The purpose of this work is to determine how much the average response time decreases as more police officers are deployed in Corvallis Oregon. The reason that it was done is that the report done by the Matrix Consulting Group (Matrix, 2008) did not determine how the response time decreases as the number of officers deployed increases. The Corvallis City Council and Corvallis Budget Commission needs to know this when making fund allocation decisions.
According to Matrix (2008), Corvallis is deploying 29 officers over three shifts. According to my calculations using the data contained in the report, the mean time a citizen waits for service is 6.6 minutes. According to an article in The Corvallis Gazette Times on January 24, 2009, the Corvallis Police Department is going to ask for four more officers. And, according to my calculations, the hiring of four more officers would result in a mean time a citizen waits for service of 3.5 minutes -- a reduction of 3.1 minutes, which is terribly important to the waiting citizen. The Council/Budget-Commission has to decide if the additional cost of four officers is worth this reduction and if this reduction is more important than other uses of the money.
This is a queueing problem. People wait in line for service. The choice of model was limited by the data. The report did not contain raw data but data summaries and averages taken from other work that the Matrix Consulting Group had done. Therefore I chose the single queue in equilibrium, identical parallel servers, unlimited queue length, first-come-first-serve queueing discipline, Poisson arrival rates, and exponential service times.Queueing is discussed in many operations research textbooks and handbooks. I used Blumenfeld (2001)
The arrival rate for each hour during each day of the week for calendar 2007 can be found on page 46 of Matrix (2008). The arrival rate varies wildly as can be seen by the graph below. 
The rate starts low on Sunday morning and increases as the week goes on. The rate is at minimums in the early mornings and at maximums in the evenings. I used an overall average arrival rate of 1.80 calls per hour on the assumption that the department would allocate officers to peak periods and because developing a computer program to allocate officers to minimize overall mean waiting time would take too much time. I would not have an answer before decisions would be made.
The mean service time -- for the primary unit -- is the sum of mean travel time to the scene, primary unit handling, report writing, and jail run times. The mean handling, report writing, and jail run time are in the table, or can be developed from the table, on page 47 and 48. Officers are deployed to three areas in Corvallis. The time to travel to the scene can be estimated with a formula (Blumenfeld, 2001) for average rectangular grid distance -- divided by an estimate of speed, 25mph -- between two random points in an area.
The service rate is the reciprocal of the mean service time. I assumed that the number of back-up units per call was Poisson distributed and used the average back-up call for service as the Poisson parameter. As per the table, the average on-scene back-up unit time was 75% of the primary unit handling time. I then computed the mean time back-up units were involved and defined one server as sum of one officer (the primary unit) and the ratio of the mean time back-up units were involved to the mean service time of the primary unit; a server is 1.6 officers. The number of officers deployed was then computed accounting for leave and administrative hours (Matrix, 2008, pages 50-51) and assuming each officer worked a 40 hour week.
The mean time a citizen waits for service as a function of the number of officers deployed is shown in the graph below. 
As one can see, the mean time a citizen waits for service with 29 deployed officers is 6.6 minutes. If the number of deployed officers is increased to 33, the waiting time decreases to about 3.5 minutes.
Proactive time -- the amount time that an officer is not answering calls and is available for other work -- is a by-product of the queue analysis. The graph below shows the average proactive fraction of time that an officer has available.
According to Matrix (2008), an officer should have 40% to 45% uncommitted time to be effective at community policing. A Corvallis officer's time is 51% uncommitted, a figure supported by the above graph. With an increase in the number of officers, the fraction of uncommitted time rises.
A table showing the plot points of the two graphs can be found here. The analytical work was done with Gauss 8.0, a product of Aptech Systems Inc.