1. A Model of Resource Sharing

The first example is a traditional resource sharing model. In the model, $N$ distinguishable processes share a certain resource. Each of these processes alternates between a sleeping state and a resource using state. However, the number of processes that may concurrently use the resource is limited to $P$ where $1 \leq P \leq N$ so that when a process wishing to move from the sleeping state to the resource using state finds $P$ processes already using the resource, that process fails to access the resource and returns to the sleeping state. Notice that when $P = 1$ this model reduces to the usual mutual exclusion problem. When $P = N$ all of the processes are independent. We shall let $\lambda^{(i)}$ be the rate at which process $i$ awakes from the sleeping state wishing to access the resource, and $\mu^{(i)}$ , the rate at which this same process releases the resource when it has possession of it. In our SAN representation, we have tow cases :