A queueing-linear programming approach to scheduling police patrol cars

In any city the arrival rate of calls for police patrol-car service varies considerably during the day. Using queuing theory and integer linear programming, we present a method for scheduling patrol cars so that specified service standards are met at each hour of the day. Two models are used. The first is an M/M/n queuing model with time-dependent parameters that is solved numerically. The second is an integer linear program in which the decision variables are the number of patrol cars working each tour and the times at which they go out of service for meals. The program's constraints are determined by the output of the queuing model. Use of the method with data from the New York City Police Department indicates that it can lead to substantial improvements in police service.