Customers arrive at a bus stop according to a Poisson process of   rate λ. Independently, buses arrive according to a renewal process with the inter-renewal interval CDF FX (x). At the epoch of a bus arrival, all waiting passengers enter the bus and the bus leaves immediately. Let R(t) be the number of customers waiting at time t.

(a) Draw a sketch of a sample function of R(t).

(b) Given that the first bus arrives at time X1 = x, find the expected number of customers picked up; then find E r(x R(t)dtl, again given the first bus arrival at X1 = x.

(c) Find limt→∞ [(0 R(τ )dτ ]/t (WP1). Assuming that FX is a non-arithmetic distribution, find limt→∞ E [R(t)]. Interpret what these quantities mean.

(d) Use (c) to find the time-average expected wait per customer.

(e) Find the fraction of time that there are no customers at the bus stop. Hint: This part is independent of (a), (b), and (c); check your answer for E [X] « 1/λ.

Text Book: Stochastic Processes: Theory for Applications By Robert G. Gallager.