Using repair-depot system reliability to determine the distribution of supportability turn-around time

Assuming constant repair times, Linton, et al. (1995) used an `expression for the reliability of the system for repairing failed units (FU) at a repair-depot´ to compute the longest repair time for a newly failed unit (NFU) which assures a given reliability level (also termed the NFU supportability turn-around time, STAT) in terms of the: (1) constant failure rate for all components, number of spares (s) on-hand; (2) number (n) of FU either `under repair´ or `scheduled to begin repair in the future´; and (3) downstream repair completion times (DRCT) for FU. Since subtraction of the repair time for a NFU from its STAT-value yields the NFU´s latest repair start-time (LRST) which assures a given repair-depot system reliability (RDSR), STAT-values are important for scheduling RST. This paper assumes that repair time is a random variable and, consequently, DRCT is a random variable. As shown in Linton, et al, (1995), STAT is the zero of a nonlinear, nonpolynomial function of DRCT; thus, STAT is also a random variable, and determining the distribution of STAT is a stochastic root-finding problem. For n=1 and s⩾0, numerical analysis and probability theory are used to find the Cdf and pdf of STAT in terms of any repair time pdf. Using the pdfs for STAT and repair time, expressions are derived for: (1) E{LRST} for a NFU; and (2) q=Pr{(repair time+c)<STAT}, c=0 and c=E{LRST}. When the repair time Cdf is exponential or 2-Erlang, numerical values are obtained for q and E(LRST), and it is shown how these values may be used by depot management to schedule RST for a NFU