Methodol Comput Appl Probab (2007) 9:521–540 DOI 10.1007/s11009-006-9007-1 Stationary Analysis of a Fluid Queue Driven by Some Countable State Space Markov Chain Fabrice Guillemin · Bruno Sericola Received: 28 December 2005 / Revised: 11 September 2006 / Accepted: 19 September 2006 / Published online: 16 February 2007 © Springer Science + Business Media, LLC 2007 Abstract Motivated by queueing systems playing a key role in the performance evaluation of telecommunication networks, we analyze in this paper the station- ary behavior of a fluid queue, when the instantaneous input rate is driven by a continuous-time Markov chain with finite or infinite state space. In the case of an infinite state space and for particular classes of Markov chains with a countable state space, such as quasi birth and death processes or Markov chains of the G/M/1 type, we develop an algorithm to compute the stationary probability distribution function of the buffer level in the fluid queue. This algorithm relies on simple recurrence relations satisfied by key characteristics of an auxiliary queueing system with normalized input rates. Keywords fluid queues · Markov chains · uniformization · stationary regime · numerical algorithm AMS 2000 Subject Classification Primary 60K25 · Secondary 60J27 1 Introduction In the performance evaluation of packet telecommunication networks, fluid flow approximations prove very useful to analyze complex systems. Among fluid flow models, fluid queues with Markov modulated input rates play a key role in the recent developments of both queueing theory and performance evaluation of packet F. Guillemin France Telecom, 2 Avenue Pierre Marzin, 22300 Lannion, France e-mail: Fabrice.Guillemin@francetelecom.com B. Sericola (B ) IRISA-INRIA, Campus de Beaulieu, 35042 Rennes Cedex, France e-mail: Bruno.Sericola@irisa.fr