Assembly of systems to maximize reliability when certain components of the systems can be bolstered in different ways is an important theme in reliability theory. This is done under assumptions of various stochastic orderings among the lifetimes of the components and the spares used to bolster them. The powerful techniques of Schur and arrangement increasing functions are used in this paper to pinpoint optimal allocation results in different settings involving active and standby redundancy allocation, minimal repair and shock-threshold models.

This article provides the theoretical basis of the virtual customer method or positive rare perturbation (RPA) method of sensitivity analysis, and in particular gives a short proof of the light traffic derivative result of Reiman and Simon [5] based on Campbell's formula. As a by-product, we obtain the archetypal H = λG formula associated with a stationary quantity of a queueing system.

A new approach is used to obtain the transient probabilities of the M/M/1 queueing system. The first step of this approach deals with the generating function of the transient probabilities of the uniformized Markov chain associated with this queue. The second step consists of the inversion of this generating function. A new analytical expression of the transient probabilities of the M/M/1 queue is then obtained.

The matrix-geometric work of Neuts could be viewed as a matrix variant of M/M/1. A 2 × 2 matrix counterpart of Neuts for M/M/∞ is introduced, the stability conditions are identified, and the ergodic solution is solved analytically in terms of the ten parameters that define it. For several cases of interest, system properties can be found from simple analytical expressions or after easy numerical evaluation of Kummer functions. When the matrix of service rates is singular, a qualitatively different solution is derived. Applications to telecommunications include some retrial models and an M/M/∞ queue with Markov-modulated input.

We propose a two-parameter family of conjugate prior distributions for the number of undiscovered objects in a class of Bayesian search models. The family contains the one-parameter Euler and Heine families as special cases. The two parameters may be interpreted respectively as an overall success rate and a rate of depletion of the source of objects. The new family gives enhanced flexibility in modelling.

Using the electric network approach, we give very simple derivations for the expected first passage from the origin to the opposite vertex in the d-cube (i.e. the Ehrenfest urn model) and the Platonic graphs.

The reliability of many stochastic repairable systems depends on several characteristics that are time dependent. In this paper, we develop general repair models for a repairable system by using auxiliary stochastic processes which describe physical characteristics of the system and derive various properties of resulting models. We also obtain an inference procedure to assess the number of failures and expected number of failures for our proposed models by observing auxiliary processes.

Clearly, our inference procedure is different from the traditional approach, where successive times of occurrence of failures are observed.

Using Palm-martingale calculus, we derive the workload characteristic function and queue length moment generating function for the BMAP/GI/1 queue with server vacations. In the queueing system under study, the server may start a vacation at the completion of a service or at the arrival of a customer finding an empty system. In the latter case we will talk of a server set-up time. The distribution of a set-up time or of a vacation period after a departure leaving a non-empty system behind is conditionally independent of the queue length and workload. Furthermore, the distribution of the server set-up times may be different from the distribution of vacations at service completion times. The results are particularized to the M/GI/1 queue and to the BMAP/GI/1 queue (without vacations).

Ball and Donnelly (1987) announced a result giving circumstances in which there is positive or negative correlation between the death times in a non-linear, Markovian death process. A proof is provided here, based on results concerning the distribution of optional random variables in terms of their conditional intensities.

Shanthikumar and Yao (1991) introduced some new stochastic order relations to compare the components of a bivariate random vector (X1, X2). As they point out in their paper, even if according to their hazard rate (or likelihood ratio) ordering, the marginal distributions may not be ordered accordingly. We introduce some new concepts where the marginal distributions preserve the corresponding stochastic orders. Also a relation between the bivariate scale model and the introduced bivariate hazard rate ordering is established.

Two devices are subjected to common shocks arriving according to two identical counting processes. Let and denote the probability of surviving k shocks for the first and the second device, respectively. We find conditions on the discrete distributions and in order to obtain the failure rate order (FR), the likelihood ratio order (LR) and the mean residual order (MR) between the random lifetimes of the two devices. We also obtain sufficient conditions under which the above mentioned relations between the discrete distributions are verified in some cumulative damage shock models.

By using large devaitions theory, we give asymptotic formulas for the transient blocking probabilities of M/M/N/N and M (with finite Poissonian sources) M/N/N queues.

We analyze a class of stochastic and dynamic vehicle routing problems in which demands arrive randomly over time and the objective is minimizing waiting time. In our previous work ([6], [7]), we analyzed this problem for the case of uniformly distributed demand locations and Poisson arrivals. In this paper, using quite different techniques, we are able to extend our results to the more realistic case where demand locations have an arbitrary continuous distribution and arrivals follow only a general renewal process. Further, we improve significantly the best known lower bounds for this class of problems and construct policies that are provably within a small constant factor relative to the optimal solution. We show that the leading behavior of the optimal system time has a particularly simple form that offers important structural insight into the behavior of the system. Moreover, by distinguishing two classes of policies our analysis shows an interesting dependence of the system performance on the demand distribution.

In this paper we study scheduling problems of multiclass customers on identical parallel processors. A new type of arrival process, called a Markov decision arrival process, is introduced. This arrival process can be controlled and allows for an indirect dependence on the numbers of customers in the queues. As a special case we show the optimality of LEPT and the µc-rule in the last node of a controlled tandem network for various cost structures. A unifying proof using dynamic programming is given.