While studying Poisson processes, I have found a problem I can't solve:
Two people, A and B, arrive at a bank and wait to be attended. A and B are in the waiting queue and A is before B. The workers attend the customers according to a Poisson process of parameter λw. A and B are willing to wait, respectively, a time TA and TB before going away. TA and TB are exponentially distributed with parameters λA and λB.
What is the probability that the person B is attended before he or she leaves?
How can we calculate the answer?
Assuming that the patience times apply equally whether a customer is in service or waiting for service, the time customer A is in the system is exponentially distributed with rate $\lambda_A+\lambda_W$. Hence the probability that customer B is attended is $$ \frac{\lambda_A+\lambda_W}{\lambda_A+\lambda_W+\lambda_B}. $$