Sticky State Banyan

A nite-buuered banyan network analysis technique designed to model networks at high traac loads is presented. The technique specially models two states of the network queues: queue empty and queue congested (roughly, zero or one slots free). A congested queue tends to stay congested because packets bound for the queue accumulate in the previous stage. The expected duration of this state is computed using a probabilistic model of a switching module feeding the congested queue. A technique for nding a lower arrival rate to an empty queue is also described. The queues themselves are modeled using independent Markov chains with an additional congested state added. The new analysis technique is novel in its modeling the higher arrival rate to a congested queue and a lower arrival rate to an empty queue. Comparison of queue state distributions obtained with the analysis and simulation show that an important feature of congestion is modeled. 1 Introduction A novel model of nite-buuered banyan networks, (multistage networks having exactly one path between each input/output pair), is presented along with an improved banyan analysis method based on the model. The distinguishing feature of the model is that the arrival rate of traac to a queue (buuer) in the network depends upon the state of the queue itself. In particular, the arrival rate to an empty queue is lower than average while the arrival rate to an almost full queue can be much higher than average. These arrival rates tend to prolong the respective queue conditions, making sticky the queue empty state and what will be called the congested state. Since traac ow in such a network depends strongly on the fraction of time a queue is empty and full, the model captures an important aspect of banyan network behavior. Many of the banyan network analysis methods appearing in the literature are based on Dias and Jump 1] and Jenq's 6] analyses of single-buuered networks. Their models are based on the observation that traac arriving at the queues in a stage would be identically distributed and the assumption that the arrivals at a queue are independent of the state of any queue. Using symmetry properties the observation is easily proven; the assumption, while clearly a simpliication, yields an easily analyzed model. Each stage is modeled by a single queue. The queue's stationary state distribution can be found in closed form although analysis of the entire network uses …