TY - JOUR
T1 - Stationary distribution convergence for generalized Jackson networks in heavy traffic
AU - Budhiraja, Amarjit
AU - Lee, Chihoon
PY - 2009/2
Y1 - 2009/2
N2 - In a recent paper, Gamarnik and Zeevi [Gamarnik, D., A. Zeevi. 2006. Validity of heavy traffic steady-state approximations in open queueing networks. Ann. Appl. Probab. 16(1) 56-90], it was shown that under suitable conditions stationary distributions of the (scaled) queue-lengths process for a generalized Jackson network converge to the stationary distribution of the associated reflected Brownian motion in the heavy traffic limit. The proof relied on certain exponential integrability assumptions on the primitives of the network. In this note we show that the above result holds under much weaker integrability conditions. We provide an alternative proof of this result assuming (in addition to natural heavy traffic and stability assumptions) only standard independence and square integrability conditions on the network primitives that are commonly used in heavy traffic analysis. Furthermore, under additional integrability conditions we establish convergence of moments of stationary distributions.
AB - In a recent paper, Gamarnik and Zeevi [Gamarnik, D., A. Zeevi. 2006. Validity of heavy traffic steady-state approximations in open queueing networks. Ann. Appl. Probab. 16(1) 56-90], it was shown that under suitable conditions stationary distributions of the (scaled) queue-lengths process for a generalized Jackson network converge to the stationary distribution of the associated reflected Brownian motion in the heavy traffic limit. The proof relied on certain exponential integrability assumptions on the primitives of the network. In this note we show that the above result holds under much weaker integrability conditions. We provide an alternative proof of this result assuming (in addition to natural heavy traffic and stability assumptions) only standard independence and square integrability conditions on the network primitives that are commonly used in heavy traffic analysis. Furthermore, under additional integrability conditions we establish convergence of moments of stationary distributions.
KW - Generalized Jackson network
KW - Heavy traffic analysis
KW - Invariant measures
KW - Reflected Brownian motion
UR - http://www.scopus.com/inward/record.url?scp=67649984883&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=67649984883&partnerID=8YFLogxK
U2 - 10.1287/moor.1080.0353
DO - 10.1287/moor.1080.0353
M3 - Article
AN - SCOPUS:67649984883
SN - 0364-765X
VL - 34
SP - 45
EP - 56
JO - Mathematics of Operations Research
JF - Mathematics of Operations Research
IS - 1
ER -