CONVENTIONAL QUEUING THEORY THAT CONCENTRATES ON MEASURES OF AVERAGE PERFORMANCE WAS FOUND NOT TO BE READILY ADAPTABLE TO THIS APPLICATION. IN A MAJOR POLICE DEPARTMENT, AVERAGE VALUES OF DELAY RESPONSE WERE FOUND TO BE ACCEPTABLE, AND PRELIMINARY ANALYSIS SHOWED THAT SUFFICIENT RESOURCES WERE AVAILABLE TO AVOID EXCESSIVE QUEUING AT PEAK LOADS. NEVERTHELESS, REPORTS OF OCCASIONAL EXCESSIVELY DELAYED RESPONSES INDICATED THE EXISTENCE OF SERIOUS PROBLEMS. EXTREME VALUE THEORY CONCENTRATES ON THE STATISTICALLY RARE LONG DELAYS THAT, BECAUSE OF THE MANY REPETITIONS OF REQUESTS FOR SERVICE, CAN OCCUR MORE FREQUENTLY IN TIME THAN IS ACCEPTABLE TO POLICE MANAGEMENT. IN THE EXAMPLE DISCUSSED, THE PREDICTED MAGNITUDES AND FREQUENCIES OF THESE LONG DELAYS PROVIDED THE JUSTIFICATION FOR AN IMMEDIATE IMPROVEMENT PROGRAM. IN GENERAL, THE STRAIGHTFORWARD METHODOLOGY DESCRIBED APPEARED TO HAVE BROAD APPLICATIONS AND PRACTICAL VALUE. SUPPORTING DATA ARE PRESENTED IN TABULAR AND GRAPHIC FORM. REFERENCES ARE PROVIDED. (AUTHOR ABSTRACT MODIFIED. (VDA)