Advertisement

The superposition of two PH-renewal processes

  • Marcel Neuts
  • Guy Latouche
Chapter

Abstract

A large variety of stochastic models may be viewed as systems of two interacting renewal processes. Unless at least one of these is a Poisson process, the analytic properties of the model are difficult to study and are often intractable. It is clear that the greater analytic tractability of such models as the M/G/1 and GI/M/1 queues is due to the memory-less property of the exponential distribution. This tractability is gained, however, at the price of a severe distributional assumption. A useful compromise is found in the introduction of the probability distributions of phase type (PH-distributions). For these, a formalism of matrix manipulations has been developed which extends the elementary properties of the exponential distribution, yet is general and versatile enough to cover the probability distributions needed in a large variety of modelling problems.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. R. Bellman, Introduction to Matrix Analysis, McGraw Hill Book Co, New York, 1960.zbMATHGoogle Scholar
  2. F.R. Gantmacher, The Theory of Matrices, Chelsea, New York, 1959.zbMATHGoogle Scholar
  3. J.F.C. Kingman, A convexity property of positive matrices, Quart. J. Math., 12, (1961), 283–284.MathSciNetzbMATHCrossRefGoogle Scholar
  4. G. Latouche, A phase-type semi-Markov point process. SIAM Journal on Algebraic and Discrete Methods, 3, (1982), 77–90.MathSciNetzbMATHCrossRefGoogle Scholar
  5. G. Latouche, An exponential semi-Markov process, with applications to queueing theory, Stochastic Models (1985), 137-170.Google Scholar
  6. G. Latouche, P.A. Jacobs and D.P. Gaver, Finite Markov chain models skip-free in one direction. Naval Research Logistics Quarterly, 31, (1984), 571–588.MathSciNetzbMATHCrossRefGoogle Scholar
  7. M.F. Neuts, Moment formulas for the Markov renewal branching process. Advances in Applied Probability, 8, (1976), 690–711.MathSciNetzbMATHCrossRefGoogle Scholar
  8. M.F. Neuts, Some explicit formulas for the steady-state behavior of the queue with semi-Markovian services times. Advances in Applied Probability, 9, (1977), 141–157.MathSciNetzbMATHCrossRefGoogle Scholar
  9. M.F. Neuts, Matric-Geometric Solutions in Stochastic Models. An Algorithm Approach. The John Hopkins University Press, Baltimore, 1981.Google Scholar
  10. M.F. Neuts, The caudal characteristic curve of queues. Technical report no 82, B, Appl. Mathematics Institute, University of Delaware, November 1982.Google Scholar
  11. M.F. Neuts, A new informative embedded Markov renewal process for the PH/G/1 queue. Technical report no 94, B, Appl. Mathematics Institute, University of Delaware, November 1983.Google Scholar
  12. M.F. Neuts, Partitioned Stochastic Matrices of M/G/1 Type and their Applications. In preparation.Google Scholar
  13. M.F. Neuts and K.S. Meier, On the use of phase type distributions in reliability modelling of systems with a small number of components. OR Spektrum, 2, (1981), 227–234.zbMATHCrossRefGoogle Scholar
  14. R. Pyke, Markov renewal processes with finitely many states. Ann. Math. Statist., 32, (1961), 1243–1259.MathSciNetzbMATHCrossRefGoogle Scholar
  15. V. Ramaswami, The N/G/1 queue and its detailed analysis. Advances in Applied Probability, 12, (1980), 222–261.MathSciNetzbMATHCrossRefGoogle Scholar
  16. V. Ramaswami, The busy period of queues which have a matrix-geometric steady state vector. Opsearch, 19, (1982), 238–261.MathSciNetzbMATHGoogle Scholar
  17. L. Takacs, Introduction to the Theory of Queues. Oxford University Press, New York, (1962).zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1986

Authors and Affiliations

  • Marcel Neuts
    • 1
  • Guy Latouche
    • 1
    • 2
  1. 1.University of DelawareUSA
  2. 2.Université Libre de BruxellesBelgium

Personalised recommendations