Queueing Systems

, Volume 38, Issue 2, pp 185–194 | Cite as

Conservation Laws for Single-Server Fluid Networks

  • Nicole Bäuerle
  • Shaler Stidham


We consider single-server fluid networks with feedback and arbitrary input processes. The server has to be scheduled in order to minimize a linear holding cost. This model is the fluid analogue of the so-called Klimov problem. Using the achievable-region approach, we show that the Gittins index rule is optimal in a strong sense: it minimizes the linear holding cost for arbitrary input processes and for all time points t≥0.

fluid network achievable-region approach conservation laws Gittins index linear program 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    D. Anick, D. Mitra and M.M. Sondhi, Stochastic theory of data handling system with multiple sources, Bell Syst. Technical J. 61 (1982) 1871-1984.Google Scholar
  2. [2]
    N. Bäuerle and U. Rieder, Optimal control of single-server fluid networks, Queueing Systems Theory Appl. 35 (2000) 185-200.Google Scholar
  3. [3]
    D. Bertsimas and J. Niño-Mora, Conservation laws, extended polymatroids and multiarmed bandit problems; a polyhedral approach to indexable systems, Math. Oper. Res. 21 (1996) 257-306.Google Scholar
  4. [4]
    H. Chen, Fluid approximations and stability of multiclass queueing networks: work-conserving disciplines, Ann. Appl. Probab. 5 (1995) 637-665.Google Scholar
  5. [5]
    H. Chen and D.D. Yao, Dynamic scheduling of a multiclass fluid network, Oper. Res. 41 (1993) 1104-1115.Google Scholar
  6. [6]
    E. Coffman and I. Mitrani, A characterization of waiting time performance realizable by single server queues, Oper. Res. 28 (1980) 810-821.Google Scholar
  7. [7]
    K.D. Dacre, K. Glazebrook and J. Niño-Mora, The achievable region approach to the optimal control of stochastic systems, J. Roy. Statist. Soc. (1999) 747-791.Google Scholar
  8. [8]
    E. Gelenbe and I. Mitrani, Analysis and Synthesis of Computer Systems (Academic Press, London, 1980).Google Scholar
  9. [9]
    T.C. Green and S. Stidham Jr., Sample-path conservation laws, with applications to scheduling queues and fluid systems, Queueing Systems Theory Appl. 36 (2000) 175-199.Google Scholar
  10. [10]
    D. Heyman and M. Sobel, Stochastic Models in Operations Research (McGraw-Hill, New York, 1982).Google Scholar
  11. [11]
    T. Kamae, U. Krengel and G.L. O'Brien, Stochastic inequalities on partially ordered spaces, Ann. Probab. 5 (1977) 899-912.Google Scholar
  12. [12]
    G.P. Klimov, Time-sharing service systems I, Theory Probab. Appl. 19 (1974) 532-551.Google Scholar
  13. [13]
    S.P. Sethi and Q. Zhang, Hierarchical Decision Making in Stochastic Manufacturing Systems (Birkhäuser, Boston, 1994).Google Scholar
  14. [14]
    J.G. Shanthikumar and D.D. Yao, Multiclass queueing systems: polymatroidal structure and optimal scheduling control, Oper. Res. 40 (Supplement 2) (1992) S293-S299.Google Scholar
  15. [15]
    P. Tsoucas, The region of achievable performance in a model of Klimov, Research Report RC16543, IBM T.J. Watson Research Center, Yorktown Heights, NY (1991).Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Nicole Bäuerle
    • 1
  • Shaler Stidham
    • 2
  1. 1.Department of Operations ResearchUniversity of UlmUlmGermany
  2. 2.Department of Operations ResearchUniversity of North CarolinaChapel HillUSA

Personalised recommendations