C. Knessl, ON THE DISTRIBUTION OF THE MAXIMUM NUMBER OF BROKEN MACHINES FOR THE REPAIRMAN PROBLEM, SIAM journal on applied mathematics, 54(2), 1994, pp. 508-547
The repairman problem is considered where a single repairman services
M identical machines that fail at exponential waiting times and have s
ervice density b(.). The distribution of the maximum number of failed
machines during a busy period is computed. When b(t) = mu(0)e(-mu 0t)
, an exact expression for this quantity is derived. This expression is
then evaluated asymptotically for M much less than 1 with mu(0) = O(M
). The case of general b() is considered, and asymptotic expansions ar
e derived for the maximum number of failed machines. These expansions
are constructed using singular perturbation techniques such as the Wen
tzel, Kramer, and Brillouin (WKB) method and the method of matched asy
mptotic expansions. Numerical comparisons show the quality of the asym
ptotic approximations.