Consider a queueing system with one server, interarrival distribution , and service distribution , where . This is the discrete analog of the queue. What can we say about its limiting behavior?
Let be the number of customers in the system at time , then is a Markov chain with transition probabilities
Suppose is an invariant measure for , that is, . A stationary distribution exists for iff , in which case . Assume WLOG that , then the global balance equations are
It follows that
Let be the generating function of , then multiplying the recurrence by and summing over yields
Solving for , we have
from which it follows that