Consider two users who arrive to a system with exponential arrival times with parameters $\lambda_a$ and $\lambda_b$. Once they arrive, the users stay in the system for an exponentially distributed time (parameters $\mu_a$ $\mu_b$) before leaving. The four exponential RVs are independent. What is the probability that A arrives before B AND leaves after B.
Letting $T_a$, $T_b$ denote the arrival times of each, and $S_a$, $S_b$ denote the service times. This probability is then $Pr(T_a < T_b \cap T_a + S_a > T_b + S_b)$ We can solve this by conditioning but it gets quite messy, is there a simpler way to tackle this? Perhaps something involving the memoryless property of the exponential distribution?