Queueing Systems 4 (1989) 69-76.
This paper presents some analytical results concerning an approximation procedure for closed queueing networks. The procedure is well known and has been found useful for product-form networks where large numbers of queues, jobs or job classes prohibit an exact analysis, as well as for networks which do not possess product-form. The procedure represents the mean sojourn time at a queue as a function of the throughput of the queue, and derives a set of fixed point equations for the throughputs of the various job classes. We begin by showing that under a mild regularity condition the fixed point equations have a unique solution. Then we show that derivatives of performance measures can be readily calculated, and that their simple form provides an interesting insight into capacity allocation in closed queueing networks.
Keywords: fixed point approximations, shadow prices, capacity allocation.