The stationary distribution of a markovian process arising in the theory of multiserver retrial queueing systems

  • Authors:
  • A. Gómez-Corral;M. F. Ramalhoto

  • Affiliations:
  • Department of Statistics and Operations Research, Mathematics Faculty University Complutense of Madrid, 28040 Madrid, Spain;Mathematics Department Instituto Superior Tecnico-Technical University of Lisbon Avenida Rovisco Pais, 1096 Lisbon, Portugal

  • Venue:
  • Mathematical and Computer Modelling: An International Journal
  • Year:
  • 1999

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Abstract

In this paper, we introduce a bivariate Markov process {X(t), t = 0} = {(C(t),Q(t)), t = 0} whose state space is a lattice semistrip E = {0,1,2,3} x Z"+. The process {X(t), t = 0} can be seen as the joint process of the number of servers and waiting positions occupied, and the number of customers in orbit of a generalized Markovian multiserver queue with repeated attempts and state dependent intensities. Using a simple approach, we derive closed form expressions for the stationary distribution of {X(t),t = 0} when a sufficient condition is satisfied. The stationary analysis of the M/M/2/2 + 1 and M/M/3/3 queues with linear retrial rates is studied as a particular case in this process.