Journal of Applied Probability

On the value function of the M/G/1 FCFS and LCFS queues

Esa Hyytiä, Samuli Aalto, Aleksi Penttinen, and Jorma Virtamo

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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.

Article information

J. Appl. Probab., Volume 49, Number 4 (2012), 1052-1071.

First available in Project Euclid: 5 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K25: Queueing theory [See also 68M20, 90B22]
Secondary: 90C40: Markov and semi-Markov decision processes

M/G/1 FCFS LCFS value function sojourn time mean delay


Hyytiä, Esa; Aalto, Samuli; Penttinen, Aleksi; Virtamo, Jorma. On the value function of the M/G/1 FCFS and LCFS queues. J. Appl. Probab. 49 (2012), no. 4, 1052--1071. doi:10.1239/jap/1354716657.

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  • Aalto, S. and Virtamo, J. (1996). Basic packet routing problem. In The Thirteenth Nordic Teletraffic Seminar (Trondheim, Norway, August 1996), pp. 85–97.
  • Adan, I. and Haviv, M. (2008). Conditional ages and residual service times in the M/G/1 queue. Tech. Rep. 2008–023, EURANDOM.
  • 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.
  • 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.
  • Becker, K. J. et al. (2000). Allocation of tasks to specialized processors: a planning approach. Europ. J. Operat. Res. 126, 80–88.
  • Bhulai, S. (2006). On the value function of the M/Cox(r)/1 queue. J. Appl. Prob. 43, 363–376.
  • Bonomi, F. (1990). On job assignment for a parallel system of processor sharing queues. IEEE Trans. Comput. 39, 858–869.
  • Conolly, B. W. (1984). The autostrada queueing problem. J. Appl. Prob. 21, 394–403.
  • 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.
  • Ephremides, A., Varaiya, P. and Walrand, J. (1980). A simple dynamic routing problem. IEEE Trans. Automatic Control 25, 690–693.
  • Fakinos, D. (1982). The expected remaining service time in a single server queue. Operat. Res. 30, 1014–1018.
  • 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.
  • 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.
  • 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.
  • 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.
  • Harchol-Balter, M., Sigman, K. and Wierman, A. (2002). Asymptotic convergence of scheduling policies with respect to slowdown. Performance Evaluation 49, 241–256.
  • 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.
  • 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.
  • 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.
  • Kelly, F. P. (1979). Reversibility and Stochastic Networks. John Wiley, Chichester.
  • Kim, J. H., Ahn, H.-S. and Righter, R. (2011). Managing queues with heterogeneous servers. J. Appl. Prob. 48, 435–452.
  • Kleinrock, L. (1975). Queueing Systems, Vol. I. Wiley-Interscience, New York.
  • Krishnan, K. R. (1987). Joining the right queue: a Markov decision rule. In Proc. 28th Conf. Decision and Control, pp. 1863–1868.
  • 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.
  • Liu, Z. and Righter, R. (1998). Optimal load balancing on distributed homogeneous unreliable processors. Operat. Res. 46, 563–573.
  • Liu, Z. and Towsley, D. (1994). Optimality of the round-robin routing policy. J. Appl. Prob. 31, 466–475.
  • Mandelbaum, A. and Yechiali, U. (1979). The conditional residual service time in the M/G/1 queue. Unpublished manuscript. Available at$\sim$uriy/Papers/conditional.pdf.
  • Mitzenmacher, M. (2001). The power of two choices in randomized load balancing. IEEE Trans. Parallel Distributed Systems 12, 1094–1104.
  • Norman, J. M. (1972). Heuristic Procedures in Dynamic Programming. Manchester University Press.
  • Puterman, M. L. (2005). Markov Decision Processes: Discrete Stochastic Dynamic Programming. John Wiley, New York.
  • Ross, S. M. (2000). Introduction to Probability Models, 7th edn. Academic Press, Burlington, MA.
  • 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.
  • Schwartz, B. L. (1974). Queuing models with lane selection: a new class of problems. Operat. Res. 22, 331–339.
  • 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.
  • Weber, R. R. (1978). On the optimal assignment of customers to parallel servers. J. Appl. Prob. 15, 406–413.
  • Whitt, W. (1986). Deciding which queue to join: some counterexamples. Operat. Res. 34, 55–62.
  • Winston, W. (1977). Optimality of the shortest line discipline. J. Appl. Prob. 14, 181–189.