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

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.

[1]  P. Naor,et al.  Queuing Problems with Heterogeneous Arrivals and Service , 1971, Oper. Res..

[2]  John Frank Charles Kingman,et al.  The single server queue in heavy traffic , 1961, Mathematical Proceedings of the Cambridge Philosophical Society.

[3]  Xin Liu,et al.  Scheduling control for Markov-modulated single-server multiclass queueing systems in heavy traffic , 2014, Queueing Syst. Theory Appl..

[4]  Urtzi Ayesta,et al.  Heavy-traffic analysis of a multi-class queue with relative priorities , 2013 .

[5]  Bert Zwart,et al.  Heavy-traffic asymptotics for networks of parallel queues with Markov-modulated service speeds , 2015, Queueing Syst. Theory Appl..

[6]  Urtzi Ayesta,et al.  Interpolation approximations for the steady-state distribution in multi-class resource-sharing systems , 2015, Perform. Evaluation.

[7]  Sem C. Borst,et al.  Sojourn time asymptotics in processor-sharing queues , 2006, Queueing Syst. Theory Appl..

[8]  Sلأren Asmussen,et al.  Applied Probability and Queues , 1989 .

[9]  Leonard Kleinrock,et al.  Time-shared Systems: a theoretical treatment , 1967, JACM.

[10]  Peter Purdue,et al.  The M/M/1 Queue in a Markovian Environment , 1974, Oper. Res..

[11]  R. Núñez Queija,et al.  Processor-Sharing Models for Integrated-Services Networks , 2000 .

[12]  R. Bass,et al.  Review: P. Billingsley, Convergence of probability measures , 1971 .

[13]  Andrei V. Zorine,et al.  On ergodicity conditions in a polling model with Markov modulated input and state-dependent routing , 2014, Queueing Syst. Theory Appl..

[14]  Toshiyuki Katsuda.  State-space collapse in stationarity and its application to a multiclass single-server queue in heavy traffic , 2010, Queueing Syst. Theory Appl..

[15]  Eitan Altman,et al.  A survey on discriminatory processor sharing , 2006, Queueing Syst. Theory Appl..

[16]  Sem C. Borst,et al.  Tail asymptotics for discriminatory processor-sharing queues with heavy-tailed service requirements , 2005, Perform. Evaluation.

[17]  Gennadi Falin,et al.  Heavy traffic analysis of M/G/1 type queueing systems with Markov-modulated arrivals , 1999 .

[18]  Isi Mitrani,et al.  Sharing a Processor Among Many Job Classes , 1980, JACM.

[19]  Tetsuya Takine.  Single-Server Queues with Markov-Modulated Arrivals and Service Speed , 2005, Queueing Syst. Theory Appl..

[20]  Eitan Altman,et al.  Markov-modulated stochastic recursive equations with applications to delay-tolerant networks , 2013, Perform. Evaluation.

[21]  R. Núñez Queija,et al.  Discriminatory processor sharing revisited , 2005, Proceedings IEEE 24th Annual Joint Conference of the IEEE Computer and Communications Societies..

[22]  Marcel F. Neuts,et al.  Matrix-geometric solutions in stochastic models - an algorithmic approach , 1982 .

[23]  Bara Kim,et al.  A single server queue with Markov modulated service rates and impatient customers , 2015, Perform. Evaluation.

[24]  Natarajan Gautam,et al.  On Queues with Markov Modulated Service Rates , 2005, Queueing Syst. Theory Appl..

[25]  Yu Wu,et al.  Heavy Traffic Approximation of Equilibria in Resource Sharing Games , 2012, IEEE Journal on Selected Areas in Communications.

[26]  Kiran M. Rege,et al.  Queue-Length Distribution for the Discriminatory Processor-Sharing Queue , 1996, Oper. Res..

[27]  G. J. K. Regterschot,et al.  The Queue M|G|1 with Markov Modulated Arrivals and Services , 1986, Math. Oper. Res..

[28]  Urtzi Ayesta,et al.  Heavy-Traffic Analysis of a Multiple-Phase Network with Discriminatory Processor Sharing , 2009, Oper. Res..

[29]  S. Grishechkin.  On a relationship between processor-sharing queues and Crump–Mode–Jagers branching processes , 1992, Advances in Applied Probability.

[30]  O. Boxma,et al.  The M/G/1 queue with two service speeds , 2001, Advances in Applied Probability.

[31]  R. J. Williams,et al.  State space collapse and diffusion approximation for a network operating under a fair bandwidth sharing policy , 2009, 0910.3821.

[32]  S. Asmussen.  The heavy traffic limit of a class of Markovian queueing models , 1987 .

[33]  I. M. Verloop,et al.  HEAVY-TRAFFIC ANALYSIS OF A NON-PREEMPTIVE MULTI-CLASS QUEUE WITH RELATIVE PRIORITIES , 2015, Probability in the Engineering and Informational Sciences.

[34]  Eitan Altman,et al.  DPS queues with stationary ergodic service times and the performance of TCP in overload , 2004, IEEE INFOCOM 2004.

[35]  Michel Mandjes,et al.  Markov-modulated infinite-server queues with general service times , 2014, Queueing Syst. Theory Appl..

[36]  Urtzi Ayesta,et al.  Sojourn time approximations in a multi-class time-sharing server , 2014, IEEE INFOCOM 2014 - IEEE Conference on Computer Communications.

[37]  Refael Hassin,et al.  To Queue or Not to Queue: Equilibrium Behavior in Queueing Systems , 2002 .