# Exact analysis of the M/M/k/setup class of Markov chains via recursive renewal reward

- 394 Downloads
- 7 Citations

## Abstract

The M/M/k/setup model, where there is a penalty for turning servers on, is common in data centers, call centers, and manufacturing systems. Setup costs take the form of a time delay, and sometimes there is additionally a power penalty, as in the case of data centers. While the M/M/1/setup was exactly analyzed in 1964, no exact analysis exists to date for the M/M/k/setup with \(k>1\). In this paper, we provide the first exact, closed-form analysis for the M/M/k/setup and some of its important variants including systems in which idle servers delay for a period of time before turning off or can be put to sleep. Our analysis is made possible by a new way of combining renewal reward theory and recursive techniques to solve Markov chains with a repeating structure. Our renewal-based approach uses ideas from renewal reward theory and busy period analysis to obtain closed-form expressions for metrics of interest such as the transform of time in system and the transform of power consumed by the system. The simplicity, intuitiveness, and versatility of our renewal-based approach makes it useful for analyzing Markov chains far beyond the M/M/k/setup. In general, our renewal-based approach should be used to reduce the analysis of any 2-dimensional Markov chain which is infinite in at most one dimension and repeating to the problem of solving a system of polynomial equations. In the case where all transitions in the repeating portion of the Markov chain are skip-free and all up/down arrows are unidirectional, the resulting system of equations will yield a closed-form solution.

## Keywords

Queueing theory Performance Resource allocation Renewal reward Repeating chains## Mathematics Subject Classification

60K25 68M20 60J25## References

- 1.Adan, I., Resing, J.: A class of Markov processes on a semi-infinite strip. Technical Report 99–03, Eindhoven University of Technology, Department of Mathematics and Computing Sciences, (1999)Google Scholar
- 2.Adan, I., van der Wal, J.: Combining make to order and make to stock. OR Spektrum
**20**, 73–81 (1998)CrossRefGoogle Scholar - 3.Artalejo, J.R., Economou, A., Lopez-Herrero, M.J.: Analysis of a multiserver queue with setup times. Queueing Syst. Theory Appl.
**51**(1–2), 53–76 (2005)CrossRefGoogle Scholar - 4.Barroso, L.A., Hölzle, U.: The case for energy-proportional computing. IEEE Comput.
**40**(12), 33–37 (2007)CrossRefGoogle Scholar - 5.Castellanos, M., Casati, F., Shan, M.-C., Dayal, U.: iBOM: A platform for intelligent business operation management. Proceedings of the 21st International Conference on Data Engineering. ICDE ’05, pp. 1084–1095. Tokyo, Japan (2005)Google Scholar
- 6.DeCandia, G., Hastorun, D., Jampani, M., Kakulapati, G., Lakshman, A., Pilchin, A., Sivasubramanian, S., Vosshall, P., Vogels, W.: Dynamo: Amazon’s highly available key-value store. Proceedings of twenty-first ACM SIGOPS Symposium on Operating Systems Principles. SOSP ’07, pp. 205–220. Stevenson, WA (2007)Google Scholar
- 7.Gandhi, A., Doroudi, S., Harchol-Balter, M., Scheller-Wolf, A.: Exact analysis of the M/M/k/setup class of Markov chains via recursive renewal reward. Technical Report CMU-CS-13-105, Carnegie Mellon University, (2013)Google Scholar
- 8.Gandhi, A., Gupta, V., Harchol-Balter, M., Kozuch, M.: Optimality analysis of energy-performance trade-off for server farm management. Perform. Eval.
**67**, 1155–1171 (2010)CrossRefGoogle Scholar - 9.Gandhi, A., Harchol-Balter, M.: How data center size impacts the effectiveness of dynamic power management. 49th Annual Allerton Conference on Communication, Control, and Computing, (2011)Google Scholar
- 10.Gandhi, A., Harchol-Balter, M., Adan, I.: Server farms with setup costs. Perform. Eval.
**67**, 1123–1138 (2010)CrossRefGoogle Scholar - 11.Gandhi, A., Harchol-Balter, M., Kozuch, M.: Are sleep states effective in data centers? 3rd IEEE International Green Computing Conference, (2012)Google Scholar
- 12.Horvath, T., Skadron, K.: Multi-mode energy management for multi-tier server clusters. Proceedings of the 17th International Conference on Parallel Architectures and Compilation Techniques. PACT ’08, pp. 270–279. Canada, Toronto (2008)Google Scholar
- 13.Keilson, J., Servi, L.: A distributional form of little’s law. Oper. Res. Lett.
**7**(5), 223–227 (1988)CrossRefGoogle Scholar - 14.Kim, J., Rosing, T.S.: Power-aware resource management techniques for low-power embedded systems. In: Son, S.H., Lee, I., Leung, J.Y.-T. (eds.) Handbook of Real-Time and Embedded Systems. Taylor-Francis Group LLC, Boca Raton (2006)Google Scholar
- 15.Kleinrock, L.: Queueing Systems, Volume I: Theory. Wiley, New York (1975)Google Scholar
- 16.Krioukov, A., Mohan, P., Alspaugh, S., Keys, L., Culler, D., Katz, R.: NapSAC: design and implementation of a power-proportional web cluster. Proceedings of the First ACM SIGCOMM Workshop on Green Networking. Green Networking ’10, pp. 15–22. New Delhi, India (2010)Google Scholar
- 17.Latouche, G., Ramaswami, V.: Introduction to Matrix Analytic Methods in Stochastic Modeling. ASA-SIAM, Philadelphia (1999)CrossRefGoogle Scholar
- 18.Levy, Y., Yechiali, U.: An M/M/s queue with servers’ vacations. INFOR
**14**, 153–163 (1976)Google Scholar - 19.Meisner, D., Gold, B.T., Wenisch, T.F.: PowerNap: eliminating server idle power. In Proceeding of the 14th international conference on Architectural Support for Programming Languages and Operating Systems, ASPLOS ’09, pages 205–216, Washington, DC, (2009)Google Scholar
- 20.Mitrani, I.: Managing performance and power consumption in a server farm. Ann. Oper. Res.
**202**(1), 121–134 (2013)Google Scholar - 21.Qin, W., Wang, Q.: Modeling and control design for performance management of web servers via an IPV approach. IEEE Trans. Control Syst. Technol.
**15**(2), 259–275 (2007)CrossRefGoogle Scholar - 22.Riska, A., Smirni, E.: M/G/1-type Markov Processes: A Tutorial. Performance Evaluation of Complex Systems: Techniques and Tools, pp. 36–63. Springer, New York (2002)CrossRefGoogle Scholar
- 23.Tian, N., Li, Q.-L., Gao, J.: Conditional stochastic decompositions in the M/M/c queue with server vacations. Stoch. Models
**15**(2), 367–377 (1999)CrossRefGoogle Scholar - 24.Van Houdt, B., van Leeuwaarden, J.: Triangular M/G/1-type and tree-like quasi-birth-death Markov chains. INFORMS J. Comput.
**23**(1), 165–171 (2011)CrossRefGoogle Scholar - 25.Van Leeuwaarden, J., Winands, E.: Quasi-birth-and-death processes with an explicit rate matrix. Stoch. Models
**22**(1), 77–98 (2006)CrossRefGoogle Scholar - 26.Welch, P.: On a generalized \(M/G/1\) queueing process in which the first customer of each busy period receives exceptional service. Oper. Res.
**12**, 736–752 (1964)CrossRefGoogle Scholar - 27.Xu, X., Tian, N.: The M/M/c queue with (e, d) setup time. J. Syst. Sci. Complex.
**21**, 446–455 (2008)CrossRefGoogle Scholar - 28.Zhang, Z.G., Tian, N.: Analysis on queueing systems with synchronous vacations of partial servers. Perform. Eval.
**52**(4), 269–282 (2003)CrossRefGoogle Scholar