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.

We consider asset allocation strategies for the case where an investor can allocate his wealth dynamically between a risky stock, whose price evolves according to a geometric Brownian motion, and a risky bond, whose price is subject to negative jumps due to its credit risk and therefore has discontinuous sample paths. We derive optimal policies for a number of objectives related to growth. In particular, we obtain the policy that minimizes the expected time to reach a given target value of wealth in an exact explicit form. We also show that this policy is exactly equivalent to the policy that is optimal for maximizing logarithmic utility of wealth and, hence, the expected average rate at which wealth grows, as well as to the policy that maximizes the actual asymptotic rate at which wealth grows. Our results generalize and unify results obtained previously for cases where the bond was risk-free in both continuous- and discrete-time.

Consider a sequence of independent and identically distributed random variables along with a specified set of k-vectors. We present an expression for E [T], the mean time until the last k observed random variables fall within this set. Not only can this expression often be used to obtain bounds on E[T], it also gives rise to an efficient way of approximating E[T] by a simulation. Specific lower and upper bounds for E[T] are also derived. These latter bounds are given in terms of a parameter, and a Markov chain Monte Carlo approach to approximate this parameter by a simulation is indicated. The results of this paper are illustrated by considering the problem of determining the mean time until a sequence of k-valued random variables has a run of size k that encompasses each value.

We study call admission rates in a linear communication network with each call identified by an arrival time, duration, bandwidth requirement, and origin-destination pair. Network links all have the same bandwidth capacity, and a call can be admitted only if there is sufficient bandwidth available on every link along the call's path. Calls not admitted are held in a queue, in contrast to the protocol of loss networks. We determine maximum admission rates (capacities) under greedy call allocation rules such as First Fit and Best Fit for several baseline models and prove that the natural necessary condition for stability is sufficient. We establish the close connections between our new problems and the classic problems of bin packing and interval packing. In view of these connections, it is surprising to find that Best Fit allocation policies are inferior to First Fit policies in the models studied.

We consider a process that starts at height y, stays there for a time X0 ∼ exp(y) when it drops to a level Z1 ∼ U(0, y). Thereafter it stays at level Zn for time Xn ∼ exp(Zn), then drops to a level Zn+1 ∼ U(0,Zn). We investigate properties of this process, as well as the Poisson hyperbolic process which is obtained by randomizing the starting point y of the above process. This process is associated with a rate 1 Poisson process in the positive quadrant: Its path is the minimal RCLL decreasing step function through Poisson points in the positive quadrant. The finite dimensional distributions are then multivariate exponential in sense of Marshall-Olkin.

This paper is devoted to the analysis of a fluid queue with a buffer content that varies linearly during periods that are governed by a three-state semi-Markov process. Two cases are being distinguished: (i) two upward slopes and one downward slope, and (ii) one upward slope and two downward slopes. In both cases, at least one of the period distributions is allowed to be completely general. We obtain exact results for the buffer content distribution, the busy period distribution, and the distribution of the maximal buffer content during a busy period. The results are obtained by establishing relations between the fluid queues and ordinary queues with instantaneous input and by using level crossing theory.

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.

In this paper we consider the use of coupling ideas in efficiently computing a certain class of transient performance measures. Specifically, we consider the setting in which the stationary distribution is unknown, and for which no exact means of generating stationary versions of the process is known. In this context, we can approximate the stationary distribution from empirical data obtained from a first-stage steady-state simulation. This empirical approximation is then used in place of the stationary distribution in implementing our coupling-based estimator. In addition to the empirically based coupling estimator itself, we also develop an associated confidence interval procedure.

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.

In this paper we consider an infinite buffer fluid model whose input is driven by independent semi-Markov processes. The output capacity of the buffer is a constant. We derive upper and lower bounds for the limiting distribution of the stationary buffer content process. We discuss examples and applications where the results can be used to determine bounds on the loss probability in telecommunication networks.

The length biased distribution occurs naturally for some sampling plans in reliability, biometry, and survival analysis. In this note, inequalities for length biased distributions are proved for monotone hazard functions and mean residual life functions. The problem of sampling and selection of experiments from the length biased distribution as opposed to the original distribution is addressed. Certain modified cross-entropy measures are also investigated.

Consider a family of distributions with survival distributions that are log concave and stochastically increasing in a parameter over which it will be mixed. It is shown that a necessary and sufficient condition for a mixture over any such family to have an increasing failure rate is that the mixing distribution have an increasing failure rate. Some observations are given on generalizing this result as well as weakening the conditions on Prekopa's Theorem.

We study the Birnbaum importance of the consecutive-k-out-of-n line and clear up confusion in some claimed but unproved results. We introduce some new techniques which not only prove these claimed results but also generalize them much further. Finally, we extend our results to the 2-out-of-m-out-of-n line.

Since callers encountering busy signals often want to redial, modern communication networks have been designed to provide automatic redialing. Redialing services commonly have two parameters: a maximum number n of retries and a total duration τ over which retries are to be made. Typically, retries are made at evenly spaced time intervals of length τ/n until either the call succeeds or n retries have failed. This rule has an obvious intuitive appeal; indeed, among the main results of this paper are proofs that τ/n-spacing is optimal in certain basic models of called-number behavior. However, it is easy to find situations where τ/n-spacing is not optimal, as the paper verifies.

All of our models assume Poisson arrivals, but different assumptions are studied for the call durations; for a given mean, these are allowed to have the relatively high-variance exponential distribution or the zero-variance distribution concentrated at a point. We approximate the probability of success for the Erlang loss model with c > 1 trunks, and we calculate exact probabilities of success for the c = 1 Erlang model and the model with one trunk and constant call durations. For the latter model, we present two intriguing conjectures, one about the optimal choice of τ when n = 1 and one about the optimality of the τ/n-spacing policy. In spite of their apparent simplicity, these conjectures seem difficult to resolve. Finally, we study policies that continue redialing until they succeed, balancing a short mean wait against a small mean number of retries to success.

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.

Consider a system consisting of n components in series, subject to failures, with one repairman. No preemption is permitted during a repair. The repairman has a list of components and when he becomes available he repairs the failed component which is closest to the top of the list. The goal is to minimize the mean time until all the components operate—the mean down time. For systems consisting of two components, we assume that the operation and repair times are generally distributed and that the operation times are stochastically ordered. For systems consisting of n components we assume exponentially distributed operation times with rate λi and fixed repair times. We prove that lists in which the components are ordered in an increasing order of their expected operation times are optimal.

In this paper, we give characterizations of nonparametric families of life distributions based on aging and variability orderings of the residual life of k-out-of-n systems. We obtain some results of increasing (decreasing) failure rate classes as well as of new classes decreasing (increasing) mean residual life of the system and new better (worse) than used in expectations of the system. Relationships among themselves and others such as new better (worse) than used, new better (worse) than used expectations, and decreasing (increasing) mean residual life are also given.

Control limit type policies are widely discussed in the literature, particularly regarding the maintenance of deteriorating systems. Previous studies deal mainly with stationary deterioration processes, where costs and transition probabilities depend only on the state of the system, regardless of its cumulative age. In this paper, we consider a nonstationary deterioration process, in which operation and maintenance costs, as well as transition probabilities “deteriorate” with both the system's state and its cumulative age. We discuss conditions under which control limit policies are optimal for such processes and compare them with those used in the analysis of stationary models.

Two maintenance models are examined: in the first (as in the majority of classic studies), the only maintenance action allowed is the replacement of the system by a new one. In this case, we show that the nonstationary results are direct generalizations of their counterparts in stationary models. We propose an efficient algorithm for finding the optimal policy, utilizing its control limit form. In the second model we also allow for repairs to better states (without changing the age). In this case, the optimal policy is shown to have the form of a 3-way control limit rule. However, conditions analogous to those used in the stationary problem do not suffice, so additional, more restrictive ones are suggested and discussed.

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

A recent interesting paper (Cooper, Niu, and Srinivasan [2]) shows how for some cyclic production systems reducing setup times can surprisingly increase work in process. There the authors show how in these situations the introduction of forced idle time can be used to optimize system performance. Here we show that introducing forced idle time at different points during the production cycle can further improve performance in these situations and also in situations where the suggestions from [2] yield no improvement.