The various duality results existing for two-barrier random walks are unified. Particular cases in the literature involving reflecting and reversing are deduced as corollaries.

Consider a network of queues in equilibrium. When are the flows observed on two given links of that network equivalent, i.e., when do they have the same law?

The verification of the equivalence of two such point processes is first reduced to an algebraic problem by a technique based on the filtering theory.

This method is then used to show that the arrival and departure processes at an M/M/1 node in a Jackson network are not always equivalent, thereby contradicting a conjecture made in [8]. An example where that equivalence holds is also given; it provides new results on the time reversibility of some familiar processes.

Using a known fact that a Galton–Watson branching process can be represented as an embedded random walk, together with a result of Heyde (1964), we first derive finite exponential moment results for the total number of descendants of an individual. We use this basic and simple result to prove analogous results for the population size at time t and the total number of descendants by time t in an age-dependent branching process. This has applications in justifying the interchange of expectation and derivative operators in simulation-based derivative estimation for generalized semi-Markov processes. Next, using the result of Heyde (1964), we show that, in a stable GI/GI/1 queue, the length of a busy period and the number of customers served in a busy period have finite exponential moments if and only if the service time does.

In this paper we investigate the characterizations of life distributions under four stochastic orderings, < p, < (p), < (p) and < L, by a unified method. Conditions for the stochastic equality of two non-negative random variables under the four stochastic orderings are derived. Many previous results are consequences. As applications, we provide characterizations of life distributions by a single value of their Laplace transforms under orderings < p and < (p) and their moment generating functions under orderings < p and < (p). Under ordering < L, a characterization is given by the expected value of a strictly completely monotone function. The conditions for the stochastic equality of two non-negative vectors under the stochastic orderings < (p), < (p) and < L are presented in terms of the Laplace transforms and moment generating functions of their extremes and sample means. Characterizations of the exponential distribution among L and L life distribution classes are also given.

We show that the waiting time in queue and the sojourn time of every customer in the G/G/1 and G/D/c queue are jointly convex in mean interarrival time and mean service time, and also jointly convex in mean interarrival time and service rate. Counterexamples show that this need not be the case, for the GI/GI/c queue or for the D/GI/c queue, for c ≧ 2. Also, we show that the average number of customers in the M/D/c queue is jointly convex in arrival and service rates.

These results are surprising in light of the negative result for the GI/GI/2 queue (Weber (1983)).

We consider the stable k-server queue with non-stationary Poisson arrivals and i.i.d. service times and show that the non-time-homogeneous Markov process Zt = (the queue length and residual service times at time t) can be subordinated to a stable time-homogeneous regenerative process. As an application we show that if the system starts from given conditions at time s then the distribution of Zt stabilizes (but depends on t) as s tends backwards to –∞. Also moment and stochastic domination results are established for the delay and recurrence times of the regenerative process leading to results on uniform rates of convergence.

We consider a production-inventory model operating in a stochastic environment that is modulated by a finite state continuous-time Markov chain. When the inventory level reaches zero, an order is placed from an external supplier. The costs (purchasing and holding costs) are modulated by the state at the order epoch time. Applying a matrix analytic approach, fluid flow techniques, and martingales, we develop methods to obtain explicit equations for these cost functionals in the discounted case and under the long-run average criterion. Finally, we extend the model to allow backlogging.

The distribution of the number of items drawn in a secretary problem, with an order s selection role and a success if any of the best s items is selected, is obtained by a probabilistic argument. Moments and asymptotics readily follow.

We consider the problem of the optimal duration of a burn-in experiment for n identical units with conditionally exponential life-times of unknown parameter Λ. The problem is formulated as an optimal stopping problem for a suitably defined two-dimensional continuous-time Markov process. By exploiting monotonicity properties of the statistical model and of the costs we prove that the optimal stopping region is monotone (according to an appropriate definition) and derive a set of equations that uniquely determine it and that can be easily solved recursively. The optimal stopping region varies monotonically with the costs. For the class of problems corresponding to a prior distribution on Λ of type gamma it is shown how the optimal stopping region varies with respect to the prior distribution and with respect to n.

A recent study on the GI/G/1 queue derives the Maclaurin series for the moments of the waiting time and the delay to respect to some parameters. By the same approach, we obtain an identity on the moments of the transient delay of the M/G/1 queue. This identity allows us to understand the transient behavior of the process better. We apply the identity with other established results to study convergence rate and stochastic concavity of the transient delay process, and to derive bounds and approximations of the moments. Our approximation and bound both have simple closed forms and are asymptotically exact as either the traffic intensity goes to zero or the process approaches stationarity. Performance of the approximation of several M/G/1 queues is illustrated by numerical experiments. It is interesting to note that our results can also help to gain variance reduction in simulation.

A martingale approach to a dynamic program with general state and action spaces is taken. Several necessary and sufficient conditions are given for a policy to be optimal. The results comprehend and modify different criteria of optimality given for dynamic programming problems. Finally, two applications are stated.

Choi and Park [2] derived an expression for the joint stationary distribution of the number of customers of k types who arrive in batches at an infinite-server system of M/M/∞ type. We propose another method of solving this problem and extend the result to the case of general service times (not necessarily independent). We also get a transient solution. Our main result states that the k- dimensional vector of the number of customers of k types in the system is a certain linear function of a (2k – 1)-dimensional vector whose coordinates are independent Poisson random variables.

Two-dimensional bounded birth and death processes are considered. State probabilities are given for the generalized case in which the transition rates are arbitrary time-independent functions of the population sizes. Some applications in epidemic theory and species interaction processes are briefly described.

In this note, we compare the arrival and time stationary distributions of the number of customers in the GI/M/c/n and GI/M/c queueing systems as n tends to infinity. We show that earlier results established for GI/M/1/n and GI/M/1 remain true. Namely, it is proved that if the interarrival time c.d.f. H is non lattice with mean value λ−1 and if the traffic intensity is strictly less than one, then the convergence rates in l1norm of the arrival and time stationary distributions of GI/M/c/n to the corresponding stationary distributions of GI/M/c are geometric and are characterized by ω, the unique solution in (0,1) of the equation z = ∫∞0 exp{-μc(1-z)t}dH(t).

The diffusions on the shape and size-and-shape spaces induced by brownian motions on the pre-size-and-shape spaces have been investigated in several papers (cf.). We here address the dual problem: the character of the diffusions on the pre-shape and pre-size-and-shape spaces which induce brownian motions on the shape and size-and-shape spaces. In particular we show that the shape and size-and-shape spaces for k labelled points in ℝm are stochastically complete if k > m and obtain the heat kernels of certain diffusions which induce brownian motions on the size-and-shape spaces.

We present general criteria for analyzing the crossing characteristics of RI, the reliability function of an m-of-n system of components operating within a laboratory (or test-bench) environment, and RO, the reliability function of the same system now operating subject to an external environment. Inside the laboratory the components' lifetimes may be dependently distributed, and the external environment is modeled using the general approach of Lindley and Singpurwalla (1986). Our techniques, which utilize results basic to the theory of order statistics, apply to broad classes of external environment models.

It has recently been shown that, for the FCFS G/G/1 queue, the workload and attained waiting time of a customer in service have the same stationary distribution. We show that, for a general queueing system setting, the workload and total attained waiting time of customers in service are identical in average but the equality of the distributions is not true in general except for the FCFS G/G/1 queue.

We consider the limit distribution of maxima and point processes, connected with crossings of an increasing level, for a sequence of Gaussian stationary processes. As an application we investigate the limit distribution of the error of approximation of Gaussian stationary periodic processes by random trigonometric polynomials in the uniform metric.

For two n-dimensional elliptical random vectors X and Y, we establish an identity for $\mathbb{E}[f({\bf Y})]- \mathbb{E}[f({\bf X})]$, where $f\,{:}\, \mathbb{R}^n \rightarrow \mathbb{R}$ satisfies some regularity conditions. Using this identity we provide a unified method to derive sufficient and necessary conditions for classifying multivariate elliptical random vectors according to several main integral stochastic orders. As a consequence we obtain new inequalities by applying the method to multivariate elliptical distributions. The results generalize the corresponding ones for multivariate normal random vectors in the literature.

Random processes of convex plates and line segments imbedded in R3 are considered in this paper, and the expected values of certain random variables associated with such processes are computed under a mean stationarity assumption, by resorting to some general formulas of integral geometry.