The arrival rate of calls for the service of police patrol cars varies considerably through the day in any city. We present a method — using queueing theory and integer linear programming — for scheduling patrol cars so that specified service standards are met at each hour of the day. Two models are used: An M/M/n queueing model with time dependent parameters which is solved numerically; and 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, and whose constraints are determined by the output of the queueing model. Applications to real situations indicate that substantially improved schedules can be derived using this methodology.
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