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On the Value Function of the M/G/1 FCFS and LCFS Queues

Part of: Special processes

Published online by Cambridge University Press:  30 January 2018

Esa Hyytiä ,
Samuli Aalto ,
Aleksi Penttinen  and
Jorma Virtamo
Esa Hyytiä*
Affiliation:
Aalto University
Samuli Aalto*
Affiliation:
Aalto University
Aleksi Penttinen*
Affiliation:
Aalto University
Jorma Virtamo*
Affiliation:
Aalto University
*
∗ Postal address: Department of Communications and Networking, Aalto University, School of Electrical Engineering, PO Box 13000, 00076 Aalto, Finland.
∗ Postal address: Department of Communications and Networking, Aalto University, School of Electrical Engineering, PO Box 13000, 00076 Aalto, Finland.
∗ Postal address: Department of Communications and Networking, Aalto University, School of Electrical Engineering, PO Box 13000, 00076 Aalto, Finland.
∗ Postal address: Department of Communications and Networking, Aalto University, School of Electrical Engineering, PO Box 13000, 00076 Aalto, Finland.
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Abstract

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We consider a single-server queue with Poisson input operating under first-come–first-served (FCFS) or last-come–first-served (LCFS) disciplines. The service times of the customers are independent and obey a general distribution. The system is subject to costs for holding a customer per unit of time, which can be customer specific or customer class specific. We give general expressions for the corresponding value functions, which have elementary compact forms, similar to the Pollaczek–Khinchine mean value formulae. The results generalize earlier work where similar expressions have been obtained for specific service time distributions. The obtained value functions can be readily applied to develop nearly optimal dispatching policies for a broad range of systems with parallel queues, including multiclass scenarios and the cases where service time estimates are available.

Keywords

M/G/1FCFSLCFSvalue functionsojourn timemean delay

MSC classification

Primary: 60K25: Queueing theory
Secondary: 90C40: Markov and semi-Markov decision processes
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 4 , December 2012 , pp. 1052 - 1071
Copyright
© Applied Probability Trust 

References

Aalto, S. and Virtamo, J. (1996). Basic packet routing problem. In The Thirteenth Nordic Teletraffic Seminar (Trondheim, Norway, August 1996), pp. 85–97.Google Scholar
Adan, I. and Haviv, M. (2008). Conditional ages and residual service times in the {M/G/1} queue. Tech. Rep. 2008–023, EURANDOM.Google Scholar
Akgun, O., Righter, R. and Wolff, R. (2011). The power of partial power of two choices. In ACM SIGMETRICS, ACM, New York, pp. 46–48.Google Scholar
Ansell, P. S., Glazebrook, K. D. and Kirkbride, C. (2003). Generalised ‘join the shortest queue’ policies for the dynamic routing of Jobs to multiclass queues. J. Operat. Res. Soc. 54, 379–389.Google Scholar
Becker, K. J. et al. (2000). Allocation of tasks to specialized processors: a planning approach. Europ. J. Operat. Res. 126, 80–88.Google Scholar
Bhulai, S. (2006). On the value function of the {M/Cox(r)/1} queue. J. Appl. Prob. 43, 363–376.Google Scholar
Bonomi, F. (1990). On Job assignment for a parallel system of processor sharing queues. IEEE Trans. Comput. 39, 858–869.Google Scholar
Conolly, B. W. (1984). The autostrada queueing problem. J. Appl. Prob. 21, 394–403.CrossRefGoogle Scholar
Crovella, M. E., Harchol-Balter, M. and Murta, C. D. (1998). Task assignment in a distributed system: improving performance by unbalancing load. In Proc. SIGMETRICS '98, ACM, New York, pp. 268–269.Google Scholar
Ephremides, A., Varaiya, P. and Walrand, J. (1980). A simple dynamic routing problem. IEEE Trans. Automatic Control 25, 690–693.CrossRefGoogle Scholar
Fakinos, D. (1982). The expected remaining service time in a single server queue. Operat. Res. 30, 1014–1018.Google Scholar
Feng, H., Misra, V. and Rubenstein, D. (2005). Optimal state-free, size-aware dispatching for heterogeneous {M/G/}-type systems. Performance Evaluation 62, 475–492.CrossRefGoogle Scholar
Gupta, V., Harchol-Balter, M., Sigman, K. and Whitt, W. (2007). Analysis of Join-the-shortest-queue routing for web server farms. Performance Evaluation 64, 1062–1081.CrossRefGoogle Scholar
Harchol-Balter, M., Crovella, M. E. and Murta, C. D. (1999). On choosing a task assignment policy for a distributed server system. J. Parallel Distributed Comput. 59, 204–228.Google Scholar
Harchol-Balter, M., Scheller-Wolf, A. and Young, A. R. (2009). Surprising results on task assignment in server farms with high-variability workloads. In Proc. of SIGMETRICS '09, ACM, New York, pp. 287–298.Google Scholar
Harchol-Balter, M., Sigman, K. and Wierman, A. (2002). Asymptotic convergence of scheduling policies with respect to slowdown. Performance Evaluation 49, 241–256.Google Scholar
Hyytiä, E., Penttinen, A. and Aalto, S. (2012). Size- and state-aware dispatching problem with queue-specific Job sizes. Europ. J. Operat. Res. 217, 357–370.Google Scholar
Hyytiä, E., Penttinen, A., Aalto, S. and Virtamo, J. (2011). Dispatching problem with fixed size Jobs and processor sharing discipline. In Proc. 23rd Internat. Teletraffic Congress, pp. 190–197.Google Scholar
Hyytiä, E., Virtamo, J., Aalto, S. and Penttinen, A. (2011). {M/M/1-PS} queue and size-aware task assignment. Performance Evaluation 68, 1136–1148.Google Scholar
Kelly, F. P. (1979). Reversibility and Stochastic Networks. John Wiley, Chichester.Google Scholar
Kim, J. H., Ahn, H.-S. and Righter, R. (2011). Managing queues with heterogeneous servers. J. Appl. Prob. 48, 435–452.Google Scholar
Kleinrock, L. (1975). Queueing Systems, Vol. I. Wiley-Interscience, New York.Google Scholar
Krishnan, K. R. (1987). Joining the right queue: a Markov decision rule. In Proc. 28th Conf. Decision and Control, pp. 1863–1868.Google Scholar
Krishnan, K. R. and Ott, T. J. (1986). State-dependent routing for telephone traffic: theory and results. In Proc. 25th Conf. Decision Control, pp. 2124–2128.Google Scholar
Liu, Z. and Righter, R. (1998). Optimal load balancing on distributed homogeneous unreliable processors. Operat. Res. 46, 563–573.Google Scholar
Liu, Z. and Towsley, D. (1994). Optimality of the round-robin routing policy. J. Appl. Prob. 31, 466–475.Google Scholar
Mandelbaum, A. and Yechiali, U. (1979). The conditional residual service time in the {M/G/1} queue. Unpublished manuscript. Available at http://www.math.tau.ac.il/∼uriy/Papers/conditional.pdf.Google Scholar
Mitzenmacher, M. (2001). The power of two choices in randomized load balancing. IEEE Trans. Parallel Distributed Systems 12, 1094–1104.Google Scholar
Norman, J. M. (1972). Heuristic Procedures in Dynamic Programming. Manchester University Press.Google Scholar
Puterman, M. L. (2005). Markov Decision Processes: Discrete Stochastic Dynamic Programming. John Wiley, New York.Google Scholar
Ross, S. M. (2000). Introduction to Probability Models, 7th edn. Academic Press, Burlington, MA.Google Scholar
Sassen, S. A. E., Tijms, H. C. and Nobel, R. D. (1997). A heuristic rule for routing customers to parallel servers. Statist. Neerlandica 51, 107–121.Google Scholar
Schwartz, B. L. (1974). Queuing models with lane selection: a new class of problems. Operat. Res. 22, 331–339.Google Scholar
Van Leeuwaarden, J., Aalto, S. and Virtamo, J. (2001). Load balancing in cellular networks using first policy iteration. Tech. Rep., Networking Laboratory, Helsinki University of Technology.Google Scholar
Weber, R. R. (1978). On the optimal assignment of customers to parallel servers. J. Appl. Prob. 15, 406–413.Google Scholar
Whitt, W. (1986). Deciding which queue to Join: some counterexamples. Operat. Res. 34, 55–62.Google Scholar
Winston, W. (1977). Optimality of the shortest line discipline. J. Appl. Prob. 14, 181–189.Google Scholar