Question

The times between the arrivals of customers at a taxi stand are independent and have a distribution F with mean F. Assume an unlimited supply of cabs, such as might occur at an airport. Suppose that each customer pays a random fare with distribution G and mean G. Let W.t/ be the total fares paid up to time t. Find limt!1EW.t/=t.

Answer 1

Check the explanation

Explanation:

Let

\(W(t) = W_1 + W_2 + ... + W_n\)

where W_i denotes the individual fare of the customer.

All W_i are independent of each other.

By formula for random sums,

E(W(t)) = E(Wi) * E(n)

\(E(Wi) = \mu_G\)

Mean inter arrival time = \(\mu_F\)

Therefore, mean number of customers per unit time = \(1 / \mu_F\)

=> mean number of customers in t time = \(t / \mu_F\)

=> \(E(n) = t / \mu_F\)