We use a Poisson imbedding technique to investigate the possibilities of generalizing some pathwise and stochastic monotonicity results from the M/M/k queues to systems with monotone failure rate service time distribution. A dichotomy between the decreasing failure rate (DFR) and the increasing failure rate (IFR) cases is revealed: In the DFR case, we achieve that the number of customers in the systems is stochastically increasing if it is idle at time 0, whereas in the IFR case, there is an alternating character that renders a bound between two identical systems that have different initial conditions. We also explore how our methods work in a comparison between systems with different numbers of stations but the same maximal capacity.

We consider a reflected independent superposition of a Brownian motion and a compound Poisson process with positive and negative jumps, which can be interpreted as a model for the content process of a storage system with different types of customers under heavy traffic. The distributions of the duration of a busy cycle and the maximum content during a cycle are determined in closed form.

For a two-node tandem fluid model with gradual input, we compute the joint steady-state buffer-content distribution. Our proof exploits martingale methods developed by Kella and Whitt. For the case of finite buffers, we use an insightful sample-path argument to extend an earlier proportionality result of Zwart to the network case.

We consider a single-server queue that is initially empty and operates under the first-in–first-out service discipline. In this system, delays (waiting times in queue) experienced by subsequent arriving customers form a transient process. We investigate its transient behavior by constructing a sample-path coupling of the transient and a general (delayed) processes. From the coupling, we obtain an identity that relates the sample paths of these two processes. This identity helps us to better understand the queue's approach to the stationary limit and to derive upper and lower bounds on the expected transient delay. In addition, we use a Brownian-motion model to approximate the identity. This produces an approximation of the expected transient delay. The approximation turns out to be identical to the corresponding first moment of a reflected Brownian motion. Thus, it is easy to compute and its accuracy is supported by numerical experiments.

This article studies the geometric convergence rate of a discrete renewal sequence with decreasing hazard rate or, more generally, new worse than used lifetimes. Several variants of these structural orderings are considered. The results are derived from power series methods; roots of generating functions are the prominent issue. Optimality of the rates are considered. Examples demonstrating the utility of the results, as well as applications to Markov chains, are presented.

We investigate the performance of channel assignment policies for cellular networks. The networks are given by an interference graph which describes the reuse constraints for the channels. In the first part, we derive lower bounds on the expected (weighted) number of blocked calls under any channel assignment policy over finite time intervals as well as in the average case. The lower bounds are solutions of deterministic control problems. As far as the average case is concerned, the control problem can be replaced by a linear program. In the second part, we consider the cellular network in the limit, when the number of available channels as well as the arrival intensities are linearly increased. We show that the network obeys a functional law of large numbers and that a fixed channel assignment policy which can be computed from a linear program is asymptotically optimal. Special networks like fully connected and star networks are considered.

Long-term-care (LTC) insurance contracts provide the insured with different benefits for several nursing care levels, for a limited number of benefit eligibility periods. A common assumption in pricing these LTC contracts is that the insured will exercise the right to claim benefits as soon as the eligibility conditions are satisfied. This assumption, however, may contradict the insured's optimization, as it might be worthwhile not to claim when in low care levels and, by doing so, save the option of claiming higher (more expensive) care levels in the future. We term this option of the insured as the deferral option. The consequence of the traditional pricing (i.e., of ignoring the deferral option) is unexpected losses to the insurer. The factors affecting the deferral option's value are the risk of death, the discount factor, the benefit levels of the different care levels, and the transition probabilities between the different care levels.

In this article, the time of the first occurrence of a rare event in a regenerating process is investigated. We obtain the bound of deviation from the distribution of the time of the first occurrence of a rare event in a regenerating process to an exponential distribution.

Barlow and Proschan presented some interesting connections between univariate classifications of life distributions and partial orderings where equivalent definitions for increasing failure rate (IFR), increasing failure rate average (IFRA), and new better than used (NBU) classes were given in terms of convex, star-shaped, and superadditive orderings. Some related results are given by Ross and Shaked and Shanthikumar. The introduction of a multivariate generalization of partial orderings is the object of the present article. Based on that concept of multivariate partial orderings, we also propose multivariate classifications of life distributions and present a study on more IFR-ness.