We consider an exponential queueing system with multiple stations, each of which has an infinite number of servers and a dedicated arrival stream of jobs. In addition, there is an arrival stream of jobs that choose a station based on the state of the system. In this paper we describe two heavy traffic approximations for the stationary joint probability mass function of the number of busy servers at each station. One of the approximations involves state-space collapse and is accurate for large traffic loads. The state-space in the second approximation does not collapse. It provides an accurate estimate of the stationary behavior of the system over a wide range of traffic loads.

In this paper, a new concept called generalized stochastic convexity is introduced as an extension of the classic notion of stochastic convexity. It relies on the well-known concept of generalized convex functions and corresponds to a stochastic convexity with respect to some Tchebycheff system of functions. A special case discussed in detail is the notion of stochastic s-convexity (s ∈ [real number symbol]), which is obtained when this system is the family of power functions {x0, x1,..., xs−1}. The analysis is made, first for totally positive families of distributions and then for families that do not enjoy that property. Further, integral stochastic orderings, said of Tchebycheff-type, are introduced that are induced by cones of generalized convex functions. For s-convex functions, they reduce to the s-convex stochastic orderings studied recently. These orderings are then used for comparing mixtures and compound sums, with some illustrations in epidemic theory and actuarial sciences.

We consider a stochastic process with an embedded point process in a stationary and ergodic context. Under a “lack of anticipation” assumption for the evolution of the process vis-à-vis the point process, a new better (worse) than used expectation property for the point process, and a monotonicity assumption for the behavior of the process between points, inequalities between event and time averages are obtained. Sample path monotonicity between points is not required (as is the case with existing approaches) and can be replaced with a simple monotonicity requirement for the expected value of the process between points. Inequalities between conditional event and time averages are also examined via a novel argument involving a conditional version of the Palm inversion formula.

We consider a finite capacity queueing system in which each arriving customer offers a reward. A gatekeeper decides based on the reward offered and the space remaining whether each arriving customer should be accepted or rejected. The gatekeeper only receives the offered reward if the customer is accepted. A traditional objective function is to maximize the gain, that is, the long-run average reward. It is quite possible, however, to have several different gain optimal policies that behave quite differently. Bias and Blackwell optimality are more refined objective functions that can distinguish among multiple stationary, deterministic gain optimal policies. This paper focuses on describing the structure of stationary, deterministic, optimal policies and extending this optimality to distinguish between multiple gain optimal policies. We show that these policies are of trunk reservation form and must occur consecutively. We then prove that we can distinguish among these gain optimal policies using the bias or transient reward and extend to Blackwell optimality.

We state a simulation metatheorem and then present an example that could be considered a counterexample to the metatheorem.

We consider a series system with p components where the failure rate of each component depends on the residual number of defects present. Successive tasks are given to the system with each task completion time independent of each component failure time. Based on the outcomes over a fixed testing period, the asymptotic properties of the estimators of the component parameters, task completion parameters, and eventual system reliability are obtained.

Let [script S] be a system that can be in one of two states, up or down. We can interpret the up state as the working state and the down state as the nonworking state for system [script S]. Assume the system [script S] is up at time t = 0. Denote the elapsed time from zero until [script S] enters down state by U1, and the elapsed time from then until [script S] is up again by D1. The interval [0,U1 + D1) is called cycle 1, and, at that point, cycle 2 starts and the sequence is repeated so on and so forth. In general, the up time and down time in the jth cycle are denoted by Uj and Dj, respectively, for j ≥ 1. We can introduce a binary process {X(t), t ≥ 0} to describe the state of [script S] at time t using X(t) = 1(0) to indicate [script S] is up (down). Many problems require finding the probability that [script S] is up at time t, that is, P(X(t) = 1). However, an even more interesting question to answer is what the probability is that [script S] is in the up state in an interval of length w starting at time t, that is, P(X(s) = 1, t ≤ s ≤ t + w). Due to the complexity of this problem, no explicit expressions are available for most systems even in the case where (Uj, Dj), j ≥ 1 are i.i.d. Fortunately, in practice engineers are more interested in the long-run properties of the above two probabilities.

Fill and Holst conjectured for the move-to-front rule that the probability that the search time is greater than c will be Schur concave in the stationary distribution for any value of c. This paper disproves the conjecture but proves some conclusions that would be implied by the conjecture.