We consider a family of M(t)/M(t)/1/1 loss systems with arrival and service intensities (λt (c), μt (c)) = (λct, μct), where (λt, μt) are governed by an irreducible Markov process with infinitesimal generator Q = (qij)m × m such that (λt, μt) = (λi, μi) when the Markov process is in state i. Based on matrix analysis we show that the blocking probability is decreasing in c in the interval [0, c∗], where c∗ = 1/maxi Σj≠iqij/(λi + μi). Two special cases are studied for which the result can be extended to all c. These results support Ross's conjecture that a more regular arrival (and service) process leads to a smaller blocking probability.

In first-passage percolation models, the passage time T(0,L) from the origin to a point L is expected to exhibit deviations of order |L|χ from its mean, while minimizing paths are expected to exhibit fluctuations of order |L|ξ away from the straight line segment . Here, for Euclidean models in dimension d, we establish the lower bounds ξ ≥ 1/(d+1) and χ ≥(1-(d-1)ξ)/2. Combining this latter bound with the known upper bound ξ ≤ 3/4 yields that χ ≥ 1/8 for d=2.

Kingman (1962) studied the effect of queue discipline on the mean and variance of the waiting time. He made no assumptions regarding the stochastic nature of the input and the service distributions, except that the input and service processes are independent of each other. When the following two conditions hold:

1. (a) no server sits idle while there are customers waiting to be served;

2. (b) the busy period is finite with probability one (i.e., the queue empties infinitely often with probability one);

A limit theorem is proved for semi-Markov processes, which depend on a small parameter, tending to 0, in the case when the processes have an absorbing state and some asymptotically non-essential states and one asymptotically essential state. The application of the theorem is illustrated by an example from reliability theory.

An algorithm is suggested for approximating the performance of a D/G/K loss system with deterministic input, generally distributed processing time, K heterogeneous servers, the random access processing discipline, and retrials in steady state. In loss systems with retrials, the units which at the instants of their arrival at the system find all the servers busy, are not lost: those units retry to be processed by merging with the incoming arrival units. In this system, a fraction of the units which have not initially been processed will be allowed to leave the system. The performance of this system in steady state is approximated by a recursive technique.

We describe a simple relation between the asymptotic behavior of the variance and of the expected number of distinct sites visited during a correlated random walk. The relation is valid for multistate random walks with finite variance in dimensions 1 and 2. A similar relation, valid in all dimensions, exists between the asymptotic behavior of the variance and of the probability of return to the origin.

We consider percolation on the graph of the product of a regular tree T with degree d and the line ℤ, in which each tree edge is open with probability 1 – exp(–JTß) and each line edge is open with probability . Let C(o, 0) be the open cluster for . Denote by θ (β) the percolation probability. Here we show that θ (β) is continuous when ß > ßc, where ßc = sup{ß : θ(β) = 0}.

Results of a previous paper (Liebetrau (1977a)) are extended to higher dimensions. An estimator V∗(t1, t2) of the variance function V(t1, t2) of a two-dimensional process is defined, and its first- and second-moment structure is given assuming the process to be Poisson. Members of a class of estimators of the form where and for 0 < α i < 1, are shown to converge weakly to a non-stationary Gaussian process. Similar results hold when the t′i are taken to be constants, when V is replaced by a suitable estimator and when the dimensionality of the underlying Poisson process is greater than two.

We consider a class of multitype Galton–Watson branching processes with a countably infinite type set $\mathcal{X}_d$ whose mean progeny matrices have a block lower Hessenberg form. For these processes, we study the probabilities $\textbf{\textit{q}}(A)$ of extinction in sets of types $A\subseteq \mathcal{X}_d$ . We compare $\textbf{\textit{q}}(A)$ with the global extinction probability $\textbf{\textit{q}} = \textbf{\textit{q}}(\mathcal{X}_d)$ , that is, the probability that the population eventually becomes empty, and with the partial extinction probability $\tilde{\textbf{\textit{q}}}$ , that is, the probability that all types eventually disappear from the population. After deriving partial and global extinction criteria, we develop conditions for $\textbf{\textit{q}} < \textbf{\textit{q}}(A) < \tilde{\textbf{\textit{q}}}$ . We then present an iterative method to compute the vector $\textbf{\textit{q}}(A)$ for any set A. Finally, we investigate the location of the vectors $\textbf{\textit{q}}(A)$ in the set of fixed points of the progeny generating vector.

In this paper we give a solution of an optimal stopping problem concerning random walks with non-zero drift, thereby proving the necessity of the existence of ESτ for Wald's equation ESτ = ES1 · Ετ to hold, even if attention is restricted to non-randomized stopping times τ. This answers a question of Robbins and Samuel (1966).

We develop an integration by parts technique for point processes, with application to the computation of sensitivities via Monte Carlo simulations in stochastic models with jumps. The method is applied to density estimation with respect to the Lebesgue measure via a modified kernel estimator which is less sensitive to variations of the bandwidth parameter than standard kernel estimators. This applies to random variables whose densities are not analytically known and requires the knowledge of the point process jump times.

A random variable Y is branching stable (B-stable) for a nonnegative integer-valued random variable J with E(J)>1 if Y*J∿cY for some scalar c, where Y*J is the sum of J independent copies of Y. We explore some aspects of this notion of stability and show that, for any Y0 with finite nonzero mean, if we define Yn+1=Yn*J/E(J) then the sequence Yn converges in law to a random variable Y∞ that is B-stable for J. Also Y∞ is the unique B-stable law with mean E(Y0). We also present results relating to random variables Y0 with zero means and infinite means. The notion of B-stability arose in a scheme for cataloguing a large network of computers.

We look at joint regular variation properties of MA(∞) processes of the form X = (Xk, k ∈ Z), where Xk = ∑j=0∞ψjZk-j and the sequence of random variables (Zi, i ∈ Z) are independent and identically distributed with regularly varying tails. We use the setup of MO-convergence and obtain hidden regular variation properties for X under summability conditions on the constant coefficients (ψj: j ≥ 0). Our approach emphasizes continuity properties of mappings and produces regular variation in sequence space.

This paper is concerned with the description of both a deterministic and stochastic branching procedure. The renewal equations for the deterministic branching population are first derived which allow for asymptotic results on the ‘number' and ‘generation' processes. A probabilistic version of these processes is then studied which presents some discrepancy with the standard Harris age-dependent branching processes.

A sequence of independent, identically distributed random variables is observed. After a sample of m has been collected attention is fixed to a particular observation. In this paper the fluctuations of the rank of this observation, as sampling continues, will be studied.

The process can be modelled by a Pólya urn scheme, and new results are obtained which are of interest in both sequential rank and Pólya urn contexts.

The equilibrium delay distribution for a system with finite-source input, constant service time, and s > 1 servers, as determined by simulation, showed the existence of values of t such that the probability of delay greater than t is a non-monotone function of the input intensity. Since repeated efforts to remove this counter-intuitive result by debugging failed, a simple case was attacked by analytical and numerical methods; and the same anomaly was revealed. The method of analysis is described and an explanation of the result is offered.

The problem of constructing an optimal coadapted coupling for a pair of symmetric random walks on Z2d was considered by Connor and Jacka (2008), and the existence of a coupling which is stochastically fastest in the class of all such coadapted couplings was demonstrated. In this paper we show how to generalise this construction to an optimal coadapted coupling for the continuous-time symmetric random walk on Knd, where Kn is the complete graph with n vertices. Moreover, we show that although this coupling is not maximal for any n (i.e. it does not achieve equality in the coupling inequality), it does tend to a maximal coupling as n → ∞.

In this paper, to model cascading failures, a new stochastic failure model is proposed. In a system subject to cascading failures, after each failure of the component, the remaining component suffers from increased load or stress. This results in shortened residual lifetimes of the remaining components. In this paper, to model this effect, the concept of the usual stochastic order is employed along with the accelerated life test model, and a new general class of stochastic failure models is generated.