Emm Erik Winands

The stochastic economic lot scheduling problem: A survey

The present literature survey focuses on the stochastic economic lot scheduling problem (SELSP). The SELSP deals with the make-to-stock production of multiple standardized products on a single machine with limited capacity under random demands, possibly random setup times and possibly random production times. The main task of a production manager in this setting is the construction of a production plan for the machine. Based on the critical elements of such a production plan, we present a classification and extensive overview of the research on the SELSP together with an indication of open research areas. By doing so, we intend to stimulate the discussion on the important problems concerning the SELSP both from a theoretical and a practical point of view.

Applications of polling systems

Since the first paper on polling systems, written by Mack in 1957, a huge number of papers on this topic has been written. A typical polling system consists of a number of queues, attended by a single server. In several surveys, the most notable ones written by Takagi, detailed and comprehensive descriptions of the mathematical analysis of polling systems are provided. The goal of the present survey paper is to complement these papers by putting the emphasis on applications of polling models. We discuss not only the capabilities, but also the limitations of polling models in representing various applications. The present survey is directed at both academicians and practitioners.

On the application of Rouché's theorem in queueing theory

The determination of the PGF of the queue length distribution often requires the zeros on and within the unit circle of functions of the type z^s-A(z), where A(z) is a PGF. We present an elementary proof of the existence of these zeros including series A(z) with a radius of convergence of one.

Scheduling in polling systems

We present a simple mean value analysis (MVA) framework for analyzing the effect of scheduling within queues in classical asymmetric polling systems with gated or exhaustive service. Scheduling in polling systems finds many applications in computer and communication systems. Our framework leads not only to unification but also to extension of the literature studying scheduling in polling systems. It illustrates that a large class of scheduling policies behaves similarly in the exhaustive polling model and the standard M/GI/1 model, whereas scheduling policies in the gated polling model behave very differently than in an M/GI/1.

Iterative approximation of k-limited polling systems

Abstract The present paper deals with the problem of calculating queue length distributions in a polling model with (exhaustive) k-limited service under the assumption of general arrival, service and setup distributions. The interest for this model is fueled by an application in the field of logistics. Knowledge of the queue length distributions is needed to operate the system properly. The multi-queue polling system is decomposed into single-queue vacation systems with k-limited service and state-dependent vacations, for which the vacation distributions are computed in an iterative approximate manner. These vacation models are analyzed via matrix-analytic techniques. The accuracy of the approximation scheme is verified by means of an extensive simulation study. The developed approximation turns out to be accurate, robust and computationally efficient.

A state-dependent polling model with k -limited service

We consider a two-queue model with state-dependent setups, in which a single server alternately serves the two queues. The high-priority queue is served exhaustively, whereas the low-priority queue is served according to the k-limited strategy. A setup at a queue is incurred only if there are customers waiting at the polled queue. We obtain the transforms of the queue length and sojourn time distributions under the assumption of Poisson arrivals, generally distributed service times, and generally distributed setup times. The interest for this model is fueled by an application in the field of logistics. It is shown how the results of this analysis can be applied in the evaluation of a stochastic two-item single-capacity production system. From these results we can conclude that significant cost reductions are possible by bounding the production runs of the low-priority item, which indicates the potential of the k-limited service discipline as priority rule in production environments.

Quasi-birth-and-death processes with an explicit rate matrix

We determine the equilibrium distribution for a class of quasi-birth-and-death (QBD) processes using the matrix-geometric method, which requires the determination of the rate matrix R. In contrast to most QBD processes, the class under consideration allows for an explicit description of R, by exploiting its probabilistic interpretation. We show that the problem of finding each element of R reduces to counting lattice paths in the transition diagram. For resolving this counting problem, we present an extended version of the classical Ballot theorem. We also present various examples of queueing models that fit into the class.

A New Method for Deriving Waiting-Time Approximations in Polling Systems with Renewal Arrivals

We study the waiting-time distributions in cyclic polling models with renewal arrivals, general service and switch-over times, and exhaustive service at each of the queues. The assumption of renewal arrivals prohibits an exact analysis and reduces the available analytic results to heavy-traffic asymptotics, limiting results for large switch-over times and large numbers of queues, and some numerical algorithms. Motivated by this, the goal of this paper is to propose a new method for deriving simple closed-form approximations for the complete waiting-time distributions that work well for arbitrary load values. Extensive simulation results show that the approximations are highly accurate over a wide range of parameter settings.

A note on polling models with renewal arrivals and nonzero switch-over times

We consider gated polling systems with general service and switch-over times and with renewal arrival processes. We derive closed-form expressions for the expected delay in heavy-traffic (HT). So far, proofs of HT limits have only been obtained for Poisson-type arrival processes, whereas for renewal arrivals limits are based on conjectures.

Branching-type polling systems with large setups

The present paper considers the class of polling systems that allow a multi-type branching process interpretation. This class contains the classical exhaustive and gated policies as special cases. We present an exact asymptotic analysis of the delay distribution in such systems, when the setup times tend to infinity. The motivation to study these setup time asymptotics in polling systems is based on the specific application area of base-stock policies in inventory control. Our analysis provides new and more general insights into the behavior of polling systems with large setup times.