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JOIN MINIMUM COST QUEUE FOR MULTICLASS CUSTOMERS: STABILITY AND PERFORMANCE BOUNDS

Published online by Cambridge University Press:  01 October 2004

Rahul Tandra ,
N. Hemachandra  and
D. Manjunath
Rahul Tandra
Affiliation:
Department of EECS, University of California, Berkeley, CA, E-mail: [email protected]
N. Hemachandra
Affiliation:
IE and OR Interdisciplinary Programme, Indian Institute of Technology, Bombay, Powai Mumbai, 400 076 India, E-mail: [email protected]
D. Manjunath
Affiliation:
Department of Electrical Engineering, Indian Institute of Technology, Bombay, Powai Mumbai, 400 076 India, E-mail: [email protected]
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Abstract

We consider a system of K parallel queues providing different grades of service through each of the queues and serving a multiclass customer population. Service differentiation is achieved by specifying different join prices to the queues. Customers of class j define a cost function ψij(ci,xi) for taking service from queue i when the join price for queue i is ci and congestion in queue i is xi and join the queue that minimizes ψij(·,·). Such a queuing system will be called the “join minimum cost queue” (JMCQ) and is a generalization of the join shortest queue (JSQ) system. Non-work-conserving (called Paris Metro pricing system) and work-conserving (called the Tirupati system) versions of the JMCQ are analyzed when the cost to an arrival of joining a queue is a convex combination of the join price for that queue and the expected waiting time in that queue at the arrival epoch. Our main results are for a two-queue system.

We obtain stability conditions and performance bounds. To obtain the lower and upper performance bounds, we propose two quasi-birth–death (QBD) processes that are derived from the original systems by suitably truncating the state space. The state space truncation in the non-work-conserving JMCQ follows the method of van Houtum and colleagues. We then show that this method is not applicable to the work-conserving JMCQ and provide sample-path-based proofs to show that the number in each queue is bounded by the number in the corresponding queues of these QBD processes. These sample-path proof techniques might also be of independent interest. We then show that the performance measures like mean queue length and revenue rate of the system are also bounded by the corresponding quantities of these QBD processes. Numerical examples show that these bounds are fairly tight. Finally, we generalize some of these results to systems with more queues.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 18 , Issue 4 , October 2004 , pp. 445 - 472
Copyright
© 2004 Cambridge University Press

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References

REFERENCES

Asmussen, S. (1987). Applied probability and queues. Chichester: Wiley.
Blake, S., Black, D., Carlson, M., Davies, E., Wang, Z., & Weiss, W. (1998). An architecture for differentiated services. Internet Engineering Task Force, Request for Comments #2475, Dec. 1998 (ftp://ftp.isi.edu/in-notes/rfc2475.txt).
Borkar, V.S. & Manjunath, D. (2002). Charge based control of DiffServ queues, submitted.
Boxma, O., Koole, G., & Liu, Z. (1996). Queueing theoretic solution methods for models of parallel and distributed systems. In O. Boxma & G. Koole (eds.), Performance evaluation of parallel and distributed systems—Solution methods, CWI Tract No. 105. Amsterdam, pp. 1–24.
Bremaud, P. (1998). Markov chains, Gibbs fields, Monte Carlo simulation, and queues. New York: Springer-Verlag.
Dube, P., Borkar, V.S., & Manjunath, D. (2002). Differential join prices for parallel queues: Social optimality, dynamic pricing algorithms and application to Internet pricing. In Proceedings of IEEE INFOCOM 2002.
Falkner, M., Devetsikiotis, M., & Lambadaris, I. (2000). An overview of pricing concepts for broadband IP networks. IEEE Communications Surveys 3: 2–13.Google Scholar
Foley, R.D. & McDonald, D.R. (2001). Join the shortest queue: Stability and exact asymptotics. Annals of Applied Probability 11(3): 569–607.Google Scholar
Gibbens, R., Mason, R., & Steinberg, R. (2000). Internet service classes under competition. IEEE Journal on Selected Areas in Communications 18(12): 2490–2498.Google Scholar
Jain, R., Mullen, R., & Hausman, R. (2001). Analysis of Paris Metro pricing strategy for QoS with a single service provider. In Proceedings of the Ninth International Workshop on Quality of Service (IWQoS 2001).
Kingman, J.F.C. (1962). Two queues in parallel. Annals of Mathematical Statistics 32: 1314–1323.Google Scholar
Manjunath, D., Goel, A., & Hemachandra, N. (2002). DiffServ node with join minimum cost queue policy: Analysis with multiclass traffic. In Proceedings of IEEE Globecom 2002 3: 2573–2577.Google Scholar
Mertens, J.F., Samuel-Cahn, E., & Zamir, S. (1978). Necessary and sufficient conditions for recurrence and transience of Markov chains, in terms of inequalities. Journal of Applied Probability 15: 848–851.Google Scholar
Neuts, M.F. (1981). Matrix geometric solutions in stochastic models. Baltimore: Johns Hopkins University Press.
Norris, J.R. (1999). Markov chains. Cambridge: Cambridge University Press.
Odlyzko, A. (1999). Paris Metro pricing for the Internet. In Proceedings of the ACM Conference on Electronic Commerce, pp. 140–147.
Parekh, A.K. & Gallager, R.G. (1993). A generalized processor sharing approach to flow control in integrated services networks: The single node case. IEEE/ACM Transactions on Networking 1(3): 344–357.Google Scholar
Serfozo, R. (1999). Introduction to stochastic networks. New York: Springer-Verlag.
Tandra, R., Hemachandra, N., & Manjunath, D. (2004). DiffServ node with join minimum cost queue policy and multiclass traffic. Performance Evaluation 55: 69–91.Google Scholar
van Houtum, G.J., Zijm, W.H.M., Adan, I.J.B.F., & Wessels, J. (1998). Bounds for performance characteristics: A systematic approach via cost structures. Stochastic models 14: 205–224.Google Scholar
van Houtum, G.J., Adan, I.J.B.F., Wessels, J., & Zijm, W.H.M. (2000). Performance analysis of parallel identical machines with a generalized shortest queue arrival mechanism, OR Spektrum 23: 411–428.Google Scholar
Walrand, J. (1988). Introduction to queueing networks. Englewood Cliffs, NJ: Prentice-Hall.
Wolff, R. (1989). Stochastic modeling and the theory of queues. Englewood Cliffs, NJ: Prentice-Hall.