Puha, Amber L.; Williams, Ruth J. Invariant states and rates of convergence for a critical fluid model of a processor sharing queue. (English) Zbl 1061.60098 Ann. Appl. Probab. 14, No. 2, 517-554 (2004). The paper investigates the fluid limit of a processor sharing queue under heavy traffic derived by H. C. Gromoll, A. L. Puha and R. J. Williams [Ann. Appl. Probab. 12, No. 3, 797–859 (2002; Zbl 1017.60092)]. Sufficient conditions are given to ensure that the fluid models converge to an invariant state and that rates of convergence can be determined. Reviewer: Hans Daduna (Hamburg) Cited in 2 ReviewsCited in 10 Documents MSC: 60K25 Queueing theory (aspects of probability theory) 68M20 Performance evaluation, queueing, and scheduling in the context of computer systems 90B22 Queues and service in operations research Keywords:processor sharing; fluid limits; heavy traffic; coupling; invariant measures Citations:Zbl 1017.60092 PDFBibTeX XMLCite \textit{A. L. Puha} and \textit{R. J. Williams}, Ann. Appl. Probab. 14, No. 2, 517--554 (2004; Zbl 1061.60098) Full Text: DOI arXiv References: [1] Bramson, M. (1996). Convergence to equilibria for fluid models of FIFO queueing networks. Queueing Systems Theory Appl. 22 5–45. · Zbl 0851.90044 · doi:10.1007/BF01159391 [2] Bramson, M. (1997). Convergence to equilibria for fluid models of head-of-the-line proportional processor sharing queueing networks. Queueing Systems Theory Appl. 23 1–26. · Zbl 0879.60087 · doi:10.1007/BF01206549 [3] Bramson, M. (1998). State space collapse with applications to heavy traffic limits for multiclass queueing networks. Queueing Systems Theory Appl. 30 89–148. · Zbl 0911.90162 · doi:10.1023/A:1019160803783 [4] Chen, H., Kella, O. and Weiss, G. (1997). Fluid approximations for a processor-sharing queue. Queueing Systems Theory Appl. 27 99–125. · Zbl 0892.90073 · doi:10.1023/A:1019105929766 [5] Durrett, R. T. (1996). Probability : Theory and Examples , 2nd ed. Duxbury, Belmont, CA. · Zbl 1202.60001 [6] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes : Characterization and Convergence . Wiley, New York. · Zbl 0592.60049 [7] Feller, W. (1971). An Introduction to Probability Theory and Its Applications 2 , 2nd ed. Wiley, New York. · Zbl 0219.60003 [8] Folland, G. (1984). Real Analysis : Modern Techniques and Their Applications . Wiley, New York. · Zbl 0549.28001 [9] Gromoll, H. C. (2004). Diffusion approximation for a processor sharing queue in heavy traffic. Ann. Appl. Probab. · Zbl 1050.60085 · doi:10.1214/105051604000000035 [10] Gromoll, H. C., Puha, A. L. and Williams, R. J. (2002). The fluid limit of a heavily loaded processor sharing queue. Ann. Appl. Probab. 12 797–859. · Zbl 1017.60092 · doi:10.1214/aoap/1031863171 [11] Heath, D., Resnick, S. and Samorodnitsky, G. (1998). Heavy tails and long range dependence in on/off processes and associated fluid models. Math. Oper. Res. 23 145–165. · Zbl 0981.60092 · doi:10.1287/moor.23.1.145 [12] Lindvall, T. (1982). Lectures on the Coupling Method . Wiley, New York. · Zbl 0850.60019 [13] Resnick, S. I. (1992). Adventures in Stochastic Processes . Birkhäuser, Boston. · Zbl 0762.60002 [14] Williams, R. J. (1998). Diffusion approximations for open multiclass queueing networks: Sufficient conditions involving state space collapse. Queueing Systems Theory Appl. 30 27–88. · Zbl 0911.90171 · doi:10.1023/A:1019108819713 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.