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**Part 6** 

**Routing** 


**Part 6** 

376 Telecommunications Networks – Current Status and Future Trends

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2010.

**Routing** 

**1. Introduction**

the duration of a busy period.

to analyze such systems.

Fluid models are powerful tools for evaluating the performance of packet telecommunication networks. By masking the complexity of discrete packet based systems, fluid models are in general easier to analyze and yield simple dimensioning formulas. Among fluid queuing systems, those with arrival rates modulated by Markov chains are very efficient to capture the burst structure of packet arrivals, notably in the Internet because of bulk data transfers. By exploiting the Markov property, very efficient numerical algorithms can be designed to estimate performance metrics such as the overflow probability, the delay of a fluid particle or

**On the Fluid Queue Driven by an Ergodic** 

**Birth and Death Process** 

Fabrice Guillemin1 and Bruno Sericola2

*2INRIA Rennes - Bretagne Atlantique, Campus de Beaulieu,* 

*1Orange Labs, Lannion* 

*35042 Rennes Cedex* 

*France* 

**16**

In the last decade, stochastic fluid models and in particular Markov driven fluid queues, have received a lot of attention in various contexts of system modeling, e.g. manufacturing systems (see Aggarwal et al. (2005)), communication systems (in particular TCP modeling; see vanForeest et al. (2002)) or more recently peer to peer file sharing process (see Kumar et al. (2007)) and economic systems (risk analysis; see Badescu et al. (2005)). Many techniques exist

The first studies of such queuing systems can be dated back to the works by Kosten (1984) and Anick et al. (1982), who analyzed fluid models in connection with statistical multiplexing of several identical exponential on-off input sources in a buffer. The above studies mainly focused on the analysis of the stationary regime and have given rise to a series of theoretical developments. For instance, Mitra (1987) and Mitra (1988) generalize this model by considering multiple types of exponential on-off inputs and outputs. Stern & Elwalid (1991) consider such models for separable Markov modulated rate processes which lead to a solution of the equilibrium equations expressed as a sum of terms in Kronecker product form. Igelnik et al. (1995) derive a new approach, based on the use of interpolating

Using the Wiener-Hopf factorization of finite Markov chains, Rogers (1994) shows that the distribution of the buffer level has a matrix exponential form, and Rogers & Shi (1994) explore algorithmic issues of that factorization. Ramaswami (1999) and da Silva Soares & Latouche (2002), Ahn & Ramaswami (2003) and da Silva Soares & Latouche (2006) respectively exhibit

polynomials, for the computation of the buffer overflow probability.
