We consider a system of N parallel queues with identical exponential service rates and a single dispatcher where tasks arrive as a Poisson process. When a task arrives, the dispatcher always assigns it to an idle server, if there is any, and to a server with the shortest queue among d randomly selected servers otherwise (1≤d≤N). This load balancing scheme subsumes the so-called join-the-idle queue policy (d=1) and the celebrated join-the-shortest queue policy (d=N) as two crucial special cases. We develop a stochastic coupling construction to obtain the diffusion limit of the queue process in the Halfin‒Whitt heavy-traffic regime, and establish that it does not depend on the value of d, implying that assigning tasks to idle servers is sufficient for diffusion level optimality.

A device is subject to damage. The damage occurs randomly in time according to a pure jump process. The device has a threshold and it fails once the damage exceeds the threshold. We show that life distribution properties of the threshold right tail probability are inherited as corresponding properties of the survival probability, under suitable conditions on the parameters of the damage process. Moreover we discuss an optimal replacement problem for such devices.

Consider a single-server queue with service times distributed as a general random variable S and with non-stationary Poisson input. It is assumed that the arrival rate function λ (·) is periodic with average value λ and that ρ = λE(S) < 1. Both weak and strong limit theorems are proved for the waiting-time process W = {W1, W2, · ··} and the server load (or virtual waiting-time process) Z = {Z(t), t ≧ 0}. The asymptotic distributions associated with Z and W are shown to be related in various ways. In particular, we extend to the case of periodic Poisson input a well-known formula (due to Takács) relating the limiting virtual and actual waiting-time distributions of a GI/G/1 queue.

In the study of dynamical systems perturbed by noise, it is important to know whether the stochastic process of interest has a stationary distribution. Four necessary and sufficient conditions are formulated for the existence of a finite invariant measure for a Feller process on a σ-compact metric (state) space. These conditions link together stability notions from several fields. The first uses a Lyapunov function reminiscent of Lagrange stability in differential equations; the second depends on Prokhorov's condition for sequential compactness of measures; the third is a recurrence condition on the ergodic averages of the transition operator; and the fourth is analogous to a condition of Ulam and Oxtoby for the nonstochastic case.

A single device shock model is studied. The device is subject to some damage process. Under the assumption that as the cumulative damage increases, the probability that any additional damage will cause failure increases, we find sufficient conditions on the shocking process so that the life distribution will be increasing failure rate.

The copula of a multivariate distribution is the distribution transformed so that one-dimensional marginal distributions are uniform. We review a different transformation of a multivariate distribution which yields standard Pareto for the marginal distributions, and we call the resulting distribution the Pareto copula. Use of the Pareto copula has a certain claim to naturalness when considering asymptotic limit distributions for sums, maxima, and empirical processes. We discuss implications for aggregation of risk and offer some examples.

We obtain explicit upper bounds in closed form for the queue length in a slotted time FCFS queue in which the service requirement is a sum of independent Markov processes on the state space {0, 1}, with integral service rate. The bound is of the form [queue length for any where c < 1 and y > 1 are given explicitly in terms of the parameters of the model. The model can be viewed as an approximation for the burst-level component of the queue in an ATM multiplexer. We obtain heavy traffic bounds for the mean queue length and show that for typical parameters this far exceeds the mean queue length for independent arrivals at the same load. We compare our results on the mean queue length with an analytic expression for the case of unit service rate, and compare our results on the full distribution with computer simulations.

A collection of jobs is to be processed by a single machine. The amount of processing required by each job is a random variable with a known probability distribution. The jobs must be processed in a manner which is consistent with a precedence relation but the machine is free to switch from one job to another at any time; such switches are costly, however. This paper discusses conditions under which there is an optimal strategy for allocating the machine to the jobs which is given by a fixed permutation of the jobs indicating in which order they should be processed. When this is so, existing algorithms may be helpful in giving the best job ordering.

In a similar spirit to the probabilistic generalization of Taylor's theorem by Massey and Whitt [13], we give a probabilistic analogue of the mean value theorem. The latter is shown to be useful in various contexts of reliability theory. In particular, we provide various applications to the evaluation of the mean total profits of devices having random lifetimes, to the mean total-time-on-test at an arbitrary order statistic of a random sample of lifetimes, and to the mean maintenance cost for the second room of queueing systems in steady state characterized by two serial waiting rooms.

Let be the kth largest among Xn1, …, Xn[nt], where Xni = (Xi – an)/bn, {Xi} is a sequence of independent random variables and bn > 0 and an are norming constants. Suppose that for each converges in distribution. Then all the finite-dimensional laws of converge. The limiting process is represented in terms of a non-homogeneous two-dimensional Poisson process.

The ‘homing' optimal control problem, described in Whittle and Gait (1970), is given a risk-sensitive formulation. It is shown that the reduction of an optimally controlled homing problem to the treatment of an uncontrolled process, demonstrated by Whittle and Gait, can be achieved in the risk-sensitive case. Two scalar problems are analyzed in detail.

The bifurcating autoregressive model has been used previously to model cell lineage data. A feature of this model is that each line of descendants from an initial cell follows an AR(1) model, and that the environmental effects on sisters are correlated. However, this model concentrates on modelling the correlations between mother and daughter cells and between sister cells, and does not explain the large correlations between more distant relatives observed by some authors. Here the model is extended, firstly by allowing lines of descent to follow an ARMA(p,q) model rather than an AR(1) model, and secondly by allowing correlations between the environmental effects of relatives more distant than sisters. The models are applied to several data sets consisting of independent cell lineage trees.

In this paper statistical properties of estimators of drift parameters for diffusion processes are studied by modern numerical methods for stochastic differential equations. This is a particularly useful method for discrete time samples, where estimators can be constructed by making discrete time approximations to the stochastic integrals appearing in the maximum likelihood estimators for continuously observed diffusions. A review is given of the necessary theory for parameter estimation for diffusion processes and for simulation of diffusion processes. Three examples are studied.

The distribution of the length of random secants through a convex region is formulated in terms of the intersection volume of the convex region with its translated self. This method allows a more straightforward approach to calculating secant-length distributions for various measures of randomness. The results are applied to calculating the transit-time distribution of particles traversing a convex region. Several examples are given.

Let U be the distribution function of the non-negative passage time of an individual edge of the square lattice, and let a0n be the minimal passage time from (0, 0) to (n, 0). The process a0n/n converges in probability as n → ∞to a finite constant μ (U) called the time constant. It is proven that μ (Uk)→ μ(U) whenever Uk converges weakly to U.

Univariate heavy-tailed analysis rests on the analytic notion of regularly varying functions. For multivariate heavy-tailed analysis, reliance on functions is awkward because multivariate distribution functions are not natural objects for many purposes and are difficult to manipulate. An approach based on vague convergence of measures makes the differences between univariate and multivariate analysis evaporate. We survey the foundations of the subject and discuss statistical attempts to assess dependence of large values. An exploratory technique is applied to exchange rate return data and shows clear differences in the dependence structure of large values for the Japanese Yen versus German Mark compared with the French Franc versus the German Mark.

In this paper the steady-state behaviour of multistate monotone systems of multistate components is considered by applying the theory for stationary and synchronous processes with an embedded point process. After reviewing some general results on stationary availability, stationary interval availability and stationary mean interval performance probabilities, we concentrate on systems with independently working and separately maintained components. For this case an explicit formula is given for the mean time which the system in steady state sojourns in states not below a fixed critical level.

In this paper we obtain a new crossing result for Brownian motion. The boundary studied is a piecewise linear function consisting of two lines. The expression obtained for the boundary crossing probability is of a simple directly computable form.

In this paper, the extremal properties of the ergodic sojourn times in G/G/1queues under various service disciplines are studied in terms of the convex ordering. It is shown that among work-conserving non-preemptive service disciplines that are service time independent, the FIFO and the LIFO service disciplines provide the minima and the maxima, respectively, of the ergodic sojourn times for any G/G/1 queue. For G/M/1 queues, this class of work-conserving service disciplines is extended to include preemptive/resume disciplines. In this case the FIFO and LIFO-P (preemptive/resume LIFO) service disciplines attain the minima and maxima, respectively. These extend results of Durr (1971), Kingman (1962) and a recent result of Ramaswami (1984). Further results are obtained for G/Em/1 and G/D/1 queues.

In this paper, we investigate k-out-of-n systems with independent and identically distributed components. Some characterizations of the IFR(2), DMRL, NBU(2) and NBUC classes of life distributions are obtained in terms of the monotonicity of the residual life given that the (n-k)th failure has occurred at time t ≥ 0. These results complement those reported by Belzunce, Franco and Ruiz (1999). Similar conclusions based on the residual life of a parallel system conditioned by the (n-k)th failure time are presented as well.