We take a new look at transient, or time-dependent Little laws for queueing systems. Through the use of Palm measures, we show that previous laws (see Bertsimas and Mourtzinou (1997)) can be generalized. Furthermore, within this framework, a new law can be derived as well, which gives higher-moment expressions for very general types of queueing system; in particular, the laws hold for systems that allow customers to overtake one another. What is especially novel about our approach is the use of Palm measures that are induced by nonstationary point processes, as these measures are not commonly found in the queueing literature. This new higher-moment law is then used to provide expressions for all moments of the number of customers in the system in an M/G/1 preemptive last-come-first-served queue at a time t > 0, for any initial condition and any of the more famous preemptive disciplines (i.e. preemptive-resume, and preemptive-repeat with and without resampling) that are analogous to the special cases found in Abate and Whitt (1987c), (1988). These expressions are then used to derive a nice structural form for all of the time-dependent moments of a regulated Brownian motion (see Abate and Whitt (1987a), (1987b)).

The game of pure strategy, sometimes called Goofspiel or Gops (see [2] and [3]), is played by two players, using a normal deck of cards, as follows. The 13 clubs are first taken out of the deck and of the remaining 39 cards the 13 hearts are given to Player I, the 13 diamonds to Player II, and the 13 spades are placed face down in the center. The spades are shuffled and one is turned face up. At this point, the two players choose one of their cards and then simultaneously discard it. The one who discards the higher card (ace being low, king high) wins from the other an amount equal to the value of the upturned spade (ace = 1, king = 13). If both players discard the same card, then neither wins. The three cards are then thrown away, a new spade upturned and the game continues. After 13 plays, there are no remaining cards and the game ends.

We prove existence and uniqueness of a stationary distribution and absolute regularity for nonlinear GARCH and INGARCH models of order (p, q). In contrast to previous work we impose, besides a geometric drift condition, only a semi-contractive condition which allows us to include models which would be ruled out by a fully contractive condition. This results in a subgeometric rather than the more usual geometric decay rate of the mixing coefficients. The proofs are heavily based on a coupling of two versions of the processes.

The discrete scan statistic in a binary (0-1) sequence of n trials is defined as the maximum number of successes within any k consecutive trials (n and k, n ≥ k, being two positive integers). It has been used in many areas of science (quality control, molecular biology, psychology, etc.) to test the null hypothesis of uniformity against a clustering alternative. In this article we provide a compound Poisson approximation and subsequently use it to establish asymptotic results for the distribution of the discrete scan statistic as n, k → ∞ and the success probability of the trials is kept fixed. An extreme value theorem is also provided for the celebrated Erdős-Rényi statistic.

Given two absolutely continuous nonnegative independent random variables, we define the reversed relevation transform as dual to the relevation transform. We first apply such transforms to the lifetimes of the components of parallel and series systems under suitably proportionality assumptions on the hazard rates. Furthermore, we prove that the (reversed) relevation transform is commutative if and only if the proportional (reversed) hazard rate model holds. By repeated application of the reversed relevation transform we construct a decreasing sequence of random variables which leads to new weighted probability densities. We obtain various relations involving ageing notions and stochastic orders. We also exploit the connection of such a sequence to the cumulative entropy and to an operator that is dual to the Dickson-Hipp operator. Iterative formulae for computing the mean and the cumulative entropy of the random variables of the sequence are finally investigated.

The species-area problem in biology and the type-token problem in literary studies are analogues of one another but have nearly disjoint literatures. Here their relationship is treated, a critique of models used in each is made and Markovian models are provided for two well-known empirical models from the species-area literature. Finally, some of the assumptions underlying each of these models are considered in the light of literary data.

We consider a discrete-time risk model which describes the evolution of the reserves of an insurance company at periodic dates fixed in advance. The amount of loss per unit of time corresponds to independent and identically distributed random variables with arithmetic distribution, and the process of the receipt of premiums is assumed to be deterministic, nonnegative but not uniform (instead of being constant and equal to 1 as in the standard, compound binomial model). For this model, we determine the probability of ruin (or of non-ruin), as well as the distribution of the severity of the eventual ruin, with some finite horizon. A compact and efficient exact expression is found by bringing up-to-date a generalised family of Appell polynomials. The method used is illustrated with some numerical examples.

In this paper we study a generalized coupon collector problem, which consists of determining the distribution and the moments of the time needed to collect a given number of distinct coupons that are drawn from a set of coupons with an arbitrary probability distribution. We suppose that a special coupon called the null coupon can be drawn but never belongs to any collection. In this context, we obtain expressions for the distribution and the moments of this time. We also prove that the almost-uniform distribution, for which all the nonnull coupons have the same drawing probability, is the distribution which minimizes the expected time to obtain a fixed subset of distinct coupons. This optimization result is extended to the complementary distribution of the time needed to obtain the full collection, proving by the way this well-known conjecture. Finally, we propose a new conjecture which expresses the fact that the almost-uniform distribution should minimize the complementary distribution of the time needed to obtain any fixed number of distinct coupons.

The sooner and later waiting time problems have been extensively studied and applied in various areas of statistics and applied probability. In this paper, we give a comprehensive study of ordered series and later waiting time distributions of a number of simple patterns with respect to nonoverlapping and overlapping counting schemes in a sequence of Markov dependent multistate trials. Exact distributions and probability generating functions are derived by using the finite Markov chain imbedding technique. Examples are given to illustrate our results.

We prove a threshold theorem for the Reed–Frost chain-binomial model which is analogous to the threshold theorem of Williams (1971) for the general stochastic epidemic. We show that when the population size is large a ‘true epidemic’ occurs with a non-zero probability if and only if an initial infective individual infects on average more than one susceptible individual.

A procedure is indicated to estimate first-passage-time p.d.f.'s through varying boundaries for a class of diffusion processes that can be transformed into the Wiener process by rather general transformations. Although this procedure is adapted to Durbin's [4] algorithm, it could be extended to other existing computation methods.

Moments of the measure of the intersection of a given set with the union of random sets may be called moments of coverage. These moments specified for the case of independent random sets in the Euclidean space, were treated by several authors. Kolmogorov (1933) and Robbins (1944), (1945), (1947) derived transformation formulas for the moments of coverage under some specified conditions. Robbins' formula was applied by Robbins (1945), Santaló (1947) and Garwood (1947) to computations of expectations and variances of coverage in the cases of spheres and intervals in 2-dimensional and in n-dimensional Euclidean space. The computations were carried out under the assumption of a simple distribution function for the centers of the covering figures (a homogeneous distribution in most of the cases and a normal one in some of them). Neyman and Bronowski (1945) computed by a different method the variance of coverage for some specific cases in the Euclidean plane.

A stationary Poisson process of hyperplanes in Rn is characterized (up to an equivalence) by the function θ such that θ(s) is the density of the Poisson point process induced on the straight lines with direction s. The set of these functions θ is a convex cone ℛ1, a basis of which is a simplex Θ, and a given function θ belongs to ℛ1 if and only if it is the supporting function of a symmetrical compact convex set which is a finite Minkowski sum of line segments or the limit of such finite sums. Another application is given concerning the tangential cone at h = 0 of a coveriance function.

We study monotone and associated Markov chains on finite partially ordered state spaces. Both discrete and continuous time, and both time-homogeneous and time-inhomogeneous chains are considered. The results are applied to binary and multistate reliability theory.

Let X1,X2,X3,··· be independent, identically distributed random variables with a continuous distribution function and let the sequence of indices {Vr} be defined as follows: and for r ≧ 1, Vr is the trial on which the rth (upper) record observation occurs. {Vr} will be an infinite sequence of random variables since the underlying distribution function of the X's is continuous. It is well known that the expected value of Vr. is infinite for every r (see, for example, Feller [1], page 15). Also define and for r > 1 δr is the number of trials between the (r - l)th and the rth record. The distributions of the random variables Vr and δr do not depend on the distribution of the original random variables. It can be shown (see Neuts [2], page 206 or Tata 1[4], page 26) that The following theorem is due to Neuts [2].

We consider a branching model, which we call the collision branching process (CBP), that accounts for the effect of collisions, or interactions, between particles or individuals. We establish that there is a unique CBP, and derive necessary and sufficient conditions for it to be nonexplosive. We review results on extinction probabilities, and obtain explicit expressions for the probability of explosion and the expected hitting times. The upwardly skip-free case is studied in some detail.

We consider a fluid queue with downward jumps, where the fluid flow rate and the downward jumps are controlled by a background Markov chain with a finite state space. We show that the stationary distribution of a buffer content has a matrix exponential form, and identify the exponent matrix. We derive these results using time-reversed arguments and the background state distribution at the hitting time concerning the corresponding fluid flow with upward jumps. This distribution was recently studied for a fluid queue with upward jumps under a stability condition. We give an alternative proof for this result using the rate conservation law. This proof not only simplifies the proof, but also explains an underlying Markov structure and enables us to study more complex cases such that the fluid flow has jumps subject to a nondecreasing Lévy process, a Brownian component, and countably many background states.

An observer watches one of a set of Poisson streams. He may switch from one stream to another instantaneously. If an arrival occurs in a stream while the observer is watching another stream, he does not see the arrival. The experiment terminates when the observer sees an arrival. We derive a formula which states essentially that the expected total time that the observer watches a stream is equal to the probability that he sees the arrival in this stream divided by the intensity of the stream. This formula is valid independently of the observation policy. We also discuss applications of this formula.

In this paper we determine the distributions of occupation times of a Markov-modulated Brownian motion (MMBM) in separate intervals before a first passage time or an exit from an interval. We derive the distributions in terms of their Laplace transforms, and we also distinguish between occupation times in different phases. For MMBMs with strictly positive variation parameters, we further propose scale functions.

We consider a classical risk model and its diffusion approximation, where the individual claims are reinsured by a reinsurance treaty with deductible b ∈ [0, b̃]. Here b = b̃ means ‘no reinsurance’ and b= 0 means ‘full reinsurance’. In addition, the insurer is allowed to invest in a riskless asset with some constant interest rate m > 0. The cedent can choose an adapted reinsurance strategy {b t }t≥0, i.e. the parameter can be changed continuously. If the surplus process becomes negative, the cedent has to inject additional capital. Our aim is to minimise the expected discounted capital injections over all admissible reinsurance strategies. We find an explicit expression for the value function and the optimal strategy using the Hamilton-Jacobi-Bellman approach in the case of a diffusion approximation. In the case of the classical risk model, we show the existence of a ‘weak’ solution and calculate the value function numerically.