Open queueing networks in heavy traffic. (English) Zbl 0549.90043

Reiman, Martin I.

Math. Oper. Res. 9, 441-458 (1984).

Summary: This paper presents heavy traffic limit theorems for the queue length and sojourn time processes associated with open queueing networks. These limit theorems state that properly normalized sequences of queue length and sojourn time processes converge weakly to a certain diffusion as the network traffic intensity converges to unity. The limit diffusion is reflected Brownian motion on the nonnegative orthant. This process behaves like Brownian motion on the interior of its state space, and reflects instantaneously on the boundaries. The reflection direction is a constant for each boundary hyperplane.

MSC:

90B22	Queues and service in operations research
60K25	Queueing theory (aspects of probability theory)
60K20	Applications of Markov renewal processes (reliability, queueing networks, etc.)

Keywords:

heavy traffic limit theorems; queue length; sojourn time; open queueing networks; limit diffusion; reflected Brownian motion