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

so that

Solving for , we have

from which it follows that

We compute

and hence

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