The Annals of Applied Probability

Heavy traffic limit for a processor sharing queue with soft deadlines

H. Christian Gromoll and Łukasz Kruk

Full-text: Open access

Abstract

This paper considers a GI/GI/1 processor sharing queue in which jobs have soft deadlines. At each point in time, the collection of residual service times and deadlines is modeled using a random counting measure on the right half-plane. The limit of this measure valued process is obtained under diffusion scaling and heavy traffic conditions and is characterized as a deterministic function of the limiting queue length process. As special cases, one obtains diffusion approximations for the lead time profile and the profile of times in queue. One also obtains a snapshot principle for sojourn times.

Article information

Source
Ann. Appl. Probab., Volume 17, Number 3 (2007), 1049-1101.

Dates
First available in Project Euclid: 22 May 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1179839182

Digital Object Identifier
doi:10.1214/105051607000000014

Mathematical Reviews number (MathSciNet)
MR2326240

Zentralblatt MATH identifier
1130.60087

Subjects
Primary: 60K25: Queueing theory [See also 68M20, 90B22]
Secondary: 60F17: Functional limit theorems; invariance principles 68M20: Performance evaluation; queueing; scheduling [See also 60K25, 90Bxx] 90B22: Queues and service [See also 60K25, 68M20]

Keywords
Processor sharing real-time queue deadlines heavy traffic measure valued process empirical process

Citation

Gromoll, H. Christian; Kruk, Łukasz. Heavy traffic limit for a processor sharing queue with soft deadlines. Ann. Appl. Probab. 17 (2007), no. 3, 1049--1101. doi:10.1214/105051607000000014. https://projecteuclid.org/euclid.aoap/1179839182


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References

  • Billingsley, P. (1986). Probability and Measure, 2nd ed. Wiley, New York.
  • Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
  • Bramson, M. (1998). State space collapse with applications to heavy traffic limits for multiclass queueing networks. Queueing Syst. 30 89–148.
  • Bramson, M. (2001). Stability of earliest-due-date, first-served queueing networks. Queueing Syst. 39 79–102.
  • Bramson, M. and Dai, J. (2001). Heavy traffic limits for some queueing networks. Ann. Appl. Probab. 11 49–90.
  • Doytchinov, B., Lehoczky, J. and Shreve, S. (2001). Real-time queues in heavy traffic with earliest-deadline-first queue discipline. Ann. Appl. Probab. 11 332–378.
  • Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • Grishechkin, S. (1994). GI/G/1 processor sharing queue in heavy traffic. Adv. in Appl. Probab. 26 539–555.
  • Gromoll, H. C. (2004). Diffusion approximation for a processor sharing queue in heavy traffic. Ann. Appl. Probab. 14 555–611.
  • 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.
  • Iglehart, D. L. and Whitt, W. (1970). Multiple channel queues in heavy traffic I. Adv. in Appl. Probab. 2 150–177.
  • Kallenberg, O. (1986). Random Measures. Academic Press, New York.
  • Kruk, \L., Lehoczky, J. and Shreve, S. (2003). Second order approximation for the customer time in queue distribution under the FIFO service discipline. Ann. Univ. Mariae Curie-Skłodowska Sect. AI Inform. 1 37–48.
  • Kruk, \L., Lehoczky, J. and Shreve, S. (2006). Accuracy of state space collapse for earliest-deadline-first queues. Ann. Appl. Probab. 16 516–561.
  • Kruk, Ł., Lehoczky, J., Shreve, S. and Yeung, S.-N. (2003). Multiple-input heavy-traffic real-time queues. Ann. Appl. Probab. 13 54–99.
  • Kruk, \L., Lehoczky, J., Shreve, S. and Yeung, S.-N. (2004). Earliest-deadline-first service in heavy-traffic acyclic networks. Ann. Appl. Probab. 14 1306–1352.
  • Kruk, Ł. and Zieba, W. (1994). On tightness of randomly indexed sequences of random variables. Bull. Pol. Acad. Sci. Math. 42 237–241.
  • Lehoczky, J. P. (1996). Real-time queueing theory. In Proceedings of the IEEE Real-Time Systems Symposium. IEEE, New York.
  • Prohorov, Y. V. (1956). Convergence of random processes and limit theorems in probability theory. Theory Probab. Appl. 1 157–214.
  • Puha, A. L., Stolyar, A. L. and Williams, R. J. (2006). The fluid limit of an overloaded processor sharing queue. Math. Oper. Res. 31 316–350.
  • Puha, A. L. and Williams, R. J. (2004). Invariant states and rates of convergence for a critical fluid model of a processor sharing queue. Ann. Appl. Probab. 14 517–554.
  • Reiman, M. I. (1984). Open queueing networks in heavy traffic. Math. Oper. Res. 9 441–458.
  • van der Vaart, A. and Wellner, J. A. (1996). Weak convergence and empirical processes. Springer, New York.
  • Williams, R. J. (1998). Diffusion approximations for open multiclass queueing networks: Sufficient conditions involving state space collapse. Queueing Syst. 30 27–88.
  • Yashkov, S. F. (1987). Processor-sharing queues: Some progress in analysis. Queueing Syst. 2 1–17.
  • Yeung, S.-N. and Lehoczky, J. P. (2004). Real-time queueing networks in heavy traffic with EDF and FIFO queue discipline. Preprint.