Is Tail-Optimal Scheduling Possible?

This paper focuses on the competitive analysis of scheduling disciplines in a large deviations setting. Although there are policies that are known to optimize the sojourn time tail under a large class of heavy-tailed job sizes e.g., processor sharing and shortest remaining processing time and there are policies known to optimize the sojourn time tail in the case of light-tailed job sizes e.g., first come first served, no policies are known that can optimize the sojourn time tail across both light-and heavy-tailed job size distributions. We prove that no such work-conserving, nonanticipatory, nonlearning policy exists, and thus that a policy must learn or know the job size distribution in order to optimize the sojourn time tail.

[1]  Linus Schrage,et al.  Letter to the Editor - A Proof of the Optimality of the Shortest Remaining Processing Time Discipline , 1968, Oper. Res..

[2]  D. Iglehart.  Extreme Values in the GI/G/1 Queue , 1972 .

[3]  J. Cohen.  SOME RESULTS ON REGULAR VARIATION FOR DISTRIBUTIONS IN QUEUEING AND FLUCTUATION THEORY , 1973 .

[4]  A. Pakes.  ON THE TAILS OF WAITING-TIME DISTRIBUTIONS , 1975 .

[5]  Aleksandr Alekseevich Borovkov,et al.  Stochastic processes in queueing theory , 1976 .

[6]  J. Teugels,et al.  On the asymptotic behaviour of the distributions of the busy period and service time in M/G/1 , 1980, Journal of Applied Probability.

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

[8]  M. Meerschaert.  Regular Variation in R k , 1988 .

[9]  J. Shanthikumar,et al.  On extremal service disciplines in single-stage queueing systems , 1990, Journal of Applied Probability.

[10]  Ward Whitt,et al.  Tail probabilities with statistical multiplexing and effective bandwidths in multi-class queues , 1993, Telecommun. Syst..

[11]  P. Glynn,et al.  Logarithmic asymptotics for steady-state tail probabilities in a single-server queue , 1994, Journal of Applied Probability.

[12]  Frank Kelly,et al.  Notes on effective bandwidths , 1994 .

[13]  Michael Pinedo,et al.  Scheduling: Theory, Algorithms, and Systems , 1994 .

[14]  F. Kelly,et al.  Stochastic networks : theory and applications , 1996 .

[15]  Fengyuan Ren,et al.  Scheduling " , , 1997 .

[16]  Venkat Anantharam,et al.  Scheduling strategies and long-range dependence , 1999, Queueing Syst. Theory Appl..

[17]  Christos H. Papadimitriou,et al.  Beyond competitive analysis [on-line algorithms] , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[18]  A. P. Zwart,et al.  Tail Asymptotics for the Busy Period in the GI/G/1 Queue , 2001, Math. Oper. Res..

[19]  Søren Asmussen,et al.  Ruin probabilities , 2001, Advanced series on statistical science and applied probability.

[20]  A. Stolyar,et al.  LARGEST WEIGHTED DELAY FIRST SCHEDULING: LARGE DEVIATIONS AND OPTIMALITY , 2001 .

[21]  P. Jelenkovic,et al.  Asymptotic loss probability in a finite buffer fluid queue with hetergeneous heavy-tailed on--off processes , 2003 .

[22]  Sem C. Borst,et al.  The impact of the service discipline on delay asymptotics , 2003, Perform. Evaluation.

[23]  A. Stolyar.  Control of end-to-end delay tails in a multiclass network: LWDF discipline optimality , 2003 .

[24]  Stan Zachary.  A Note on Veraverbeke's Theorem , 2004, Queueing Syst. Theory Appl..

[25]  Damon Wischik,et al.  Big queues , 2004, Lecture notes in mathematics.

[26]  M. Mandjes,et al.  SOJOURN TIMES IN THE M/G/1 FB QUEUE WITH LIGHT-TAILED SERVICE TIMES , 2003, Probability in the Engineering and Informational Sciences.

[27]  Nick McKeown,et al.  Part I: buffer sizes for core routers , 2005, CCRV.

[28]  Michel Mandjes,et al.  Large deviations of sojourn times in processor sharing queues , 2006, Queueing Syst. Theory Appl..

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

[30]  Bert Zwart,et al.  A large-deviations analysis of the GI/GI/1 SRPT queue , 2005, Queueing Syst. Theory Appl..

[31]  Z. Palmowski,et al.  On the exact asymptotics of the busy period in GI/G/1 queues , 2006, Advances in Applied Probability.

[32]  Bert Zwart,et al.  Tails in scheduling , 2007, PERV.

[33]  Mor Harchol-Balter.  Special Issue on New Perspectives in Scheduling , 2007 .

[34]  Predrag R. Jelenkovic,et al.  Adaptive and scalable comparison scheduling , 2007, SIGMETRICS '07.

[35]  Adam Wierman,et al.  Preventing Large Sojourn Times Using SMART Scheduling , 2008, Oper. Res..

[36]  Adam Wierman,et al.  Scheduling despite inexact job-size information , 2008, SIGMETRICS '08.

[37]  Adam Wierman,et al.  The Foreground-Background queue: A survey , 2008, Perform. Evaluation.

[38]  Adam Wierman,et al.  Tail-robust scheduling via limited processor sharing , 2010, Perform. Evaluation.