Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-24T04:29:22.003Z Has data issue: false hasContentIssue false
  • English
  • Français

A new ordering for stochastic majorization: theory and applications

Part of: Special processes Distribution theory - Probability

Published online by Cambridge University Press:  01 July 2016

Cheng-Shang Chang
Cheng-Shang Chang*
Affiliation:
IBM Thomas J. Watson Research Center
*
∗Postal address: IBM Research Division, T. J. Watson Research Center, P.O. Box 704, Yorktown Heights, NY 10598, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we develop a unified approach for stochastic load balancing on various multiserver systems. We expand the four partial orderings defined in Marshall and Olkin, by defining a new ordering based on the set of functions that are symmetric, L-subadditive and convex in each variable. This new partial ordering is shown to be equivalent to the previous four orderings for comparing deterministic vectors but differs for random vectors. Sample-path criteria and a probability enumeration method for the new stochastic ordering are established and the ordering is applied to various fork-join queues, routing and scheduling problems. Our results generalize previous work and can be extended to multivariate stochastic majorization which includes tandem queues and queues with finite buffers.

Keywords

LOAD BALANCINGSTOCHASTIC CONVEXITYMAJORIZATION ORDERINGROUTINGSCHEDULINGFORK-JOIN QUEUES

MSC classification

Primary: 60E05: Distributions
Secondary: 60K25: Queueing theory
Type
Research Article
Information
Advances in Applied Probability , Volume 24 , Issue 3 , September 1992 , pp. 604 - 634
Copyright
Copyright © Applied Probability Trust 1992 

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] Baccelli, F. and Makowski, A. M. (1989) Queueing models for systems with synchronization constraints. Proc. IEEE 77, 138–161.Google Scholar
[2] Baccelli, F., Makowski, A. M. and Shwartz, A. (1989) The fork-join queue and related systems with synchronization constraints: stochastic ordering and computable bounds. Adv. Appl. Prob. 21, 629–660.Google Scholar
[3] Barlow, R. E. and Proschan, F. (1975) Stochastic Theory of Reliability and Life Testing. Holt, Rinehart and Winston, Reading, MA.Google Scholar
[4] Bruno, J., Downey, P. and Frederickson, G. N. (1981) Sequencing tasks with exponential service times to minimize the expected flow time or makespan. J. Assoc. Comput. Mach 28, 100–113.CrossRefGoogle Scholar
[5] Chang, C. S. and Pinedo, M. (1990) Bounds and inequalities for single server loss systems. QUESTA 6, 425–436.Google Scholar
[6] Chang, C. S., Chao, X. L. and Pinedo, M. (1990) Integration of discrete-time correlated Markov processes in a TDM system: structural results. Prob. Eng. Inf. Sci. 4, 29–56.Google Scholar
[7] Chang, C. S., Chao, X. L., Pinedo, M. and Weber, R. R. (1992) On the optimality of the LEPT rule and cµ rules for machines in parallel. J. Appl. Prob. 29, 667–681.Google Scholar
[8] Chang, C. S., Chao, X. L. and Pinedo, M. (1991) Monotonicity results for queues with doubly stochastic Poisson arrivals: Ross's conjecture. Adv. Appl. Prob. 23, 210–228.Google Scholar
[9] Chang, C. S., Chao, X. L., Pinedo, M. and Shanthikumar, J. G. (1991) Stochastic convexity for multidimensional processes and its applications. IEEE Trans. Autom. Control 36, 1341–1355.Google Scholar
[10] Chow, Y.-S. and Teicher, H. (1988) Probability Theory, Independence, Interchangeability, Martingales. Springer-Verlag, New York.Google Scholar
[11] Ephremides, A., Varaiya, P. and Walrand, J. (1980) A simple dynamic routing problem. IEEE Trans. Autom. Control. 25, 690–693.Google Scholar
[12] Esary, J. D., Proschan, F. and Walkup, D. W. (1967) Association of random variables with applications. Ann. Math. Statist 38, 1466–1474.Google Scholar
[13] Gun, L. (1989) Performance Evaluation and Optimization of Parallel Systems with Synchronization. Ph.D dissertation, University of Maryland, College Park.Google Scholar
[14] Gun, L., Jean-Marie, A., Makowski, A. M. and Tedijanto, (1990) Convexity results for queues with Bernoulli mechanisms. Preprint.Google Scholar
[15] Hardy, G. H., Littlewood, J. E. and Pólya, G. (1952) Inequalities. Cambridge University Press.Google Scholar
[16] Kampke, T. (1987) On the optimality of static priority policies in stochastic scheduling on parallel machines. J. Appl. Prob. 24, 430–448.Google Scholar
[17] Kampke, T. (1989) Optimal scheduling of jobs with exponential service times on identical parallel processors. Operat. Res. 37, 126–133.Google Scholar
[18] Keilson, J. (1979) Markov Chain Models—Rarity and Exponentiality. Springer-Verlag, New York.Google Scholar
[19] Marshall, A. W. and Olkin, I. (1979) Inequalities: Theory of Majorization and Its Applications. Academic Press, New York.Google Scholar
[20] Meester, L. E. and Shanthikumar, J. G. (1990) Stochastic convexity on general space. Preprint.Google Scholar
[21] Menich, R. (1987) Optimality of shortest queue routing for dependent service stations. Proc. 26th Conf Decision and Control , pp. 1069–1072.Google Scholar
[22] Nelson, R. and Tantawi, A. N. (1988) Approximation analysis of fork/join synchronization in parallel queues. IEEE Trans. Comput. 37, 739–743.Google Scholar
[23] Niu, S. C. (1980) Bounds for the expected delays in some tandem queues. J. Appl. Prob. 17, 831–838.Google Scholar
[24] Rinott, Y. (1973) Multivariate majorization and rearrangement inequalities with some applications to probability and statistics. Israel J. Math. 15, 60–77.Google Scholar
[25] Rolski, T. (1981) Queues with non-stationary input stream: Ross's conjecture. Adv. Appl. Prob. 13, 603–618.Google Scholar
[26] Rolski, T. (1986) Upper bounds for single server queues with doubly stochastic Poisson arrivals. Math. Operat. Res. 11, 442–450.Google Scholar
[27] Ross, S. M. (1983) Stochastic Processes. Wiley, New York.Google Scholar
[28] Shaked, M. and Shanthikumar, J. G. (1988) Temporal stochastic convexity and concavity. Stoch. Proc. Appl. 27, 1–20.Google Scholar
[29] Shaked, M. and Shanthikumar, J. G. (1988) Stochastic convexity and its applications. Adv. Appl. Prob. 20, 427–446.Google Scholar
[30] Shaked, M. and Tong, Y. L. (1985) Some partial orderings of exchangeable random variables by positive dependence. J Multivariate Anal. 17, 333–349.Google Scholar
[31] Shanthikumar, J. G. (1987) Stochastic majorization of random variables with proportional equilibrium rates. Adv. Appl. Prob. 19, 854–872.Google Scholar
[32] Shanthikumar, J. G. and Yao, D. D. (1992) Spatiotemporal convexity of stochastic processes and applications. Prob. Eng. Inf. Sci. 6, 1–16.Google Scholar
[33] Shanthikumar, J. G. and Yao, D. D. (1991) Strong stochasticity convexity: closure properties and application. J. Appl. Prob. 28, 131–145.Google Scholar
[34] Stoyan, D. (1983) Comparison Methods for Queues and Other Stochastic Models. Wiley, New York.Google Scholar
[35] Tsoucas, P. (1989) Stochastic monotonicity of the output process of parallel queues. Preprint.Google Scholar
[36] Weber, R. R. (1978) On the optimal assignment of customers to parallel servers. J. Appl. Prob. 15, 406–413.Google Scholar
[37] Weber, R. R. (1979) On the marginal benefit of adding servers to G/GI/m queues. Management Sci. 25, 946–951.Google Scholar
[38] Weber, R. R. (1983) A note on waiting time in single server queues. Operat. Res. 31, 950–951.Google Scholar
[39] Weiss, G. (1982) Multiserver stochastic scheduling. In Deterministic and Stochastic Scheduling , ed. Dempster, M. A. H., Lenstra, J. K. and Kan, Rinnooy, pp. 157–179, Reidel, Dordrecht.Google Scholar
[40] Weiss, G. and Pinedo, M. (1980) Scheduling tasks with exponential service times on non-identical processors to minimize various cost functions. J. Appl. Prob. 17, 187–202.Google Scholar
[41] Winston, W. (1977) Optimality of the shortest line discipline. J. Appl. Prob. 14, 181–189.Google Scholar
[42] Whitt, W. (1986) Deciding which queue to join: some counterexamples. Operat. Res. 34, 55–62.Google Scholar
[43] Van Der Heyden, L. (1981) Scheduling jobs with exponential processing and arrival times on identical processors so as to minimize the expected makespan. Math. Operat. Res. 6, 305–312.Google Scholar