This paper is concerned with the class of d-dimensional diffusion processes known as semi-martingales reflecting Brownian motions (SRBMs). Such processes arise as approximations for open d-station queueing networks. The data for such a process are a drift vector θ, a non-singular d×d covariance matrix Σ, and a d×d reflection matrix R. A standard problem is to determine under what conditions the process is positive recurrent. Necessary and sufficient conditions for positive recurrence are easy to formulate for d = 2, but not for d > 2. Associated with the pair (θ,R) are fluid paths, which are solutions of deterministic equations corresponding to the random equations of the SRBM. A standard result in literature states that when every fluid path associated with the SRBM is attracted to the origin, the SRBM is positive recurrent. In this article the author provides a family of examples, in d = 6, for which the SRBM is positive recurrent, but possesses a divergent linear fluid path. Leonardo Pasini
Recensione dell'articolo: (Bramson, Maury - " A positive recurrent reflecting Brownian motion with divergent fluid patch" - Ann.Appl.Probab. 21 (2011), no.3, 951–986.)
PASINI, Leonardo
2012-01-01
Abstract
This paper is concerned with the class of d-dimensional diffusion processes known as semi-martingales reflecting Brownian motions (SRBMs). Such processes arise as approximations for open d-station queueing networks. The data for such a process are a drift vector θ, a non-singular d×d covariance matrix Σ, and a d×d reflection matrix R. A standard problem is to determine under what conditions the process is positive recurrent. Necessary and sufficient conditions for positive recurrence are easy to formulate for d = 2, but not for d > 2. Associated with the pair (θ,R) are fluid paths, which are solutions of deterministic equations corresponding to the random equations of the SRBM. A standard result in literature states that when every fluid path associated with the SRBM is attracted to the origin, the SRBM is positive recurrent. In this article the author provides a family of examples, in d = 6, for which the SRBM is positive recurrent, but possesses a divergent linear fluid path. Leonardo PasiniI documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
