A full-information version of the secretary problem in which the objective is to maximize the time of possession of a relatively best object is considered both in the case in which only the most recently observed object may be chosen and in the case in which any of the previously observed objects may be chosen. The main purpose of this paper is to obtain, under an optimal rule, both the asymptotic proportional durations when the number of objects tends to infinity, and the expected durations when the number of objects remains finite. The integral expressions for these asymptotic values are derived in both cases: the approximate numerical values are 0.435 in the former (no-recall) case and 0.449 in the latter (recall) case, indicating a surprisingly small difference. The expected values of the optimal stopping times are also obtained from the planar Poisson process analysis.

In this article we study stochastic perturbations of partial differential equations describing forced-damped vibrations of a string. Two models of such stochastic disturbances are considered; one is triggered by an initial white noise, and the other is in the form of non-Gaussian random forcing. Let uε (t, x) be the displacement at time t of a point x on a string, where the time variable t ≧ 0, and the space variable . The small parameter ε controls the intensity of the random fluctuations. The random fields uε (t, x) are shown to satisfy a large deviations principle, and the random deviations of the unperturbed displacement function are analyzed as the noise parameter ε tends to zero.

Interacting particle systems provide an attractive framework for modelling the growth and spread of biological populations and diseases. One problem with their use in applications is that in most cases the existing information about their critical values and equilibrium densities is too crude to be useful. In this paper we describe a method for estimating these quantities that does not require very much computer time and produces fairly accurate results.

By analogy with statistical mechanics random collision processes are considered. In two cases we derive their limiting distributions using discrimination information. In one case the limiting distribution is unique and in the other one the family of limiting distributions is an exponential family.

A new procedure that generates the transient solution of the first moment of the state of a Markovian queueing network with state-dependent arrivals, services, and routeing is developed. The procedure involves defining a partial differential equation that relates an approximate multivariate cumulant generating function to the intensity functions of the network. The partial differential equation then yields a set of ordinary differential equations which are numerically solved to obtain the first moment.

Consider a queueing system in which customers arrive in accordance with a renewal process having an interarrival distribution F, and in which the service times of customers are independent and have distribution G. Moreover, suppose that there are k (k ≦ ∞) servers and that an arriving customer is immediately served if he finds one of the servers free, and if not then he joins the queue.

A diffusion model is used to find heavy traffic approximate autocorrelation functions for several variables in queueing systems (i.e. for waiting time, system time, number in system and unfinished work). A table is given by which correlations can easily be found for each variable in any GI/G/1 queue. Further, the infinite sum or integral of the autocorrelations is also found, and the spectral density function. The sum has applications in statistical analysis of queues.

Extensive comparisons of approximate and exact correlations and their sums are reported, particularly for waiting times and system times, but also including number in system in M/M/1 queues. In general the correlations have similar accuracy to the probability distributions found by diffusion approximations. The percentage error is less for number in system than for waiting time.

Using martingale methods, we derive a set of theorems of boundary crossing probabilities for a Brownian motion with different kinds of stochastic boundaries, in particular compound Poisson process boundaries. We present both the numerical results and simulation experiments. The paper is motivated by limits on exposure of UK banks set by CHAPS. The central and participating banks are interested in the probability that the limits are exceeded. The problem can be reduced to the calculation of the boundary crossing probability from a Brownian motion with stochastic boundaries. Boundary crossing problems are also very popular in many fields of statistics.

Christ and Avi-Itzhak (2002) analyzed a queueing system with two competing servers who determine their service rates so as to optimize their individual utilities. The system is formulated as a two-person game; Christ and Avi-Itzhak proved the existence of a unique Nash equilibrium which is symmetric. In this paper, we explore globally optimal solutions. We prove that the unique Nash equilibrium is generally strictly inferior to a globally optimal solution and that optimal solutions are symmetric and require the servers to adopt service rates that are smaller than those occurring in equilibrium. Furthermore, given a symmetric globally optimal solution, we show how to impose linear penalties on the service rates so that the given optimal solution becomes a unique Nash equilibrium. When service rates are not observable, we show how the same effect is achieved by imposing linear penalties on a corresponding signal.

We study the asymptotic behaviors of estimators of the parameters in an Ornstein–Uhlenbeck process with linear drift, such as the law of the iterated logarithm (LIL) and Berry–Esseen bounds. As an application of the Berry–Esseen bounds, the precise rates in the LIL for the estimators are obtained.

For stationary sequences X = {Xn}n≥1 we relate τ, the limiting mean number of exceedances of high levels un by X1,…,Xn, and ν, the limiting mean number of upcrossings of the same level, through the expression θ = (ν/τ)η, where θ is the extremal index of X and η is a new parameter here called the upcrossings index. The upcrossings index is a measure of the clustering of upcrossings of u by variables in X, and the above relation extends the known relation θ = ν/τ, which holds under the mild-oscillation local restriction D″(u) on X. We present a new family of local mixing conditions D̃(k)(u) under which we prove that (a) the intensity of the limiting point process of upcrossings and η can both be computed from the k-variate distributions of X; and (b) the cluster size distributions for the exceedances are asymptotically equivalent to those for the lengths of one run of exceedances or the lengths of several consecutive runs which are separated by at most k − 2 nonexceedances and, except for the last one, each contain at most k − 2 exceedances.

For a given k ≥ 1, subintervals of a given interval [0, X] arrive at random and are accepted (allocated) so long as they overlap fewer than k subintervals already accepted. Subintervals not accepted are cleared, while accepted subintervals remain allocated for random retention times before they are released and made available to subsequent arrivals. Thus, the system operates as a generalized many-server queue under a loss protocol. We study a discretized version of this model that appears in reference theories for a number of applications, including communication networks, surface adsorption-desorption processes, and reservation systems. Our primary interest is in steady-state estimates of the vacant space, i.e. the total length of available subintervals kX - ∑ℓ i , where the ℓ i are the lengths of the subintervals currently allocated. We obtain explicit results for k = 1 and for general k with all subinterval lengths equal to 2, the classical dimer case of chemical applications. Our focus is on the asymptotic regime of large retention times.

This paper is concerned with the preservation of unimodality under coherent structures of independent components having a common life distribution function. This result in a way generalizes a result of Alam [1], as Alam's result indirectly also deals with preservation of unimodality for (n – i + 1)-out-of-n systems of independent and identically distributed components. The usefulness of this property of coherent systems in obtaining sharper upper bounds on the reliability of the concerned system has been illustrated below for a bridge structure with components having a gamma life distribution function.

Wahba (1978) and Weinert et al. (1980), using different models, show that an optimal smoothing spline can be thought of as the conditional expectation of a stochastic process observed with noise. This observation leads to efficient computational algorithms. By going back to the Hilbert space formulation of the spline minimization problem, we provide a framework for linking the two different stochastic models. The last part of the paper reviews some new efficient algorithms for spline smoothing.

The paper deals with the asymptotic analysis of waiting time distribution for service systems with finite queue capacity. First an M/M/m system is considered and the rate of approximation is given. Then the case of the M/G/1 system is studied for traffic intensity ρ > 1. In the last section a condition is given under which an estimate can be derived for the remainder term in central limit theorems for randomly stopped sums.

We consider a repairable system with a finite state space which evolves in time according to a Markov process as long as it is working. We assume that this system is getting worse and worse while running: if the up-states are ranked according to their degree of increasing degradation, this is expressed by the fact that the Markov process is assumed to be monotone with respect to the reversed hazard rate and to have an upper triangular generator. We study this kind of process and apply the results to derive some properties of the stationary availability of the system. Namely, we show that, if the duration of the repair is independent of its completeness degree, then the more complete the repair, the higher the stationary availability, where the completeness degree of the repair is measured with the reversed hazard rate ordering.

The use of a branching process argument in complex queueing situations often leads to a discussion of a non-linear matrix integral equation of Volterra type. By the use of a fixed point theorem we show these equations have a unique solution.

We estimate the limiting availability of a system when the operating and repair times form a stationary bivariate sequence. These estimators are shown to be consistent and asymptotically normal under certain conditions. In particular, we estimate the limiting availability for a bivariate exponential autoregressive process.

This paper proposes a class of complete financial market models, the benchmark models, with security price processes that exhibit intensity-based jumps. The benchmark or reference unit is chosen to be the growth-optimal portfolio. Primary security account prices, when expressed in units of the benchmark, turn out to be local martingales. In the proposed framework an equivalent risk-neutral measure need not exist. Benchmarked fair derivative prices are obtained as conditional expectations of future benchmarked prices under the real-world probability measure. This concept of fair pricing generalizes the classical risk-neutral approach and the actuarial present-value pricing methodology.