Three customers A, B, and C, simultaneously arrive at a bank with two tellers on duty. The two tellers were idle when the three customers arrived, and A goes directly to one teller, B goes to the other teller, and C waits until either A or B leaves before she can begin receiving service. If the service times provided by the tellers are exponentially distributed with a mean of 4 minutes, what is the probability that customer A is still in the bank after the other two customers leave
probability on Poisson process
0
$\begingroup$
probability
poisson-process
1 Answers
1
Hint:
Denoting the service times by $A,B,C$ you are asked to find:$$\Pr(A>B+C)=\Pr(A>B)\Pr(A-B>C\mid A>B)$$ Use symmetry to find the first factor. Exponential distribution is memoryless and this fact (again together with symmetry) can be used to find the second factor.
Other route is working out:$$\Pr(A>B+C)=\int\Pr(A>b+c)f_B(b)f_C(c)dbdc$$