Random Fluid Limit of an Overloaded Polling Model

In the present paper, we study the evolution of an overloaded cyclic polling model that starts empty. Exploiting a connection with multitype branching processes, we derive fluid asymptotics for the joint queue length process. Under passage to the fluid dynamics, the server switches between the queues infinitely many times in any finite time interval causing frequent oscillatory behavior of the fluid limit in the neighborhood of zero. Moreover, the fluid limit is random. In addition, we suggest a method that establishes finiteness of moments of the busy period in an M/G/1 queue.

[1]  高木 英明,et al.  Analysis of polling systems , 1986 .

[2]  Dimitri Petritis,et al.  A Markov chain model of a polling system with parameter regeneration , 2007 .

[3]  Robert D. van der Mei,et al.  Towards a unifying theory on branching-type polling systems in heavy traffic , 2007, Queueing Syst. Theory Appl..

[4]  Serguei Foss,et al.  A stability criterion via fluid limits and its application to a polling system , 1999, Queueing Syst. Theory Appl..

[5]  Jacques Resing,et al.  Polling systems and multitype branching processes , 1993, Queueing Syst. Theory Appl..

[6]  J. W. Cohen,et al.  QUEUEING, PERFORMANCE AND CONTROL IN ATM , 1991 .

[7]  Robert D. van der Mei,et al.  Polling Models with Two-Stage Gated Service: Fairness Versus Efficiency , 2007, International Teletraffic Congress.

[8]  T. E. Harris,et al.  The Theory of Branching Processes. , 1963 .

[9]  C. Mack,et al.  THE EFFICIENCY OF N MACHINES UNI-DIRECTIONALLY PATROLLED BY ONE OPERATIVE WHEN WALKING TIME AND REPAIR TIMES ARE CONSTANTS , 1957 .

[11]  Eitan Altman,et al.  Control of Polling in Presence of Vacations in Heavy Traffic with Applications to Satellite and Mobile Radio Systems , 2002, SIAM J. Control. Optim..

[12]  Hideaki Takagi,et al.  Queueing analysis of polling models: progress in 1990-1994 , 1998 .

[13]  Adam Wierman,et al.  Fairness and efficiency for polling models with the k-gated service discipline , 2012, Perform. Evaluation.

[14]  Russell Lyons,et al.  A Conceptual Proof of the Kesten-Stigum Theorem for Multi-Type Branching Processes , 1997 .

[15]  S. G. Foss,et al.  On the Stability of a Queueing System with Uncountably Branching Fluid Limits , 2005, Probl. Inf. Transm..

[16]  C. Graham.  Functional central limit theorems for a large network in which customers join the shortest of several queues , 2004, math/0403538.

[17]  H. Kesten,et al.  A Limit Theorem for Multidimensional Galton-Watson Processes , 1966 .

[18]  Edward G. Coffman,et al.  Polling Systems in Heavy Traffic: A Bessel Process Limit , 1998, Math. Oper. Res..

[19]  Robert D. van der Mei,et al.  Polling models with multi-phase gated service , 2012, Ann. Oper. Res..

[20]  Onno Boxma.  Analysis and optimization of polling systems , 1991 .

[21]  Dirk P. Kroese.  HEAVY TRAFFIC ANALYSIS FOR CONTINUOUS POLLING MODELS , 1995 .

[22]  Vladimir Vatutin,et al.  Multitype Branching Processes and Some Queueing Systems , 2002 .

[23]  Hideaki Takagi,et al.  Stochastic Analysis of Computer and Communication Systems , 1990 .