We consider the optimal control of two parallel servers in a two-stage tandem queuing system with two flexible servers. New jobs arrive at station 1, after which a series of two operations must be performed before they leave the system. Holding costs are incurred at rate h1 per unit time for each job at station 1 and at rate h2 per unit time for each job at station 2.

The system is considered under two scenarios; the collaborative case and the noncollaborative case. In the prior, the servers can collaborate to work on the same job, whereas in the latter, each server can work on a unique job although they can work on separate jobs at the same station. We provide simple conditions under which it is optimal to allocate both servers to station 1 or 2 in the collaborative case. In the noncollaborative case, we show that the same condition as in the collaborative case guarantees the existence of an optimal policy that is exhaustive at station 1. However, the condition for exhaustive service at station 2 to be optimal does not carry over. This case is examined via a numerical study.

At a buffered switch in an ATM (asynchronous transfer mode) network it is important to know what combinations of different types of traffic can be carried simultaneously without risking more than a very small probability of overflowing the buffer. We show that a simple and serviceable measure of effective bandwidths may be computed for stationary traffic sources. For large buffers the effective bandwidth of a source is a function only of its mean rate, index of dispersion, and the size of the buffer.

We describe a stochastic dynamic programming approach for “real option”-based valuation of electricity generation capacity incorporating operational constraints and start-up costs. Stochastic prices of electricity and fuel are represented by recombining multinomial trees. Generators are modeled as a strip of cross-commodity call options with a delay and a cost imposed on each option exercise. We illustrate implications of operational characteristics on the valuation of generation assets under different modeling assumptions about the energy commodity prices. We find that the impacts of operational constraints on real asset valuation are dependent on both the model specification and the nature of operating characteristics.

In this paper we present a simple method to fit a discrete distribution on the first two moments of a given random variable. With the Fitted distribution we solve approximately Lindley's equation for the D/G/1 queue with discrete service-time distribution using a moment-iteration method. Numerical results show excellent performance of the method.

Monte Carlo simulation is one alternative for analyzing options markets when the assumptions of simpler analytical models are violated. We introduce techniques for the sensitivity analysis of option pricing, which can be efficiently carried out in the simulation. In particular, using these techniques, a single run of the simulation would often provide not only an estimate of the option value but also estimates of the sensitivities of the option value to various parameters of the model. Both European and American options are considered, starting with simple analytically tractable models to present the idea and proceeding to more complicated examples. We then propose an approach for the pricing of options with early exercise features by incorporating the gradient estimates in an iterative stochastic approximation algorithm. The procedure is illustrated in a simple example estimating the option value of an American call. Numerical results indicate that the additional computational effort required over that required to estimate a European option is relatively small.

This article deals with the effect of information and uncertainty on profits in an unobservable single-server queuing system. We consider scenarios in which the service rate, the service quality, or the waiting conditions are random variables that are known to the server but not to the customers. We ask whether the server is motivated to reveal these parameters. We investigate the structure of the profit function and its sensitivity to the variance of the random variable. We consider and compare variations of the model according to whether the server can modify the service price after observing the realization of the random variable.

In this article, we identify conditions under which the epoch times and the interepoch intervals of a nonhomogeneous Poisson process have logconcave densities. The results are extended to relevation counting processes. We also study discrete-time counting processes and find conditions under which the epoch times and the interepoch intervals of these discrete-time processes have logconcave discrete probability densities. The results are interpreted in terms of minimal repair and record values. Several examples illustrate the theory.

Two well-known orders that have been introduced and studied in reliability theory are defined via stochastic comparison of inactivity time: the reversed hazard rate order and the mean inactivity time order. In this article, some characterization results of those orders are given. We prove that, under suitable conditions, the reversed hazard rate order is equivalent to the mean inactivity time order. We also provide new characterizations of the decreasing reversed hazard rate (increasing mean inactivity time) classes based on variability orderings of the inactivity time of k-out-of-n system given that the time of the (n − k + 1)st failure occurs at or sometimes before time t ≥ 0. Similar conclusions based on the inactivity time of the component that fails first are presented as well. Finally, some useful inequalities and relations for weighted distributions related to reversed hazard rate (mean inactivity time) functions are obtained.

This article deals with estimations of probabilities of rare events using fast simulation based on the splitting method. In this technique, the sample paths are split into multiple copies at various stages in the simulation. Our aim is to optimize the algorithm and to obtain a precise confidence interval of the estimator using branching processes. The numerical results presented suggest that the method is reasonably efficient.

An expression of the extropy of a mixed system's lifetime was given firstly. Based on this expression, two mixed systems with same signature but with different components were compared. It was shown that the extropy of lifetime of a mixed system equals to that of its dual system if the lifetimes of the components have symmetric probability density function. Moreover, some bounds of the extropy of lifetimes of mixed systems were obtained and the concept of Jensen–extropy (JE) divergence of mixed systems was proposed. The JE divergence is non-negative and it can be used as an alternative information criteria for comparing mixed systems with homogeneous components. To illustrate the applications of JE divergence, some examples are addressed at the end of this paper.

It is well known that mixtures of decreasing failure rate (DFR) distributions are always DFR. It turns out that, very often, mixtures of increasing failure rate (IFR) distributions can decrease at least in some intervals of time. Usually, this property can be observed asymptotically as t → ∞. In this article, several types of underlying continuous IFR distribution are considered. Two models of mixing are studied: additive and multiplicative. The limiting behavior of a mixture failure rate function is analyzed. It is shown that the conditional characteristics (expectation and variance) of the mixing parameter are crucial for the limiting behavior. Several examples are presented and possible generalizations are discussed.

We consider a network of N identical /M/l or /M/∞ queues. There are two types of arriving customers, those that have no routing choice, and those that first pick r queues at random, and are then routed to the least busy of those queues. We derive the limiting distribution of queue lengths as N→∞, and investigate how this distribution varies with r. We show that even a small amount of routing choice can lead to substantial gains in performance through resource pooling. We corroborate these conclusions by carrying out some simulations of a related model, from which the previous model can be derived by an exchangeable queue simplification. We also observe that the exchangeable queue simplification results in a performance gain for some parameters, in contrast to earlier work.

Data on the Internet is sent by packets that go through a network of routers. A router drops packets either when its buffer is full or when it uses the Active Queue Management. Currently, the majority of the Internet routers use a simple Drop Tail strategy. The rate at which a user injects the data into the network is determined by transmission control protocol (TCP). However, most connections in the Internet consist only of few packets, and TCP does not really have an opportunity to adjust the sending rate. Thus, the data flow generated by short TCP connections appears to be some uncontrolled stochastic process. In the present work we try to describe the interaction of the data flow generated by short TCP connections with a network of finite buffers. The framework of retrial queues and networks seems to be an adequate approach for this problem. The effect of packet retransmission becomes essential when the network congestion level is high. We consider several benchmark retrial network models. In some particular cases, an explicit analytic solution is possible. If the analytic solution is not available or too entangled, we suggest using a fixed-point approximation scheme. In particular, we consider a network of one or two tandem M/M/1/K-type queues with blocking and with an M/M/1/∞-type retrial (orbit) queue. We explicitly solve the models with particular choices of K, derive stability conditions for K≥1, and present several graphs based on numerical results.

We consider a storage model that can be on or off. When on, the content increases at some state-dependent rate and the system can switch to the off state at a state-dependent rate as well. When off, the content decreases at some state-dependent rate (unless it is at zero) and the system can switch to the on position at a state-dependent rate. This process is a special case of a piecewise deterministic Markov process. We identify the stationary distribution and conditions for its existence and uniqueness.

We consider an M/G/1 queue with the special feature of additional negative customers, who arrive according to a Poisson process. Negative customers require no service, but at their arrival a stochastic amount of work is instantaneously removed from the system. We show that the workload distribution in this M/G/1 queue with negative customers equals the waiting time distribution in a GI/G/1 queue with ordinary customers only; the effect of the negative customers is incorporated in the new arrival process.

In this article, we study the problem of finding tight bounds on the expected value of the kth-order statistic E [Xk:n] under first and second moment information on n real-valued random variables. Given means E [Xi] = μi and variances Var[Xi] = σi2, we show that the tight upper bound on the expected value of the highest-order statistic E [Xn:n] can be computed with a bisection search algorithm. An extremal discrete distribution is identified that attains the bound, and two closed-form bounds are proposed. Under additional covariance information Cov[Xi,Xj] = Qij, we show that the tight upper bound on the expected value of the highest-order statistic can be computed with semidefinite optimization. We generalize these results to find bounds on the expected value of the kth-order statistic under mean and variance information. For k < n, this bound is shown to be tight under identical means and variances. All of our results are distribution-free with no explicit assumption of independence made. Particularly, using optimization methods, we develop tractable approaches to compute bounds on the expected value of order statistics.

The depth of insertion and the internal path length of recursive trees are studied. Luc Devroye has recently shown that the depth of insertion in recursive trees is asymptotically normal. We give a direct alternative elementary proof of this fact. Furthermore, via the theory of martingales, we show that In, the internal path length of a recursive tree of order n, converges to a limiting distribution. In fact, we show that there exists a random variable I such that (In – n In n)/n→I almost surely and in quadratic mean, as n → α. The method admits, in passing, the calculation of the first two moments of In.

We study first-passage percolation on the random graph Gp(N) with exponentially distributed weights on the links. For the special case of the complete graph, this problem can be described in terms of a continuous-time Markov chain and recursive trees. The Markov chain X(t) describes the number of nodes that can be reached from the initial node in time t. The recursive trees, which are uniform trees of N nodes, describe the structure of the cluster once it contains all the nodes of the complete graph. From these results, the distribution of the number of hops (links) of the shortest path between two arbitrary nodes is derived.

We generalize this result to an asymptotic result, as N → ∞, for the case of the random graph where each link is present independently with a probability pN as long as NpN/(log N)3 → ∞. The interesting point of this generalization is that (1) the limiting distribution is insensitive to p and (2) the distribution of the number of hops of the shortest path between two arbitrary nodes has a remarkable fit with shortest path data measured in the Internet.

This note illustrates that a combination of the approach in our previous papers (Boucherie and Boxma, 1996, Probability in the Engineering and Informational Sciences10: 261–277; Jain and Sigman, 1996, Probability in the Engineering and Informational Sciences 10: 519–531) directly leads to a Pollaczek-Khintchine form for the workload in a queue with negative customers. The same technique is also applied to risk processes with lump additions.

Simulated annealing is a class of sequential search techniques for solving continuous global optimization problems. In this paper we attempt to help explain the success of simulated annealing for this class of problems by studying an idealized version of this algorithm, which we call adaptive search. The prototypical adaptive search algorithm generates a sequence of improving points drawn conditionally from samples from a corresponding sequence of probability distributions. Under the condition that the sequence of distributions stochastically dominate in objective function value the uniform distribution, we show that the expected number of improving points required to achieve the global optimum within a prespecified error grows at most linearly in the dimension of the problem for a large class of global optimization problems. Moreover, we derive a cooling schedule for simulated annealing, which follows in a natural way from the definition of the adaptive search algorithm.