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Fluid Limit for the Machine Repairman Model with Phase-Type Distributions

  • Laura Aspirot
  • Ernesto Mordecki
  • Gerardo Rubino
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8054)

Abstract

We consider the Machine Repairman Model with N working units that break randomly and independently according to a phase-type distribution. Broken units go to one repairman where the repair time also follows a phase-type distribution. We are interested in the behavior of the number of working units when N is large. For this purpose, we explore the fluid limit of this stochastic process appropriately scaled by dividing it by N.

This problem presents two main difficulties: two different time scales and discontinuous transition rates. Different time scales appear because, since there is only one repairman, the phase at the repairman changes at a rate of order N, whereas the total scaled number of working units changes at a rate of order 1. Then, the repairman changes N times faster than, for example, the total number of working units in the system, so in the fluid limit the behavior at the repairman is averaged. In addition transition rates are discontinuous because of idle periods at the repairman, and hinders the limit description by an ODE.

We prove that the multidimensional Markovian process describing the system evolution converges to a deterministic process with piecewise smooth trajectories. We analyze the deterministic system by studying its fixed points, and we find three different behaviors depending only on the expected values of the phase-type distributions involved. We also find that in each case the stationary behavior of the scaled system converges to the unique fixed point that is a global attractor. Proofs rely on martingale theorems, properties of phase-type distributions and on characteristics of piecewise smooth dynamical systems. We also illustrate these results with numerical simulations.

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References

  1. 1.
    Lavenberg, S.: Computer performance modeling handbook. Notes and reports in computer science and applied mathematics. Academic Press (1983)Google Scholar
  2. 2.
    Haque, L., Armstrong, M.J.: A survey of the machine interference problem. European Journal of Operational Research 179(2), 469–482 (2007)zbMATHCrossRefGoogle Scholar
  3. 3.
    Robert, P.: Stochastic Networks and Queues. Stochastic Modelling and Applied Probability Series. Springer, New York (2003)zbMATHCrossRefGoogle Scholar
  4. 4.
    Kurtz, T.G.: Representation and approximation of counting processes. In: Fleming, W.H., Gorostiza, L.G. (eds.) Advances in Filtering and Optimal Stochastic Control. LNCIS, vol. 42, pp. 177–191. Springer, Berlin (1982)CrossRefGoogle Scholar
  5. 5.
    Iglehart, D.L., Lemoine, A.J.: Approximations for the repairman problem with two repair facilities, I: No spares. Advances in Applied Probability 5(3), 595–613 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Iglehart, D.L., Lemoine, A.J.: Approximations for the repairman problem with two repair facilities, II: Spares. Advances in Appl. Probability 6, 147–158 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Ethier, S.N., Kurtz, T.G.: Markov processes: Characterization and convergence. Wiley Series in Probability and Statistics. John Wiley & Sons Inc., New York (1986)zbMATHCrossRefGoogle Scholar
  8. 8.
    Darling, R.W.R., Norris, J.R.: Differential equation approximations for Markov chains. Probab. Surv. 5, 37–79 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Shwartz, A., Weiss, A.: Large deviations for performance analysis. Stochastic Modeling Series. Chapman & Hall, London (1995)zbMATHGoogle Scholar
  10. 10.
    Ayesta, U., Erausquin, M., Jonckheere, M., Verloop, I.M.: Scheduling in a random environment: Stability and asymptotic optimality. IEEE/ACM Trans. Netw. 21(1), 258–271 (2013)CrossRefGoogle Scholar
  11. 11.
    Ball, K., Kurtz, T.G., Popovic, L., Rempala, G.: Asymptotic analysis of multiscale approximations to reaction networks. The Annals of Applied Probability 16(4), 1925–1961 (2005)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Bortolussi, L.: Hybrid limits of continuous time markov chains. In: Proceedings of Eighth International Conference on Quantitative Evaluation of Systems (QEST), pp. 3–12 (September 2011)Google Scholar
  13. 13.
    Bortolussi, L., Tribastone, M.: Fluid limits of queueing networks with batches. In: Proceedings of the Third Joint WOSP/SIPEW International Conference on Performance Engineering, ICPE 2012, pp. 45–56. ACM, New York (2012)CrossRefGoogle Scholar
  14. 14.
    Houdt, B.V., Bortolussi, L.: Fluid limit of an asynchronous optical packet switch with shared per link full range wavelength conversion. In: SIGMETRICS 2012, pp. 113–124 (2012)Google Scholar
  15. 15.
    Benaïm, M., Le Boudec, J.Y.: A class of mean field interaction models for computer and communication systems. Perform. Eval. 65(11-12), 823–838 (2008)CrossRefGoogle Scholar
  16. 16.
    Le Boudec, J.Y.: The Stationary Behaviour of Fluid Limits of Reversible Processes is Concentrated on Stationary Points. Technical report (2010)Google Scholar
  17. 17.
    Asmussen, S., Albrecher, H.: Ruin probabilities, 2nd edn. Advanced Series on Statistical Science & Applied Probability, vol. 14. World Scientific Publishing Co. Pte. Ltd., Hackensack (2010)zbMATHGoogle Scholar
  18. 18.
    Bortolussi, L.: Supplementary material of Hybrid Limits of Continuous Time Markov Chains, http://www.dmi.units.it/~bortolu/files/qest2011supp.pdf (2011) (accesed March 15, 2013)

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Laura Aspirot
    • 1
  • Ernesto Mordecki
    • 1
  • Gerardo Rubino
    • 2
  1. 1.Universidad de la RepúblicaMontevideoUruguay
  2. 2.INRIARennesFrance

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