A bookmaker makes a book on a horse race: he offers odds against the various horses winning the race, and gamblers accept bets at those odds when they find the odds attractive. The book at a particular time consists of the bookmaker's winnings according to the different outcomes of the race if the race were run at that time. We consider strategies the bookmaker might adopt when deciding how to alter his quoted odds as bets accumulate. The bookmaker is assumed to behave conservatively in the sense that he tries to minimise his expected maximum loss over all possible outcomes of the race.

This paper considers a class of epidemic models in which susceptibles may enter or leave the population according to a general continuous time density dependent Markov chain. A sequence of such epidemics indexed by N, the initial number of susceptibles, is constructed on the same probability space as a time-inhomogeneous birth-and-death process. A coupling argument is then used to demonstrate the strong convergence of the sequence of infectives to the birth-and-death process. This result is used to provide a threshold analysis of the epidemic model in question.

We explore a dynamic approach to the problems of call admission and resource allocation for communication networks with connections that are differentiated by their quality of service requirements. In a dynamic approach, the amount of spare resources is estimated on-line based on feedbacks from the network's quality of service monitoring mechanism. The schemes we propose remove the dependence on accurate traffic models and thus simplify the tasks of supplying traffic statistics required of network users. In this paper we present two dynamic algorithms. The objective of these algorithms is to find the minimum bandwidth necessary to satisfy a cell loss probability constraint at an asynchronous transfer mode (ATM) switch. We show that in both schemes the bandwidth chosen by the algorithm approaches the optimal value almost surely. Furthermore, in the second scheme, which determines the point closest to the optimal bandwidth from a finite number of choices, the expected learning time is finite.

We solve the Fokker-Planck equation for the Wiener process with drift in the presence of elastic boundaries and a fixed start point. An explicit expression is obtained for the first passage density. The cases with pure absorbing and/or reflecting barriers arise for a special choice of a parameter constellation. These special cases are compared with results in Darling and Siegert [5] and Sweet and Hardin [15].

In this article we define the s-order equilibrium distribution of a life distribution and establish some interesting properties regarding moments for s-order equilibrium distributions. Results are expected to be useful in reliability and renewal processes.

In this paper we study the problem of computing the downtime distribution of a parallel system comprising stochastically identical components. It is assumed that the components are independent, with an exponential life-time distribution and an arbitrary repair time distribution. An exact formula is established for the distribution of the system downtime given a specific type of system failure scenario. It is shown by performing a Monte Carlo simulation that the portion of the system failures that occur as described by this scenario is close to one when we consider a system with quite available components, the most common situation in practice. Thus we can use the established formula as an approximation of the downtime distribution given system failure. The formula is compared with standard Markov expressions. Some possible extensions of the formula are presented.

An exponential inequality is generalized to one involving the tail of a decreasing failure rate distribution. The results are then applied in various situations, notably when the exponential inequality does not apply.

The multihit–one target model induces a stochastic ordering of cell survival with respect to the cell sensitivity characteristics. This property can be used for a description of cell killing effects in heterogeneous populations of cells on the basis of randomized versions of the model. In such versions, either the critical number of lesions or the mean number of hits per unit dose (sensitivity), or both, are assumed to be random. We give some new results specifying conditions under which the randomized multihit models are identifiable, with a focus on the following cases: (1) the critical number of radiation-induced lesions, m, is random; (2) the sensitivity parameter, x, is random given m is known or otherwise; (3) x and m form a pair of independent random variables.

Given a finite collection of strings of letters from a fixed alphabet, it is of interest, in the contexts of data compression and DNA sequencing, to find the length of the shortest string which contains each of the given strings as a consecutive substring. In order to analyze the average behavior of the optimal superstring length, substrings of specified lengths are considered with the letters selected independently at random. An asymptotic expression is obtained for the savings from compression, i.e. the difference between the uncompressed (concatenated) length and the optimal superstring length.

In this paper we study continuous flow finite buffer systems with input rates modulated by Markov chains. Discrete event simulations are applied for estimating loss probabilities. The simulations are executed under a twisted version of the original probability measure (importance sampling). We present a simple rule for determining a new measure, then show that the new measure matches the ‘most likely' empirical measure that we expect from large deviations arguments, and finally prove optimality of the new measure.

In 1979, Melamed proved that, in an open migration process, the absence of ‘loops' is necessary and sufficient for the equilibrium flow along a link to be a Poisson process. In this paper, we prove approximation theorems with the same flavour: the difference between the equilibrium flow along a link and a Poisson process with the same rate is bounded in terms of expected numbers of loops. The proofs are based on Stein's method, as adapted for bounds on the distance of the distribution of a point process from a Poisson process in Barbour and Brown (1992b). Three different distances are considered, and illustrated with an example consisting of a system of tandem queues with feedback. The upper bound on the total variation distance of the process grows linearly with time, and a lower bound shows that this can be the correct order of approximation.

Bounds on the tail of compound distributions are considered. Using a generalization of Wald's fundamental identity, we derive upper and lower bounds for various compound distributions in terms of new worse than used (NWU) and new better than used (NBU) distributions respectively. Simple bounds are obtained when the claim size distribution is NWUC, NBUC, NWU, NBU, IMRL, DMRL, DFR and IFR. Examples on how to use these bounds are given.

It is known that a generalized open Jackson queueing network after appropriate scaling (in both time and space) converges almost surely to a fluid network under the uniform topology. Under the same topology, we show that the distance between the scaled queue length process of the queueing network and the fluid level process of the corresponding fluid network converges to zero in probability at an exponential rate.

In this paper conditions for the convergence of a class of simulated annealing algorithms for continuous global optimization are given. The previous literature about the subject gives results for the convergence of algorithms in which the next candidate point is generated according to a probability distribution whose support is the whole feasible set. A class of possible cooling schedules has been introduced in order to remove this restriction.

A Cox risk process with a piecewise constant intensity is considered where the sequence (Li) of successive levels of the intensity forms a Markov chain. The duration σi of the level Li is assumed to be only dependent via Li. In the small-claim case a Lundberg inequality is obtained via a martingale approach. It is shown furthermore by a Lundberg bound from below that the resulting adjustment coefficient gives the best possible exponential bound for the ruin probability. In the case where the stationary distribution of Li contains a discrete component, a Cramér–Lundberg approximation can be obtained. By way of example we consider the independent jump intensity model (Björk and Grandell 1988) and the risk model in a Markovian environment (Asmussen 1989).

In this paper, transient characteristics related to excursions of the occupation process of M/M/∞ queues are studied, when the excursion level is large and close to the mean offered load. We show that the classical diffusion approximation by an Ornstein–Uhlenbeck (OU) process captures well the average values of the transient variables considered, while the asymptotic distributions of these variables depart from those corresponding to the OU process. They exhibit, however, equivalent tail behaviour at infinity and numerical evidence shows that they are amazingly close to each other over the whole half-line.

Busy period problems in infinite server queues are studied systematically, starting from the batch service time. General relations are given for the lengths of the busy cycle, busy period and idle period, and for the number of customers served in a busy period. These relations show that the idle period is the most difficult while the busy cycle is the simplest of the four random variables. Renewal arguments are used to derive explicit results for both general and special cases.

Vasicek (1977) proved that among all queueing disciplines that do not change the departure process of the queue, FIFO and LIFO yield, respectively, the smallest and the largest expectation of any given convex function of the service delay. In this note we further show that, if arriving customers join the queue stochastically ‘closer' to the server(s), then the expected value of any convex function of service delay is larger. As a more interesting result, we also show that if the function under consideration is concave, then the conclusion will be exactly the opposite. This result indicates that LIFO will be the best discipline if the delay cost is an increasing function but at a diminishing rate.

A random vibration model is investigated in this paper. The model is formulated as a cosine function with a constant frequency and a random walk phase. We show that this model is second-order stationary and can be rewritten as a vector-valued AR(1) model as well as a scalar ARMA(2, 1) model. The linear innovation sequence of the AR(1) model is shown to be a martingale difference sequence while the linear innovation sequence of the ARMA(2, 1) model is only an uncorrelated sequence. A non-linear predictor is derived from the AR(1) model while a linear predictor is derived from the ARMA(2, 1) model. We deduce that the non-linear predictor of this model has less mean square error than that of the linear predictor. This has significance, for example, for predicting seasonal phenomena with this model. In addition, the limit distributions of the sample mean, the finite Fourier transforms and the autocovariance functions are derived using a martingale approach. The limit distribution of autocovariance functions differs from the classical result given by Bartlett's formula.

We obtain a necessary and sufficient condition for the interaction between a service system and an external environment under which the stationary joint distribution of the set of busy channels and the state of the external environment is given by a product-form formula.