Losses in M/GI/m/n Queues

The $M/GI/m/n$ queueing system with $m$ homogeneous servers and the finite number $n$ of waiting spaces is studied. Let $λ$ be the customers arrival rate, and let $μ$ be the reciprocal of the expected service time of a customer. Under the assumption $λ=mμ$ it is proved that the expected number of losses during a busy period is the same value for all $n\geq1$, while in the particular case of the Markovian system $M/M/m/n$ the expected number of losses during a busy period is $\frac{m^m}{m!}$ for all $n\geq0$. Under the additional assumption that the probability distribution function of a service time belongs to the class NBU or NWU, the paper establishes simple inequalities for those expected numbers of losses in $M/GI/m/n$ queueing systems.

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