Avi Mandelbaum

Discrete flow networks: bottleneck analysis and fluid approximations

We conduct bottleneck analysis of a deterministic dynamic discrete-flow network. The analysis presupposes only the existence of long-run averages, and is based on a continuous fluid approximation to the network in terms of these averages. The results provide functional strong laws-of-large-numbers for stochastic Jackson queueing networks since they apply to their sample paths with probability one.

Strong approximations for time-dependent queues

A time-dependent M t / M t /1 queue alternates through periods of under-, over-, and critical loading. We derive period-dependent, pathwise asymptotic expansions for its queue length, within the framework of strong approximations. Our main results include time-dependent fluid approximations, supported by a functional strong law of large numbers, and diffusion approximations, supported by a functional central limit theorem. This complements and extends previous work on asymptotic expansions of the queue-length transition probabilities.

Scheduling a multi class queue with many exponential servers: asymptotic optimality in heavy traffic

We consider the problem of scheduling a queueing system in which many statistically identical servers cater to several classes of impatient customers. Service times and impatience clocks are exponential while arrival processes are renewal. Our cost is an expected cumulative discounted function, linear or nonlinear, of appropriately normalized performance measures. As a special case, the cost per unit time can be a function of the number of customers waiting to be served in each class, the number actually being served, the abandonment rate, the delay experienced by customers, the number of idling servers, as well as certain combinations thereof. We study the system in an asymptotic heavy-traffic regime where the number of servers n and the offered load r are simultaneously scaled up and carefully balanced: n\approx r+\beta \sqrtr for some scalar \beta. This yields an operation that enjoys the benefits of both heavy traffic (high server utilization) and light traffic (high service levels.)

The impact of customers’ patience on delay and abandonment: some empirically-driven experiments with the M/M/n + G queue

Abstract.Our research is motivated by a phenomenon that has been observed in telephone call center data: a clear linear relation between the probability to abandon and average waiting time. Such a relation is theoretically justifiable when customers’ patience is memoryless, but it lacks an explanation in general. We thus analyze its robustness within the framework of the M/M/n + G queue, which gives rise to further theory and empirically-driven experiments. In the theoretical part of the paper, we establish order relations for performance measures of the M/M/n + G queues, and some light-traffic results. In particular, we prove that, with

Linear estimators and measurable linear transformations on a Hilbert space

\lambda, \mu, n

Discrete multi-armed bandits and multi-parameter processes

, and average patience time fixed, deterministic patience minimizes the probability to abandon and maximizes the average wait in queue. In the experimental part, we describe the behavior of M/M/n + G performance measures for different patience distributions. The findings are then related to our theoretical results and some observed real-data phenomena. In particular, clear non-linear relations (convex, concave and mixed) emerge between the probability to abandon and average wait. However, when restricted over low to moderate abandonment rates, approximate linearity prevails, as observed in practice. Note to the Reader: A color-version of the present paper is downloadable from http://iew3.technion.ac.il/serveng/References/references.html.

Regenerative closed queueing networks

SummaryWe consider the problem of estimating the mean of a Gaussian random vector with values in a Hilbert space. We argue that the natural class of linear estimators for the mean is the class of measurable linear transformations. We give a simple description of all measurable linear transformations with respect to a Gaussian measure. If X and θ are jointly Gaussian then E[θ¦X] is a measurable linear transformation. As an application of the general theory we describe all measurable linear transformations with respect to the Wiener measure in terms of Wiener integrals.

On Harris Recurrence in Continuous Time

SummaryThe general multi-armed bandit problem is reformulated and solved as a control problem over a partially ordered set. The approach taken provides a technically convenient framework for bandit-like problems. It also adds insight to the structure of strategies over partially ordered sets.

Coalescence of Skew Brownian Motions

Consider a closed queueing network of single-server stations, as introduced by Jackson, Gordon-Newell and Whittle, but with service times that are arbitrarily distributed with finite means. We verify that such a network has a unique stationary distribution if one of its service times is unbounded. The proof is carried out by modeling the network as a piecewise-deterministic Markov process which is shown to be regenerative. If, furthermore, the unbounded service-time distribution is non-lattice, then the stationary distribution is also the unique equilibrium. When the service-times enjoy finite second moments, and the number of customers in the network grows indefinitely, this equilibrium (properly normalized) converges to the equilibrium of a reflected Brownian motion on a simplex

A Brownian control problem for a simple queueing system in the Halfin–Whitt regime

We show that a continuous-time Markov process X is Harris recurrent if and only if there exists a nonzero α-finite measure I½ on its state space such that X surely hits sets with positive I½-measure. This simple criterion is applied to some nonparametric closed queueing networks.