A heavy traffic analysis for semi-open networks

Systems arising in the performance modelling of data communication networks and manufacturing job shops have flow constraints that occur in neither purely open nor purely closed queueing network models. In this article, we analyze a class of queueing networks that are semi-open. One node is identified as the feeder node, like a CPU or a schedular. It receives all the external arrivals to the system, and has an infinite buffer capacity. Jobs then leave to enter a larger flow controlled network. When the number in this part of the system equals K, the server for the feeder queue shuts down, and no more customers are admitted into this controlled network. Semi-open networks have been analyzed exactly for one or two nodes in the controlled subnetwork by Konheim and Reiser [A.G. Konheim, M. Reiser, J. ACM vol. 23, no. 2, 328-341; and SIAM J. Computing vol. 7, no. 2 (1978) 210-219], and asymptotically for a one node subnetwork by Kogan and Pukhalskii [Y.A. Kogan, A.A. Pukhalskii, in Performance '84, ed. E. Gelenbe (Elsevier, 1984) pp. 549-558]. The latter set of authors analyzed this system by obtaining a heavy traffic limit for the feeder queue, and the subnetwork had the limiting distribution of the M/M/1/K queue. In this article, we generalize the results of Kogan and Pukhalskii to the case of the subnetwork being a flow controlled Jackson network with an arbitrary number of nodes. To obtain the heavy traffic limit, we need to compute the variance for the effective service times of the feeder queue. This variance can be solved in terms of covariances for the departure process of the flow controlled network with a Poisson source, and the event that the network is in some state. In turn, we show that these covariances can be computed by solving a finite set of linear equations. We derive these equations by using Kronecker products to decompose the Markov generator for the queueing network into right and left shift operators.