Answer:
a) [tex]\pi(0)=\frac{1}{3}[/tex]
b) [tex]X=\frac{2}{3}[/tex]
Step-by-step explanation:
From the question we are told that:
Arrival time with Poisson's process =[tex]t_p=2-3minutes[/tex]
where
[tex]\lambda_1=2\\\lambda_2=3[/tex]
a)
Generally the equation Poisson's process is mathematically given by
[tex]\pi()n*\lambda_1=\pi(n+1*) \lambda_2[/tex]
[tex]\pi()n*2=\pi(n+1*)*3[/tex]
[tex]\pi(n+1)=\frac{2}{3} \pi+1*\pi(0)[/tex]
Therefore
[tex]\pi(0)(1+2/3+(2/3)^2+...)=1[/tex]
[tex]\pi(0)=\frac{1}{3}[/tex]
The proportion of arriving customers that get taxis.
[tex]\pi(0)=\frac{1}{3}[/tex]
b)
Generally the average number of taxis X waiting is mathematically given by
[tex]X=1-\pi(0)\\X=1-(1/3)[/tex]
[tex]X=\frac{2}{3}[/tex]
