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

Answer:

a) $\pi(0)=\frac{1}{3}$

b) $X=\frac{2}{3}$

Step-by-step 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

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

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

$\pi(n+1)=\frac{2}{3} \pi+1*\pi(0)$

Therefore

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

$\pi(0)=\frac{1}{3}$

The proportion of arriving customers that get taxis.

$\pi(0)=\frac{1}{3}$

b)

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

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

$X=\frac{2}{3}$