The Annals of Applied Probability

Earliest-deadline-first service in heavy-traffic acyclic networks

Łukasz Kruk, John Lehoczky, Steven Shreve, and Shu-Ngai Yeung

Full-text: Open access


This paper presents a heavy traffic analysis of the behavior of multi-class acyclic queueing networks in which the customers have deadlines. We assume the queueing system consists of J stations, and there are K different customer classes. Customers from each class arrive to the network according to independent renewal processes. The customers from each class are assigned a random deadline drawn from a deadline distribution associated with that class and they move from station to station according to a fixed acyclic route. The customers at a given node are processed according to the earliest-deadline-first (EDF) queue discipline. At any time, the customers of each type at each node have a lead time, the time until their deadline lapses. We model these lead times as a random counting measure on the real line. Under heavy traffic conditions and suitable scaling, it is proved that the measure-valued lead-time process converges to a deterministic function of the workload process. A two-station example is worked out in detail, and simulation results are presented to illustrate the predictive value of the theory. This work is a generalization of Doytchinov, Lehoczky and Shreve [Ann. Appl. Probab. 11 (2001) 332–379], which developed these results for the single queue case.

Article information

Ann. Appl. Probab., Volume 14, Number 3 (2004), 1306-1352.

First available in Project Euclid: 13 July 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

Acyclic networks due dates heavy traffic queueing diffusion limits random measures.


Kruk, Łukasz; Lehoczky, John; Shreve, Steven; Yeung, Shu-Ngai. Earliest-deadline-first service in heavy-traffic acyclic networks. Ann. Appl. Probab. 14 (2004), no. 3, 1306--1352. doi:10.1214/105051604000000314.

Export citation


  • Billingsley, P. (1986). Probability and Measure, 2nd ed. Wiley, New York.
  • Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
  • Bourbaki, N. (1966). General Topology 1. Addison–Wesley, Reading, MA.
  • Bramson, M. (2004). Stability of earliest-due-date, first-served queueing networks. Queueing Syst. Theory Appl. To appear.
  • 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. 11 332–379.
  • Ethier, S. N. and Kurtz, T. G. (1985). Markov Processes: Characterization and Convergence. Wiley, New York.
  • Harrison, J. M. (1995). Balanced fluid models of multiclass queueing networks: A heavy traffic conjecture. In Stochastic Networks (F. P. Kelly and R. Williams, eds.) 1–20. Springer, New York.
  • Kruk, Ł., Lehoczky, J. P., Shreve, S. E. and Yeung, S. N. (2003). Multiple-input heavy-traffic real-time queues. Ann. Appl. Probab. 13 54–99.
  • Lehoczky, J. P. (1997). Using real-time queueing theory to control lateness in real-time systems. Performance Evaluation Review 25 158–168.
  • Lehoczky, J. P. (1998). Real-time queueing theory. In Proceedings of the IEEE Real-Time Systems Symposium 186–195.
  • Lehoczky, J. P. (1998). Scheduling communication networks carrying real-time traffic. In Proceedings of the IEEE Real-Time Systems Symposium 470–479.
  • Markowitz, D. M. and Wein, L. M. (2001). Heavy traffic analysis of dynamic cyclic policies: A unified treatment of the single machine scheduling problem. Oper. Res. 49 246–270.
  • Peterson, W. P. (1991). A heavy traffic limit theorem for networks of queues with different customer types. Math. Oper. Res. 16 90–118.
  • Prokhorov, Yu. (1956). Convergence of random processes and limit theorems in probability theory. Theory Probab. Appl. 1 157–214.
  • Van Mieghem, J. A. (1995). Dynamic scheduling with convex delay costs: The generalized $c\mu$ rule. Ann. Appl. Probab. 5 809–833.
  • Williams, R. J. (1998). Diffusion approximations for open multiclass queueing networks: Sufficient conditions involving state space collapse. Queueing Systems Theory Appl. 30 27–88.
  • 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.