Abstract
In this paper we derive a multidimensional version of the rate conservation law (RCL) for càdlàg processes of bounded variation. These results are then used to analyze queueing models which have a natural multidimensional characterization, such as priority queues. In particular the RCL is used to establish certain conservation laws between the idle probabilities for such queues. We use the relations to provide a detailed analysis of preemptive resume priority queues with M/G inputs. Special attention is paid to the validity of the so-called reduced service rate approximation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Abate and W. Whitt, “Asymptotics for m/g/1 low-priority waiting time tail probabilities,” Queueing Systems vol. 25 pp. 173–223, 1997.
A. W. Berger and W. Whitt, “Effective bandwidths with priorities,” IEEE/ACM Trans. on Networking vol. 6 pp. 447–460, 1998.
A. A. Borovkov, Stochastic Processes in Queueing Theory, Applications of Mathematics, Springer Verlag: NY, 1976.
P. Brémaud and F. Bacelli, Palm Probabilities and Stationary Queues, Lecture Notes in Statistics 41, Springer Verlag: Berlin, 1987.
J. W. Cohen, The Single Server Queue, North Holland Series in Applied Mathematics and Mechanics, North Holland, 1982.
D. J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes, Springer Series in Statistics, Springer Verlag, 1988.
S. Delas, R. R. Mazumdar, and C. Rosenberg, “Cell loss asymptotics in buffers handling a large number of sources with HOL service priorities.” In Proc. of INFOCOM'99, New York, 1999. Longer version submitted to Queueing Systems.
M. H. A. Davis, Markov Models and Optimization, Monographs on Statistics and Applied Probability, Chapman & Hall, 49, 1999.
F. Guillemin, R. Mazumdar, A. Dupuis, and J. Boyer, “Analysis of fluid weighted fair queueing systems,” Journal of Applied Probability vol. 40 pp. 180–199, 2003.
J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, Comprehensive Studies in Mathematics, Springer Verlag, 1987.
N. K. Jaiswal, Priority Queues, volume 50 of Mathematics in Science and Engineering, Academic Press, 1968.
N. El Karaoui, “Un problème de, réflexion et ses applications au temps local et aux équations différentielles stochastiques sur ℝ. Cas continu.” In J. Azema and M. Yor (eds.), Temps locaux, number 52-53 in Astérisques, pp. 117–144. Société Mathématique de France, 1978.
L. Kleinrock, Queueing Systems, Vol. I: Theory, J. Wiley, 1975.
R. Mazumdar, V Badrinath, F. Guillemin, and C. Rosenberg, “On pathwise rate conservation for a class of semi-martingales,” Stochastic Processes and Their Applications vol. 47 pp. 119–130, 1993.
R. Mazumdar and F. Guillemin, “Forward equations for reflected diffusions with jumps,” Applied Mathematics and Optimization vol. 33 pp. 81–102, 1996.
R. Mazumdar, R. Kannurpatti, and C. Rosenberg, “On rate conservation for nonstationary processes,” J. Appl. Prob. vol. 28 pp. 762–770, 1991.
M. Miyazawa, “The derivation of invariance relations in complex queueing systems with stationary inputs,” Adv. App. Prob. pp. 875–885, 1983.
M. Miyazawa, “Time-dependent rate conservation laws for a process defined with a stationary marked point process and their applications,” J. Appl. Prob. vol. 31 pp. 114–129, 1994.
M. Miyazawa, “Rate conservation laws: A survey,” Queueing Systems Theory Appl. vol. 15 pp. 1–58, 1994.
W. Rudin, Real and complex analysis, McGraw Hill: New York, 1965.
H. Takagi, Queueing analysis, Vol 1: A Foundation of Performance Evaluation, North-Holland, 1991.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Guillemin, F., Mazumdar, R. Rate Conservation Laws for Multidimensional Processes of Bounded Variation with Applications to Priority Queueing Systems. Methodology and Computing in Applied Probability 6, 135–159 (2004). https://doi.org/10.1023/B:MCAP.0000017710.98631.46
Issue Date:
DOI: https://doi.org/10.1023/B:MCAP.0000017710.98631.46