Extreme Values in the GI/G/1 Queue

Abstract

Consider a GI/G/1 queue in which Wn is the waiting time of the nth customer, W(t) is the virtual waiting time at time t, and Q(t) is the number of customers in the system at time t. We let the extreme values of these processes be {$}W{_}n{^}\backslashast = \backslashmax {{}W{_}j: 0 \backslashleqq j \backslashleqq n{}}, W{^}\backslashast(t) = \backslashsup {{}W(s): 0 \backslashleqq s \backslashleqq t{}}{$}, and {$}Q{^}\backslashast(t) = \backslashsup {{}Q(s): 0 \backslashleqq s \backslashleqq t{}}{$}. The asymptotic behavior of the queue is determined by the traffic intensity $ρ$, the ratio of arrival rate to service rate. When {$}ρ1{$}. For the case {$}\rho{\textless}1{$}, it is necessary to obtain the tail behavior of the maximum of a random walk with negative drift before it first enters the set (-∞, 0].