In Example 7.20, let denote the proportion of passengers that wait less than x for a

In Example 7.20, let π denote the proportion of passengers that wait less than x for a bus to arrive. That is, with Wi equal to the waiting time of passenger i, if we define Xi =



1, if Wi < x 0, if Wi x then π = limn→∞ n i=1 Xi /n.

(a) With N equal to the number of passengers that get on the bus, use renewal reward process theory to argue that

π = E[X1 +···+ XN ]

E[N]

(b) With T equal to the time between successive buses, determine E[X1 +···+ XN |T = t].

(c) Show that E[X1 +···+ XN ] = λE[min (T, x)].

(d) Show that

π =

 x 0 P(T > t) dt E[T ] = Fe(x)

= 2 3 = 1 − P32, m12 = 10, m23 = 20, m31 = 15, m32 = 25. Define an appropriate semi-Markov process and determine

(a) the proportion of time the taxi is waiting at location i, and

(b) the proportion of time the taxi is on the road from i to j, i, j = 1, 2, 3.