Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-26T07:11:41.010Z Has data issue: false hasContentIssue false
  • English
  • Français

OPTIMAL CONTROL POLICIES FOR AN M/M/1 QUEUE WITH A REMOVABLE SERVER AND DYNAMIC SERVICE RATES

Published online by Cambridge University Press:  29 July 2019

Pamela Badian-Pessot ,
Mark E. Lewis Open the ORCID record for Mark E. Lewis [Opens in a new window]  and
Douglas G. Down
Pamela Badian-Pessot
Affiliation:
School of Operations Research and Information Engineering, Cornell University, Ithaca, New York, United States E-mails: plb93@cornell.edu; mark.lewis@cornell.edu
Mark E. Lewis
Affiliation:
School of Operations Research and Information Engineering, Cornell University, Ithaca, New York, United States E-mails: plb93@cornell.edu; mark.lewis@cornell.edu
Douglas G. Down
Affiliation:
Department of Computing and Software, McMaster University, Hamilton, Ontario, Canada E-mail: downd@mcmaster.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider an M/M/1 queue with a removable server that dynamically chooses its service rate from a set of finitely many rates. If the server is off, the system must warm up for a random, exponentially distributed amount of time, before it can begin processing jobs. We show under the average cost criterion, that work conserving policies are optimal. We then demonstrate the optimal policy can be characterized by a threshold for turning on the server and the optimal service rate increases monotonically with the number in system. Finally, we present some numerical experiments to provide insights into the practicality of having both a removable server and service rate control.

Keywords

applied probabilityqueueing theorystochastic dynamic programming
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 35 , Issue 2 , April 2021 , pp. 189 - 209
Copyright
Copyright © Cambridge University Press 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Baker, K. (1973). A note on operating policies for the queue M/M/1 with exponential startups. INFOR: Information Systems and Operational Research 11(1): 71–72.Google Scholar
2.Barr, J. (2015). Cloud computing, server utilization, and the environment. https://aws.amazon.com/blogs/aws/cloud-computing-server-utilization-the-environment/, June 2015. Last Accessed :11 September2018.Google Scholar
3.Barroso, L.A. & Hölzle, U. (2007). The case for energy-proportional computing. IEEE Computer 40(12): 33–37.CrossRefGoogle Scholar
4.Bell, C.E. (1971). Characterization and computation of optimal policies for operating an M/G/1 queuing system with removable server. Operations Research 19(1): 208–218.CrossRefGoogle Scholar
5.Borthakur, A., Medhi, J., & Gohain, R. (1987). Poisson input queueing system with startup time and under control-operating policy. Computers & Operations Research 14(1): 33–40.CrossRefGoogle Scholar
6.Chen, Y., Das, A., Qin, W., Sivasubramaniam, A., Wang, Q., & Gautam, N. (2005). Managing server energy and operational costs in hosting centers. ACM SIGMETRICS performance evaluation review 33(1): 303–314.CrossRefGoogle Scholar
7.Chong, K.C., Henderson, S.G., & Lewis, M.E. (2018). Two-class routing with admission control and strict priorities. Probability in the Engineering and Informational Sciences 32(2): 163–178.CrossRefGoogle Scholar
8.Crabill, T. B. (1974). Optimal control of a maintenance system with variable service rates. Operations Research 22(4): 736–745.CrossRefGoogle Scholar
9.Delforge, P. & Whitney, J. (2014). Issue paper: data center efficiency assessment scaling up energy efficiency across the data center industry: evaluating key drivers and barriers. Natural Resource Defense Council (NRDC), August 2014.Google Scholar
10.Dimitrakopoulos, Y. & Burnetas, A. (2017). The value of service rate flexibility in an M/M/1 queue with admission control. IISE Transactions 49(6): 603–621.CrossRefGoogle Scholar
11.Federgruen, A. & So, K.C. (1991). Optimality of threshold policies in single-server queueing systems with server vacations. Advances in Applied Probability 23(2): 388–405.CrossRefGoogle Scholar
12.Feinberg, E.A. & Kella, O. (2002). Optimality of D-policies for an M/G/1 queue with a removable server. Queueing Systems 42(4): 355–376.CrossRefGoogle Scholar
13.Gandhi, A., Gupta, V., Harchol-Balter, M., & Kozuch, M.A. (2010). Optimality analysis of energy-performance trade-off for server farm management. Performance Evaluation 67(11): 1155–1171.CrossRefGoogle Scholar
14.Gandhi, A., Harchol-Balter, M., & Adan, I. (2010). Server farms with setup costs. Performance Evaluation 67(11): 1123–1138.CrossRefGoogle Scholar
15.Gebrehiwot, M.E., Aalto, S.A., & Lassila, P. (2014). Optimal sleep-state control of energy-aware M/G/1 queues. Proceedings of the 8th International Conference on Performance Evaluation Methodologies and Tools, pp. 82–89.Google Scholar
16.George, J.M. & Harrison, J.M. (2001). Dynamic control of a queue with adjustable service rate. Operations Research 49(5): 720–731.CrossRefGoogle Scholar
17.Heyman, D.P. (1968). Optimal operating policies for M/G/1 queuing systems. Operations Research 16(12): 362–382.CrossRefGoogle Scholar
18.Ke, J.-C. (2003). The optimal control of an M/G/1 queueing system with server startup and two vacation types. Applied Mathematical Modelling 27(6): 437–450.CrossRefGoogle Scholar
19.Koole, G. (1998). Structural results for the control of queueing systems using event-based dynamic programming. Queueing Systems 30(3-4): 323–339.CrossRefGoogle Scholar
20.Kumar, R., Lewis, M.E., & Topaloglu, H. (2013). Dynamic service rate control for a single-server queue with markov-modulated arrivals. Naval Research Logistics (NRL) 60(8): 661–677.CrossRefGoogle Scholar
21.Lippman, S.A. (1975). Applying a new device in the optimization of exponential queuing systems. Operations Research 23(4): 687–710.CrossRefGoogle Scholar
22.Maccio, V.J. & Down, D.G. (2015). On optimal control for energy-aware queueing systems. In Teletraffic Congress (ITC 27), 27th International, pages 98–106. IEEE, 2015.CrossRefGoogle Scholar
23.Maccio, V.J. & Down, D.G. (2015). On optimal policies for energy-aware servers. Performance Evaluation 90: 36–52.CrossRefGoogle Scholar
24.Rudin, W. (1976). Principles of mathematical analysis, Vol. 3, New York: McGraw-Hill.Google Scholar
25.Sennott, L.I. (1999). Stochastic dynamic programming and the control of queueing systems. New York: John Wiley & Sons.Google Scholar
26.Shehabi, A., Smith, S.J., Sartor, D.A., Brown, R.E., Herrlin, M., Koomey, J.G., Masanet, E.R., Horner, N., Azevedo, I.L., & Lintner, W. (2016). United states data center energy usage report. Lawrence Berkeley National Laboratory, June 2016.CrossRefGoogle Scholar
27.Wang, K.-H., Wang, T.-Y., & Pearn, W.L. (2007). Optimal control of the N-policy M/G/1 queueing system with server breakdowns and general startup times. Applied Mathematical Modelling 31(10): 2199–2212.CrossRefGoogle Scholar
28.Yadin, M., & Naor, P. (1963). Queueing systems with a removable service station. Journal of the Operational Research Society 14(4): 393–405.CrossRefGoogle Scholar
29.Yoon, S., & Lewis, M.E. (2004). Optimal pricing and admission control in a queueing system with periodically varying parameters. Queueing Systems 47(3): 177–199.CrossRefGoogle Scholar