Annals of Operations Research

, Volume 9, Issue 1, pp 531–543 | Cite as

Majorization and arrangement orderings in open queueing networks

  • D. D. Yao
Real World Issues in Probability Modeling


Consider an open Jackson network of queues. Majorization and arrangement orderings are studied to order, respectively, various loading and server-assignment policies. It is shown that under these order relations, stochastic and likelihood ratio orderings can be established for the maximum and the minimum queue lengths and for the total number of jobs in the network. Stochastic majorization and stochastic orderings are also established, respectively, for the queue-length vector and the associated order-statistic vector. Implications of the results on loading and assignment decisions are discussed.

Keywords and phrases

Jackson network majorization arrangement ordering stochastic ordering likelihood ratio ordering 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    W.J. Gordon and G.F. Newell, Closed queueing networks with exponential servers, Oper. Res. 15(1967)252.Google Scholar
  2. [2]
    J.R. Jackson, Networks of waiting lines, Oper. Res. 5(1957)518.Google Scholar
  3. [3]
    F. Kelly,Reversibility and Stochastic Networks (Wiley, New York, 1979).Google Scholar
  4. [4]
    A.W. Marshall and I. Olkin,Inequalities: Theory of Majorization and Its Application (Academic Press, New York, 1979).Google Scholar
  5. [5]
    T.G. Price, Probability models of multi-programmed computer systems. Ph.D. dissertation, Department of Electrical Engineering, Stanford University, Stanford, California (1974).Google Scholar
  6. [6]
    F. Proschan and J. Sethuraman, Stochastic comparisons of order statistics from heterogeneous populations, with applications in reliability, J. Multivariate Analysis 6(1976)608.Google Scholar
  7. [7]
    S. Ross,Stochastic Processes (Wiley, New York, 1983).Google Scholar
  8. [8]
    J.G. Shamthikumar, On the superiority of balanced load in a flexible manufacturing system, Department of Industrial Engineering and Operations Research, Syracuse University, Syracuse, New York (1982).Google Scholar
  9. [9]
    J.G. Shanthikumar and K.E. Stecke, Reducing work-in-process inventory in certain classes of flexible manufacturing systems, Eur. J. Oper. Res. 26(1986)266.Google Scholar
  10. [10]
    D.D. Yao, Some properties of the throughput function of closed networks of queues, Oper. Res. Lett. 3, 6(1985)313.Google Scholar

Copyright information

© J.C. Baltzer A.G., Scientific Publishing Company 1987

Authors and Affiliations

  • D. D. Yao
    • 1
  1. 1.Department of Industrial Engineering and Operations ResearchColumbia UniversityNew YorkUSA

Personalised recommendations