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We describe the use of analytical and numerical models of a multi-gain-stage single-carrier multiplication (SCM) avalanche photodiode (APD) to generate time-resolved receiver operating-characteristic (ROC) curves. First, pseudo-DC analytic models of discrete multi-stage APDs are used to generate the statistical properties of the SCM APD necessary for ROC analysis. Next, numerical models are used to develop the joint probability density function (PDF) of the SCM APD gain and avalanche buildup time as a function of the device structure, material properties and local electric fields. The instantaneous (time-resolved) carrier count distributions resulting from photon- and dark-initiated impact ionization chains are used to calculate the mean and variance of the currents induced in the circuits of the photoreceiver over the integration times of the detection event. Last, autocorrelation functions are generated to allow the parameters of the signal, the noise and the signal embedded in noise-which are necessary for ROC hypothesis testing-to be calculated for the instances of the impulse response. It is shown that time-resolved ROC analysis of an APD photoreceiver, which includes the instantaneous properties of photon- and dark-initiated avalanche events, allows for better optimization of photoreceiver performance than does non-time-resolved ROC analysis.