Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-12T04:54:30.183Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic majorization of random variables by proportional equilibrium rates

Published online by Cambridge University Press:  01 July 2016

J. George Shanthikumar
J. George Shanthikumar*
Affiliation:
University of California, Berkeley
*
∗Postal address: School of Business Administration, University of California, Berkeley, CA 94720, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

The equilibrium rate rY of a random variable Y with support on non-negative integers is defined by rY(0) = 0 and rY(n) = P[Y = n – 1]/P[Y – n], Let (j = 1, …, m; i = 1,2) be 2m independent random variables that have proportional equilibrium rates with (j = 1, …, m; i = 1, 2) as the constant of proportionality. When the equilibrium rate is increasing and concave [convex] it is shown that , …, ) majorizes implies , …, for all increasing Schur-convex [concave] functions whenever the expectations exist. In addition if , (i = 1, 2), then

Keywords

CONVEXITYBIRTH-DEATH QUEUEQUEUEING NETWORKS
Type
Research Article
Information
Advances in Applied Probability , Volume 19 , Issue 4 , December 1987 , pp. 854 - 872
Copyright
Copyright © Applied Probability Trust 1987 

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

Birnbaum, Z. W. (1948) On random variables with comparable peakedness. Ann. Math. Statist. 19, 76–81.Google Scholar
Buzacott, J. A. and Shanthikumar, J. G. (1980) Models for understanding flexible manufacturing systems. AIIE Trans. 12, 339–350.Google Scholar
Buzacott, J. A. and Shanthikumar, J. G. (1985) On approximate queueing models of dynamic job shops. Management Sci. 31, 870–887.CrossRefGoogle Scholar
Efron, B. (1965) Increasing properties of Pólya frequency functions. Ann. Math. Statist. 36, 272–279.Google Scholar
Grassmann, W. (1983) The convexity of the mean queue size of the M/M/c queue with respect to the traffic intensity. J. Appl. Prob. 20, 916–919.CrossRefGoogle Scholar
Gordon, W.J. and Newell, G.F. (1967) Closed queueing networks with exponential servers. Operat. Res. 15, 252–267.Google Scholar
Hollander, M., Proschan, F. and Sethuraman, J. (1981) Decreasing in transposition property of overlapping sums, and applications. J. Mult. Anal. 11, 50–57.Google Scholar
Jackson, J. R. (1963) Jobshop-like queueing systems. Management Sci. 10, 131–142.CrossRefGoogle Scholar
Kamae, T., Krengel, U. and O'Brien, G.L. (1977) Stochastic inequalities on partially ordered spaces. Ann. Prob. 5, 899–912.Google Scholar
Karlin, S. and Proschan, F. (1960) Pólya-type distributions of convolutions. Ann. Math. Statist. 31, 721–736.CrossRefGoogle Scholar
Keilson, J. and Sumita, U. (1982) Uniform stochastic ordering and related inequalities. Canad. J. Statist. 10, 181–188.Google Scholar
Kelly, F. (1979) Reversibility and Stochastic Networks. Wiley, New York.Google Scholar
Lee, H. L. and Cohen, M. A. (1983) A note on the convexity of performance measures of M/M/c queueing systems. J. Appl. Prob. 20, 920–923.CrossRefGoogle Scholar
Marshall, A. W. and Olkin, I. (1979) Inequalities: Theory of Majorizations and its Applications. Academic Press, New York.Google Scholar
Proschan, F. and Sethuraman, J. (1976) Stochastic comparisons of order statistics from heterogeneous populations, with applications in reliability. J. Mult. Anal. 6, 608–616.CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J. G. (1988) Stochastic convexity and its applications. Adv. Appl. Prob. 20.Google Scholar
Shanthikumar, J. G. (1982) On the superiority of balanced load in a flexible manufacturing system. Technical Report, Department of IE & OR, Syracuse University.Google Scholar
Shanthikumar, J. G. (1987) On stochastic comparison of random vectors. J. Appl. Prob. 24, 123–136.Google Scholar
Shanthikumar, J. G. and Stecke, K. E. (1986) Reducing work-in-process inventory in certain classes of flexible manufacturing systems. Eur. J. Operat. Res. 26, 266–271.CrossRefGoogle Scholar
Shanthikumar, J. G. and Yao, D. D. (1985) Second-order properties of the throughput of a closed queueing network. Math. Operat. Res. To appear.Google Scholar
Shanthikumar, J. G. and Yao, D. D. (1986a) The preservation of likelihood ratio order under convolution. Stoch. Proc. Appl. 23, 259–267.CrossRefGoogle Scholar
Shanthikumar, J. G. and Yao, D. D. (1986b) The effect of increasing service rates in a closed queueing network. J. Appl. Prob. 23, 474–483.Google Scholar
Stecke, K. E. and Morin, T. (1985) Optimality of balanced workloads in flexible manufacturing systems. Eur. J. Operat. Res. 20, 68–82.Google Scholar
Veinott, A. F. (1965) Optimal policy in a dynamic, single product, non-stationary inventory model with several demand classes. Operat. Res. 13, 761–778.Google Scholar
Yao, D. D. (1985a) Some properties of the throughput function of closed networks of queues. Operat. Res. Letters 3, 313–317.Google Scholar
Yao, D. D. (1985b) Majorization and arrangement orderings in open queueing networks. Ann. Operat. Res. To appear.Google Scholar
Yao, D. D. and Kim, S. C. (1985) Reducing the congestion in a class of job shops.Google Scholar
Yao, D. D. and Kim, S. C. (1987) Some order relations in closed networks of queues with multi-server stations. Naval Res. Logist. Quart. 34, 53–66.Google Scholar
Yao, D. D. and Shanthikumar, J. G. (1987) The optimal input rates to a system of manufacturing cells. INFOR, Canad. J. Operat. Res. Inf. Proc. 25, 57–65.Google Scholar