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.

Question

Mathematics

Toodles

4 March, 09:49

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.

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Toby Franco · Answer 1

Check the explanation

Step-by-step 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/)