On the Integrodifferential Equation of Takacs. II

and X(t) = A'(t) > 0 are given. It is assumed that there exists a c > 0 such that e- h(x) E L2(O, oo). The moment fWxkh(x) dx, if it exists, is denoted by k. We put i,(s) = !oh(x)e-x dx . Furthermore, we suppose that f0 [X(t)]2 dt exists as a possibly improper Riemann integral for all T > 0. The stochastic process q(t) represents the waiting time of a customer arriving at time t in a queue with Poisson arrivals of variable density X(t), with H(x) the distribution of service times. F(t) is the probability that the counter is unoccupied at time t. Our present purpose is to study the behavior of F(t) for large t, especially under conditions that turn out to guarantee that F(t) does not approach zero. Previous knowledge ([2], [3], [4]) in this direction appears to be restricted essentially to the case X(t) = const. The following was proved in [1], although it does not appear as an explicit