A queueing model having a nonstationary Interrupted Poisson arrival process (IPP(t)), s time-dependent exponential unreliable/repairable servers and finite capacity c is introduced, and an approximation method for analysis of it is developed and tested. Approximations are developed for the time-dependent queue length moments and the system viewpoint waiting time distributions and moments. The approximation involves state- space partitioning and numerically integrating partial-moment differential equations (PMDEs). Surrogate distribution approximations (SDA's) are used to close the system of PMDEs. The approximations allow for analysis using only \((s+1)(s+6)\) differential equations for the queue length moments rather than the \(2(c+1)(s+1)\) equations required by the classic method of numerically integrating the full set of Kolmogorov-forward equations. Effectively hours of cpu time are reduced to minutes for even modest capacity systems. Approximations for waiting time distributions and moments are developed.