Abstract
General exact light traffic limit theorems are given for the distribution of steadystate workloadV, in open queueing networks having as input a general stationary ergodic marked point process {(t n ,K n )n≥0 (where tn denotes the arrival time and Kn the routing and service times of the nth customer). No independence assumptions of any kind are required of the input. As the light traffic regime, it is only required that the Palm distribution for the exogenous interarrival time converges weakly to infinity (while the service mechanism is not allowed to change much). As is already known in the context of a single-server queue, work is much easier to deal with mathematically in light traffic than is customer delayD, and consequently, our results are far more general than existing results forD. We obtain analogous results for multi-channel and infinite-channel queues. In the context of open queueing networks, we handle both the total workload in the network as well as the workload at isolated nodes.
This is a preview of subscription content, log in via an institution to check access.
References
S. Asmussen, Light traffic equivalence in single-server queues, Ann. Appl. Prob. 2 (1992).
A.A. Borovkov,Asymptotic Methods in Queueing Theory (Wiley, 1984).
F. Baccelli and P. Brémaud,Palm Probabilities and Stationary Queues, Lecture Notes in Statistics, Vol. 41 (Springer, Berlin, 1987).
F. Baccelli and P. Brémaud, Virtual customers in sensitivity and light traffic analysis via Campbell's formula for point processes, Ann. Appl. Prob. (1992), to appear.
D. Daley and T. Rolski, A light traffic approximation for a single-server queue, Math. Oper. Res. 9(1984)624–628.
D. Daley and T. Rolski, Light traffic approximations in queues, Math. Oper. Res. 16(1991)57–71.
K.W. Fendick and W. Whitt, Measurements and approximation to describe the offered traffic and predict the average workload in a single-server queue, Proc. IEEE 77(1989)171–194.
P. Franken, D. König, U. Arndt and V. Schmidt,Queues and Point Processes (Akademie Verlag, Berlin, 1982).
J. Kiefer and J. Wolfowitz, On the theory of queues with many servers, Trans. Amer. Math. Soc. 78(1955)1–18.
M. Miyazawa, The derivation of invariance relations in complex queueing systems with stationary inputs, Adv. Appl. Prob. 15(1982)874–885.
G.F. Newell,Applications of Queueing Theory, 2nd Ed. (Chapman and Hall, New York, 1982).
K. Sigman, A note on a sample-path rate conservation law and its relationship withH=λG, Adv. Appl. Prob. (1991), to appear.
K. Sigman and G. Yamazaki, Heavy and light traffic in fluid models with burst arrivals, Ann. Inst. Statist. Math. (1992), to appear.
M.I. Reiman and B. Simon, Light traffic limits of sojourn time distributions in Markovian queueing networks, Stoch. Models 4(1988)191–233.
M.I. Reiman and B. Simon, Open queueing systems in light traffic, Math. Oper. Res.14(1989) 26–59.
W. Whitt, An interpolation approximation for the mean workload in aGI/G/1 queue, Oper. Res. 37(1989)936–952.
R.W. Wolff, Tandem queues with dependent service times in light traffic, Oper. Res. 30(1982)619–635.
R.W. Wolff,Stochastic Modeling and the Theory of Queues (Prentice-Hall, Englewood Cliffs, NJ, 1989).
Author information
Authors and Affiliations
Additional information
Research supported in part by the Japan Society for the Promotion of Science during the author's fellowship in Tokyo, and by NSF Grant DDM 895 7825.
Rights and permissions
About this article
Cite this article
Sigman, K. Light traffic for workload in queues. Queueing Syst 11, 429–442 (1992). https://doi.org/10.1007/BF01163865
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01163865