Sample Path Analysis of Token Rings

Abstract We consider two models for a token ring with a general (in particular, non-Poisson) arrival process. In both models the arrival process is characterized by pathwise conditions: in the first case, by the average rate and the burstiness (Cruz bounds) and in the second case by the existence of a long-run average rate. For both types of input process, we show that a sufficient condition for stability is that the rate at which work arrives to the system is smaller than one. We allow the service policy to be quite general, requiring only that the amount of work that leaves the system in a “cycle” is at least the amount of work that was present there in the beginning of the cycle. This includes the standard gated and exhaustive policies. For the first model we obtain deterministic upper bounds on the workload and the cycle times. Our results show that guarantees of no loss and/or service quality can be enforced by shaping the input flows to a token ring using, for example, leaky buckets, without the necessity for additional control mechanisms. We also show that the output process satisfies a Cruz type bound. This is useful in analyzing networks where the output from one token ring can be the input to another element of the network.