Abstract
We investigate the tail behavior of the sojourn-time distribution for a request of a given length in an M/G/1 Processor-Sharing (PS) queue. An exponential asymptote is proven for general service times in two special cases: when the traffic load is sufficiently high and when the request length is sufficiently small. Furthermore, using the branching process technique we derive exact asymptotics of exponential type for the sojourn time in the M/M/1 queue. We obtain an equation for the asymptotic decay rate and an exact expression for the asymptotic constant. The decay rate is studied in detail and is compared to other service disciplines. Finally, using numerical methods, we investigate the accuracy of the exponential asymptote.
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J. Abate and W. Whitt, Asymptotics for M/G/1 low-priority waiting-time tail probabilities. Queueing Systems 25 (1997) 173–233.
J. Abate and W. Whitt, The Fourier-series method for inverting transforms of probability distributions. Queueing Systems 10 (1992) 5–88.
S.C. Borst, O.J. Boxma, J.A. Morrison, and R. Núñez-Queija, The equivalence between processor sharing and service in random order. Operations Research Letters 31 (2003) 254–262.
S.C. Borst, R. Núñez-Queija, and A.P. Zwart, Sojourn time asymptotics in Processor-Sharing queues. Queueing Systems 53 (2006) 31–51.
L. Breiman, On some limit theorems similar to the arc-sin law. Theory of Probability and its Applications 10 (1965) 323–331.
E.G. Coffman, R. Muntz, and H. Trotter, Waiting time distributions for processor-sharing systems. Journal of the ACM 17 (1970) 123–130.
D. Denisov and A.P. Zwart, On a theorem of Breiman and a class of random difference equations. EURANDOM report 2005-039 (2005), http://www.eurandom.nl/reports/2005/039DDreport.pdf.
R. Egorova, A.P. Zwart, and O.J. Boxma, Sojourn time tails in the M/D/1 Processor Sharing queue. Probability in the Engineering and Informational Sciences 20 (2006) 492–446.
L. Flatto, The waiting time distribution for the random order of service M/M/1 queue. Annals of Applied Probability 7 (1997) 382–409.
S. Grishechkin, On a relationship between processor-sharing queues and Crump-Mode-Jagers branching processes. Advances in Applied Probability 24 (1992) 653–698.
P.R. Jelenković and P. Momčilović, Large deviation analysis of subexponential waiting times in a processor-sharing queue. Mathematics of Operations Research 28 (2004) 587–608.
V. Kalashnikov and G. Tsitsiashvili, Tail of waiting times and their bounds. Queueing Systems 32 (1999) 257–283.
M. Mandjes and M. Nuyens, Sojourn times in the M/G/1 FB queue with light-tailed service times. Probability in the Engineering and Informational Sciences 19 (2004) 351–361.
M. Mandjes and A.P. Zwart, Large deviations for sojourn times in Processor Sharing queues. Queueing Systems 52 (2006) 237–250.
R. Núñez-Queija, Processor-sharing models for integrated-services networks. PhD thesis, Eindhoven University of Technology (2000).
M. Nuyens and A.P. Zwart, A large-deviations analysis of the GI/GI/1 SRPT queue. Queueing Systems 54 (2005) 85–97.
T.J. Ott, The sojourn-time distribution in the M/G/1 queue with processor sharing. Journal of Applied Probability 21 (1984) 360–378.
R. Schassberger, A new approach to the M/G/1 processor sharing queue. Advances in Applied Probability 16 (1984) 802–813.
B. Sengupta, An approximation for the sojourn-time distribution for the GI/G/1 processor-sharing queue. Stochastic Models 8 (1992) 35–57.
S.F. Yashkov, A derivation of response time distribution for a M/G/1 Processor-Sharing queue. Problems of Control and Information Theory 12 (1983) 133–148.
S.F. Yashkov, Processor-sharing queues: some progress in analysis. Queueing Systems 2 (1987) 1–17.
S.F. Yashkov, On a heavy-traffic limit theorem for the M/G/1 processor-sharing queue. Stochastic Models 9 (1993) 467–471.
A.P. Zwart and O.J. Boxma, Sojourn time asymptotics in the M/G/1 processor sharing queue. Queueing Systems 35 (2000) 141–166.
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AMS 2000 Subject Classifications Primary:60K25,Secondary: 60F10,68M20,90B22
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Egorova, R., Zwart, B. Tail behavior of conditional sojourn times in Processor-Sharing queues. Queueing Syst 55, 107–121 (2007). https://doi.org/10.1007/s11134-006-9007-4
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DOI: https://doi.org/10.1007/s11134-006-9007-4