The problem of characterizing detection system performance presents the following dilemma: On the one hand, if the system is good, then performance failures will be extremely rare events. But on the other hand, if the occurrence of rare events is to be characterized, then reliance entirely on Monte Carlo simulation would require an enormous number of runs, and the expense would be prohibitive. Algebraic models contribute a complementary approach to circumvent the problem. Such models can augment the simulation approach in a variety of ways. This paper discusses several such models, focussing particularly on models for detection probability and false alarm rate. Numerical results confirming specific models are presented.