The Annals of Applied Probability

Strong approximations for multiclass feedforward queueing networks

Hong Chen and Xinyang Shen

Full-text: Open access


This paper derives the strong approximation for a multiclass queueing network,where jobs after service completion can only move to a downstream service station. Job classes are partitioned into groups. Within a group, jobs are served in the order of arrival; that is, a first-in first-out (FIFO) discipline is in force, and among groups, jobs are served under a preassigned preemptive priority discipline. We obtain the strong approximation for the network through an inductive application of an input–output analysis for a single-station queue. Specifically, we show that if the input data (i.e., the arrival and the service processes) satisfy an approximation (such as the functional law-of-iterated logarithm approximation or the strong approximation), then the output data (i.e., the departure processes) and the performance measures (such as the queue length, the workload and the sojourn time processes) satisfy a similar approximation. Based on the strong approximation, some procedures are proposed to approximate the stationary distribution of various performance measures of the queueing network. Our work extends and complements the existing work of Peterson and Harrison and Williams on the feedforward queueing network. The numeric examples show that strong approximation provides a better approximation than that suggested by a straightforward interpretation of the heavy traffic limit theorem.

Article information

Ann. Appl. Probab., Volume 10, Number 3 (2000), 828-876.

First available in Project Euclid: 22 April 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles 60K25: Queueing theory [See also 68M20, 90B22] 60G17: Sample path properties
Secondary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 90B10: Network models, deterministic 90B22: Queues and service [See also 60K25, 68M20]

Multiclass queueing network diffusion approximations fluid approximations heavy traffic reflected Brownian motion and strong approximation


Chen, Hong; Shen, Xinyang. Strong approximations for multiclass feedforward queueing networks. Ann. Appl. Probab. 10 (2000), no. 3, 828--876. doi:10.1214/aoap/1019487511.

Export citation


  • Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • Chen, H. (1996). A sufficient condition for the positive recurrence of a semimartingale reflecting Brownian motion in an orthant. Ann.Appl.Probab. 6 758-765.
  • Chen, H. and Mandelbaum, A. (1994). Hierarchical modeling of stochastic networks II. Strong approximations. In Stochastic Modeling and Analysis of Manufacturing Systems (D. D. Yao, ed.) 107-131. Springer, Berlin.
  • Cs ¨org o, M. and Horv´ath, L. (1993). Weighted Approximations in Probability and Statistics. Wiley, New York.
  • Dai, J. G. and Harrison, J. M. (1992). Reflected Brownian motion in an orthant: numerical methods for steady-state analysis. Ann.Appl.Probab. 2 65-86.
  • Fendick, K. W., Saksena, V. R. and Whitt, W. (1989). Dependence in packet queues. IEEE Trans. Communications 37 1173-1183.
  • Glynn, P. W. (1990). Diffusion approximations. In Handbooks in Operations Research and Management Science 2.Stochastic Models (D. P. Heyman and M. J. Sobel, eds.) 145-198. North-Holland, Amsterdam.
  • Harrison, J. M. (1985). Brownian Motion and Stochastic Flow Systems. Wiley, New York.
  • Harrison, J. M. and Nguyen, V. (1993). Brownian models of multiclass queueing networks: current status and open problems. Queueing Systems Theory Appl. 13 5-40.
  • Harrison, J. M. and Williams, R. J. (1992). Brownian models of feedforward queueing networks: quasireversibility and product form solutions. Ann.Appl.Probab.2 263-293.
  • Horv´ath, L. (1990). Strong approximations of open queueing networks. Math Oper.Res. 17 487-508.
  • Kleinrock, L. (1976). Queueing Systems 2.Computer Applications. Wiley, New York.
  • Lemoine, A. J. (1978). Network of queues-a survey of weak convergence results. Management Sci. 24 1175-1193.
  • Mandelbaum, A. and Massey, W. A. (1995). Strong approximation for time-dependent queues. Math.Oper.Res. 20 33-64.
  • Mandelbaum, A., Massey, W. A. and Reiman, M. (1998). Strong approximation for Markov service networks. Queueing Systems Theory Appl. 30 149-201.
  • Mandelbaum, A. and Pats, G. (1998). Stochastic networks I. Approximations and applications with continuous diffusion limits. Ann.Appl.Probab. 8 569-646.
  • Peterson, W. P. (1991). A heavy trafficlimit theorem for networks of queues with multiple customer types. Math.Oper.Res. 16 90-118.
  • Shen, X. (2000). Performance evaluation of multiclass queueing networks via Brownian motions. Ph.D. thesis, Univ. British Columbia.
  • Whitt, W. (1974). Heavy traffictheorems for queues: a survey. In Mathematical Methods in Queueing Theory (A. B. Clarke, ed.) 307-350. Springer, New York.
  • Whitt, W. (1980). Some useful functions for functional limit theorems. Math.Oper.Res. 5 67-85.
  • Williams, R. J. (1996). On the approximation of queueing networks in heavy traffic. In Stochastic Networks: Theory and Applications (F. P. Kelly, S. Zachary and I. Ziedins, eds.) 35-56. Oxford Univ. Press.
  • Zhang, H. (1997). Strong approximations for irreducible closed queueing networks. Appl. Probab. 29 498-522.
  • Zhang, H., Hsu, G. and Wang, R. (1990). Strong approximations for multiple channel queues in heavy traffic. J.Appl.Probab. 28 658-670.