# The Waiting-Time Distribution for the *GI/G/1* Queue under the *D*-Policy

We study a generalization of the *GI/G/1* queue in which the
server is turned off at the end of each busy period and is reactivated only
when the sum of the service times of all waiting customers exceeds a given
threshold of size *D*. Using the concept of a "randomly-selected" arriving
customer, we obtain as our main result a relation that expresses the
waiting-time distribution of customers in this model in terms of
characteristics associated with a corresponding standard *GI/G/1* queue,
obtained by setting *D=0*. If either the arrival process is Poisson or the
service times are exponentially distributed, then this representation of the
waiting-time distribution can be specialized to yield explicit,
transform-free formulas; we also derive, in both of these cases, the
expected customer waiting times. Our results are potentially useful, for
example, for studying optimization models in which the threshold *D* can be
controlled.