are powerful tools for used to characterizing various physical systems. In practice, measurement errors are often present and probability models are employed to account for such uncertainties. In this paper we present a Monte Carlo scheme that yields unbiased estimators for expectations of random elliptic partial differential equations. This algorithm combines a multilevel Monte Carlo method (Giles (2008)) and a randomization scheme proposed by Rhee and Glynn (2012), (2013). Furthermore, to obtain an estimator with both finite variance and finite expected computational cost, we employ higher-order approximations.

In this paper we show that the waiting and the sojourn times of a customer in a single-stage, multiple-server, G/G/c queueing system are increasing and starshaped with respect to the mean service time. Usefulness of this result in the design of the optimal service speed in the G/G/c queueing system is also demonstrated.