Consider a single-server queueing system in which customers arrive in accordance with a renewal process. Each customer brings in a random amount of work, chosen independently according to the distribution G. The server serves one customer at a time. However, the server processes work at rate i per unit time whenever there are i customers in the system. For instance, if a customer with workload 8 enters service when there are three other customers waiting in line, then if no one else arrives that customer will spend 2 units of time in service. If another customer arrives after 1 unit of time, then our customer will spend a total of 1.8 units of time in service provided no one else arrives.

Let Wi denote the amount of time customer i spends in the system. Also, define E[W] by E[W] = lim n→∞ (W1 +···+ Wn)/n and so E[W] is the average amount of time a customer spends in the system.

Let N denote the number of customers that arrive in a busy period.

(a) Argue that?

*30. For a renewal process, let A(t) be the age at time t. Prove that if μ < ∞, then with probability 1 A(t)
t → 0 as t → ∞