It has been argued recently that steady-state parameters (like cell loss probability) have important drawbacks when used as quality of service criteria of telecommunication networks. Therefore the first buffer overflow time has been proposed as an alternative, transient characteristic. In this paper the first buffer overflow time in a constant service rate queue is studied. In particular, the formulas for Laplace transform and expected value of its distribution for any initial queue size are proven. Moreover, numerical illustrations of analytical results are presented.