TY - JOUR

T1 - A positive recurrent reflecting brownian motion with divergent fluid path

AU - Bramson, Maury

PY - 2011/6

Y1 - 2011/6

N2 - Semimartingale reflecting Brownian motions (SRBMs) are diffusion processes with state space the d-dimensional nonnegative orthant, in the interior of which the processes evolve according to a Brownian motion, and that reflect against the boundary in a specified manner. The data for such a process are a drift vector θ, a nonsingular 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 of Dupuis and Williams [Ann. Probab. 22 (1994) 680-702] states that when every fluid path associated with the SRBM is attracted to the origin, the SRBM is positive recurrent. Employing this result, El Kharroubi, Ben Tahar and Yaacoubi [Stochastics Stochastics Rep. 68 (2000) 229-253, Math. Methods Oper. Res. 56 (2002) 243-258] gave sufficient conditions on (θ,σ,R) for positive recurrence for d = 3; Bramson, Dai and Harrison [Ann. Appl. Probab. 20 (2009) 753-783] showed that these conditions are, in fact, necessary. Relatively little is known about the recurrence behavior of SRBMs for d > 3. This pertains, in particular, to necessary conditions for positive recurrence. Here, we provide a family of examples, in d = 6, with θ = (-1,-1, ⋯ ,-1)T , σ = I and appropriate R, that are positive recurrent, but for which a linear fluid path diverges to infinity. These examples show in particular that, for d ≥ 6, the converse of the Dupuis-Williams result does not hold.

AB - Semimartingale reflecting Brownian motions (SRBMs) are diffusion processes with state space the d-dimensional nonnegative orthant, in the interior of which the processes evolve according to a Brownian motion, and that reflect against the boundary in a specified manner. The data for such a process are a drift vector θ, a nonsingular 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 of Dupuis and Williams [Ann. Probab. 22 (1994) 680-702] states that when every fluid path associated with the SRBM is attracted to the origin, the SRBM is positive recurrent. Employing this result, El Kharroubi, Ben Tahar and Yaacoubi [Stochastics Stochastics Rep. 68 (2000) 229-253, Math. Methods Oper. Res. 56 (2002) 243-258] gave sufficient conditions on (θ,σ,R) for positive recurrence for d = 3; Bramson, Dai and Harrison [Ann. Appl. Probab. 20 (2009) 753-783] showed that these conditions are, in fact, necessary. Relatively little is known about the recurrence behavior of SRBMs for d > 3. This pertains, in particular, to necessary conditions for positive recurrence. Here, we provide a family of examples, in d = 6, with θ = (-1,-1, ⋯ ,-1)T , σ = I and appropriate R, that are positive recurrent, but for which a linear fluid path diverges to infinity. These examples show in particular that, for d ≥ 6, the converse of the Dupuis-Williams result does not hold.

KW - Fluid model

KW - Heavy traffic

KW - Queueing networks

KW - Reflecting Brownian motion

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U2 - 10.1214/10-AAP713

DO - 10.1214/10-AAP713

M3 - Article

AN - SCOPUS:79958226955

VL - 21

SP - 951

EP - 986

JO - Annals of Applied Probability

JF - Annals of Applied Probability

SN - 1050-5164

IS - 3

ER -