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.

Relational structures offer a common framework for handling graphs and hypergraphs of various kinds. Operations like disjoint union, the creation of new relations by means of quantifier-free formulas, and relabellings of relations make it possible to denote them using algebraic expressions. It is known that every monadic second-order property of a structure is verifiable in time proportional to the size of such an algebraic expression defining it. We prove here that this result remains true if we also use in these algebraic expressions a fusion operation that fuses all elements of the domain satisfying some unary predicate. The value mapping from these algebraic expressions to the structures they denote is a monadic second-order definable transduction, which means that the structure is definable inside the tree representing the algebraic expression by monadic second-order formulas. It follows (by using results of other articles) that, with this fusion operation, we cannot generate more graph families, but we can generate them with less unary auxiliary predicates. We also obtain clear-cut characterizations of Vertex Replacement and Hyperedge Replacement context-free graph grammars in terms of four types of operations, amongst which is the fusion of vertices satisfying a specified predicate.

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.

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.

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.

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.

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.

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.

In this article, we develop methods for estimating the expected time to the first loss in an Erlang loss system. We are primarily interested in estimating this quantity under light traffic conditions. We propose and compare three simulation techniques as well as two Markov chain approximations. We show that the Markov chain approximations proposed by us are asymptotically exact when the load offered to the system goes to zero. The article also serves to highlight the fact that efficient estimation of transient quantities of stochastic systems often requires the use of techniques that combine analytical results with simulation.

We prove that the effective resistances of spherically symmetric random trees dominate in mean the effective resistances of random trees corresponding branching processes in varying environments and having the same growth law of spherically symmetric trees. We conjecture that the statement does not necessarily hold true in the case of stochastic domination and give an idea of constructing a counterexample.

In this article, we propose a new continuous-time stochastic inventory model with deterioration and stock-dependent demand items. We then formulate the problem of finding the optimal impulse control schedule that minimizes the total expected return over an infinite horizon, as a quasivariational inequality (QVI) problem. The QVI is shown to lead to an (s, S) policy, where s and S are determined uniquely as a solution of some algebraic equations.

This article deals with Markovian models defined on a finite-dimensional discrete state space and possess a stationary state distribution of a product-form. We view the space of such models as a mathematical object and explore its structure. We focus on models on an orthant [script Z]+n, which are homogeneous within subsets of [script Z]+n called walls, and permit only state transitions whose ∥ ∥∞-length is 1. The main finding is that the space of such models exhibits a decoupling principle: In order to produce a given product-form distribution, the transition rates on distinct walls of the same dimension can be selected without mutual interference. This principle holds also for state spaces with multiple corners (e.g., bounded boxes in [script Z]+n).

In addition, we consider models which are homogeneous throughout a finite-dimensional grid [script Z]n, now without a fixed restriction on the length of the transitions. We characterize the collection of product-form measures which are invariant for a model of this kind. For such models with bounded transitions, we prove, using Choquet's theorem, that the only possible invariant measures are product-form measures and their combinations.

We study systems of parallel queues with finite buffers, a single server with random connectivity to each queue, and arriving job flows with random or class-dependent accessibility to the queues. Only currently connected queues may receive (preemptive) service at any given time, whereas an arriving job can only join one of its accessible queues. Using the coupling method, we study three key models, progressively building from simpler to more complicated structures.

In the first model, there are only random server connectivities. It is shown that allocating the server to the Connected queue with the Fewest Empty Spaces (C-FES) stochastically minimizes the number of lost jobs due to buffer overflows, under conditions of independence and symmetry.

In the second model, we additionally consider random accessibility of queues by arriving jobs. It is shown that allocating the server to the C-FES and routing each arriving job to the currently Accessible queue with the Most Empty Spaces (C-FES/A-MES) minimizes the loss flow stochastically, under similar assumptions.

In the third model (addressing a target application), we consider multiple classes of arriving job flows, each allowed access to a deterministic subset of the queues. Under analogous assumptions, it is again shown that the C-FES/A-MES policy minimizes the loss flow stochastically.

The random connectivity/accessibility aspect enhances significantly the structure and application scope of the classical parallel queuing model. On the other hand, it introduces essential additional dynamics and considerable complications. It is interesting that a simple policy like FES/MES, known to be optimal for the classical model, extends to the C-FES/A-MES in our case.

We consider a queue fed by a large number, say n, on–off sources with generally distributed on- and off-times. The queueing resources are scaled by n: The buffer is B ≡ nb and the link rate is C ≡ nc. The model is versatile. It allows one to model both long-range-dependent traffic (by using heavy-tailed on-periods) and short-range-dependent traffic (by using light-tailed on-periods). A crucial performance metric in this model is the steady state buffer overflow probability.

This probability decays exponentially in n. Therefore, if n grows large, naive simulation is too time-consuming and fast simulation techniques have to be used. Due to the exponential decay (in n), importance sampling with an exponential change ofmeasure goes through, irrespective of the on-times being heavy or light tailed. An asymptotically optimal change of measure is found by using large deviations arguments. Notably, the change of measure is not constant during the simulation run, which is different from many other studies (usually relying on large buffer asymptotics).

Numerical examples show that our procedure improves considerably over naive simulation. We present accelerations, we discuss the influence of the shape of the distributions on the overflow probability, and we describe the limitations of our technique.