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.

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.

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.

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.

In this article, some results on stochastic comparisons of the inspection paradox introduced by Ross [Probability in the Engineering and Informational Sciences 17: 47–51 (2003)] are established in the sense of the likelihood ratio order.

We obtain some techniques to study the shape of reliability functions (failure rate, mean residual life, etc.) by using the s-equilibrium distribution of a renewal process defined by Fagiuoli and Pellerey (Naval Res. Logist., 1993). We apply these techniques to study how to obtain distributions with bathtub shaped failure rate (BFR) from mixtures of two positive truncated normal distributions.

We consider the problem of routing incoming airplanes to two runways of an airport. Due to air turbulence, the necessary separation time between two successive landing operations depends on the type of airplane. When viewed as a queuing problem, this means that we have dependent service times. The aim is to minimize the waiting times of aircrafts. We consider here a model in which arrivals form a stochastic process and the decision-maker does not know anything about future arrivals. We formulate this as a problem of stochastic dynamic programming and investigate the monotonicity of optimal routing strategies with respect to the workload of the runways, for example. We show that an optimal strategy is monotone (i.e., of switching type) only in a restricted case where decisions depend on the state of the runways only and not on the type of the arriving aircraft. Surprisingly, in the more realistic case where this type is also known to the decision-maker, monotonicity need not hold.

We consider shot-noise and max-shot-noise processes driven by spatial stationary Cox (doubly stochastic Poisson) processes. We derive their upper and lower bounds in terms of the increasing convex order, which is known as the order relation to compare the variability of random variables. Furthermore, under some regularity assumption of the random intensity fields of Cox processes, we show the monotonicity result which implies that more variable shot patterns lead to more variable shot noises. These are direct applications of the results obtained for so-called Ross-type conjectures in queuing theory.