Encyclopedia of Operations Research and Management Science

2001 Edition
| Editors: Saul I. Gass, Carl M. Harris

Matrix-analytic stochastic models

Reference work entry
DOI: https://doi.org/10.1007/1-4020-0611-X_598

A rich class of models for queues, dams, inventories, and other stochastic processes has arisen out of matrix/vector generalizations of classical approaches. We present three specific examples in the following, namely, matrix-analytic solutions for M/G/1-type queueing problems, matrix-geometric solutions to GI/ M/1-type queueing problems, and finally, the Markov arrival process (MAP) generalization of the renewal point process.


The unifying structure that underlies these models is an imbedded Markov renewal process whose transition probability matrix is of the form:
This is a preview of subscription content, log in to check access.


  1. [1]
    Abate, J., Choudhury, G.L., and Whitt, W. (1994). “Asymptotics for steady-state tail probabilities in structured Markov queueing models,” Stochastic Models, 10, 99–143.Google Scholar
  2. [2]
    Asmussen, S. and Perry, D. (1992). “On cycle maxima, first passage problems and extreme value theory for queues,” Stochastic Models, 8, 421–458.Google Scholar
  3. [3]
    Asmussen, S. and Ramaswami, V. (1990). “Probabilistic interpretation of some duality results for the matrix paradigms in queueing theory,” Stochastic Models, 6, 715–733.Google Scholar
  4. [4]
    Falkenberg, E. (1994). “On the asymptotic behavior of the stationary distribution of Markov chains of M/ G/1-type,” Stochastic Models, 10, 75–97.Google Scholar
  5. [5]
    Gail, H.R., Hantler, S.L., and Taylor, B.A. (1994). “Solutions of the basic matrix equations for the M/G/1 and G/M/1 Markov chains,” Stochastic Models, 10, 1–43.Google Scholar
  6. [6]
    Latouche, G. (1985). “An exponential semi-Markov Process, with applications to queueing theory,” Stochastic Models, 1, 137–169.Google Scholar
  7. [7]
    Latouche, G. (1993). “Algorithms for infinite Markov chains with repeating columns,” in Linear Algebra, Markov Chains and Queueing Models, Meyer, C.D. and Plemmons, R.J., eds., Springer-Verlag, New York, 231–265.Google Scholar
  8. [8]
    Latouche, G. and Ramaswami, V. (1993). “A logarithmic reduction algorithm for quasi-birth-and-death processes,” Jl. Appl. Prob., 30, 650–674.Google Scholar
  9. [9]
    Lucantoni, D.M. (1991). “New results on the single server queue with a batch Markovian arrival process,” Stochastic Models, 7, 1–46.Google Scholar
  10. [10]
    Lucantoni, D.M. (1993). “The BMAP/G/1 queue: a tutorial,” in Models and Techniques for Performance Evaluation of Computer and Communications Systems, L. Donatiello and R. Nelson, eds., Springer-Verlag, New York.Google Scholar
  11. [11]
    Lucantoni, D.M., Meier-Hellstern, K.S., and Neuts, M.F. (1990). “A single server queue with server vacations and a class of non-renewal arrival processes,” Adv. Appl. Prob., 22, 676–705.Google Scholar
  12. [12]
    Neuts, M.F. (1979). “A versatile Markovian point process,” Jl. Appl. Prob., 16, 764–779.Google Scholar
  13. [13]
    Neuts, M.F. (1981). Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach, The Johns Hopkins University Press, Baltimore. Re-printed by Dover Publications, 1994.Google Scholar
  14. [14]
    Neuts, M.F. (1986). “The caudal characteristic curve of queues,” Adv. Appl. Prob., 18, 221–54.Google Scholar
  15. [15]
    Neuts, M.F. (1986). “Generalizations of the Pollaczek-Khinchin integral equation in the theory of queues,” Adv. Appl. Prob., 18, 952–990.Google Scholar
  16. [16]
    Neuts, M.F. (1989). Structured Stochastic Matrices of M/G/1 Type and Their Applications, Marcel Dekker, New York.Google Scholar
  17. [17]
    Neuts, M.F. (1992). “Models Based on the Markovian Arrival Process,” IEEE Trans. Communications, Special Issue on Teletraffic, E75-B, 1255–1265.Google Scholar
  18. [18]
    Neuts, M.F. and Narayana, S. (1992). “The first two moment matrices of the counts for the Markovian arrival process,” Stochastic Models, 8, 459–477.Google Scholar
  19. [19]
    Neuts, M.F. and Takahashi, Y. (1981). “Asymptotic behavior of the stationary distributions in the GI/ PH/c queue with heterogeneous servers,” Z. f. Wahrschein-lichkeitstheorie, 57, 441–452.Google Scholar
  20. [20]
    Ramaswami, V. (1988). “A stable recursion for the steady state vector in Markov chains of M/G/1 type,” Stochastic Models, 4, 183–188.Google Scholar
  21. [21]
    Ramaswami, V. (1990). “A duality theorem for the matrix paradigms in queueing theory, Stochastic Models, 6, 151–161. Google Scholar
  22. [22]
    Ramaswami, V. (1990). “From the matrix-geometric to the matrix-exponential,” Queueing Systems, 6, 229–260.Google Scholar
  23. [23]
    Sengupta, B. (1989). “Markov processes whose steady state distribution is matrix-exponential with an application to the GI/PH/1 queue,” Adv. Appl. Prob., 21, 159–180.Google Scholar
  24. [24]
    Schellhaas, H. (1990). “On Ramaswami's algorithm for the computation of the steady state vector in Markov chains of M/G/1-type,” Stochastic Models, 6, 541–550.Google Scholar
  25. [25]
    Tweedie, R.L. (1982). “Operator-geometric stationary distributions for Markov chains with application to queueing models,” Adv. Appl. Prob., 14, 368–91.Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  1. 1.The University of ArizonaTucsonUSA