We analyze Energy Packet Networks (EPNs) in which the service centers consist of multiclass queues and the Data Packets (DPs) initiate the transfer (i.e., the arrival of a DP at the battery triggers the movement) with multiple energy packet requirements. In other words, a class-k DP in cell i is sent successfully to the next cell if there are $c_i^{(k)}$ energy packets and it is dropped otherwise. Besides, we consider that the queues handling DPs operate under one of the following disciplines: First-Input-First-Output (FIFO), Processor Sharing (PS), or Preemptive Last-Input-First-Output (LIFO-PR). This model is an extension of previously studied EPNs [6,16] where the steady-state distribution of the number of jobs in the queues has a product form. In our model, we show the existence of a product form of the steady-state stationary distribution, where the load of the servers is given by a fixed point expression. We study the existence of a solution to the derived fixed point problem and we provide sufficient conditions for the stability of our model. Finally, we show that, for feed forward EPNs, the load of all the queues can be fully characterized.

An important problem in reliability theory is to determine the reliability of a system from the reliability of its components. If E is a finite set of components, then certain subsets of E are prescribed to be the operating states of the system. A formation is any collection F of minimal operating states whose union is E. Reliability domination is defined as the total number of odd cardinality formations minus the total number of even cardinality formations. The purpose of this paper is to establish some new results concerning reliability domination. In the special case where the system can be identified with a graph or digraph, these new results lead to some new graph-theoretic properties and to simple proofs of certain known theorems. The pertinent graph-theoretic properties include spanning trees, acyclic orientations, Whitney's broken cycles, and Tutte's internal activity associated with the chromatic polynomial.

We consider a system of K parallel queues providing different grades of service through each of the queues and serving a multiclass customer population. Service differentiation is achieved by specifying different join prices to the queues. Customers of class j define a cost function ψij(ci,xi) for taking service from queue i when the join price for queue i is ci and congestion in queue i is xi and join the queue that minimizes ψij(·,·). Such a queuing system will be called the “join minimum cost queue” (JMCQ) and is a generalization of the join shortest queue (JSQ) system. Non-work-conserving (called Paris Metro pricing system) and work-conserving (called the Tirupati system) versions of the JMCQ are analyzed when the cost to an arrival of joining a queue is a convex combination of the join price for that queue and the expected waiting time in that queue at the arrival epoch. Our main results are for a two-queue system.

We obtain stability conditions and performance bounds. To obtain the lower and upper performance bounds, we propose two quasi-birth–death (QBD) processes that are derived from the original systems by suitably truncating the state space. The state space truncation in the non-work-conserving JMCQ follows the method of van Houtum and colleagues. We then show that this method is not applicable to the work-conserving JMCQ and provide sample-path-based proofs to show that the number in each queue is bounded by the number in the corresponding queues of these QBD processes. These sample-path proof techniques might also be of independent interest. We then show that the performance measures like mean queue length and revenue rate of the system are also bounded by the corresponding quantities of these QBD processes. Numerical examples show that these bounds are fairly tight. Finally, we generalize some of these results to systems with more queues.

The concept of Energy Packet Network (EPN) proposed by Gelenbe, is a new framework for modeling power grids that takes distributed energy generation such as renewable energy sources into consideration, and which contributes to modeling the smart grid. Based on G-network theory, this paper presents a simplified model of EPN and formulates energy-distribution as an optimization problem. We analyze it theoretically, and detail its optimal solutions. In addition to using existing optimization algorithms, a heuristic algorithm is proposed to solve for EPN optimization. The optimal solutions and efficacy of the algorithm are illustrated with numerical experiments. Further, we present an EPN with disconnections and a similar optimization problem is investigated. Optimal solutions are presented, and numerical results using the analytic optimal solutions, random solutions, a cooperative particle swarm optimizer and a heuristic algorithm illustrate the power of different approaches for solving energy-distribution problems using the EPN formalism.

In this paper, we review different definitions that multi-state k-out-of-n systems have received along the literature and study them in a unified way using the algebra of monomial ideals. We thus obtain formulas and algorithms to compute their reliability and bounds for it. We provide formulas and computer experiments for simple and generalized multi-state k-out-of-n systems and for binary k-out-of-n systems with multi-state components.

This article presents a new iterative method for computing the equilibrium distribution of a finite Markov chain, which has the significant advantage of providing good upper and lower bounds for the equilibrium probabilities. The method approximates the expected number of visits to each state between two successive visits to a given reference state. Numerical examples indicate that the performance of this method is quite good.

We show that the time to extinction in critical or subcritical Galton–Watson and Markov branching processes has the antiaging property of log-convex density, and therefore has the decreasing failure rate (DFR) property of reliability theory. Apart from providing new insights into the structure of such extinction time distributions, which cannot generally be expressed in a closed form, a consequence of our result is that one can invoke sharp reliability bounds to provide very simple bounds on the tail and other characteristics of the extinction time distribution. The limit distribution of the residual time to extinction in the subcritical case also follows as a direct consequence. A sequel to this paper will further consider the critical case and other ramifications of the log-convexity of the extinction time distribution.

Some properties of repair age (the elapsed time since the last repair) and the residual repair life (the time until the next repair) are studied for the minimal repair process under various assumptions on the aging characteristics of the equipment under maintenance. Bounds on expected repair age and expected residual repair life are obtained. Some characterizations of the Poisson process are also proved.

Many modern networks grow from blocks. We study the probabilistic behavior of parameters of a blocks tree, which models several kinds of networks. It grows from building blocks that are themselves rooted trees. We investigate the number of leaves, depth of nodes, total path length, and height of such trees. We use methods from the theory of Pólya urns and martingales.

This article derives amazingly accurate approximations to the state probabilities and waiting-time probabilities in the M/D/1 queue using a two-phase process with negative probabilities to approximate the deterministic service time. The approximations are in the form of explicit expressions involving geometric and exponential terms. The approximations extend to the finite-capacity M/D/1/N + 1 queue.

In this paper, we discuss the problem of pricing discretely sampled variance swaps under a hybrid stochastic model. Our modeling framework is a combination with a double Heston stochastic volatility model and a Cox–Ingersoll–Ross stochastic interest rate process. Due to the application of the T-forward measure with the stochastic interest process, we can only obtain an efficient semi-closed form of pricing formula for variance swaps instead of a closed-form solution based on the derivation of characteristic functions. The practicality of this hybrid model is demonstrated by numerical simulations.

In this paper, we study some stochastic comparisons of the maxima in two multiple-outlier geometric samples based on the likelihood ratio order, hazard rate order, and usual stochastic order. We establish a sufficient condition on parameter vectors for the likelihood ratio ordering to hold. For the special case when n = 2, it is proved that the p-larger order between the two parameter vectors is equivalent to the hazard rate order as well as usual stochastic order between the two maxima. Some numerical examples are presented for illustrating the established results.

This paper describes a generalization of the classical regenerative method of simulation output analysis. Instead of blocking a generated sample path on returns to a fixed return state, a more general scheme to randomly decompose the path is used. In some cases, this decomposition scheme results in regeneration times that are a supersequence of the classical regeneration times. This “accelerated” regeneration is advantageous in several simulation contexts. It is shown that when this decomposition scheme accelerates regeneration relative to the classical regenerative method, it also yields a smaller asymptotic variance of the regenerative variance estimator than the classical method. Several other contexts in which increased regeneration frequency is beneficial are also discussed.

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.

In this paper, we consider a history-dependent mixed shock model which is a combination of the history-dependent extreme shock model and the history-dependent $\delta$-shock model. We assume that shocks occur according to the generalized Pólya process that contains the homogeneous Poisson process, the non-homogeneous Poisson process and the Pólya process as the particular cases. For the defined survival model, we derive the corresponding survival function, the mean lifetime and the failure rate. Further, we study the asymptotic and monotonicity properties of the failure rate. Finally, some applications of the proposed model have also been included with relevant numerical examples.

Three new notions of positive dependence (positive regression dependence, positive left-tail regression dependence, and positive right-tail regression dependence) are studied in this article. Consider a latent variable model where the manifest random variables T1,T2,…,Tn given latent random variable/vector (Θ1,…,Θm) are conditional independent. Conditions are identified under which T1,…,Tn possesses the new dependence notions for different types of latent variable model. Applications of the results are also provided.

We consider control policies for perishable inventory systems with random input whose purpose is to mitigate the effects of unavailability. In the basic uncontrolled system, the arrival times of the items to be stored and the ones of the demands for those items form independent Poisson processes. The shelf lifetime of every item is finite and deterministic. Every demand is for a single item and is satisfied by the oldest item on the shelf, if available. The first controlled model excludes the possibility of unsatisfied demands by introducing a second source of fresh items that is completely reliable and delivers without delay whenever the system becomes empty. In the second model, there is no additional ordering option by outsourcing. However, to avoid the most adverse effects of unavailability, the demands are classified into different categories of urgency. An incoming demand is satisfied or not according to its category and the current state of the system. For both models, we determine the steady-state distribution of the virtual outdating process, which is then used to derive the relevant cost functionals: the steady-state distribution and expected value of the number of items in the system, the rate of outdatings, as well as, for model 1, the rate of special orders from the external source and, for model 2, the rate of unsatisfied demands.

Carousels are rotatable closed-loop storage systems for small items, where items are stored in bins along the loop. An order at a carousel consists of (say) n different items stored there. We analyze two problems: (1) minimizing the total time to fill an order (travel time) and (2) order delays as they arrive, are filled, and depart. We define clumpy orders and the nearest-end-point heuristic (NEPH) for picking them. We determine conditions for NEPH to be optimal for problem (1), and under a weak stochastic assumption, we derive the distribution of travel time. We compare NEPH with the nearest-item heuristic. Under Poisson arrivals and assumptions much weaker than in the literature, we show that problem (2) may be modeled as an M/G/1 queue.

This paper describes techniques to characterize the number of singletons in the setting of the generalized birthday problem, that is, the birthday problem in which the birthdays are non-uniformly distributed over the year. Approximations for the mean and variance presented which explicitly indicate the impact of the heterogeneity (expressed in terms of the Kullback–Leibler distance with respect to the homogeneous distribution). Then an iterative scheme is presented for determining the distribution of the number of singletons. The approximations are validated by experiments with demographic data.

Consider a single machine that can process multiple jobs in batch mode. We have n jobs and the processing time of job j is a random variable Xj with distribution Fj. Up to b jobs can be processed simultaneously by the machine. The jobs in a batch all have to start at the same time and the batch is completed when all jobs have finished their processing (i.e., at the maximum of the processing times of the jobs in that batch). We are interested in two objective functions, namely the minimization of the expected makespan and the minimization of the total expected completion time. We first show that under certain fairly general conditions, the minimization of the expected makespan is equivalent to specific deterministic combinatorial problems, namely the Weighted Matching problem and the Set Partitioning problem. We then consider the case when all jobs have the same mean processing time but different variances. We show that for certain special classes of processing time distributions the Smallest Variance First rule minimizes the expected makespan as well as the total expected completion time. In our conclusions we present various general rules that are suitable for the minimization of the expected makespan and the total expected completion time in batch scheduling.