In both cyclic and finite-horizon contexts, piecewise-constant rate functions are commonly encountered in models with nonhomogeneous Poisson processes. We develop an algorithm, with no user-specified parameters, that returns a smoother rate function that maintains the expected number of arrivals. The algorithm proceeds in two steps: PQRS (Piecewise-Quadratic Rate Smoothing) returns a continuous and differentiable piecewise-quadratic function without regard to negativity. If negative rates occur, then MNO (Max Nonnegativity Ordering) returns the maximum of zero and another piecewise-quadratic function. MNO maintains continuity of rates and first derivatives, but with some exceptions. Our analysis allows fitting the MNO-PQRS function to require storage complexity of the order of the number of intervals and computational complexity of the order of the number of intervals squared. MNO-PQRS can be used as a stand-alone routine, or as an endgame for the authors' earlier algorithm, I-SMOOTH.
