THIS PAPER DEVELOPS MODELS THAT MAXIMIZE THE EXPECTED NUMBER OF OCCASIONS PER UNIT OF TIME THAT A POLICE PATROL UNIT ENTERS A STREET SEGMENT DURING THE TIME THAT A CRIME IS VISIBLE. CONSTRAINTS ARE ADDED THAT INSURE A MINIMUM PATROL COVERAGE TO ALL STREETS. THE SUCCESSIVE VISITS FROM STREET-TO-STREET FORM A MARKOV CHAIN. THE SOLUTION THAT MAXIMIZES THE OBJECTIVE FUNCTION GIVES A STOCHASTIC DECISION RULE WHICH IS USED WITH MONTE CARLO TECHNIQUES TO GENERATE A RANDOM PATROL SCHEDULE. THE PROBLEM IS POSED WITH ONE CAR AND SEVERAL CARS PATROLLING THE SAME REGION. (AUTHOR ABSTRACT)