Assume I have service times distributed according to a poisson process. There are 3 servers with service rates $\lambda_1$, $\lambda_2$ and $\lambda_3$ and each customer needs to visit them all in the same order. If I already have e.g. a customer $C_1$ at server 2 and customer $C_2$ at server 3 and a new customer $C_{new}$ enters the system, I want to know the expected time until he will be at server 3.
The intuitiv approach would be to add up the service times at the desks and also add the remainder of service time of a customer already at a desk, when the new customer arrives there. But that does not sound correct since the poisson process is memoryless.
Can I combine expected waiting times here, weighted by their probability? So when $C_{new}$ leaves server 1, I would now add $P(C_1\text{ already left})\frac{1}{\lambda_2} + P(C_1\text{ still at server 2})(2\frac{1}{\lambda_2})$ and similarly at server 3.