Working in the infinite plane R2, consider a Poisson process of black points with intensity 1, and an independent Poisson process of red points with intensity λ. We grow a disc around each black point until it hits the nearest red point, resulting in a random configuration Aλ, which is the union of discs centered at the black points. Next, consider a fixed disc of area n in the plane. What is the probability pλ(n) that this disc is covered by Aλ? We prove that if λ3nlogn = y then, for sufficiently large n, e-8π2y ≤ pλ(n) ≤ e-2π2y/3. The proofs reveal a new and surprising phenomenon, namely, that the obstructions to coverage occur on a wide range of scales.

Queues with advanced reservations are endemic in the real world. In such a queue, the 'arrival' process is an incoming stream of customer 'booking requests', rather than actual customers requiring immediate service. We consider a model with a Poisson booking request process with rate λ. Associated with each request is a pair of independent random variables (Ri, Si) constituting a request for service over a period Si, starting at a time Ri into the future. Our interest is in the probability that a customer will be rejected due to capacity constraints. We present a simulation of a finite-capacity queue in which we record the proportion of rejected customers, and then move to an analysis of a queue with infinitely-many servers. Obviously no customers are rejected in the latter case. However, the event that the arrival of the extra customer will cause the number of customers in the queue to exceed C at some point during its service can be used as a proxy for the event that the customer would have been rejected in a system with finite capacity C. We start by calculating the transient and stationary distributions for some performance measures for the infinite-server queue. By observing that the stationary measure for the bookings diary (that is, the list of customers currently on hand, together with their start times and service times) is the same as the law for the entire sample path of an infinite server queue with a specified nonhomogenous Poisson input process, which we call the bookings queue, we are able to write down expressions for the abovementioned probability that, at some time during a requested service, the number of customers exceeds C. This measure serves as a bound for the probability that an incoming arrival would be refused admission in a system with C servers and, for a well-dimensioned system, it is to be hoped that it is a good approximation. We test the quality of this approximation by comparing our analytical results for the infinite-server case against simulation results for the finite-server case.

We consider a class of multi-type particle systems whose structure is similar to that of a contact process and show that additivity is equivalent to the existence of a dual process, extending a result of Harris. We prove a necessary and sufficient condition for the model to preserve positive correlations. We then show that complete convergence on Zd holds for a large subclass of models including the two-stage contact process and a household model, and give examples.

The full-information duration problem with a random number N of objects is considered. These objects appear sequentially and their values Xk are observed, where Xk, independent of N, are independent and identically distributed random variables from a known continuous distribution. The objective of the problem is to find a stopping rule that maximizes the duration of holding a relative maximum (e.g. the kth object is a relative maximum if Xk = max{X1, X2, . . ., Xk}). We assume that N is a random variable with a known upper bound n, so two models, Model 1 and Model 2, can be considered according to whether the planning horizon is N or n. The structure of the optimal rule, which depends on the prior distribution assumed on N, is examined. The monotone rule is defined and a necessary and sufficient condition for the optimal rule to be monotone is given for both models. Special attention is paid to the class of priors such that N / n converges, as n → ∞, to a random variable Vm having density fVm(v) = m(1 - v)m-1, 0 ≤ v ≤ 1 for a positive integer m. An interesting feature is that the optimal duration (relative to n) for Model 2 is just (m + 1) times as large as that for Model 1 asymptotically.

We study the asymptotic behavior of a new particle filter approach for the estimation of hidden Markov models. In particular, we develop an algorithm where the latent-state sequence is segmented into multiple shorter portions, with an estimation technique based upon a separate particle filter in each portion. The partitioning facilitates the use of parallel processing, which reduces the wall-clock computational time. Based upon this approach, we introduce new estimators of the latent states and likelihood which have similar or better variance properties compared to estimators derived from standard particle filters. We show that the likelihood function estimator is unbiased, and show asymptotic normality of the underlying estimators.

In this paper we investigate different methods that may be used to compare coherent systems having heterogeneous components. We consider both the case of systems with independent components and the case of systems with dependent components. In the first case, the comparisons are based on the new concept of the survival signature due to Coolen and Coolen-Maturi (2012) which extends the well-known concept of system signatures to the case of components with lifetimes that need not be independent and identically distributed. In the second case, the comparisons are based on the concept of distortion functions. A graphical procedure (called an RR-plot) is proposed as an alternative to the analytical methods when there are two types of components.

We propose a novel theoretical characterization of the optimal 'Gittins index' policy in multi-armed bandit problems with non-Gaussian, infinitely divisible reward distributions. We first construct a continuous-time, conditional Lévy process which probabilistically interpolates the sequence of discrete-time rewards. When the rewards are Gaussian, this approach enables an easy connection to the convenient time-change properties of a Brownian motion. Although no such device is available in general for the non-Gaussian case, we use optimal stopping theory to characterize the value of the optimal policy as the solution to a free-boundary partial integro-differential equation (PIDE). We provide the free-boundary PIDE in explicit form under the specific settings of exponential and Poisson rewards. We also prove continuity and monotonicity properties of the Gittins index in these two problems, and discuss how the PIDE can be solved numerically to find the optimal index value of a given belief state.

We propose a class of random scale-free spatial networks with nested community structures called SHEM and analyze Reed–Frost epidemics with community related independent transmissions. We show that in a specific example of the SHEM the epidemic threshold may be trivial or not as a function of the relation among community sizes, distribution of the number of communities, and transmission rates.

We investigate the degree profile and total weight in Apollonian networks. We study the distribution of the degrees of vertices as they age in the evolutionary process. Asymptotically, the (suitably-scaled) degree of a node with a fixed label has a Mittag-Leffler-like limit distribution. The degrees of nodes of later ages have different asymptotic distributions, influenced by the time of their appearance. The very late arrivals have a degenerate distribution. The result is obtained via triangular Pólya urns. Also, via the Bagchi–Pal urn, we show that the number of terminal nodes asymptotically follows a Gaussian law. We prove that the total weight of the network asymptotically follows a Gaussian law, obtained via martingale methods. Similar results carry over to the sister structure of the k-trees, with minor modification in the proof methods, done mutatis mutandis.

We consider Markov processes, which describe, e.g. queueing network processes, in a random environment which influences the network by determining random breakdown of nodes, and the necessity of repair thereafter. Starting from an explicit steady-state distribution of product form available in the literature, we note that this steady-state distribution does not provide information about the correlation structure in time and space (over nodes). We study this correlation structure via one-step correlations for the queueing-environment process. Although formulas for absolute values of these correlations are complicated, the differences of correlations of related networks are simple and have a nice structure. We therefore compare two networks in a random environment having the same invariant distribution, and focus on the time behaviour of the processes when in such a network the environment changes or the rules for travelling are perturbed. Evaluating the comparison formulas we compare spectral gaps and asymptotic variances of related processes.

We provide exact computations for the drift of random walks in dependent random environments, including k-dependent and moving average environments. We show how the drift can be characterized and evaluated using Perron–Frobenius theory. Comparing random walks in various dependent environments, we demonstrate that their drifts can exhibit interesting behavior that depends significantly on the dependency structure of the random environment.

Let N and M be positive integers satisfying 1≤ M≤ N, and let 0< p0 < p1 < 1. Define a process {Xn}n=0∞ on ℤ as follows. At each step, the process jumps either one step to the right or one step to the left, according to the following mechanism. For the first N steps, the process behaves like a random walk that jumps to the right with probability p0 and to the left with probability 1-p0. At subsequent steps the jump mechanism is defined as follows: if at least M out of the N most recent jumps were to the right, then the probability of jumping to the right is p1; however, if fewer than M out of the N most recent jumps were to the right then the probability of jumping to the right is p0. We calculate the speed of the process. Then we let N→ ∞ and M/N→ r∈[0,1], and calculate the limiting speed. More generally, we consider the above questions for a random walk with a finite number l of threshold levels, (Mi,pi) i=1l, above the pre-threshold level p0, as well as for one model with l=N such thresholds.

In this paper we consider an Ornstein–Uhlenbeck (OU) process (M(t))t≥0 whose parameters are determined by an external Markov process (X(t))t≥0 on a finite state space {1, . . ., d}; this process is usually referred to as Markov-modulated Ornstein–Uhlenbeck. We use stochastic integration theory to determine explicit expressions for the mean and variance of M(t). Then we establish a system of partial differential equations (PDEs) for the Laplace transform of M(t) and the state X(t) of the background process, jointly for time epochs t = t1, . . ., tK. Then we use this PDE to set up a recursion that yields all moments of M(t) and its stationary counterpart; we also find an expression for the covariance between M(t) and M(t + u). We then establish a functional central limit theorem for M(t) for the situation that certain parameters of the underlying OU processes are scaled, in combination with the modulating Markov process being accelerated; interestingly, specific scalings lead to drastically different limiting processes. We conclude the paper by considering the situation of a single Markov process modulating multiple OU processes.

In this paper we provide a perturbation analysis of finite time-inhomogeneous Markov processes. We derive closed-form representations for the derivative of the transition probability at time t, with t > 0. Elaborating on this result, we derive simple gradient estimators for transient performance characteristics either taken at some fixed point in time t, or for the integrated performance over a time interval [0 , t]. Bounds for transient performance sensitivities are presented as well. Eventually, we identify a structural property of the derivative of the generator matrix of a Markov chain that leads to a significant simplification of the estimators.

We study the distribution Ex[exp(-q∫0t1(a,b)(Xs)ds); Xt ∈ dy], where -∞ ≤ a < b < ∞, and where q, t > 0 and x ∈ R for a spectrally negative Lévy process X. More precisely, we identify the Laplace transform with respect to t of this measure in terms of the scale functions of the underlying process. Our results are then used to price step options and the particular case of an exponential spectrally negative Lévy jump-diffusion model is discussed.

We derive a new equation for the optimal investment boundary of a general irreversible investment problem under exponential Lévy uncertainty. The problem is set as an infinite time-horizon, two-dimensional degenerate singular stochastic control problem. In line with the results recently obtained in a diffusive setting, we show that the optimal boundary is intimately linked to the unique optional solution of an appropriate Bank–El Karoui representation problem. Such a relation and the Wiener–Hopf factorization allow us to derive an integral equation for the optimal investment boundary. In case the underlying Lévy process hits any point in R with positive probability we show that the integral equation for the investment boundary is uniquely satisfied by the unique solution of another equation which is easier to handle. As a remarkable by-product we prove the continuity of the optimal investment boundary. The paper is concluded with explicit results for profit functions of Cobb–Douglas type and CES type. In the former case the function is separable and in the latter case nonseparable.