We consider the problem of scheduling the transmissions of multiple data users (flows) sharing the same wireless channel (server). The unique feature of this problem is the fact that the capacity (service rate) of the channel varies randomly with time and asynchronously for different users. We study a scheduling policy called the exponential scheduling rule, which was introduced in an earlier paper. Given a system with N users, and any set of positive numbers {an}, n = 1, 2,…, N, we show that in a heavy-traffic limit, under a nonrestrictive ‘complete resource pooling’ condition, this algorithm has the property that, for each time t, it (asymptotically) minimizes maxnanq̃n(t), where q̃n(t) is the queue length of user n in the heavy-traffic regime.

Sums of log-normals frequently appear in a variety of situations, including engineering and financial mathematics. In particular, the pricing of Asian or basket options is directly related to finding the distributions of such sums. There is no general explicit formula for the distribution of sums of log-normal random variables. This paper looks at the limit distributions of sums of log-normal variables when the second parameter of the log-normals tends to zero or to infinity; in financial terms, this is equivalent to letting the volatility, or maturity, tend either to zero or to infinity. The limits obtained are either normal or log-normal, depending on the normalization chosen; the same applies to the reciprocal of the sums of log-normals. This justifies the log-normal approximation, much used in practice, and also gives an asymptotically exact distribution for averages of log-normals with a relatively small volatility; it has been noted that all the analytical pricing formulae for Asian options perform poorly for small volatilities. Asymptotic formulae are also found for the moments of the sums of log-normals. Results are given for both discrete and continuous averages. More explicit results are obtained in the case of the integral of geometric Brownian motion.

A class of haploid population models with population size N, nonoverlapping generations and exchangeable offspring distribution is considered. Based on an analysis of the discrete ancestral process, we present solutions, algorithms and strong upper bounds for the expected time back to the most recent common ancestor which hold for arbitrary sample size n ∈ {1,…,N}. New insights into the asymptotic behaviour of the expected time back to the most recent common ancestor for large population size are presented relating the results to coalescent theory.

The full-information best-choice problem, as posed by Gilbert and Mosteller in 1966, asks us to find a stopping rule which maximizes the probability of selecting the largest of a sequence of n i.i.d. standard uniform random variables. Porosiński, in 1987, replaced a fixed n by a random N, uniform on {1,2,…,n} and independent of the observations. A partial-information problem, imbedded in a 1980 paper of Petruccelli, keeps n fixed but allows us to observe only the sequence of ranges (max - min), as well as whether or not the current observation is largest so far. Recently, Porosiński compared the solutions to his and Petruccelli's problems and found that the two problems have identical optimal rules as well as risks that are asymptotically equal. His discovery prompts the question: why? This paper gives a good explanation of the equivalence of the optimal rules. But even under the lens of a planar Poisson process model, it leaves the equivalence of the asymptotic risks as somewhat of a mystery. Meanwhile, two other problems have been shown to have the same limiting risks: the full-information problem with the (suboptimal) Porosiński-Petruccelli stopping rule, and the full-information ‘duration of holding the best’ problem of Ferguson, Hardwick and Tamaki, which turns out to be nothing but the Porosiński problem in disguise.

Within reliability theory, identifiability problems arise through competing risks. If we have a series system of several components, and if that system is replaced or repaired to as good as new on failure, then the different component failures represent competing risks for the system. It is well known that the underlying component failure distributions cannot be estimated from the observable data (failure time and identity of failed component) without nontestable assumptions such as independence. In practice many systems are not subject to the ‘as good as new’ repair regime. Hence, the objective of this paper is to contrast the identifiability issues arising for different repair regimes. We consider the problem of identifying a model within a given class of probabilistic models for the system. Different models corresponding to different repair strategies are considered: a partial-repair model, where only the failing component is repaired; perfect repair, where all components are as good as new after a failure; and minimal repair, where components are only minimally repaired at failures. We show that on the basis of observing a single socket, the partial-repair model is identifiable, while the perfect- and minimal-repair models are not.

The maximum-entropy formalism developed by E. T. Jaynes is applied to the breaking strain of a bundle of fibers of various cross-sectional areas. When the bundle is subjected to a tensile load, and it is assumed that Hooke's law applies up to the breaking strain of the fibers, it is proved that the survival strain distribution for a fiber in the bundle is restricted to a certain class consisting of generalizations of the log-logistic distribution. Since Jaynes's formalism is a generalization of statistical thermodynamics, parallels are drawn between concepts in thermodynamics and in the theory of inhomogeneous bundles of fibers. In particular, heat transfer corresponds to damage to the bundle in the form of broken fibers, and the negative reciprocal of the parameter corresponding to thermodynamic temperature is the resistance of the bundle to damage.

Stephens and Donnelly (2000) constructed an efficient sequential importance-sampling proposal distribution on coalescent histories of a sample of genes for computing the likelihood of a type configuration of genes in the sample. In the current paper a characterization of their importance-sampling proposal distribution is given in terms of the diffusion-process generator describing the distribution of the population gene frequencies. This characterization leads to a new technique for constructing importance-sampling algorithms in a much more general framework when the distribution of population gene frequencies follows a diffusion process, by approximating the generator of the process.

The third-generation (3G) mobile communication system uses a technique called code division multiple access (CDMA), in which multiple users use the same frequency and time domain. The data signals of the users are distinguished using codes. When there are many users, interference deteriorates the quality of the system. For more efficient use of resources, we wish to allow more users to transmit simultaneously, by using algorithms that utilize the structure of the CDMA system more effectively than the simple matched filter (MF) system used in the proposed 3G systems. In this paper, we investigate an advanced algorithm called hard-decision parallel interference cancellation (HD-PIC), in which estimates of the interfering signals are used to improve the quality of the signal of the desired user. We compare HD-PIC with MF in a simple case, where the only two parameters are the number of users and the length of the coding sequences. We focus on the exponential rate for the probability of a bit-error, explain the relevance of this parameter, and investigate how it scales when the number of users grows large. We also review extensions of our results, proved elsewhere, showing that in HD-PIC, more users can transmit without errors than in the MF system.

We study a family of locally self-similar stochastic processes Y = {Y(t)}t∈ℝ with α-stable distributions, called linear multifractional stable motions. They have infinite variance and may possess skewed distributions. The linear multifractional stable motion processes include, in particular, the classical linear fractional stable motion processes, which have stationary increments and are self-similar with self-similarity parameter H. The linear multifractional stable motion process Y is obtained by replacing the self-similarity parameter H in the integral representation of the linear fractional stable motion process by a deterministic function H(t). Whereas the linear fractional stable motion is always continuous in probability, this is not in general the case for Y. We obtain necessary and sufficient conditions for the continuity in probability of the process Y. We also examine the effect of the regularity of the function H(t) on the local structure of the process. We show that under certain Hölder regularity conditions on the function H(t), the process Y is locally equivalent to a linear fractional stable motion process, in the sense of finite-dimensional distributions. We study Y by using a related α-stable random field and its partial derivatives.

De Iorio and Griffiths (2004) developed a new method of constructing sequential importance-sampling proposal distributions on coalescent histories of a sample of genes for computing the likelihood of a type configuration of genes in the sample by simulation. The method is based on approximating the diffusion-process generator describing the distribution of population gene frequencies, leading to an approximate sample distribution and finally to importance-sampling proposal distributions. This paper applies that method to construct an importance-sampling algorithm for computing the likelihood of samples of genes in subdivided population models. The importance-sampling technique of Stephens and Donnelly (2000) is thus extended to models with a Markov chain mutation mechanism between gene types and migration of genes between subpopulations. An algorithm for computing the likelihood of a sample configuration of genes from a subdivided population in an infinitely-many-alleles model of mutation is derived, extending Ewens's (1972) sampling formula in a single population. Likelihood calculation and ancestral inference in gene trees constructed from DNA sequences under the infinitely-many-sites model are also studied. The Griffiths-Tavaré method of likelihood calculation in gene trees of Bahlo and Griffiths (2000) is improved for subdivided populations.

We introduce a class of discrete-time stochastic processes generated by interacting systems of reinforced urns. We show that such processes are asymptotically partially exchangeable and we prove a strong law of large numbers. Examples and the analysis of particular cases show that interacting reinforced-urn systems are very flexible representations for modelling countable collections of dependent and asymptotically exchangeable sequences of random variables.

An R-out-of-N repairable system, consisting of N independent components, is operating if at least R components are functioning. The system fails whenever the number of good components decreases from R to R-1. A failed component is sent to a repair facility. After a failed component has been repaired it is as good as new. Formulae for the availability of the system using Markov renewal and semi-regenerative processes are derived. We assume that either the repair times of the components are generally distributed and the components' lifetimes are phase-type distributed or vice versa. Some duality results between the two systems are obtained. Numerical examples are given for several distributions of lifetimes and of repair times.

Let Xn,…,X1 be independent, identically distributed (i.i.d.) random variables with distribution function F. A statistician, knowing F, observes the X values sequentially and is given two chances to choose Xs using stopping rules. The statistician's goal is to stop at a value of X as small as possible. Let equal the expectation of the smaller of the two values chosen by the statistician when proceeding optimally. We obtain the asymptotic behaviour of the sequence for a large class of Fs belonging to the domain of attraction (for the minimum) 𝒟(Gα), where Gα(x) = [1 - exp(-xα)]1(x ≥ 0) (with 1(·) the indicator function). The results are compared with those for the asymptotic behaviour of the classical one-choice value sequence , as well as with the ‘prophet value’ sequence

We study the suprema over compact time intervals of stationary locally bounded α-stable processes. The behaviour of these suprema as the length of the time interval increases turns out to depend significantly on the ergodic-theoretical properties of a flow generating the stationary process.

We consider the dynamic scheduling of a multiclass queueing system with two servers, one dedicated (server 1) and one flexible (server 2), with no arrivals. Server 1 is dedicated to processing type-1 jobs while server 2 is primarily responsible for processing type-2 jobs but can also aid server 1 with its work. We address when it is optimal for server 2 to aid server 1 with type-1 jobs rather than process type-2 jobs. The objective is to minimize the total holding costs incurred until all jobs in the system are processed and leave the system. We show that the optimal policy can exhibit one of three possible structures: (i) an exhaustive policy for type-2 jobs, (ii) a nonincreasing switching curve in the number of type-1 jobs and (iii) a nondecreasing switching curve in the number of type-1 jobs. We characterize the necessary and sufficient conditions under which each policy will be optimal. We also explore the use of the optimal policy for the problem with no arrivals as a heuristic for the problem with dynamic arrivals.

In this paper, we consider a failure point process related to the Markovian arrival process defined by Neuts. We show that it converges in distribution to a homogeneous Poisson process. This convergence takes place in the context of rare occurrences of failures. We also provide a convergence rate of the convergence in total variation of this point process using an approach developed by Kabanov, Liptser and Shiryaev for the doubly stochastic Poisson process driven by a finite Markov process.

In this paper we derive some of the main ergodicity properties of a class of Markov renewal processes and the associated marked point processes. This class represents a generic model of applied probability and is of importance in earthquake modeling, reliability theory and queueing.

We consider a sequence of random graphs constructed by a hierarchical procedure. The construction replaces existing edges by pairs of edges in series or parallel with probability p. We investigate the effective resistance across the graphs, first-passage percolation on the graphs and the Cheeger constants of the graphs as the number of edges tends to infinity. In each case we find a phase transition at

This paper introduces a method of generating real harmonizable multifractional Lévy motions (RHMLMs). The simulation of these fields is closely related to that of infinitely divisible laws or Lévy processes. In the case where the control measure of the RHMLM is finite, generalized shot-noise series are used. An estimation of the error is also given. Otherwise, the RHMLM Xh is split into two independent RHMLMs, Xε,1 and Xε,2. More precisely, Xε,2 is an RHMLM whose control measure is finite. It can then be rewritten as a generalized shot-noise series. The asymptotic behaviour of Xε,1 as ε → 0+ is further elaborated. Sufficient conditions to approximate Xε,1 by a multifractional Brownian motion are given. The error rate in terms of Berry-Esseen bounds is then discussed. Finally, some examples of simulation are given.