Define a renewal process and a renewal reward process.

State and prove the strong law of large numbers for these processes.

[You may assume the strong law of large numbers for independent, identically-distributed random variables.

State and prove Little's formula.

Customers arrive according to a Poisson process with rate $\nu$ at a single server, but a restricted waiting room causes those who arrive when $n$ customers are already present to be lost. Accepted customers have service times which are independent and identicallydistributed with mean $\alpha$ and independent of the arrival process. Let $P_{j}$ be the equilibrium probability that an arriving customer finds $j$ customers already present.

Using Little's formula, or otherwise, determine a relationship between $P_{0}, P_{n}, \nu$ and

Part II