Abstract
Over the past few decades, the Processor-Sharing (PS) discipline has attracted a great deal of attention in the queueing literature. While the PS paradigm emerged in the sixties as an idealization of round-robin scheduling in time-shared computer systems, it has recently captured renewed interest as a useful concept for modeling the flow-level performance of bandwidth-sharing protocols in communication networks. In contrast to the simple geometric queue length distribution, the sojourn time lacks such a nice closed-form characterization, even for exponential service requirements. In case of heavy-tailed service requirements however, there exists a simple asymptotic equivalence between the sojourn time and the service requirement distribution, which is commonly referred to as a reduced service rate approximation. In the present survey paper, we give an overview of several methods that have been developed to obtain such an asymptotic equivalence under various distributional assumptions. We outline the differences and similarities between the various approaches, discuss some connections, and present necessary and sufficient conditions for an asymptotic equivalence to hold. We also consider the generalization of the reduced service rate approximation to several extensions of the M/G/1 PS queue. In addition, we identify a relationship between the reduced service rate approximation and a queue length distribution with a geometrically decaying tail, and extend it to so-called bandwidth-sharing networks. The state-of-the-art with regard to sojourn time asymptotics in PS queues with light-tailed service requirements is also briefly described. Last, we reflect on some possible avenues for further research.
Similar content being viewed by others
References
E. Altman, K. Avrachenkov, and U. Ayesta, A survey on Discriminatory Processor Sharing (2006). In this special issue.
S. Asmussen, Applied Probability and Queues (Springer-Verlag, New York, 2003).
S. Asmussen, C. Klüppelberg, and K. Sigman, Sampling at subexponential times, with queueing applications, Stoch. Proc. Appl. 79 (1999) 265–286.
R. Bekker, S.C. Borst, and R. Nú nez-Queija, Performance of TCP-friendly streaming sessions in the presence of heavy-tailed elastic flows, Perf. Evaluation 61 (2005) 143–162.
J.L. van den Berg and O.J. Boxma, The M/G/1 queue with processor sharing and its relation to a feedback queue, Queueing Systems 9 (1991) 365–402.
N.H. Bingham and R.A. Doney, Asymptotic properties of supercritical branching processes I: the Galton-Watson process, Adv. Appl. Prob. 6 (1974) 711–731.
N.H. Bingham, C.M. Goldie, and J.L. Teugels, Regular Variation (Cambridge University Press, 1987).
T. Bonald and L. Massoulié, Impact of fairness on Internet performance, in: Proc. ACM Sigmetrics/Performance (2001).
T. Bonald and A. Proutière, Insensitivity in processor sharing networks, Perf. Evaluation 49 (2002) 193–209.
T. Bonald and A. Proutière, Insensitive bandwidth sharing in data networks, Queueing Systems 44 (2003) 69–100.
T. Bonald and A. Proutière, On stochastic bounds for monotonic processor sharing networks, Queueing Systems 47 (2004) 81–106.
S.C. Borst, O.J. Boxma, J.A. Morrison, and R. Nú nez-Queija, The equivalence between processor sharing and service in random order, Oper. Res. Lett. 31 (2003) 254–262.
S.C. Borst, O.J. Boxma, R. Nú nez-Queija, and A.P. Zwart, The impact of the service discipline on delay asymptotics, Perf. Evaluation 54 (2003) 177–206.
S.C. Borst, R. Nú nez-Queija, and M.J.G. van Uitert, User-level performance of elastic traffic in a differentiated-services environment, Proc. Performance (2002) 507–519.
S.C. Borst, R. Nú nez-Queija, and A.P. Zwart, Bandwidth sharing with heterogeneous service requirements, Ann. Telecommun. 59 (2004) 1297–1311.
S.C. Borst, D.T.M.B. van Ooteghem, and A.P. Zwart, Tail asymptotics for discriminatory processor-sharing queues with heavy-tailed service requirements, Perf. Evaluation 61 (2005) 281–298.
O.J. Boxma and V. Dumas, The busy period in the fluid queue, Perf. Eval. Rev. 26 (1998) 100–110.
L. Breiman. On some limit theorems similar to the arc-sin law, Theory of Prob. Appl. 10 (1965) 351–360.
E.G. Coffman Jr., R.R. Muntz, and H. Trotter, Waiting time distribution for processor-sharing systems, J. ACM 17 (1970) 123–130.
J.W. Cohen, The multiple phase service network with generalised processor sharing, Acta Informatica 12 (1979) 245–284.
M. Crovella and A. Bestavros, Self-similarity in World Wide Web traffic: Evidence and possible causes, in: Proc. ACM Sigmetrics (1996) pp. 160–169.
F. Delcoigne, A. Proutière, and G. Régnié, Modelling integration of streaming and data traffic, in: Proc. ITC Specialist Seminar (Würzburg, Germany, 2002).
R. Egorova, A.P. Zwart, and O.J. Boxma, Sojourn time tails in the M/D/1 processor sharing queue. SPOR-Report 2005-05, Eindhoven University of Technology, Prob. Eng. Inf. Sciences (2006) to appear.
P. Embrechts, C. Klüppelberg, and T. Mikosch, Modeling Extremal Events (Springer, 1997).
G. Fayolle, I. Mitrani, and R. Iasnogorodski, Sharing a processor among many job classes, J. ACM 27 (1980) 519–532.
L. Flatto, The waiting time distribution for the random order service M/M/1 queue, Ann. Appl. Prob. 7 (1997) 382–409.
S. Foss and D.A. Korshunov, Sampling at random time with a heavy-tailed distribution, Markov Proc. Rel. Fields 6 (2000) 543–568.
F. Guillemin, Ph. Robert, and A.P. Zwart, Tail asymptotics for processor sharing queues, Adv. Appl. Prob. 36 (2004) 525–543.
D.P. Heyman, T.V. Lakshman, and A.L. Neidhardt, A new method for analysing feedback-based protocols with applications to engineering Web traffic over the Internet, in: Proc. ACM Sigmetrics (1997) 24–38.
D.L. Jagerman and B. Sengupta, The GI/M/1 processor-sharing queue and its heavy traffic analysis, Comm. Statist.—Stoch. Models 7 (1991) 379–395.
P.R. Jelenković, Network multiplexer with truncated heavy-tailed arrival streams, in: Proc. IEEE Infocom (New York, NY, USA, 1999) pp. 625–633.
P.R. Jelenković and P. MomČilović, Large deviation analysis of subexponential waiting times in a processor sharing queue, Math. Oper. Res. 28 (2003) 587–608.
P.R. Jelenković, P. MomČilović, and A.P. Zwart, Reduced load equivalence under subexponentiality, Queueing Systems 46 (2004) 97–112.
L. Kleinrock, Analysis of a time-shared processor, Nav. Res. Log. Quarterly 11 (1964) 59–73.
L. Kleinrock, Time-shared systems: A theoretical treatment, J. ACM 14 (1967) 242–261.
M.R.H. Mandjes and A.P. Zwart, Large deviations for sojourn times in processor sharing queues, Queueing Systems (to appear).
L. Massoulié and J.W. Roberts, Bandwidth sharing: Objectives and algorithms, in: Proc. IEEE Infocom (New York, NY, USA, 1999) 1395–1403.
D. Mitra, Waiting time distributions from closed queueing network models of shared-processor systems, in: F.J. Kylstra (ed.), Proc. Performance, North-Holland, Amsterdam (1981) 113–131.
D. Mitra and J.A. Morrison, Asymptotic expansions of moments of the waiting time in closed and open processor-sharing systems with multiple job classes, Adv. Appl. Prob. 15 (1983) 813–839.
J.A. Morrison, Response-time distribution for a processor-sharing system, SIAM J. Appl. Math. 45 (1985) 152–167.
J.A. Morrison and D. Mitra, Heavy-usage asymptotic expansions for the waiting time in closed processor-sharing systems with multiple classes, Adv. Appl. Prob. 17 (1985) 163–185.
M. Nabe, M. Murata, and H. Miyahara, Analysis and modelling of World Wide Web traffic for capacity dimensioning of Internet access lines, Perf. Evaluation 34 (1998) 249–271.
R. Nú nez-Queija, Processor-Sharing Models for Integrated-Services Networks, Ph.D. thesis, Eindhoven University of Technology (2000) ISBN 90-646-4667-8 (also available from the author upon request).
R. Nú nez-Queija, Queues with equally heavy sojourn time and service requirement distributions, Ann. Oper. Res. 113 (2002) 101–117.
M. Nuyens, A. Wierman, and A.P. Zwart, Preventing large sojourn times with SMART scheduling, (SPOR Report 2005-13, Eindhoven University of Technology, 2005). Submitted for publication.
M. Nuyens and A.P. Zwart, A large deviations analysis of the GI/GI/1 SRPT queue (SPOR Report 2005–06, Eindhoven University of Technology, 2005). Submitted for publication.
D.T.M.B. van Ooteghem, A.P. Zwart, and S.C. Borst, A stochastic mean-value method for the derivation of delay asymptotics in heavy-tailed processor-sharing systems (SPOR Report 2004-11, Eindhoven University of Technology, 2004).
T.J. Ott, The sojourn time distributions in the M/G/1 queue with processor sharing, J. Appl. Prob. 21 (1984) 360–378.
Z. Palmowksi and T. Rolski, On busy period asymptotics in the GI/GI/1 queue, Available at http://www.math.uni.wroc.pl~zpalma/publication.html. Submitted for publication.
F. Pollaczek, La loi d'attente des appels téléphoniques, Comptes Rendus Acad. Sci. Paris 222 (1946) 352–355.
V. Ramaswami, The sojourn time in the GI/M/1 queue with processor sharing, J. Appl. Prob. 21 (1984) 437–442.
M. Sakata, S. Noguchi, and J. Oizumi, Analysis of a processor-shared queueing model for time-sharing systems, in: Proc. 2nd Hawaii Int. Conf. System Sciences (1969) pp. 625–628.
M. Sakata, S. Noguchi, and J. Oizumi, An analysis of the M/G/1 queue under round-robin scheduling, Oper. Res. 19 (1971) 371–385.
B. Sengupta and D.L. Jagerman, A conditional response time of the M/M/1 processor-sharing queue, AT&T Techn. J. 64 (1985) 409–421.
S.F. Yashkov, Processor-sharing queues: Some progress in analysis, Queueing Systems 2 (1987) 1–17.
S.F. Yashkov, Mathematical problems in the theory of processor-sharing queueing systems, J. Soviet Math. 58 (1992) 101–147.
A.P. Zwart, Sojourn times in a multi-class processor sharing queue, in: P. Key and D. Smith (eds.), Teletraffic Engineering in a Competitive World, Proc. ITC-16, Edinburgh, UK, (North-Holland, Amsterdam) (1999) 335–344.
A.P. Zwart, Tail asymptotics for the busy period in the GI/G/1 queue, Math. Oper. Res. 26 (2001) 475–483.
A.P. Zwart and O.J. Boxma, Sojourn time asymptotics in the M/G/1 processor sharing queue, Queueing Systems 35 (2000) 141–166.
Author information
Authors and Affiliations
Additional information
AMS Subject Classification 60K25 (primary), 60F10, 68M20, 90B18, 90B22 (secondary).
Rights and permissions
About this article
Cite this article
Borst, S., Núñez-Queija, R. & Zwart, B. Sojourn time asymptotics in processor-sharing queues. Queueing Syst 53, 31–51 (2006). https://doi.org/10.1007/s11134-006-7585-9
Issue Date:
DOI: https://doi.org/10.1007/s11134-006-7585-9