INTEGRAL-REPRESENTATIONS AND ASYMPTOTICS FOR INFINITE-CAPACITY AND FINITE-CAPACITY QUEUES DESCRIBED BY THE UNFINISHED WORK .1.

We consider the M/M/1 queue described by the unfinished work and also a finite-capacity version of the model, where customers who would caus e the unfinished work to exceed a given capacity are rejected and lost . We derive integral representations for the time-dependent distributi on functions. Then we evaluate these integrals using the saddle point method. This yields a new set of approximations for describing the tra nsient behavior of the M/M/1 queue. We show that our formulas yield so me interesting insights into how the process settles to its steady-sta te distribution.