Motivated by recent development in high speed networks, in this paper we study two types of stability problems: (i) conditions for queueing networks that render bounded queue lengths and bounded delay for customers, and (ii) conditions for queueing networks in which the queue length distribution of a queue has an exponential tail with rate . To answer these two types of stability problems, we introduce two new notions of tra c characterization: minimum envelope rate (MER) and minimum envelope rate with respect to . Based on these two new notions of tra c characterization, we develop a set of rules for network operations such as superposition, input-output relation of a single queue, and routing. Speci cally, we show that (i) the MER of a superposition process is less than or equal to the sum of the MER of each process, (ii) a queue is stable in the sense of bounded queue length if the MER of the input tra c is smaller than the capacity, (iii) the MER of a departure process from a stable queue is less than or equal to that of the input process (iv) the MER of a routed process from a departure process is less than or equal to the MER of the departure process multiplied by the MER of the routing process. Similar results hold for MER with respect to under a further assumption of independence. These rules provide a natural way to analyze feedforward networks with multiple classes of customers. For single class networks with nonfeedforward routing, we provide a new method to show that similar stability results hold for such networks under the FCFS policy. Moreover, when restricting to the family of two-state Markov modulated arrival processes, the notion of MER with respect to is shown IEEE Trans. Automatic Control, Vol. 39, pp. 913-931, 1994. 0 to be equivalent to the recently developed notion of e ective bandwidth in communication networks.
In REF , the authors provide some results regarding some conditions that render bounded queue lengths both for a single queue and for feedforward networks.
Stability, queue length and delay of deterministic and stochastic queueing networks
IEEE Transactions on Automatic Control
Mathematics