In this article we characterize l∞-spherical density functions by means of epoch times of nonhomogeneous pure birth processes. Some further properties of l∞-spherical densities, such as Schur-concavity, positive dependence, and stochastic comparisons, are also given. The relationships of l∞-spherical densities to notions of interest in reliability theory are highlighted.

We consider the problem of dynamic allocation of a single server with batch processing capability to a set of parallel queues. Jobs from different classes cannot be processed together in the same batch. The arrival processes are mutually independent Poisson flows with equal rates. Batches have independent and identically distributed exponentially distributed service times, independent of the batch size and the arrival processes. It is shown that for the case of infinite buffers, allocating the server to the longest queue, stochastically maximizes the aggregate throughput of the system. For the case of equal-size finite buffers the same policy stochastically minimizes the loss of jobs due to buffer overflows. Finally, for the case of unequal-size buffers, a threshold-type policy is identified through an extensive simulation study and shown to consistently outperform other conventional policies. The good performance of the proposed threshold policy is confirmed in the heavy-traffic regime using a fluid model.

Two discrete time risk models under rates of interest are introduced. Ruin probabilities in the two risk models are discussed. Stochastic inequalities for the ruin probabilities are derived by martingales and renewal recursive techniques. The inequalities can be used to evaluate the ruin probabilities as upper bounds. Numerical illustrations for these results are given.

We consider two model variants of a production-inventory system. The system is characterized by a producing machine which is susceptible to failure following which it must be repaired to make it operative again. The machine's production can also be stopped deliberately because of stocking capacity limitations. During ON periods the input into the buffer is continuous and uniform (until a threshold is reached), whereas during OFF periods the output from the buffer is a compound Poisson process. We are interested in computing the equilibrium content level process under the assumption that full backlogging is allowed. In the first model, variant OFF periods are independent of the demand process, and in the second variant, they are determined and controlled in accordance with a certain level crossing stopping rule.

We consider a system with heterogeneous unreliable components that requires only one component to be turned on in order for it to operate. Repair workers may have different skills and may be unavailable for random periods of time. The problem is to determine a usage and repair policy to maximize system availability. We give conditions under which the optimal usage policy is to always use, or turn on, the component with the shortest repair time, and the optimal repair policy is to always repair the most reliable component (with the smallest failure rate). We fully characterize the optimal policy when there are only two components. Our system is equivalent to a closed system with multiple single-server queues, where the objective is to minimize server idle time at one of the queues.

The deviation matrix of an ergodic, continuous-time Markov chain with transition probability matrix P(·) and ergodic matrix Π is the matrix D ≡ ∫0∞(P(t) − Π) dt. We give conditions for D to exist and discuss properties and a representation of D. The deviation matrix of a birth–death process is investigated in detail. We also describe a new application of deviation matrices by showing that a measure for the convergence to stationarity of a stochastically increasing Markov chain can be expressed in terms of the elements of the deviation matrix of the chain.

In this article we consider a finite queue with its arrivals controlled by the random early detection algorithm. This is one of the most prominent congestion avoidance schemes in the Internet routers. The aggregate arrival stream from the population of transmission control protocol sources is locally considered stationary renewal or Markov modulated Poisson process with general packet length distribution. We study the exact dynamics of this queue and provide the stability and the rates of convergence to the stationary distribution and obtain the packet loss probability and the waiting time distribution. Then we extend these results to a two traffic class case with each arrival stream renewal. However, computing the performance indices for this system becomes computationally prohibitive. Thus, in the latter half of the article, we approximate the dynamics of the average queue length process asymptotically via an ordinary differential equation. We estimate the error term via a diffusion approximation. We use these results to obtain approximate transient and stationary performance of the system. Finally, we provide some computational examples to show the accuracy of these approximations.