In 1990, Ramaswami proved that, given a Markov renewal process of M/G/1 type, it is possible to construct a Markov renewal process of GI/M/1 type such that the matrix transforms G(z, s) for the M/G/1-type process and R(z, s) for the GI/M/1-type process satisfy a duality relationship. In his 1996 PhD thesis, Bright used time reversal arguments to show that it is possible to define a different dual for positive-recurrent and transient processes of M/G/1 type and GI/M/1 type. Here we compare the properties of the Ramaswami and Bright dual processes and show that the Bright dual has desirable properties that can be exploited in the design of algorithms for the analysis of Markov chains of GI/M/1 type and M/G/1 type.

We study a class of optimal allocation problems, including the well-known bomber problem, with the following common probabilistic structure. An aircraft equipped with an amount x of ammunition is intercepted by enemy airplanes arriving according to a homogeneous Poisson process over a fixed time duration t. Upon encountering an enemy, the aircraft has the choice of spending any amount 0 ≤ y ≤ x of its ammunition, resulting in the aircraft's survival with probability equal to some known increasing function of y. Two different goals have been considered in the literature concerning the optimal amount K(x, t) of ammunition spent: (i) maximizing the probability of surviving for time t, which is the so-called bomber problem; and (ii) maximizing the number of enemy airplanes shot down during time t, which we call the fighter problem. Several authors have attempted to settle the following conjectures about the monotonicity of K(x, t): (A) K(x, t) is decreasing in t; (B) K(x, t) is increasing in x; and (C) the amount x - K(x, t) held back is increasing in x. Conjectures (A) and (C) have been shown for the bomber problem with discrete ammunition, while (B) is still an open question. In this paper we consider both time and ammunition to be continuous, and, for the bomber problem, we prove (A) and (C), while, for the fighter problem, we prove (A) and (C) for one special case and (B) and (C) for another. These proofs involve showing that the optimal survival probability and optimal number shot down are totally positive of order 2 (TP2) in the bomber and fighter problems, respectively. The TP2 property is shown by constructing convergent sequences of approximating functions through an iterative operation which preserves TP2 and other properties.

The numerical analysis of quasi-birth-and-death processes rests on the resolution of a matrix-quadratic equation for which efficient algorithms are known when the matrices have finite order, that is, when the number of phases is finite. In this paper we consider the case of infinitely many phases from the point of view of theoretical convergence of truncation and augmentation schemes, and we develop four different methods. Two methods rely on forced transitions to the boundary. In one of these methods, the transitions occur as a result of the truncation itself, while in the other method, they are artificially introduced so that the augmentation may be chosen to be as natural as possible. Two other methods rely on forced transitions within the same level. We conclude with a brief numerical illustration.

This paper gives easy proofs of conditional limit laws for the population size Zt of a critical Markov branching process whose offspring law is attracted to a stable law with index 1 + α, where 0 ≤ α ≤ 1. Conditioning events subsume the usual ones, and more general initial laws are considered. The case α = 0 is related to extreme value theory for the Gumbel law.

On the boundary of a Galton-Watson tree we can define the visibility measure by splitting mass equally between the children of each vertex, and the branching measure by splitting unit mass equally between all vertices in the nth generation and then letting n go to infinity. The multifractal structure of each of these measures is well studied. In this paper we address the question of a joint multifractal spectrum, i.e. we ask for the Hausdorff dimension of the boundary points which simultaneously have an unusual local dimension for both these measures. The resulting two-parameter spectrum exhibits a number of surprising new features, among them the emergence of a swallowtail-shaped spectrum for the visibility measure in the presence of a nontrivial condition on the branching measure.

Consider a continuous-time renewal risk model with a constant force of interest. We assume that claim sizes and interarrival times correspondingly form a sequence of independent and identically distributed random pairs and that each pair obeys a dependence structure described via the conditional tail probability of a claim size given the interarrival time before the claim. We focus on determining the impact of this dependence structure on the asymptotic tail probability of discounted aggregate claims. Assuming that the claim size distribution is subexponential, we derive an exact locally uniform asymptotic formula, which quantitatively captures the impact of the dependence structure. When the claim size distribution is extended regularly varying tailed, we show that this asymptotic formula is globally uniform.

The stochastic Lagrange wave model is a realistic alternative to the Gaussian linear wave model, which has been successfully used in ocean engineering for more than half a century. In this paper we present the slope distributions and other characteristic distributions at level crossings for asymmetric Lagrange time waves, i.e. what can be observed at a fixed measuring station, thereby extending results previously given for space waves. The distributions are given as expectations in a multivariate normal distribution, and they have to be evaluated by simulation or numerical integration. Interesting characteristic variables are the slope in time, the slope in space, and the vertical particle velocity when the waves are observed close to instances when the water level crosses a predetermined level. The theory has been made possible by recent generalizations of Rice's formula for the expected number of marked crossings in random fields.

We introduce a formal limit, which we refer to as a fluid limit, of scaled stochastic models for a cache managed with the least-recently-used algorithm when requests are issued according to general stochastic point processes. We define our fluid limit as a superposition of dependent replications of the original system with smaller item sizes when the number of replications approaches ∞. We derive the average probability that a requested item is not in a cache (average miss probability) in the fluid limit. We show that, when requests follow inhomogeneous Poisson processes, the average miss probability in the fluid limit closely approximates that in the original system. Also, we compare the asymptotic characteristics, as the cache size approaches ∞, of the average miss probability in the fluid limit to those in the original system.

File-sharing networks are distributed systems used to disseminate files among nodes of a communication network. The general simple principle of these systems is that once a node has retrieved a file, it may become a server for this file. In this paper, the capacity of these networks is analyzed with a stochastic model when there is a constant flow of incoming requests for a given file. It is shown that the problem can be solved by analyzing the asymptotic behavior of a class of interacting branching processes. Several results of independent interest concerning these branching processes are derived and then used to study the file-sharing systems.

Contemporaneous aggregation of N independent copies of a random-coefficient AR(1) process with random coefficient a ∈ (−1, 1) and independent and identically distributed innovations belonging to the domain of attraction of an α-stable law (0 < α < 2) is discussed. We show that, under the normalization N1/α, the limit aggregate exists, in the sense of weak convergence of finite-dimensional distributions, and is a mixed stable moving average as studied in Surgailis, Rosiński, Mandrekar and Cambanis (1993). We focus on the case where the slope coefficient a has probability density vanishing regularly at a = 1 with exponent b ∈ (0, α − 1) for α ∈ (1, 2). We show that in this case, the limit aggregate {X̅t} exhibits long memory. In particular, for {X̅t}, we investigate the decay of the codifference, the limit of partial sums, and the long-range dependence (sample Allen variance) property of Heyde and Yang (1997).

We consider a system with Poisson arrivals and independent and identically distributed service times, where requests in the system are served according to the state-dependent (Cohen's generalized) processor-sharing discipline, where each request receives a service capacity that depends on the actual number of requests in the system. For this system, we derive expressions as well as tight insensitive upper bounds for the moments of the conditional sojourn time of a request with given required service time. The bounds generalize and extend corresponding results, recently given for the single-server processor-sharing system in Cheung et al. (2006) and for the state-dependent processor-sharing system with exponential service times by the authors (2008). Analogous results hold for the waiting times. Numerical examples for the M/M/m-PS and M/D/m-PS systems illustrate the given bounds.

We describe the processes obtained by time reversal of a class of stationary jump diffusion processes that model the dynamics of genetic variation in populations subject to repeated bottlenecks. Assuming that only one lineage survives each bottleneck, the forward process is a diffusion on [0,1] that jumps to the boundary before diffusing back into the interior. We show that the behavior of the time-reversed process depends on whether the boundaries are accessible to the diffusive motion of the forward process. If a boundary point is inaccessible to the forward diffusion then time reversal leads to a jump diffusion that jumps immediately into the interior whenever it arrives at that point. If, instead, a boundary point is accessible then the jumps off of that point are governed by a weighted local time of the time-reversed process.

We develop classification results for max-stable processes, based on their spectral representations. The structure of max-linear isometries and minimal spectral representations play important roles. We propose a general classification strategy for measurable max-stable processes based on the notion of co-spectral functions. In particular, we discuss the spectrally continuous-discrete, the conservative-dissipative, and the positive-null decompositions. For stationary max-stable processes, the latter two decompositions arise from connections to nonsingular flows and are closely related to the classification of stationary sum-stable processes. The interplay between the introduced decompositions of max-stable processes is further explored. As an example, the Brown-Resnick stationary processes, driven by fractional Brownian motions, are shown to be dissipative.

We study a continuous-time stochastic process on strings made of two types of particle, whose dynamics mimic the behaviour of microtubules in a living cell; namely, the strings evolve via a competition between (local) growth/shrinking as well as (global) hydrolysis processes. We give a complete characterization of the phase diagram of the model, and derive several criteria of the transient and recurrent regimes for the underlying stochastic process.

We propose a model for the presence/absence of a population in a collection of habitat patches. This model assumes that colonisation and extinction of the patches occur as distinct phases. Importantly, the local extinction probabilities are allowed to vary between patches. This permits an investigation of the effect of habitat degradation on the persistence of the population. The limiting behaviour of the model is examined as the number of habitat patches increases to ∞. This is done in the case where the number of patches and the initial number of occupied patches increase at the same rate, and for the case where the initial number of occupied patches remains fixed.

Consider a family of random ordered graph trees (Tn)n≥1, where Tn has n vertices. It has previously been established that if the associated search-depth processes converge to the normalised Brownian excursion when rescaled appropriately as n → ∞, then the simple random walks on the graph trees have the Brownian motion on the Brownian continuum random tree as their scaling limit. Here, this result is extended to demonstrate the existence of a diffusion scaling limit whenever the volume measure on the limiting real tree is nonatomic, supported on the leaves of the limiting tree, and satisfies a polynomial lower bound for the volume of balls. Furthermore, as an application of this generalisation, it is established that the simple random walks on a family of Galton-Watson trees with a critical infinite variance offspring distribution, conditioned on the total number of offspring, can be rescaled to converge to the Brownian motion on a related α-stable tree.

A useful result when dealing with backward stochastic differential equations is the comparison theorem of Peng (1992). When the equations are not based on Brownian motion, the comparison theorem no longer holds in general. In this paper we present a condition for a comparison theorem to hold for backward stochastic differential equations based on arbitrary martingales. This theorem applies to both vector and scalar situations. Applications to the theory of nonlinear expectations are also explored.

Consider the following classical problem in ad-hoc networks. Suppose that n devices are distributed uniformly at random in a given region. Each device is allowed to choose its own transmission radius, and two devices can communicate if and only if they are within the transmission radius of each other. The aim is to (quickly) establish a connected network of low average and maximum degree. In this paper we present the first efficient distributed protocols that, in poly-logarithmically many rounds and with high probability, set up a connected network with O(1) average degree and O(log n) maximum degree. Our algorithms are based on the following result, which is a nontrivial consequence of classical percolation theory. Suppose that each device sets up its transmission radius in order to reach the K closest devices. There exists a universal constant K (independent of n) such that, with high probability, there will be a unique giant component (i.e. a connected component of size Θ(n)). Furthermore, all remaining components will be of size O(log2n). This leads to an efficient distributed probabilistic test for membership in the giant component, which can be used in a second phase to achieve full connectivity.

We investigate the final size distribution of the SIR (susceptible-infected-recovered) epidemic model in the critical regime. Using the integral representation of Martin-Löf (1998) for the hitting time of a Brownian motion with parabolic drift, we derive asymptotic expressions for the final size distribution that capture the effect of the initial number of infectives and the closeness of the reproduction number to zero. These asymptotics shed light on the bimodularity of the limiting density of the final size observed in Martin-Löf (1998). We also discuss the connection to the largest component in the Erdős-Rényi random graph, and, using this connection, find an integral expression of the Laplace transform of the normalized Brownian excursion area in terms of Airy functions.