### Heavy-traffic limits for polling models with exhaustive service and non-FCFS service order policies

#### Abstract

We study cyclic polling models with exhaustive service at each queue under a variety of non-FCFS (first-come-first-served) local service orders, namely last-come-first-served with and without preemption, random-order-of-service, processor sharing, the multi-class priority scheduling with and without preemption, shortest-job-first, and the shortest remaining processing time policy. For each of these policies, we first express the waiting-time distributions in terms of intervisit-time distributions. Next, we use these expressions to derive the asymptotic waiting-time distributions under heavy-traffic assumptions, i.e. when the system tends to saturate. The results show that in all cases the asymptotic waiting-time distribution at queue i is fully characterized and of the form Γ Θi, with Γ and Θi independent, and where Γ is gamma distributed with known parameters (and the same for all scheduling policies). We derive the distribution of the random variable Θi which explicitly expresses the impact of the local service order on the asymptotic waiting-time distribution. The results provide new fundamental insight into the impact of the local scheduling policy on the performance of a general class of polling models. The asymptotic results suggest simple closed-form approximations for the complete waiting-time distributions for stable systems with arbitrary load values.

#### Article information

Source
Adv. in Appl. Probab., Volume 47, Number 4 (2015), 989-1014.

Dates
First available in Project Euclid: 11 December 2015

https://projecteuclid.org/euclid.aap/1449859797

Digital Object Identifier
doi:10.1239/aap/1449859797

Mathematical Reviews number (MathSciNet)
MR3433293

Zentralblatt MATH identifier
1333.60196

#### Citation

Vis, P.; Bekker, R.; van der Mei, R. D. Heavy-traffic limits for polling models with exhaustive service and non-FCFS service order policies. Adv. in Appl. Probab. 47 (2015), no. 4, 989--1014. doi:10.1239/aap/1449859797. https://projecteuclid.org/euclid.aap/1449859797

#### References

• Ayesta, U., Boxma, O. J. and Verloop, I. M. (2012). Sojourn times in a processor sharing queue with multiple vacations. Queueing Systems 71, 53–78.
• Baker, K. R. (1984). Sequencing rules and due-date assignments in a job shop. Manag. Sci. 30, 1093–1104.
• Bansal, N. and Gamarnik, D. (2006). Handling load with less stress. Queueing Systems 54, 45–54.
• Bekker, R. \et (2015). The impact of scheduling policies on the waiting-time distributions in polling systems. Queueing Systems 79, 145–172.
• Boon, M. A. A. (2011). Polling models: from theory to traffic intersections. Doctoral thesis, Eindhoven University of Technology.
• Boon, M. A. A., Adan, I. J. B. F. and Boxma, O. J. (2010). A polling model with multiple priority levels. Performance Evaluation 67, 468–484.
• Boon, M. A. A., Adan, I. J. B. F. and Boxma, O. J. (2010). A two-queue polling model with two priority levels in the first queue. Discrete Event Dynamic Systems 20, 511–536.
• Boon, M. A. A., van der Mei, R. D. and Winands, E. M. M. (2011). Applications of polling systems. Surveys Operat. Res. Manag. Sci. 16, 67–82.
• Borst, S. C., Boxma, O. J., Morrison, J. A. and Núñez Queija, R. (2003). The equivalence between processor sharing and service in random order. Operat. Res. Lett. 31, 254–262.
• Boxma, O., Bruin, J. and Fralix, B. (2009). Sojourn times in polling systems with various service disciplines. Performance Evaluation 66, 621–639.
• Coffman, E. G., Jr, Puhalskii, A. A. and Reiman, M. I. (1995). Polling systems with zero switchover times: a heavy-traffic averaging principle. Ann. Appl. Prob. 5, 681–719.
• Coffman, E. G., Jr, Puhalskii, A. A. and Reiman, M. I. (1998). Polling systems in heavy traffic: a Bessel process limit. Math. Operat. Res. 23, 257–304.
• Ehrlich, W. K., Hariharan, R., Reeser, P. K. and van der Mei, R. D. (2001). Performance of web servers in a distributed computing environment. In Teletraffic Engineering in the Internet Era, Elsevier, Amsterdam, pp. 137–148.
• Harchol-Balter, M. (2009). Queueing Disciplines. In Wiley Encyclopedia of Operations Research and Management Science, John Wiley, New York, 13pp.
• Kawasaki, N. \et (2000). Waiting time analysis of $M^X/G/1$ queues with/without vacations under random order of service discipline. J. Operat. Res. Soc. Japan 43, 455–468.
• Kella, O. and Yechiali, U. (1988). Priorities in M/G/1 queue with server vacations. Naval Res. Logistics 35, 23–34.
• Kingman, J. F. C. (1962). On queues in which customers are served in random order. Proc. Cambridge Phil. Soc. 58, 79–91.
• Olsen, T. L. and van der Mei, R. D. (2003). Polling systems with periodic server routeing in heavy traffic: distribution of the delay. J. Appl. Prob. 40, 305–326.
• Scholl, M. and Kleinrock, L. (1983). On the $M$/$G$/1 queue with rest periods and certain service-independent queueing disciplines. Operat. Res. 31, 705–719.
• Schrage, L. E. and Miller, L. W. (1966). The queue $M$/$G$/1 with the shortest remaining processing time discipline. Operat. Res. 14, 670–684.
• Takagi, H. (1986). Analysis of Polling Systems. MIT Press, Cambridge, MA.
• Takagi, H. and Kudoh, S. (1997). Symbolic higher-order moments of the waiting time in an $M$/$G$/1 queue with random order of service. Commun. Statist. Stoch. Models 13, 167–179.
• Tijms, H. C. (2003). A First Course in Stochastic Models. John Wiley, Chichester.
• Van der Mei, R. D. (1999). Distribution of the delay in polling systems in heavy traffic. Performance Evaluation 38, 133–148.
• Van der Mei, R. D. (2000). Polling systems with switch-over times under heavy load: moments of the delay. Queueing Systems 36, 381–404.
• Van der Mei, R. D. (2007). Towards a unifying theory on branching-type polling systems in heavy traffic. Queueing Systems 57, 29–46.
• Van der Mei, R. D., Hariharan, R. and Reeser, P. K. (2001). Web server performance modeling. Telecommun. Systems 16, 361–378.
• Van Dorp, J. R. and Kotz, S. (2003). Generalized trapezoidal distributions. Metrika 58, 85–97.
• Vis, P., Bekker, R. and van der Mei, R. D. (2014). Heavy-traffic limits for polling models with exhaustive service and non-FCFS service order policies. Tech. Rep. ST-1401, CWI.
• Vishnevskii, V. M. and Semenova, O. V. (2006). Mathematical methods to study the polling systems. Automation Remote Control 67, 173–220.
• Wierman, A., Winands, E. M. M. and Boxma, O. J. (2007). Scheduling in polling systems. Performance Evaluation 64, 1009–1028. \enlargethispage1.4pt
• Winands, E. M. M., Adan, I. J. B. F. and van Houtum, G. J. (2006). Mean value analysis for polling systems. Queueing Systems 54, 35–44.