## Stability, queue length and delay of deterministic and stochastic queueing networks (1994)

Venue: | IEEE Transactions on Automatic Control |

Citations: | 171 - 20 self |

### BibTeX

@ARTICLE{Chang94stability,queue,

author = {Cheng-Shang Chang},

title = {Stability, queue length and delay of deterministic and stochastic queueing networks},

journal = {IEEE Transactions on Automatic Control},

year = {1994},

volume = {39},

pages = {913--931}

}

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### OpenURL

### Abstract

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 traffic characterization: minimum envelope rate (MER) and minimum envelope rate with respect to `. Based on these two new notions of traffic characterization, we develop a set of rules for network operations such as superposition, input-output relation of a single queue, and routing. Specifically, 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 traffic 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 to be