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Consider an $M/GI/ \infty $ queue with the following service time distribution: the service time is $1/\mu_i$ with probabbility $p_i$, and $\sum_{i=1}^kp_i=1$ and $\sum_{i=1}^kp_i/\mu_i=1/\mu$. In other words the service time consists of a mixture of $K$ deterministic service times. I am trying to understand if the departure process of the model is Poisson? Does anyone have any ideas? Thank you in advance !

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It seems that the departure process is indeed a Poisson process. See for example the first line of the paper: Newell, G. F. "The $M/G/\infty$ Queue." SIAM Journal on Applied Mathematics 14.1 (1966): 86-88.

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