Volume 1, Number 2 / August issue 2015
Manuel Alberto M. Ferreira
A differential equation occurring in the M|M|∞ queue parameters study using a Markov renewal process

This work main objective is to present a process to compute the Markov renewal matrix R(t) for Markov renewal processes with infinite countable states space, which semi-Markov matrixes Q(t) are immigration and death type and assume a tridiagonal form. These processes occur often in practical applications. And the difficulty in obtaining friendly results for R(t) is a great obstacle to its application in practical cases modelling. It is exemplified the application to the M|M ∞ queue particular case. The work presented in (5) is resumed but now the emphasis is put in the determination of a pfaffian solution that must be considered in this example.
Keywords: Markov renewal process, tridiagonal matrix, M|M ∞ queue, pfaffian, exact differential equation

Cite this article:
Manuel Alberto M. Ferreira. A differential equation occurring in the M|M|∞ queue parameters study using a Markov renewal process. Acta Scientiae et Intellectus, 1(2), 10-15.


  1. A. Erdélyi et al, Higher Transcendental Functions, vol. 1, New York, Toronto, London: McGraw-Hill Book Company, Inc., 1953.
  2. E. Çinlar, Markov renewal theory, Advances in Applied Probability, 1(1969), 123-187.
  3. M. A. M. Ferreira, Some considerations about the queue approximation by a Markov renewal process, Information Sciences and Computing, Volume 2013, Number 1, Article ID ISC040713, 12 pages.
  5. M.A.M. Ferreira. Remarks on the approximation of a M|G|∞ queue system by a Markov renewal process. Journal of Mathematics and Technology 2014; 5(1), 5-9. DOI:
  6. M. A. M. Ferreira. Some Notes on a Semi-MARKOV Matrix Occurring in the Queue Parameters Study. Journal of Mathematics and Technology 2014; 5(2), 9-12. DOI:
  7. M. A. M. Ferreira, R. Menezes, Equações com Diferenças, Lisboa: Edições Sílabo, 1992.
  8. M. A. M. Ferreira, I. Amaral, MATEMÁTICA-Integrais Múltiplos, Equações Diferenciais (4ª Edição), Lisboa: Edições Sílabo, 1994.
  9. M. A. M. Ferreira, M. Andrade, an infinite servers queue systems with Poisson and non Poisson arrival busy period simulation, Emerging Issues in the Natural and Applied Sciences 2013; 3(1), 38-58. DOI: 10. 7813/einas.2013/3-1/4.
  10. M. A. M. Ferreira, M. Andrade, J.A. Filipe, M. P. Coelho, Statistical Queuing Theory with Some Applications, International Journal of Latest Trends in Finance and Economic Sciences 2011;1(4), 190-195.
  11. M. F. Ramalhoto, D. H. Girmes, Markov renewal approach to counter theory in: J.P. Barra et al (Eds), Recent Developments in Statistics, North Holland, 581-590, 1977.
  12. M. F. Ramalhoto, Breves considerações sobre aproximações e limites em sistemas do tipo , IST, CEAUL, Lisboa, 1985.