# The fluid limit of the multiclass processor sharing queue

- 170 Downloads
- 2 Citations

## Abstract

Consider a single server queueing system with several classes of customers, each having its own renewal input process and its own general service times distribution. Upon completing service, customers may leave, or re-enter the queue, possibly as customers of a different class. The server is operating under the egalitarian processor sharing discipline. Building on prior work by Gromoll et al. (Ann. Appl. Probab. 12:797–859, 2002) and Puha et al. (Math. Oper. Res. 31(2):316–350, 2006), we establish the convergence of a properly normalized state process to a fluid limit characterized by a system of algebraic and integral equations. We show the existence of a unique solution to this system of equations, both for a stable and an overloaded queue. We also describe the asymptotic behavior of the trajectories of the fluid limit.

## Keywords

Fluid limit Fluid model Measure valued process Multiclass networks Processor sharing## Mathematics Subject Classification (2000)

68M20 90B22## References

- 1.Altman, E., Jiménez, T., Kofman, D.: DPS queues with stationary ergodic service times and the performance of TCP in overload. In: Proc. IEEE INFOCOM’04, Hong-Kong (2004) Google Scholar
- 2.Altman, E., Avrachenkov, K., Ayesta, U.: A survey on discriminatory processor sharing. Queueing Syst.
**53**, 53–63 (2006) CrossRefGoogle Scholar - 3.Athreya, K.B., Rama Murthy, K.: Feller’s renewal theorem for systems of renewal equations. J. Indian Inst. Sci.
**58**(10), 437–459 (1976) Google Scholar - 4.Ben Tahar, A., Jean-Marie, A.: Population effects in multiclass processor sharing queues. In: Proc. Valuetools 2009, Fourth International Conference on Performance Evaluation Methodologies and Tools, Pisa, October 2009 Google Scholar
- 5.Ben Tahar, A., Jean-Marie, A.: The fluid limit of the multiclass processor sharing queue. INRIA research report RR 6867, version 2, April 2009 Google Scholar
- 6.Berman, A., Plemmons, A.J.: Nonnegative Matrices in the Mathematical Sciences. SIAM Classics in Applied Mathematics, vol. 9 (1994) CrossRefGoogle Scholar
- 7.Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968) Google Scholar
- 8.Bramson, M.: Convergence to equilibria for fluid models of FIFO queueing networks. Queueing Syst., Theory Appl.
**22**(1–2), 5–45 (1996) CrossRefGoogle Scholar - 9.Bramson, M.: Convergence to equilibria for fluid models of head-of-the-line proportional processor sharing queueing networks. Queueing Syst., Theory Appl.
**23**(1–4), 1–26 (1997) Google Scholar - 10.Bramson, M.: State space collapse with application to heavy traffic limits for multiclass queueing networks. Queueing Syst., Theory Appl.
**30**(1–2), 89–148 (1998) CrossRefGoogle Scholar - 11.Chen, H., Kella, O., Weiss, G.: Fluid approximations for a processor sharing queue. Queueing Syst., Theory Appl.
**27**, 99–125 (1997) CrossRefGoogle Scholar - 12.Dawson, D.A.: Measure-valued Markov processes, école d’été de probabilités de Saint Flour. Lecture Notes in Mathematics NO 1541, vol. XXI. Springer, Berlin (1993) Google Scholar
- 13.Durrett, R.T.: Probability: Theory and Examples, 2nd edn.. Duxbury, Belmont (1996) Google Scholar
- 14.Egorova, R., Borst, S., Zwart, B.: Bandwidth-sharing networks in overload. Perform. Eval.
**64**, 978–993 (2007) CrossRefGoogle Scholar - 15.Gromoll, H.C.: Diffusion approximation for a processor sharing queue in heavy traffic. Ann. Appl. Probab.
**14**, 555–611 (2004) CrossRefGoogle Scholar - 16.Gromoll, H.C., Kruk, L.: Heavy traffic limit for a processor sharing queue with soft deadlines. Ann. Appl. Probab.
**17**(3), 1049–1101 (2007) CrossRefGoogle Scholar - 17.Gromoll, H.C., Williams, R.: Fluid Limits for Networks with Bandwidth Sharing and General Document Size Distributions. Ann. Appl. Probab.
**10**(1), 243–280 (2009) CrossRefGoogle Scholar - 18.Gromoll, H.C., Puha, A.L., Williams, R.J.: The fluid limit of a heavily loaded processor sharing queue. Ann. Appl. Probab.
**12**, 797–859 (2002) CrossRefGoogle Scholar - 19.Gromoll, H.C., Robert, Ph., Zwart, B.: Fluid Limits for Processor-Sharing Queues with Impatience. Math. Oper. Res.
**33**(2), 375–402 (2008) CrossRefGoogle Scholar - 20.Horn, R., Johnson, C.: Matrix Analysis. Cambridge University Press, Cambridge (1985) Google Scholar
- 21.Jean-Marie, A., Robert, P.: On the transient behavior of the processor sharing queue. Queueing Syst., Theory Appl.
**17**, 129–136 (1994) CrossRefGoogle Scholar - 22.Puha, A.L., Williams, R.J.: Invariant states and rates of convergence for the fluid limit of a heavily loaded processor sharing queue. Ann. Appl. Probab.
**14**, 517–554 (2004) CrossRefGoogle Scholar - 23.Puha, A.L., Stolyar, A.L., Williams, R.J.: The fluid limit of an overloaded processor sharing queue. Math. Oper. Res.
**31**(2), 316–350 (2006) CrossRefGoogle Scholar - 24.de Saporta, B.: Étude de la solution stationnaire de l’équation
*Y*(*n*+1)=*a*(*n*)*Y*(*n*)+*b*(*n*) à coefficients aléatoires. PhD thesis, University of Rennes 1 (2004) Google Scholar - 25.Williams, R.J.: Diffusion approximation for open multiclass queueing networks: sufficient conditions involving state space collapse. Queueing Syst., Theory Appl.
**30**, 27–88 (1998) CrossRefGoogle Scholar - 26.Yashkov, S.F., Yashkova, A.S.: Processor sharing: a survey of the mathematical theory. Autom. Remote Control
**68**(9), 1662–1731 (2007) CrossRefGoogle Scholar - 27.Zhang, J., Dai, J.G., Zwart, B.: Law of large number limits of limited processor-sharing queues. Math. Oper. Res.
**34**(4), 937–970 (2009) CrossRefGoogle Scholar