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This is a special case of a $M/M/c$ queuing system with $c=2$. Here, there is a single customer queue that feeds into two servers, one of which has a higher rate of service than the other. Call them Able and Baker respectively. If a customer arrives and both Able and Baker are free, he preferentially chooses Able, otherwise he chooses Baker. The customer enqueues if both the servers are busy. My question is are there closed form expressions for various performance metrics for this system? For example, mean sojourn time in system, probability of customer being busy etc.

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  • $\begingroup$ This is readily simulated with server priorities in Ciw, but I don't know a closed form for it. $\endgroup$
    – Galen
    Dec 20, 2023 at 3:37

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