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I have one queue with two servers $S_1$ and $S_2$.The serving times are modeled $\sim exp(\mu_1)$ and $\sim exp(\mu_2)$ respectively.

The first server is free while the second has two clients, $A$ whose service is in progress and $B$ that is waiting. There is a path for every client so each of them must be served first by $S_1$ and then by $S_2$.
Then another client $X$ arrives. Now, help me developing this concept:

  1. $P_b$, the probability that customer B is still in the system when $S_1$ finishes to serve $X$.

Talking in terms of times, let $T_1$ the time used to serve $X$ and let $T_2$ the time used to serve $B$ after $S_2$ finished to serve A. Then, that means that I want $T_A+T_B>T_1$.

But I don't know how to transpose this into probability.

Thank you

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  • $\begingroup$ Does this answer your question? Probability of service in a queue theory problem with exponential random variable $\endgroup$
    – fblundun
    Commented Nov 28, 2020 at 23:21
  • $\begingroup$ No, here I'm asking another probability. I want to know the probability that the customer B is still in the system when $S_1$ finishes to serve $X$. In that question we wondered the $P(T_1 < T_2)$, here we are asking for $P(T_A+ T_B > T_1 )$ $\endgroup$
    – docdev
    Commented Nov 29, 2020 at 10:02

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