Question

Consider a taxi station where taxis and customers arrive in accordance with Poisson processes with respective rates of one and two per minute. A taxi will wait no matter how many other taxis are present. However, an arriving customer that does not find a taxi waiting leaves. Find a. the average number of taxis waiting. b. the proportion of arriving customers that get taxis.

Answer 1

a. Average number of taxis waiting = 1.

b. the proportion of arriving customers that get taxis = 1/2

Step-by-step explanation:

Let the state be the number of taxis waiting. Then we get a birth-death process with λ= 1 and μ= 2.

Also, this can be thought of as an M/M/1 system where being serviced is equivalent to waiting for a customer.

Therefore:

(a) Average number of taxis waiting = 1μ-λ = 1

(b) The proportion of arriving customers that get a taxi is the proportion of arriving customers that find at least one taxi waiting.

This is equivalent to the proportion of time the system is not in state 0. This is equal to 1-Po = 1 - (1-λμ) = 1/2.