This paper considers ergodic, continuous-time Markov chains $\{X(t)\}_{t \in (\!-\infty,\infty)}$ on $\mathbb{Z}^+=\{0,1,\ldots\}$ . For an arbitrarily fixed $N \in \mathbb{Z}^+$ , we study the conditional stationary distribution $\boldsymbol{\pi}(N)$ given the Markov chain being in $\{0,1,\ldots,N\}$ . We first characterize $\boldsymbol{\pi}(N)$ via systems of linear inequalities and identify simplices that contain $\boldsymbol{\pi}(N)$ , by examining the $(N+1) \times (N+1)$ northwest corner block of the infinitesimal generator $\textbf{\textit{Q}}$ and the subset of the first $N+1$ states whose members are directly reachable from at least one state in $\{N+1,N+2,\ldots\}$ . These results are closely related to the augmented truncation approximation (ATA), and we provide some practical implications for the ATA. Next we consider an extension of the above results, using the $(K+1) \times (K+1)$ ( $K > N$ ) northwest corner block of $\textbf{\textit{Q}}$ and the subset of the first $K+1$ states whose members are directly reachable from at least one state in $\{K+1,K+2,\ldots\}$ . Furthermore, we introduce new state transition structures called (K, N)-skip-free sets, using which we obtain the minimum convex polytope that contains $\boldsymbol{\pi}(N)$ .

We study networks of interacting queues governed by utility-maximising service-rate allocations in both discrete and continuous time. For finite networks we establish stability and some steady-state moment bounds under natural conditions and rather weak assumptions on utility functions. These results are obtained using direct applications of Lyapunov–Foster-type criteria, and apply to a wide class of systems, including those for which fluid-limit-based approaches are not applicable. We then establish stability and some steady-state moment bounds for two classes of infinite networks, with single-hop and multi-hop message routes. These results are proved by considering the infinite systems as limits of their truncated finite versions. The uniform moment bounds for the finite networks play a key role in these limit transitions.

Whole genome amplification is important for multipoint mapping by sperm or oocyte typing and genetic disease diagnosis. Polymerase chain reaction is not suitable for amplifying long DNA sequences. This paper studies a new technique, designated PEP-primer-extension-preamplification, for amplifying long DNA sequences using the theory of branching processes. A mathematical model for PEP is constructed and a closed formula for the expected target yield is obtained. A central limit theorem and a strong law of large numbers for the number of kth generation target sequences are proved.

We use a simple coupling to prove the classical result that the renewal function U of a zero-delayed renewal process satisfies U(t) – λ . t→λ2μ2/2 as t→∞ if the life-length distribution is of non-lattice type and has finite first and second moments μ and μ2 respectively; λ is the renewal intensity, and is equal to 1/μ.

While the convex hull of n d-dimensional balls is not a polytope, it does have an underlying combinatorial structure similar to that of a polytope. In the worst case, its combinatorial complexity can be of order Ω(n[d/2]). The thrust of this work is to show that its complexity is typically much smaller, and that it can therefore be constructed more quickly on average than in the worst case. To this end, four models of the random d-ball are developed, and the expected combinatorial complexity of the convex hull of n independent random d-balls is investigated for each. As n grows without bound, these expectations are O(1), O(n(d–1)/(d+4)), O(1) (for d = 2 only), and O(n(d–1)/(d+3)).

We consider the stationary solution Z of the Markov chain {Zn}n∈ℕ defined by Zn+1=ψn+1(Zn), where {ψn}n∈ℕ is a sequence of independent and identically distributed random Lipschitz functions. We estimate the probability of the event {Z>x} when x is large, and develop a state-dependent importance sampling estimator under a set of assumptions on ψn such that, for large x, the event {Z>x} is governed by a single large jump. Under natural conditions, we show that our estimator is strongly efficient. Special attention is paid to a class of perpetuities with heavy tails.

Three different stereological methods for the determination of second-order quantities of planar fibre processes which have been suggested in the literature are considered. Proofs of the formulae are given (also by using a new integral geometric formula), relations between the methods are derived and the prerequisites are discussed. Furthermore, edge-corrected unbiased estimators for the second-order quantities are given.

This paper establishes structural properties for the throughput of a large class of queueing networks with i.i.d. new-better-than-used service times. The main result obtained in this paper is applied to a wide range of networks, including tandems, cycles and fork-join networks with general blocking and starvation (as well as certain networks with splitting and merging of traffic streams), to deduce the concavity of their throughput as a function of system parameters, such as buffer and initial job configurations, and blocking and starvation parameters. These results have important implications for the optimal design and control of such queueing networks by providing exact solutions, reducing the search space over which optimization need be performed, or establishing the convergence of optimization algorithms. In order to obtain results for such disparate networks in a unified manner, we introduce the framework of constrained discrete event systems (CDES), which enables us to characterize any permutable and non-interruptive queueing network through its constraint set. The main result of this paper establishes comparison properties of the event occurrence processes of CDES as a function of the constraint sets, which are then translated into the above-mentioned concavity of the throughput as a function of system parameters in the context of queueing networks.

We consider a controlled diffusion process, the description of which depends on an unknown parameter α, and investigate the following control policy. To each α an optimal stationary control is associated. α is estimated recurrently from the trajectory by Bayes' method, and the optimal stationary control corresponding to the estimate is used. We establish the consistency of the estimate, and present asymptotic properties of the criterion function. They follow from the central limit theorem, from the law of large numbers and from the law of the iterated logarithm for local martingales.

In this paper we develop methods for reducing the study, the computation, and the construction of stochastic functionals to those of standard concepts such as the moments of the pertinent random variables. Principally, our methods are based on the notion of ladder height densities and their Laguerre expansions, and our results provide a unifying framework for the distinct approaches of Dufresne (2000) and Schröder (2005).

For the n-dimensional birth, death and migration process with constant rates, Aksland (1975) found a necessary and sufficient condition that the extinction-probability be one. This condition is translated into the form of direct relations between the parameters of the model.

The primary objective in the present paper is to gain fundamental understanding of the performance achievable in ATM networks as a function of the various system characteristics. We derive limit theorems that characterize the achievable performance in terms of the offered traffic, the admissible region, and the revenue measure. The insights obtained allow for substantial simplifications in the design of real-time connection admission control algorithms. In particular, we describe how the boundaries of admissible regions with convex complements may be linearized - thus reducing the admissible region - so as to obtain a convenient loss network representation. The asymptotic results for the achievable performance suggest that the potential reduction in revenue is immaterial in high-capacity networks. Numerical experiments confirm that the actual reduction is typically negligible, even in networks of moderate capacity.

We consider a queue which is obtained by slightly modifying the M/PH/1 queue, so that weak correlations are introduced among the interarrival times. We show how the stationary probability distribution may be studied by a perturbation analysis.

We propose and study a random crystalline algorithm (a discrete approximation) of the Gauss curvature flow of smooth simple closed convex curves in ℝ2 as a stepping stone to the full understanding of such phenomena as the wearing process of stones on a beach.

Consider a bundle that has been constructed by placing n fibers in parallel between two clamps. The simplest assumption on fiber load sharing is that all non-failed fibers share the load applied to the bundle equally. Phoenix (1978b) shows that, as n tends to ∞, the time to failure of such a bundle is asymptotically normally distributed under certain assumptions on the stochastic nature of the fibers. In this paper we show that this asymptotic normality is preserved but with changed mean and variance if random slacks are introduced, as caused by random variations in the lengths of the fibers. A theorem on random linear combinations of order statistics, which is given in the appendix, is used to prove the result.

Let X1, X2,… be i.i.d. random points in ℝ2 with distribution ν, and let Nn denote the number of points spanning the convex hull of X1, X2,…,Xn. We obtain lim infn→∞E(Nn)n-1/3 ≥ γ1 and E(Nn) ≤ γ2n1/3(logn)2/3 for some positive constants γ1, γ2 and sufficiently large n under the assumption that ν is a certain self-similar measure on the unit disk. Our main tool consists in a geometric application of the renewal theorem. Exactly the same approach can be adopted to prove the analogous result in ℝd.

We consider a birth and death process with Q-matrix of rates qm,m + 1 = mβ, qm,m − 1 = mδ, qm,m = – m(β + δ) and qm,n = 0 otherwise. We assume that 0 < δ < β and β – δ = 1. The asymptotic behavior of first-arrival time at state n given that the process is at state m at time zero is expressed in terms of polynomials and it is shown that if m < n and m is large that the first-arrival time is close to log n/m with high probability.