Asymptotic Analysis of Losses in the $GI\mathrm{/}M\mathrm{/}m\mathrm{/}n$ Queueing System as $n$ Increases to Infinity

The paper studies asymptotic behavior of the loss probability for the $GI\mathrm{/}M\mathrm{/}m\mathrm{/}n$ queueing system as $n$ increases to infinity. The approach of the paper is based on applications of classic results of Tak'acs (1967) and the Tauberian theorem with remainder of Postnikov (1979-1980) associated with the recurrence relation of convolution type. The main result of the paper is associated with asymptotic behavior of the loss probability. Specifically it is shown that in some cases (precisely described in the paper) where the load of the system approaches 1 from the left and $n$ increases to infinity, the loss probability of the $GI\mathrm{/}M\mathrm{/}m\mathrm{/}n$ queue becomes asymptotically independent of the parameter $m$.