An invariance principle for semimartingale reflecting Brownian motions in an orthant

Semimartingale reflecting Brownian motions in an orthant (SRBMs) are of interest in applied probability because of their role as heavy traffic approximations for open queueing networks. It is shown in this paper that a process which satisfies the definition of an SRBM, except that small random perturbations in the defining conditions are allowed, is close in distribution to an SRBM. This perturbation result is called an invariance principle by analogy with the invariance principle of Stroock and Varadhan for diffusions with boundary conditions. A crucial ingredient in the proof of this result is an oscillation inequality for solutions of a perturbed Skorokhod problem. In a subsequent paper, the invariance principle is used to give general conditions under which a heavy traffic limit theorem holds for open multiclass queueing networks.

[1]  A. Skorokhod Limit Theorems for Stochastic Processes , 1956 .

[2]  S. Varadhan,et al.  Diffusion processes with boundary conditions , 1971 .

[3]  D. W. Stroock,et al.  Multidimensional Diffusion Processes , 1979 .

[4]  J. Harrison,et al.  Reflected Brownian Motion on an Orthant , 1981 .

[5]  Martin I. Reiman,et al.  Open Queueing Networks in Heavy Traffic , 1984, Math. Oper. Res..

[6]  Ruth J. Williams,et al.  A boundary property of semimartingale reflecting Brownian motions , 1988 .

[7]  Lawrence M. Wein,et al.  Scheduling networks of queues: Heavy traffic analysis of a simple open network , 1989, Queueing Syst. Theory Appl..

[8]  A. Bernard,et al.  Regulations dÉterminates et stochastiques dans le premier “orthant” de RN , 1991 .

[9]  P. Dupuis,et al.  On Lipschitz continuity of the solution mapping to the Skorokhod problem , 1991 .

[10]  William P. Peterson,et al.  A Heavy Traffic Limit Theorem for Networks of Queues with Multiple Customer Types , 1991, Math. Oper. Res..

[11]  F. P. Kelly,et al.  Dynamic routing in open queueing networks: Brownian models, cut constraints and resource pooling , 1993, Queueing Syst. Theory Appl..

[12]  R. J. Williams,et al.  Existence and uniqueness of semimartingale reflecting Brownian motions in an orthant , 1993 .

[13]  Yang Wang,et al.  Nonexistence of Brownian models for certain multiclass queueing networks , 1993, Queueing Syst. Theory Appl..

[14]  W. Whitt Large Fluctuations in a Deterministic Multiclass Network of Queues , 1993 .

[15]  Hong Chen,et al.  Diffusion approximations for re-entrant lines with a first-buffer-first-served priority discipline , 1996, Queueing Syst. Theory Appl..

[16]  Wanyang Dai Brownian approximations for queueing networks with finite buffers :modeling, heavy traffic analysis and numerical implementations , 1996 .

[17]  Ruth J. Williams,et al.  Existence and Uniqueness of Semimartingale Reflecting Brownian Motions in Convex Polyhedrons , 1996 .

[18]  Ruth J. Williams,et al.  Diffusion approximations for open multiclass queueing networks: sufficient conditions involving state space collapse , 1998, Queueing Syst. Theory Appl..

[19]  Maury Bramson,et al.  State space collapse with application to heavy traffic limits for multiclass queueing networks , 1998, Queueing Syst. Theory Appl..

[20]  Wanyang Dai,et al.  A heavy traffic limit theorem for a class of open queueing networks with finite buffers , 1999, Queueing Syst. Theory Appl..