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.

“Guess Who?” is a popular two player game where players ask “Yes”/“No” questions to search for their opponent's secret identity from a pool of possible candidates. This is modeled as a simple stochastic game. Using this model, the optimal strategy is explicitly found. Contrary to popular belief, performing a binary search is not always optimal. Instead, the optimal strategy for the player who trails is to make certain bold plays in an attempt catch up. This is discovered by first analyzing a continuous version of the game where players play indefinitely and the winner is never decided after finitely many rounds.

Stochastic clearing theory has wide-spread applications in the context of supply chain and service operations management. Historical application domains include bulk service queues, inventory control, and transportation planning (e.g., vehicle dispatching and shipment consolidation). In this paper, motivated by a fundamental application in shipment consolidation, we revisit the notion of service performance for stochastic clearing system operation. More specifically, our goal is to evaluate and compare service performance of alternative operational policies for clearing decisions, as quantified by a measure of timely service referred to as Average Order Delay ($AOD$). All stochastic clearing systems are subject to service delay due to the inherent clearing practice, and $\textrm {AOD}$ can be thought of as a benchmark for evaluating timely service. Although stochastic clearing theory has a long history, the existing literature on the analysis of $\textrm {AOD}$ as a service measure has several limitations. Hence, we extend the previous analysis by proposing a more general method for a generic analytical derivation of $\textrm {AOD}$ for any renewal-type clearing policy, including but not limited to alternative shipment consolidation policies in the previous literature. Our proposed method utilizes a new martingale point of view and lends itself for a generic analytical characterization of $\textrm {AOD}$, leading to a complete comparative analysis of alternative renewal-type clearing policies. Hence, we also close the gaps in the literature on shipment consolidation via a complete set of analytically provable results regarding $\textrm {AOD}$ which were only illustrated through numerical tests previously.

We consider open networks of queues with Processor-Sharing discipline and signals. The signals deletes all the customers present in the queues and vanish instantaneously. The customers may be usual customers or inert customers. Inert customers do not receive service but the servers still try to share the service capacity between all the customers (inert or usual). Thus a part of the service capacity is wasted. We prove that such a model has a product-form steady-state distribution when the signal arrival rates are positive.

New computing and communications paradigms will result in traffic loads in information server systems that fluctuate over much broader ranges of time scales than current systems. In addition, these fluctuation time scales may only be indirectly known or even be unknown. However, we should still be able to accurately design and manage such systems. This paper addresses this issue: we consider an M/M/1 queueing system operating in a random environment (denoted M/M/1(R)) that alternates between HIGH and LOW phases, where the load in the HIGH phase is higher than in the LOW phase. Previous work on the performance characteristics of M/M/1(R) systems established fundamental properties of the shape of performance curves. In this paper, we extend monotonicity results to include convexity and concavity properties, provide a partial answer to an open problem on stochastic ordering, develop new computational techniques, and include boundary cases and various degenerate M/M/1(R) systems. The basis of our results are novel representations for the mean number in system and the probability of the system being empty. We then apply these results to analyze practical aspects of system operation and design; in particular, we derive the optimal service rate to minimize mean system cost and provide a bias analysis of the use of customer-level sampling to estimate time-stationary quantities.

This article discusses a real-world application of a terminating two-person stochastic game. The problem comes from a Dutch television game show in which two finalists play a dice game. Each player chooses a number of dice to be rolled. The score of the roll is added to the player's total provided that none of the dice showed the outcome one. The first player reaching a prespecified number of points is the winner. This article discusses the computation and the structure of an optimal strategy.

Bayesian networks are convenient graphical expressions for high-dimensional probability distributions which represent complex relationships between a large number of random variables. They have been employed extensively in areas such as bioinformatics, artificial intelligence, diagnosis, and risk management. The recovery of the structure of a network from data is of prime importance for the purposes of modeling, analysis, and prediction. There has been a great deal of interest in recent years in the NP-hard problem of learning the structure of a Bayesian network from observed data. Typically, one assigns a score to various structures and the search becomes an optimization problem that can be approached with either deterministic or stochastic methods. In this paper, we introduce a new search strategy where one walks through the space of graphs by modeling the appearance and disappearance of edges as a birth and death process. We compare our novel approach with the popular Metropolis–Hastings search strategy and give empirical evidence that the birth and death process has superior mixing properties.

A general type of model that results in components having a negative type of dependence is proposed. Specifically, a system of dependent binary components, in which component i is said to function if at least one of a given number of multinomial trials results in that outcome, is considered. The multinomial probabilities that maximize the probability of a k-of-n system are determined. Some approximations for the system reliability are proposed first for a system that functions unless k consecutive components are failed and then for a general system. A simulation method that uses a variance reduction technique to estimate the reliability is presented.

We consider a single-server queuing system with two job classes under service policies of threshold type. The server switches from type 1 to type 2 when either the former queue is empty or the latter reaches size T; it switches from type 2 to type 1 when the former queue size drops below T and the latter is not empty. The joint queue-length distribution is determined for preemptive and nonpreemptive implementations using both analytic techniques and the power series algorithm.

Cloud storage faces many problems in the storage process which badly affect the system's efficiency. One of the most problems is insufficient buffer space in cloud storage. This means that the packets of data wait to have storage service which may lead to weakness in performance evaluation of the system. The storage process is considered a stochastic process in which we can determine the probability distribution of the buffer occupancy and the buffer content and predict the performance behavior of the system at any time. This paper modulates a cloud storage facility as a fluid queue controlled by Markovian queue. This queue has infinite buffer capacity which determined by the M/M/1/N queue with constant arrival and service rates. We obtain the analytical solution of the distribution of the buffer occupancy. Moreover, several performance measures and numerical results are given which illustrate the effectiveness of the proposed model.

In this paper, we continue our investigation of the shape of the failure rate of a mixture of two densities. In our recent paper, Block, Langberg and Savits [2], we introduced a variation of Glaser's method in which we emphasized the role of the mixing parameter q. There we determined all possible shapes of the failure rate function for a mixture of one exponential and one gamma density. Here we classify all possible shapes for a mixture of two gamma densities having shape parameters b, c>0 and scale parameters λ, μ>0.

We consider an M/G/1 retrial queue with finite capacity of the retrial group. We derive the Laplace transform of the busy period using the catastrophe method. This is the key point for the numerical inversion of the density function and the computation of moments. Our results can be used to approach the corresponding descriptors of the M/G/1 queue with infinite retrial group, for which direct analysis seems intractable.

In this paper, we study the bivariate Birnbaum–Saunders (BVBS) distribution from a reliability point of view. The monotonicity of the hazard rates of the univariate as well as the conditional distributions is discussed. Clayton's association measure is obtained in terms of the hazard gradient and its value in the case of the BVBS distribution is derived. The probability distributions, in the case of series and parallel systems, are derived and the monotonicity of the failure rate, in the case of series system, is discussed.

In order to understand queueing performance given only partial information about the model, we propose determining intervals of likely values of performance measures given that limited information. We illustrate this approach for the mean steady-state waiting time in the $GI/GI/K$ queue. We start by specifying the first two moments of the interarrival-time and service-time distributions, and then consider additional information about these underlying distributions, in particular, a third moment and a Laplace transform value. As a theoretical basis, we apply extremal models yielding tight upper and lower bounds on the asymptotic decay rate of the steady-state waiting-time tail probability. We illustrate by constructing the theoretically justified intervals of values for the decay rate and the associated heuristically determined interval of values for the mean waiting times. Without extra information, the extremal models involve two-point distributions, which yield a wide range for the mean. Adding constraints on the third moment and a transform value produces three-point extremal distributions, which significantly reduce the range, producing practical levels of accuracy.

In the paper, we study general Markovian models of loss systems with random resource requirements, in which customers at arrival occupy random quantities of various resources and release them at departure. Customers may request negative quantities of resources, but total amount of resources allocated to customers should be nonnegative and cannot exceed predefined maximum levels. Allocating a negative volume of a resource to a customer leads to a temporary increase in its volume in the system. We derive necessary and sufficient conditions for the product-form of the stationary probability distribution of the Markov jump process describing the system.

In this study, we consider option pricing under a Markov regime-switching GARCH-jump (RS-GARCH-jump) model. More specifically, we derive the risk neutral dynamics and propose a lattice algorithm to price European and American options in this framework. We also provide a method of parameter estimation in our RS-GARCH-jump setting using historical data on the underlying time series. To measure the pricing performance of the proposed algorithm, we investigate the convergence of the tree-based results to the true option values and show that this algorithm exhibits good convergence. By comparing the pricing results of RS-GARCH-jump model with regime-switching GARCH (RS-GARCH) model, GARCH-jump model, GARCH model, Black–Scholes (BS) model, and Regime-Switching (RS) model, we show that accommodating jump effect and regime switching substantially changes the option prices. The empirical results also show that the RS-GARCH-jump model performs well in explaining option prices and confirm the importance of allowing for both jump components and regime switching.

Parallel processing systems with jobs structured as random graphs, where the nodes correspond to executable tasks and the directed edges to precedence constraints, are studied from a queueing theoretic point of view under general stationarity assumptions on the job flows. Jobs need to have their tasks processed non-preemptively by a set of uniform processors. Simple, natural greedy schemes of allocating processors to tasks are shown to asymptotically minimize the long-term average execution time per job. The stability condition for this queueing system is specified, and greedy allocation schemes are shown to stabilize the system under the maximum possible job arrival rate. Some recurrence properties of the system state are also established.

Directed acyclic independence graphs (DAIGs) play an important role in recent developments in probabilistic expert systems and influence diagrams (Chyu [1]). The purpose of this note is to show that DAIGs can usefully be grouped into equivalence classes where the members of a single class share identical Markov properties. These equivalence classes can be identified via a simple graphical criterion. This result is particularly relevant to model selection procedures for DAIGs (see, e.g., Cooper and Herskovits [2] and Madigan and Raftery [4]) because it reduces the problem of searching among possible orientations of a given graph to that of searching among the equivalence classes.

Maymin proved that for binary systems, the path-cut lower bound is sharper than the minimax lower bound for highly reliable components, which was originally conjectured by Barlow and Proschan. In this note, an example is constructed to illustrate that this, in general, is not the case for multistate systems. However, an affirmative result is obtained under mild conditions we imposed. Examples are given to illustrate the applications of our results.

We study a globally gated polling system with a dormant server, which makes a halt at its home base when there are no customers present in the system. We derive an explicit expression for the cycle time distribution as well as for the waiting-time distribution at each of the queues. As a justification of the dormant server policy, we show the waiting time at each of the queues to be smaller (in the increasing-convex-ordering sense) than in the ordinary nondormant server case.