Exact Results for Nonsymmetric Token Ring Systems

This paper derives exact results for a token ring system with exhaustive or gated service. There are N nodes on the ring and control is passed sequentially from one to the next. Messages with random lengths arrive at each node and are placed on the ring when the control arrives at that node. Exhaustive service means that the queue at a node is empty before the token is released and gated means that only those messages in the queue at the arrival of the token are served at that cycle. Generating function recursions for the terminal service time (the total sojourn time of a token at a node) and, from this, joint cycle and intervisit times are derived. Using known results relating the marginal generating functions of the waiting time and the cycle and intervisit time, it is shown that the N mean waiting times at the nodes require the solution of N(N - 1) and N 2 equations for the exhaustive and gated cases, respectively. The arrival processes are assumed to be Poisson with different rates and the service processes are general and different at each node. In addition the token overhead is allowed to have an arbitrary but independent distribution at each node. Explicit, simply programmed equations are given. It is shown, arguing from the form of the equations, that there is a conservation law in effect in this system. If the nodal mean waiting times are weighted by the relative intensity, defined here as the intensity weighted mean, then the sum takes on a particularly simple form and is independent of the placement of the nodes on the ring. When the service means at each node are equal, this quantity is just the system mean waiting time.