On the distribution of the maximum number of broken machines for the repairman problem

The repairman problem is considered where a single repairman services M identical machines that fail at exponential waiting times and have service density. The distribution of the maximum number of failed machines during a busy period is computed. When ,an exact expression for this quantity is derived. This expression is then evaluated asymptotically for with . The case of general is considered, and asymptotic expansions are derived for the maximum number of failed machines. These expansions are constructed using singular perturbation techniques such as the Wentzel, Kramer, and Brillouin method and the method of matched asymptotic expansions. Numerical comparisons show the quality of the asymptotic approximations.