The Annals of Applied Probability

Accuracy of state space collapse for earliest-deadline-first queues

Łukasz Kruk, John Lehoczky, and Steven Shreve

Full-text: Open access

Abstract

This paper presents a second-order heavy traffic analysis of a single server queue that processes customers having deadlines using the earliest-deadline-first scheduling policy. For such systems, referred to as real-time queueing systems, performance is measured by the fraction of customers who meet their deadline, rather than more traditional performance measures, such as customer delay, queue length or server utilization. To model such systems, one must keep track of customer lead times (the time remaining until a customer deadline elapses) or equivalent information. This paper reviews the earlier heavy traffic analysis of such systems that provided approximations to the system’s behavior. The main result of this paper is the development of a second-order analysis that gives the accuracy of the approximations and the rate of convergence of the sequence of real-time queueing systems to its heavy traffic limit.

Article information

Source
Ann. Appl. Probab., Volume 16, Number 2 (2006), 516-561.

Dates
First available in Project Euclid: 29 June 2006

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1151592242

Digital Object Identifier
doi:10.1214/105051605000000809

Mathematical Reviews number (MathSciNet)
MR2244424

Zentralblatt MATH identifier
1129.60084

Subjects
Primary: 60K25: Queueing theory [See also 68M20, 90B22]
Secondary: 60G57: Random measures 60J65: Brownian motion [See also 58J65] 68M20: Performance evaluation; queueing; scheduling [See also 60K25, 90Bxx]

Keywords
State space collapse due dates heavy traffic queueing diffusion limits random measures

Citation

Kruk, Łukasz; Lehoczky, John; Shreve, Steven. Accuracy of state space collapse for earliest-deadline-first queues. Ann. Appl. Probab. 16 (2006), no. 2, 516--561. doi:10.1214/105051605000000809. https://projecteuclid.org/euclid.aoap/1151592242


Export citation

References

  • Bickel, P. J. and Wichura, M. J. (1971). Convergence for multiparameter stochastic processes and some applications. Ann. Math. Statist. 42 1656–1670.
  • Billingsley, P. (1986). Probability and Measure, 2nd ed. Wiley, New York.
  • Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
  • Doytchinov, B., Lehoczky, J. P. and Shreve, S. E. (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. (1985). Markov Processes: Characterization and Convergence. Wiley, New York.
  • Iglehart, D. and Whitt, W. (1970). Multiple channel queues in heavy traffic. I. Adv. in Appl. Probab. 2 150–177.
  • Iglehart, D. and Whitt, W. (1971). The equivalence of functional central limit theorems for counting processes and associated partial sums. Ann. Math. Statist. 42 1372–1378.
  • Karatzas, I. and Shreve, S. E. (1988). Brownian Motion and Stochastic Calculus. Springer, New York.
  • Kruk, \L., Lehoczky, J. P. and Shreve, S. E. (2003). Second-order approximation for the customer time in queue distribution under the FIFO service discipline. Annales UMCS Informatica AI 1 37–48.
  • Kruk, \L., Lehoczky, J. P., Shreve, S. E. and Yeung, S.-N. (2003). Multiple-input heavy-traffic real-time queues. Ann. Appl. Probab. 13 54–99.
  • Kruk, \L., Lehoczky, J. P., Shreve, S. E. and Yeung, S.-N. (2004). Earliest-deadline-first service in heavy traffic acyclic networks. Ann. Appl. Probab. 14 1306–1352.
  • Panwar, S. S. and Towsley, D. (1992). Optimality of the stochastic earliest deadline policy for the G$/$M$/$c queue serving customers with deadlines. Second ORSA Telecommunications Conference. Boca Raton.
  • Prokhorov, Yu. (1956). Convergence of random processes and limit theorems in probability theory. Theory Probab. Appl. 1 157–214.
  • Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. Wiley, New York.
  • Stankovic, J. A., Spuri, M., Ramamritham, K. and Buttazzo, G. C. (1998). Deadline Scheduling for Real-Time Systems. Springer, Berlin.
  • Whitt, W. (2002). Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues. Springer, New York.
  • Yeung, S. N. and Lehoczky, J. P. (2002). Real-time queueing networks in heavy traffic with EDF and FIFO queue discipline. Working paper, Dept. Statistics, Carnegie Mellon Univ.