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.

Linear birth–death processes with immigration and emigration are major models in the study of population processes of biological and ecological systems, and their transient analysis is important in the understanding of the structural behavior of such systems. The spectral method has been widely used for solving these processes; see, for example, Karlin and McGregor [11]. In this article, we provide an alternative approach: the method of characteristics. This method yields a Volterra-type integral equation for the chance of extinction and an explicit formula for the z-transform of the transient distribution. These results allow us to obtain closed-form solutions for the transient behavior of several cases that have not been previously explicitly presented in the literature.

In this article, we observe that processes with the uniform order statistics property (UOSP) can be characterized by the condition that their first n epoch times have a joint [ell ]∞≤-spherical density, n ≥ 1. Some related results, and some further properties of [ell ]∞≤-spherical densities, are also given. We also extend some of the results regarding the UOSP to the more general (not necessarily uniform) order statistics property. Finally, we develop a theory of discrete-time discrete-state processes with the UOSP, where the need to consider multiple jumps, at a single time point, arises.

Recent advances from the theory of multivariate stochastic orderings can be used to formalize the “folk theorem” that positive correlations lead to larger buffer levels at a discrete-time infinite capacity multiplexer queue. In particular, it is known that if the input traffic is larger than its independent version in the supermodular (sm) ordering, then their corresponding buffer contents are similarly ordered in the increasing convex (icx) ordering.

A sufficient condition for the aforementioned sm comparison is the stochastic increasingness in sequence (SIS) property of the input traffic. In this article, we provide conditions for the stationary on–off source to be SIS. We then use this result to find conditions under which the superposition of independent on–off sources and the M|G|∞ input model are each sm greater than their respective independent version. Similar but weaker SIS conditions are also obtained for renewal on–off processes.

In this article, we study a single-server queue with FIFO service and cyclic interarrival and service times. An efficient approximative algorithm is developed for the first two moments of the waiting time. Numerical results are included to demonstrate that the algorithm yields accurate results. For the special case of exponential interarrival times, we present a simple exact analysis.

We derive convexity results and related properties for the value functions of tandem queuing systems. The results for standard multiserver queues are new. For completeness, we also prove and generalize existing results on tandems of controllable queues. The results can be used to compare queuing systems. This is done for systems with and without batch arrivals and for systems with different numbers of on–off sources.

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.

In this paper, we develop a general approach to compute the probabilities of unresolved clones in random pooling designs. This unified and systematic approach gives better insight for handling the dependency issue among the columns and among the rows. Consequently, we identify some faster computation formulas for four random pooling designs proposed in the literature, and we derive some probability distribution functions of the number of unresolved clones that were not available before.

In connection with the classical insurance risk problem, Ross [2] develops a recursive formula for approximating a tail probability in a geometrically compounded distribution. Here, we consider rates of convergence for this approximation.

We give a probabilistic proof of an identity concerning the expectation of an arbitrary function of a compound random variable and then use this identity to obtain recursive formulas for the probability mass function of compound random variables when the compounding distribution is Poisson, binomial, negative binomial random, hypergeometric, logarithmic, or negative hypergeometric. We then show how to use simulation to efficiently estimate both the probability that a positive compound random variable is greater than a specified constant and the expected amount by which it exceeds that constant.

Supply interruptions resulting from unpredictable events, such as machine breakdowns, order cancellations, unscheduled maintenance, and labor strikes can produce adverse effects on the production/inventory system. In this article, we consider a periodic-review inventory system subject to random demand and unreliable supply. The availability of supply is modeled as an alternating renewal process with general distributions for the durations of the UP and DOWN cycles. We consider the lost-sales case and also discuss the backorder case, for both the discounted and long-run average cost criteria. For the linear cost model, we derive the structural properties and bounds of the optimal policy. We also propose the “end-of-cycle” inventory return contract and show that it may be mutually beneficial to both the firm and the supplier.

We show that winning the game New Age Solitaire only depends on the number of rising sequences in the deck used. The probability of winning for the special case of a new deck that is shuffled using the GSR shuffle (and two variants) are studied. We show that this game pinpoints the Achilles' heel of the GSR shuffle as is demonstrated using the variation distance.

In a system with one queue and several service stations, it is a natural principle to route a customer to the idle station with the distributionwise shortest service time. For the case with exponentially distributed service times, we use a coupling to give strong support to that principle. We also treat another topic. A modified version of our methods brings support to the design principle: It is better with few but quick servers.

We consider the following queuing system which arises as a model of a wireless link shared by multiple users. There is a finite number N of input flows served by a server. The system operates in discrete time t = 0,1,2,…. Each input flow can be described as an irreducible countable Markov chain; waiting customers of each flow are placed in a queue. The sequence of server states m(t), t = 0,1,2,…, is a Markov chain with finite number of states M. When the server is in state m, it can serve μim customers of flow i (in one time slot).

The scheduling discipline is a rule that in each time slot chooses the flow to serve based on the server state and the state of the queues. Our main result is that a simple online scheduling discipline, Modified Largest Weighted Delay First, along with its generalizations, is throughput optimal; namely, it ensures that the queues are stable as long as the vector of average arrival rates is within the system maximum stability region.

In this article, we consider a compound Poisson insurance risk model with a random discount factor. This model is also known as the compound filtered Poisson model. By using some stochastic analysis techniques, a convergence result for the discounted surplus process, an expression for the ruin probability, and the upper bounds for the ruin probability are obtained.

It is well known that for a stochastically monotone Markov chain {Jn}n≥1 a function γ(n) = Cov[f (J1),g(Jn)] is decreasing if f and g are increasing. We prove this property for a special subclass of nonmonotone double stochastic Markov chains.

We derive the distribution of the time-to-empty for an open tandem Jackson network assuming that while in equilibrium at time 0, the arrival stream is suddenly shut off. The analysis is based on analogous results regarding the distribution of the time-to-empty for the corresponding closed tandem Jackson network. The results obtained are used in the analysis of a two-class tandem Jackson network with FIFO discipline in which customers of the second class have negligible service times.

We investigate fast simulation techniques for estimating the unreliability in large Markovian models of highly reliable systems for which analytical/numerical techniques are difficult to apply. We first show mathematically that for “small” time horizons, the relative simulation error, when using the importance sampling techniques of failure biasing and forcing, remains bounded as component failure rates tend to zero. This is in contrast to naive simulation where the relative error tends to infinity. For “large” time horizons where these techniques are not efficient, we use the approach of first bounding the unreliability in terms of regenerative-cycle-based measures and then estimating the regenerative-cycle-based measures using importance sampling; the latter can be done very efficiently. We first use bounds developed in the literature for the asymptotic distribution of the time to hitting a rare set in regenerative systems. However, these bounds are “close” to the unreliability only for a certain range of time horizons. We develop new bounds that make use of the special structure of the systems that we consider and are “close” to the unreliability for a much wider range of time horizons. These techniques extend to non-Markovian, highly reliable systems as long as the regenerative structure is preserved.

The aim of this article is to derive the income and cost functionals required to determine the actuarial value of certain types of perishable inventory system. In the basic model, 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 price of an item depends on its shelf age. For an actuarial valuation, it is important to know the distribution of the total value of the items in the system and the expected (discounted) total income and cost generated by the system when in steady state. All of these functionals are determined explicitly. As extensions of the original model, we also deal with the case of batch arrivals and general renewal interdemand times; in both cases, closed-form solutions are obtained.

Let F be the common distribution function of the increments of a random walk {Sn, n ≥ 0} with S0 = 0 and a negative drift and let {N(t), t ≥ 0} be a general counting process, independent of {Sn, n ≥ 0}. This article investigates the tail probability, denoted by ψ(x; t), of the maximum of SN(v) over a finite horizon 0 ≤ v ≤ t. When F is strongly subexponential, some asymptotics for ψ(x; t) are derived as x → ∞. The merit is that all of the obtained asymptotics are uniform for t in a finite or infinite time interval.