Question

2. A taxi company has three cabs. Calls come in to the dispatcher at times of a Poisson process with rate 2 per hour. Suppose that each requires an exponential amount of time with mean 20 minutes, and that callers will hang up if they hear there are no cabs available. (a) What is the probability all three cabs are busy when a call comes in

Answer 1

0.406

Explanation:

Let us make the number of cabs available to be x.

Number of calls is distributed as Poisson(\lambda =2) and

the availability of cab is distributed as Exponential(\theta =3).

Therefore, the probability all three cabs are busy when a call comes in is given by;

P(X=0)=\frac{e^{-\lambda}\lambda^x}{x!}\theta e^{-\theta x}=\frac{e^{-2}2^0}{0!}3e^{-3\times 0}= 0.406