Branching-type polling systems with large setups

Abstract

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.

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

“…A strong conjecture is presented in Ref. [20] that in this case the distribution of W i S tends to a uniform distribution on [0, 1− i 1− ] as S → ∞; for the case of Poisson arrivals, a rigorous proof of this result was given in Ref. [11] .…”

“…The most general results known for renewal-driven polling models are asymptotic results, such as heavy-traffic asymptotics [14] or asymptotics for large switch-over times [20] . Closed-form approximations are available for the mean waiting time [2] , but today there is no simple approximation available for the tail probabilities of the waiting times at each of the queues.…”

Polling systems with batch service

“…These studies mostly assume that the server can take in any number of products for service at a time and that products arrive according to Poisson arrival processes. For polling models with renewal arrivals, hardly any exact results are known, except for asymptotic regimes for heavy traffic (Olsen and Van der Mei 2005) or large switch-over times (Winands 2011). Faced by this, approximations have been developed for the mean waiting time (Boon et al 2011), recently extended to the complete waiting-time distribution (Dorsman et al 2011).…”

Polling Systems With Two-Phase Gated Service

“…One of the most remarkable results is that there appears to be a striking difference in complexity between polling models. For MTBP-type polling models, a number of solution techniques have been proposed, including the classical buffer-occupancy and stationtime techniques [29], the Descendant Set Approach [11], and the recently proposed mean value analysis [45]. Models that satisfy this MTBP structure allow for an exact analysis, whereas models that violate the MTBP structure are often more intricate and require heavy-weight numerical techniques to obtain the queue-length and waiting-time distributions [4,5].…”

“…They also gave conjectures for the HT limits of the first two moments of the waiting times for systems with an arbitrary number of queues. Van der Mei and Winands [40] use the mean value analysis (MVA) [45] to derive HT limits for the expected delay for cyclic Poisson-driven polling models with exhaustive and gated service at all queues. Van der Mei [36] considered the general class of polling models that can be described by MTBPs [28] and used the theory of critical MTBP [24] to obtain a framework for deriving HT limits for the waiting-time and queue-length disributions.…”