A carousel is a computer-controlled warehousing system, which is widely used to store small- and medium-sized goods. One of the most important performance characteristics of such systems is the pick time of an order, which mostly depends on the travel time of the carousel. In this article, we consider some reasonable heuristics for order picking. In particular, we establish properties of the Nearest Item (NI) heuristic. This one is frequently used in practice. We derive tight upper bounds for the travel time under the NI heuristic and closed-form expressions for its mean and variance. We also present a simple two-moment approximation for the distribution of the travel time. In addition, we find the mean, variance, and distribution for the number of turns.

In this article, we study a method to compare queueing systems and their fluid limits. For a certain class of queueing systems, it is shown that the expected workload (and certain functions of the workload) is higher in the queueing system than in the fluid approximation. This class is characterized by convexity of the value function in the state component(s) where external arrivals occur. The main example that we consider is a tandem of multiserver queues with general service times and Markov-modulated arrivals. The analysis is based on dynamic programming and the use of phase-type distributions. Numerical examples to illustrate the results are also given.

Consider the GI/GI/1 queue with the Last-Come First-Served Preemptive-Resume service discipline. We give intuitive explanations for (1) the geometric nature of the stationary queue length distribution and (2) the mutual independence of the residual service requirements of the customers in the queue, both considered at arbitrary time points. These distributions have previously been established in the literature by either first considering the system at arrival instants or using balance equations. Our direct arguments provide further understanding of properties 1 and 2.

We consider a fluid system in which during off-times the buffer content increases as a piecewise linear process according to some general semi-Markov process, and during on-times, it decreases with a state-dependent rate (or remains at zero). The lengths of off-times are exponentially distributed. We show that such a system has a stationary distribution which satisfies a decomposition property where one component in the decomposition is associated with some dam process and the other with a clearing process. For the cases of constant and linear decrease rates, the steady-state Laplace–Stieltjes transform and moments of the buffer content are computed explicitly.

The purpose of this article is to describe various conditions on the parameters of pairs of nonhomogeneous Poisson or pure birth processes under which the corresponding epoch times or interepoch intervals are stochastically ordered in various senses. We derive results involving the usual stochastic order, the multivariate hazard rate order, the multivariate likelihood ratio order, as well as the dispersive and the mean residual life orders. A sample of applications involving generalized Yule processes, load-sharing models, and minimal repairs in reliability theory illustrate the usefulness of the new results.

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.

We study deviation matrices of birth–death processes. This is relevant to the control of multidimensional queueing systems. We give an algorithm for computing deviation matrices for birth–death processes. As an application, we compute them explicitly for the M/M/s/N and M/M/s/∞ queues.

Let X1:n ≤ X2:n ≤ ··· ≤ Xn:n denote the order statistics of a set of independent and not necessarily identically distributed random variables X1,..., Xn. Under mild assumptions, it is shown that Xk−1:n−1 ≤lrXk:n for k = 2,..., n if X1 ≤lrX2 ≤lr ··· ≤lrXn and that Xk:n ≤lrXk:n−1 for k = 1,..., n − 1 if X1 ≥lrX2 ≥lr ··· ≥lrXn, where ≤lr denotes the likelihood ratio order. Concerning the mean residual life order (≤mrl), it is shown that Xn−1:n−1 ≤mrlXn:n if Xj ≤mrlXn for j = 1,..., n − 1. Two counterexamples are also given to illustrate that Xk−1:n−1 ≤mrlXk:n in this case is, in general, not true for k < n.

In this article, we give some results on the preservation of orderings between the components under the formation of coherent systems with different structures. We consider the stochastic, failure rate, reversed failure rate, and likelihood ratio orderings to compare two coherent structures formed from a set of components and from two sets of components, in both cases, when components are independent and either identically distributed or not necessarily identically distributed. As a consequence, we get new stochastic comparisons of systems with different k-out-of-n structures, and some results of partial orderings given by other authors may be also obtained from ours.