Markov-modulated M/G/1-type queue in heavy traffic and its application to time-sharing disciplines

Abstract

This paper deals with a single-server queue with modulated arrivals, service requirements and service capacity. In our first result, we derive the mean of the total workload assuming generally distributed service requirements and any service discipline which does not depend on the modulating environment. We then show that the workload is exponentially distributed under heavy-traffic scaling. In our second result, we focus on the discriminatory processor sharing (DPS) discipline. Assuming exponential, class-dependent service requirements, we show that the joint distribution of the queue lengths of different customer classes under DPS undergoes a state-space collapse when subject to heavy-traffic scaling. That is, the limiting distribution of the queue length vector is shown to be exponential, times a deterministic vector. The distribution of the scaled workload, as derived for general service disciplines, is a key quantity in the proof of the state-space collapse.Acknowledgements: The authors would like to thank Urtzi Ayesta, Joke Blom and Michel Mandjes for helpful discussions.

Single-server queues under overdispersion in the heavy-traffic regime

“…Upon inspecting the above derivation, we see that we have in addition proven the asymptotic independence of Q and J in the regime where q " 1: More precisely, as q " 1, using that PGFs uniquely characterize their underlying (joint) distribution, we have that the bivariate random vector ðð1 À qÞQ, JÞ converges to ð Q, JÞ, where Q is exponentially distributed with the mean we identified above and J such that Pð J ¼ iÞ ¼ p i , where, remarkably, Q and J are independent; cf. the results in e.g., [6,16,40] . Hence we have the following result.…”

“…V] . Heavy-traffic analyses for Markov-modulated single-server queues and their QBD counterparts are given in e.g., [6,9,16,40] .…”

Optimal Control of Parallel Queues for Managing Volunteer Convergence

“…Random server arrivals and departures means that our model is an example of a queueing system with a controlled Markov‐modulated service capacity: service capacity can change according to a controlled external environment that is governed by a Markov process. Markov‐modulated queues also have been well studied in the literature Neuts (1981), Purdue (1974), Regterschot and De Smit (1986), Mahabhashyam and Gautam (2005), Perel and Yechiali (2008), Thorsdottir and Verloop (2016). We refer the reader to the many references there.…”