We analyze the smoothing effect of superposing homogeneous sources in a network. We consider a tandem queueing network representing the nodes that customers generated by these sources pass through. The servers in the tandem queues have different time varying service rates. In between the tandem queues there are propagation delays. We show that for arbitrary arrival and service processes which are mutually independent, the sum of unfinished works in the tandem queues is monotone in the number of homogeneous sources in the increasing convex order sense, provided the total intensity of the foreground traffic is constant. The results hold for both fluid and discrete traffic models.

Classic works of Karlin and McGregor and Jones and Magnus have established a general correspondence between continuous-time birth-and-death processes and continued fractions of the Stieltjes-Jacobi type together with their associated orthogonal polynomials. This fundamental correspondence is revisited here in the light of the basic relation between weighted lattice paths and continued fractions otherwise known from combinatorial theory. Given that sample paths of the embedded Markov chain of a birth-and-death process are lattice paths, Laplace transforms of a number of transient characteristics can be obtained systematically in terms of a fundamental continued fraction and its family of convergent polynomials. Applications include the analysis of evolutions in a strip, upcrossing and downcrossing times under flooring and ceiling conditions, as well as time, area, or number of transitions while a geometric condition is satisfied.

A curious connection exists between the theory of optimal stopping for independent random variables, and branching processes. In particular, for the branching process Z n with offspring distribution Y, there exists a random variable X such that the probability P(Z n = 0) of extinction of the nth generation in the branching process equals the value obtained by optimally stopping the sequence X 1,…, X n , where these variables are i.i.d. distributed as X. Generalizations to the inhomogeneous and infinite horizon cases are also considered. This correspondence furnishes a simple ‘stopping rule’ method for computing various characteristics of branching processes, including rates of convergence of the nth generation's extinction probability to the eventual extinction probability, for the supercritical, critical and subcritical Galton-Watson process. Examples, bounds, further generalizations and a connection to classical prophet inequalities are presented. Throughout, the aim is to show how this unexpected connection can be used to translate methods from one area of applied probability to another, rather than to provide the most general results.

This paper studies the connection between the dynamical and equilibrium behaviour of large uncontrolled loss networks. We consider the behaviour of the number of calls of each type in the network, and show that, under the limiting regime of Kelly (1986), all trajectories of the limiting dynamics converge to a single fixed point, which is necessarily that on which the limiting stationary distribution is concentrated. The approach uses Lyapunov techniques and involves the evolution of the transition rates of a stationary Markov process in such a way that it tends to reversibility.

Consider a situation where a known number, n, of objects appear sequentiallyin a random order. At each stage, the present object is presented to d ≥ 2 different selectors, who must jointly decide whether to select or reject it, irrevocably. Exactly one object must be chosen. The observation at stage j is a d-dimensional vector R (j) = (R 1(j),…, R d (j)), where R i (j) is the relative rank of the jth object, by the criterion of the ith selector. The decision whether to stop or not at time j is based on the d-dimensional random vectors R (1),…, R (j). The criteria according to which each selector ranks the objects can either be dependent or independent. Although the goal of each selector is to maximize the probability of choosing the best object from his/her point of view, all d selectors must cooperate and chose the same object. The objective studied here is the maximization of the minimum over the d individual probabilities of choosing the best object. We exhibit the structure of the optimal rule. For independent criteria we give a full description of the rule and show that the optimal value tends to d -d/(d-1), as n → ∞. Furthermore, we show that as n → ∞, the liminf of the values under negatively associated criteria is bounded below by d -d/(d-1).

Consider the following birth-growth model in ℝ. Seeds are born randomly according to an inhomogeneous space-time Poisson process. A newly formed point immediately initiates a bi-directional coverage by sending out a growing branch. Each frontier of a branch moves at a constant speed until it meets an opposing one. New seeds continue to form on the uncovered parts on the line. We are interested in the time until a bounded interval is completely covered. The exact and limiting distributions as the length of interval tends to infinity are obtained for this completion time by considering a related Markov process. Moreover, some strong limit results are also established.

This paper studies a revenue management problem in which a finite number of substitutable commodities are sold to two different market segments at respective prices. It is required that a certain number of commodities are reserved for the high-price segment to ensure a minimum service level. The two segments are served concurrently at the beginning of the season. To improve revenues, management may choose to close the low-price segment at a time when the chance of selling all items at the high price is promising. The difficulty is determining when such a decision should be made. We derive the exact solution in closed form using the theory of optimal stopping time. We show that the optimal decision is made in reference to a sequence of thresholds in time. These time thresholds take both remaining sales season and inventory into account and exhibit a useful monotone property.

A unifying technology is introduced for finding explicit closed form expressions for joint moment generating functions of various random quantities associated with some waiting time problems. Sooner and later waiting times are covered for general discrete- and continuous-time models. The models are either Markov chains or semi-Markov processes with a finite number of states. Waiting times associated with generalized phase-type distributions, that are of interest in survival analysis and other areas, are also covered.

We introduce the notion of weakly approaching sequences of distributions, which is a generalization of the well-known concept of weak convergence of distributions. The main difference is that the suggested notion does not demand the existence of a limit distribution. A similar definition for conditional (random) distributions is presented. Several properties of weakly approaching sequences are given. The tightness of some of them is essential. The Cramér-Lévy continuity theorem for weak convergence is generalized to weakly approaching sequences of (random) distributions. It has several applications in statistics and probability. A few examples of applications to resampling are given.

A (generalized) stochastic fluid system Q is defined as the one-dimensionalSkorokhod reflection of a finite variation process X (with possibly discontinuous paths). We write X as the (not necessarily minimal) difference of two positive measures, A, B, and prove an alternative ‘integral representation’ for Q. This representation forms the basis for deriving a ‘Little's law’ for an appropriately constructed stationary version of Q. For the special case where B is the Lebesgue measure, a distributional version of Little's law is derived. This is done both at the arrival and departure points of the system. The latter result necessitates the consideration of a ‘dual process’ to Q. Examples of models for X, including finite variation Lévy processes with countably many jumps on finite intervals, are given in order to illustrate the ideas and point out potential applications in performance evaluation.

We consider a probabilistic model of a heterogeneous population P subdivided into homogeneous sub-cohorts. A main assumption is that the frailties give rise to a discrete, exchangeable random vector. We put ourselves in the framework of stochastic filtering to derive the conditional distribution of residual lifetimes of surviving individuals, given an observed history of failures and survivals. As a main feature of our approach, this study is based on the analysis of behaviour of the vector of ‘occupation numbers’.

A duality is presented for continuous-time, real-valued, monotone, stochastic recursions driven by processes with stationary increments. A given recursion defines the time evolution of a content process (such as a dam or queue), and it is shown that the existence of the content process implies the existence of a corresponding dual risk process that satisfies a dual recursion. The one-point probabilities for the content process are then shown to be related to the one-point probabilities of the risk process. In particular, it is shown that the steady-state probabilities for the content process are equivalent to the first passage time probabilities for the risk process. A number of applications are presented that flesh out the general theory. Examples include regulated processes with one or two barriers, storage models with general release rate, and jump and diffusion processes.

A martingale is used to study extinction probabilities of the Galton-Watson process using a stopping time argument. This same martingale defines a martingale function in its argument s; consequently, its derivative is also a martingale. The argument s can be classified as regular or irregular and this classification dictates very different behavior of the Galton-Watson process. For example, it is shown that irregularity of a point s is equivalent to the derivative martingale sequence at s being closable, (i.e., it has limit which, when attached to the original sequence, the martingale structure is retained). It is also shown that for irregular points the limit of the derivative is the derivative of the limit, and two different types of norming constants for the asymptotics of the Galton-Watson process are asymptotically equivalent only for irregular points.

Let I 1,I 2,…,I n be a sequence of independent indicator functions defined on a probability space (Ω, A, P). We say that index k is a success time if I k = 1. The sequence I 1,I 2,…,I n is observed sequentially. The objective of this article is to predict the lth last success, if any, with maximum probability at the time of its occurrence. We find the optimal rule and discuss briefly an algorithm to compute it in an efficient way. This generalizes the result of Bruss (1998) for l = 1, and is equivalent to the problem of (multiple) stopping with l stops on the last l successes. We then extend the model to a larger class allowing for an unknown number N of indicator functions, and present, in particular, a convenient method for an approximate solution if the success probabilities are small. We also discuss some applications of the results.

We study a random record model where the observation X i has continuous distribution function F α i (αi > 0) and the number of available observations is random and independent of the observations. We obtain the joint distribution of the record values and inter-record times for our model. We investigate the distribution of the number of records when the number of observations has one of the common distributions and the α's increase geometrically or linearly. A particularly interesting case arises when the observations arrive at time points paced by a Poisson point process. For this model we obtain distributional results for the inter-arrival times of records for a large class of combinations of α structures and intensity functions.

Lamperti's transformation, an isometry between self-similar and stationary processes, is used to solve some problems of linear estimation of continuous-time, self-similar processes. These problems include causal whitening and innovations representations on the positive real line, as well as prediction from certain finite and semi-infinite intervals. The method is applied to the specific case of fractional Brownian motion (FBM), yielding alternate derivations of known prediction results, along with some novel whitening and interpolation formulae. Some associated insights into the problem of discrete prediction are also explored. Closed-form expressions for the spectra and spectral factorization of the stationary processes associated with the FBM are obtained as part of this development.

We study the following stochastic investment model: opportunities occur randomly over time, following a renewal process with mean interarrival time d, and at each of them the decision-maker can choose a distribution for an instantaneous net gain (or loss) from the set of all probability measures that have some prespecified expected value e and for which his maximum possible loss does not exceed his current capital. Between the investments he spends money at some constant rate. The objective is to avoid bankruptcy as long as possible. For the case e>d we characterize a strategy maximizing the probability that ruin never occurs. It is proved that the optimal value function is a concave function of the initial capital and uniquely determined as the solution of a fixed point equation for some intricate operator. In general, two-point distributions suffice; furthermore, we show that the cautious strategy of always taking the deterministic amount e is optimal if the interarrival times are hyperexponential, and, in the case of bounded interarrival times, is optimal ‘from some point on’, i.e. whenever the current capital exceeds a certain threshold. In the case e = 0 we consider a class of natural objective functions for which the optimal strategies are non-stationary and can be explicitly determined.

The paper compares non-parametric (design-based) and parametric (model-based) approaches to the analysis of data in the form of replicated spatial point patterns in two or more experimental groups. Basic questions for data of this kind concern estimating the properties of the underlying spatial point process within each experimental group, and comparing the properties between groups. A non-parametric approach, building on work by Diggle et. al. (1991), summarizes each pattern by an estimate of the reduced second moment measure or K-function (Ripley (1977)) and compares mean K-functions between experimental groups using a bootstrap testing procedure. A parametric approach fits particular classes of parametric model to the data, uses the model parameter estimates as summaries and tests for differences between groups by comparing fits with and without the assumption of common parameter values across groups. The paper discusses how either approach can be implemented in the specific context of a single-factor replicated experiment and uses simulations to show how the parametric approach can be more efficient when the underlying model assumptions hold, but potentially misleading otherwise.

In, Tory and Pickard show that a simple subclass of unilateral AR processes identifies with Gaussian Pickard random fields on Z 2. First, we extend this result to the whole class of unilateral AR processes, by showing that they all satisfy a Pickard-type property, under which correlation matching and maximum entropy properties are assessed. Then, it is established that the Pickard property provides the ‘missing’ equations that complement the two-dimensional Yule-Walker equations, in the sense that the conjunction defines a one-to-one mapping between the set of AR parameters and a set of correlations. It also implies Markov chain conditions that allow exact evaluation of the likelihood and an exact sampling scheme on finite lattices.

Based on remote sensing of a potential minefield, point locations are identified, some of which may not be mines. The mines and mine-like objects are to be distinguished based on their point patterns, although it must be emphasized that all one sees is the superposition of their locations. In this paper, we construct a hierarchical spatial point-process model that accounts for the different patterns of mines and mine-like objects and uses posterior analysis to distinguish between them. Our Bayesian approach is applied to minefield data obtained from a multispectral video remote-sensing system.