Queueing Systems

, Volume 37, Issue 1–3, pp 233–257 | Cite as

Skorohod–Loynes Characterizations of Queueing, Fluid, and Inventory Processes

  • William L. Cooper
  • Volker Schmidt
  • Richard F. Serfozo

Abstract

We consider queueing, fluid and inventory processes whose dynamics are determined by general point processes or random measures that represent inputs and outputs. The state of such a process (the queue length or inventory level) is regulated to stay in a finite or infinite interval – inputs or outputs are disregarded when they would lead to a state outside the interval. The sample paths of the process satisfy an integral equation; the paths have finite local variation and may have discontinuities. We establish the existence and uniqueness of the process based on a Skorohod equation. This leads to an explicit expression for the process on the doubly-infinite time axis. The expression is especially tractable when the process is stationary with stationary input–output measures. This representation is an extension of the classical Loynes representation of stationary waiting times in single-server queues with stationary inputs and services. We also describe several properties of stationary processes: Palm probabilities of the processes at jump times, Little laws for waiting times in the system, finiteness of moments and extensions to tandem and treelike networks.

Skorohod equation Loynes' representation fluid models Little's law Palm probability queues inventory 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    F. Baccelli and P. Bremaud, Elements of Queueing Theory (Springer, New York, 1994).Google Scholar
  2. [2]
    M. Bramson, State space collapse with application to heavy traffic limits for multiclass queueing networks, Queueing Systems 30 (1997) 80–148.Google Scholar
  3. [3]
    P. J. Brockwell, S.I. Resnick and R.L. Tweedie, Storage processes with general release rule and additive inputs, Adv. in Appl. Probab. 14 (1982) 392–433.Google Scholar
  4. [4]
    E. Çinlar, Theory of continuous storage with Markov additive inputs and a general release rule, J. Math. Anal. Appl. 43 (1973) 207–231.Google Scholar
  5. [5]
    E. Çinlar and M. Pinsky, On dams with additive inputs and a general release rule, J. Appl. Probab. 9 (1972) 422–429.Google Scholar
  6. [6]
    J.D. Dai and W. Dai, A heavy traffic limit theorem for a class of open queueing networks with finite buffers, Queueing Systems 32 (1999) 5–40.Google Scholar
  7. [7]
    D.J. Daley and T. Rolski, Finiteness of waiting-time moments in general stationary single-server queues, Ann. Appl. Probab. 2 (1992) 987–1008.Google Scholar
  8. [8]
    P. Franken, D. König, U. Arndt and V. Schmidt, Point Processes and Queues (Akademie-Verlag, Berlin, 1981 and Wiley, Chichester, 1982).Google Scholar
  9. [9]
    J.M. Harrison, Brownian Motion and Stochastic Flow Systems (Wiley, New York, 1985).Google Scholar
  10. [10]
    D.P. Heyman, A performance model of the credit manager algorithm, Comp. Networks and ISDN Systems 24 (1992) 81–91.Google Scholar
  11. [11]
    I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus (Springer, New York, 1991).Google Scholar
  12. [12]
    O. Kella and W. Whitt, Stability and structural properties of stochastic storage networks, J. Appl. Probab. 33 (1996) 1169–1180.Google Scholar
  13. [13]
    D. König and V. Schmidt, Zufällige Punktprozesse (Teubner-Verlag, Stuttgart, 1992).Google Scholar
  14. [14]
    T. Konstantopoulos and G. Last, On the dynamics and performance of stochastic fluid systems, J. Appl. Probab. 37 (2000) 652–667.Google Scholar
  15. [15]
    T. Konstantopoulos, M. Zazanis and G. de Veciana, Conservation laws and reflection mappings with an application to multiclass mean value analysis for stochastic fluid queues, Stochastic Process. Appl. 65 (1997) 139–146.Google Scholar
  16. [16]
    R. Loynes, The stability of a queue with non-independent interarrival and service times, Proc. Cambridge Phil. Soc. 58 (1962) 497–520.Google Scholar
  17. [17]
    R.B. Lund, A dam with seasonal input, J. Appl. Probab. 31 (1994) 526–541.Google Scholar
  18. [18]
    A. Mandelbaum and G. Pats, State-dependent stochastic networks. I. Approximations and applications with continuous diffusion limits, Ann. Appl. Probab. 8 (1998) 569–646.Google Scholar
  19. [19]
    D. Mitra, Stochastic theory of a fluid model of producers and consumers coupled by a buffer, Adv. in Appl. Probab. 20 (1988) 646–676.Google Scholar
  20. [20]
    M. Miyazawa, A formal approach to queueing processes in the steady state and their applications, J. Appl. Probab. 16 (1979) 332–346.Google Scholar
  21. [21]
    P.A.P. Moran, The Theory of Storage (Wiley, New York, 1959).Google Scholar
  22. [22]
    D. Perry, A double band control policy of a Brownian perishable inventory system, Probab. Engrg. Inform. Sci. 11 (1997) 361–373.Google Scholar
  23. [23]
    M.I. Reiman, Open queueing networks in heavy traffic, Math. Oper. Res. 9 (1984) 441–458.Google Scholar
  24. [24]
    V. Schmidt and R.F. Serfozo, Campbell's formula and applications to queueing, in: Advances in Queueing: Theory, Methods and Open Problems, ed. J.H. Dshalalow (CRC Press, Boca Raton, FL, 1995) pp. 225–242.Google Scholar
  25. [25]
    R.F. Serfozo, Introduction to Stochastic Networks (Springer, New York, 1999).Google Scholar
  26. [26]
    A.V. Skorohod, Stochastic equations for diffusion processes in a bounded region, Theory Probab. Appl. 6 (1961) 264–274.Google Scholar
  27. [27]
    R.J. Williams, Diffusion approximations for open multiclass queueing networks: sufficient conditions involving state space collapse, Queueing Systems 30 (1997) 27–88.Google Scholar
  28. [28]
    R.W. Wolff, Stochastic Modeling and the Theory of Queues (Prentice-Hall, Englewood Cliffs, NJ, 1989).Google Scholar
  29. [29]
    K. Yamada, Two limit theorems for queueing systems around the convergence of stochastic integrals with respect to renewal processes, Stochastic Process. Appl. 80 (1999) 103–128.Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • William L. Cooper
    • 1
  • Volker Schmidt
    • 2
  • Richard F. Serfozo
    • 3
  1. 1.Department of Mechanical EngineeringUniversity of MinnesotaMinneapolisUSA
  2. 2.Department of StochasticsUniversity of UlmUlmGermany
  3. 3.School of Industrial and Systems EngineeringGeorgia Institute of TechnologyAtlantaUSA

Personalised recommendations