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There are a couple of ways to derive the steady state probabilities for a $M/M/1/k$ queuing system with Markovian* arrivals (the first $M$), exponential service time distribution (the second $M$), a single server (the 1), and a finite total system capacity of $k$. Note this implies the queue can be at most $k-1$. *Recall a system with $M$ arrivals has ...


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