Question

Consider a taxi station at an airport where taxis and (groups of) customers arrive at times of Poisson processes with rates 2 and 3 per minute. Suppose that a taxi will wait no matter how many other taxis are present. However, if an arriving person does not find a taxi waiting he leaves to find alternative transportation. (a) Find the proportion of arriving customers that get taxis. (b) Find the average number of taxis waiting

Answer 1

a)
`\pi(0)=(1)/(3)`

b)
`X=(2)/(3)`

Explanation:

From the question we are told that:

Arrival time with Poisson's process =
`t_p=2-3minutes`

where

`\lambda_1=2\\\lambda_2=3`

a)

Generally the equation Poisson's process is mathematically given by

`\pi()n*\lambda_1=\pi(n+1*) \lambda_2`

`\pi()n*2=\pi(n+1*)*3`

`\pi(n+1)=(2)/(3) \pi+1*\pi(0)`

Therefore

`\pi(0)(1+2/3+(2/3)^2+...)=1`

`\pi(0)=(1)/(3)`

The proportion of arriving customers that get taxis.

`\pi(0)=(1)/(3)`

b)

Generally the average number of taxis X waiting is mathematically given by

`X=1-\pi(0)\\X=1-(1/3)`

`X=(2)/(3)`