A new closed-form explicit expression is derived for the probability den sity function of the length of a busy period starting with i customers in an M/M/1/K queue, where K is the capacity of the system. The density function is given as a weighted sum of K negative exponential distributions with coefficients calculated from K distinct zeros of a polynomial that involves Chebyshev polynomials of the second kind. The mean and second moment of the busy period are also shown ex plicitly. In addition, the symmetric results for the first passage time from state i to state K are presented. We also consider the regeneration cycle of state i.