Extreme Values of Queue Lengths in M/G/1 and GI/M/1 Systems

We study the limiting behavior of maximum queue lengths in the M/G/1 and GI/M/1 service systems. When the systems are positive recurrent, the distributions of their maximum queue lengths, under standard linear normalizations, either do not converge or they converge to degenerate limits. Consequently, one cannot use classical extreme value theory to characterize their limiting behavior. We show, however, that by varying the system parameters in a certain way as the time interval grows, these maxima do indeed have three possible limit distributions. Two of them are classical extreme value distributions and the third one is a new distribution. The latter distribution is the best one for practical approximations.