a computer simulation model for the study of police patrol deployment

8/17/2019 A Computer Simulation Model for the Study of Police Patrol Deployment

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TECHNICAL ARTICLE

A computer simulation model for the

study of police patrol deployment

Gary M. Kern

Department of ManagementCollege of Business Administration

University of Notre Dame

Simulation models have been

suggested as a means of assist-

ing patrol administrators

making deployment decisions.The simulation model reportedhere allows the administrator to

study the performance of several

complex dispatching tactics

including multiple unit dis-patching, preemption, and in-tersector dispatching.

Environmental variables,such as interarrival and service

times, can be set to reflect anynumber ofpossible probabilitydistributions. Sample reportsgenerated by the model are

presented, as is informationregarding the model’s

verification.

Keywords: discrete-eventsimulation, police patrol deploy-ment, dispatching rules, SLAM

Introduction

The police patrol administrator is faced with the difficulttask of using scarce resources to serve an uncertain de-mand. The administrator must be concerned with the

performance of his or her scarce resources. However, unlike

private service operations, this performance is oftenmeasured in terms of some value other than dollar profit.

Patrol administrators seek to improve the performanceof their scarce resources (patrol cars and sworn officers) byimproving patrol deployment. Deployment can be dividedinto three highly interrelated decisions: patrol sector design,initial patrol unit geographic allocation, and dispatching.Numerous management scientists have developedmodels to aid the patrol administrator (see, for example, [5],[14], [8], [9]). The models help administrators improvepatrol deployment. Many are static models. The models aremathematicalprogramming formulations which attempt todescribe the performance of a given patrol deployment in agiven patrol environment.

Many of these previously-developed static mathematicalmodels had to rely on simplifying assumptions to makethem tractable. For instance, Chaiken and Dormont’s PatrolCar Allocation Model (PCAM) cannot describe the perform-ance of certain dispatching rules that include intersector

dispatching [3], [4], [6].

Some researchers have designed simulation models ofpolice patrol operations. Kolesar and Walker created such asimulation model [11]. Their model did not allow for the

preemption of low-priority calls so that high-priority callscould be served. Larson developed a discrete-event simula-tion that was quite sophisticated [12]. It could simulate the

performance of deployments which included both preemp-tion and intersector dispatching.

This paper reports the development of a discrete-eventcomputer simulation model that extends upon Larson’swork. The current simulation model is able to assign calls


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for service (CFS) to more than one patrol unit. Such a

dispatching rule is often employed for dangerous incidents.The model is also capable of simulating sophisticateddispatching rules.

The model allows the patrol administrator to define hisor her patrol environment. The administrator then enters a

deployment. The model reports the simulated performanceof the deployment given the environment.

The model is currently designed to operate on a main-frame computer, processing in batch mode. The simulation

language SLAM provides the basis for the model [18]. Themodel’s design also makes liberal use of FORTRANsubroutines.

The simulation model has been used to develop aca-demic research in the area of patrol deployment [10]. Thesestudies investigated the relative effectiveness of variousdeployments in common patrol settings. Simulated per-formancesof the deployments were collected and statisti-cally compared. Such experiments are helpful for support-ing patrol administration.

This paper reports key features of the simulation modeland its use. The following sections of the paper willdescribe:

~ the event structure that was modeled,~ the general design of the model,~ input required of the model,~ simulated performance results reported by the model,~ assumptions and limitations of the model, and~ verification of the model.

Event structure

Police patrol forces are expected to investigate criminalincidents, intercede in civil disputes, and prevent theoccurrence of future incidents. Most requests for service arehandled in a common manner. Figure A displays the typicalprocess used by patrol forces in response to a call forservice.

(Figure A is

adaptedfrom a time-frame

developedin Larson, [12].)This framework was used to identify the important

events for the simulation model. Using the terms that

appear in Figure A, the four basic events modeled by thesimulator are:

1) arrival of a CFS (t5)2) arrival ofan assigned patrol unit at the CFS location (t9)3) completion of service of a CFS (tid4) return of patrol unit to patrol area (til).

The model initiates the processing of a CFS with thearrival of the call. Call arrivals are generated by the modelbased upon the user’s description of demand. When a callarrives, the model uses the selected dispatching logic to

assign patrol unit(s) to serve the call. The patrol unit(s)

begin travel to the location of the CFS. An assigned patrolunit then arrives at the scene of the CFS. Service of the CFS

is initiated.

The next event is the completion of service of a CFS.

Upon completion of service, the unit(s) involved must

return to the patrolling area to which they are assigned. Thefinal event in the process is the arrival of a patrol unit to its

patrol area. The unit is again available to respond to a CFS.

General model designThe model was designed to follow the common service

procedure described above. A general flowchart of themodel is shown in Figure B. After the occurrence of any of

’

the four major events, a set of decision rules supplied by theuser (researcher or patrol administrator) is activated. Thedecision rules govern dispatching and the assessment of

patrol unit availability.For the purposes of the model, deployment has been

divided into five components. Three of the componentsdescribe the dispatching decision rules. This is not to beconstrued as in some way &dquo;biasing&dquo; the relative importanceof dispatching to overall deployment. Rather, it allows themodel to recognize that dispatching is a dynamic decision-

making process.Deployment is expressed as the combination of sector

design, initial allocation, queued call selection, intersector

dispatching, and preemption. Sector design and initialallocation are static decisions; they are made once perdecision period (be it a scheduled work day, a patrol shift,or an hour). However, dispatching is performed throughoutthe time period in light of the current patrol unit availabil-

ity. Thus, the model incorporates the dynamic nature of

dispatching.

The features used to describe a dispatching rule havebeen suggested in the previous research on deployment.The inclusion of factors like intersector dispatching and

preemption have been shown in some cases to improveperformance of patrol operations [10].

The model allows the dispatching rule to incorporateintersector dispatching. Intersector dispatching permits theassignment of a patrol unit available in sector A to a CFSlocated in sector B. Whether to include intersector assign-ment in the dispatching rule should be a decision made bythe patrol administrator.

Figure A. Sequence of events for a call for service.


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Figure B. System flowchart.

Preemption, if included in a dispatching rule, allows the

dispatcher to suspend current service of a low-priority CFS.Such a suspension of service would make available a patrolunit to serve an urgent call that otherwise would have been

queued.

When a CFS arrives, the dispatcher must decide whetherthere are units available to serve the call. There may beunits patrolling at the time; if not, the dispatching rule maydefine no patrol unit as available. In such a case, the call is

queued and awaits future assignment. A call selection rulemust be established for the order in which queued calls willbe served.

InputThe model requires a description of the patrol situation it isto simulate. As mentioned above, the deployment is input

in terms of five features. The user must also state a profile ofthe CFS demand expected to be encountered. This sectiondescribes the nature of these divisions of the input.

Demand is described by a series of characteristics.

Typically, dispatchers categorize CFS into one of several

incident types. Each incident type is given a code. Incidenttype maps to a degree of severity or priority. High-prioritycalls (those CFS that are more urgent in nature) receive twoor more patrol units for service. It is assumed that safeservice of the call cannot be completed by a single patrolunit.

The arrival rate for each incident defines the frequencywith which the incident occurs. The service rate describes

the time required to complete service of a given typeof incident. The distribution of incident location fre-

quency (the sector in which each CFS originates) must also


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be described.

Input for the environmentalfactors discussed above is

expected to include an average measure and the expecteddistribution for the factor’s values. For instance, interarrival

times are usually modeled as exponentially distributedrandom variables with a given average value. The simula-tion model is designed so that a specific subroutine gener-ates a specific environmentalfactor’s values. This makes itvery simple to change the model to reflect a change in theassumed distribution for a given factor. Again, such

flexibility is not always possible in a static descriptivemodel.Sector design is described in terms of average travel time

from one sector to each of the other sectors. A matrix of

these average travel times is developed to express eachsector’s relative location in the general geographic region.Travel time measures the distance from one sector’s

centroid to another. The user may assume any means of

measuring this distance (e.g., euclidian distance, rectangu-lar distance).

Travel time is the manner in which sector design affectsthe service provided to CFS. Similar to the structure usedfor generating demand measures, a travel time distributionis also the basis for a subroutine which generates randomvalues.

Initial allocation is expressed as the assignment of patrolunits to the sectors. The user simply inputs the total numberof units and then assigns each one to a specific sector.

Dispatching, as mentioned earlier, is described as achosen value for each of three features (queued call selec-tion, intersector dispatching, and preemption). This is not toimply that intersector dispatching must be included. It

Figure G Input values.

implies that the user must decide whether it should beincluded. For instance, intersector dispatching may not beallowed at all, or it could be allowed for only emergencycalls, or it may be allowed for all calls.

The simulation prepares a report describing the inputvalues. Please refer to Figure C. This report is a goodsummary of the values that the user typically must provide.The current version of the model assumes that all ran-

domly-generated values are exponentially distributed. But,as mentioned earlier, it is a simple adjustment to change

this assumption.Based upon the input, the computer model is able to

simulate the arrival of events, processing events accordingto the deployment in effect, and collect performancemeasures. The following section will describe the outputreported by the model.

z

Output’

The simulation model has been designed to allow forvariable replication lengths and number of replications. The _scheduling of an initial transient state, after which perform-ance collectors are cleared, is also possible. The transient -state length is also a variable. Although many researchers have studied deployment

and suggested numerous models to support the develop-ment of deployment, no one has reported a universally-accepted means of expressinghow well a deployed patrolforce has performed its collective job. No satisfactory meansexists for expressing how many criminal incidents a patrolforce prevents. Expressions of incidents &dquo;solved&dquo; are also

questionable in that the patrol force (and the deployment ofthis force) do not have total control over eventual incident

resolution.

For these reasons, alternative measures of overall

performance have often been used. These measures addressthe efficiency with which patrol service is delivered.

Efficiency relates to the degree of customer (citizen)satisfaction with the service rendered [17], [2].

The simulation model reports a collection of theseefficiency measures. They can be divided into response timeand dispatch delay. A complete sample of the outputreported by the model is provided as Figure D(1)-D(3).These reports were generated by a FORTRAN subroutinebuilt into the simulation. The format of the reports can berevised as the user desires by altering this single subroutine.

Dispatch delay, in terms of Figure A, is the amount oftime that elapses between when a CFS arrives and when thedispatcher assigns the CFS to patrol units (t to t ). It reflects .the time a call remains queued before it is dispatched.Figure D(1) shows the output displayed about dispatchdelay. The averagedispatch delay is reported for eachincident code and sector. A histogram reflecting the numberof calls that suffer

long delaysis

alsoshown.

Response time (also shown on Figure DO) is the lengthof time that a citizen must wait before service of their

incident begins. This would reflect not only the dispatchdelay but also the travel time required for assigned patrolunits to reach the CFS location. Referring to Figure A, ,response time is the time from ts to tg. Response time is animportant measure because a deployment may not onlyaffect dispatch delay but also the patrol unit selected toserve a given call. The deployment therefore affects the


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Figure D(l). Report of dispatch delayand response time.

travel time in addition to the dispatch delay. Again,response time averagesand histograms are reported foreach incident code and for each sector.

Dispatch delay and response time measure how quicklythe patrol system reacts to CFS. But the patrol administratormust also be concerned with other factors. Patrol officers

are expected to patrol in expectation that this activity willreduce the occurrence of future incidents. Patrol officers areconcerned with the relative amount of patrol time they login relation to other officers on duty [1]. Therefore, workload

Figure D(2). Patrol time by sector and Patrol Unit.

equity is also a deployment performance issue. Figure D(2)presents the relative amount of time each patrol unit spendson patrol. Units are arranged by the sector to which they areallocated, and a patrol time average is reported for each

sector.The possible inclusion of intersector dispatching into a

dispatching rule presents another performance issue. A

patrol administrator may be concerned with the amount oftime a given patrol unit spends working outside the sectorto which it has been allocated. Figure D(3) displays theinformation reported addressing this issue. Each patrolunit’s time during the shift is shown as percentages of time

spent in each sector. This gives some indication of theinteraction between initial allocation and intersector

dispatching in a given deployment.

Assumptions and limitationsThe nature of the patrol deployment situation required the

forming of some assumptions during the design of themodel. These assumptions do not reduce the viability of themodel as a decision-support tool. However, the user must

recognize these assumptions when interpreting the model’sresults.

The model assumes that dispatchers are perfect andmake their decisions instantaneously. Human error is not afactor in dispatcher decision-making. Once again referringto Figure A, the simulator assumes that no time elapsesfrom t3 to t6. (The model does not consider times for ti to t3’)These factors would not affect the relative performance of a


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Figure D(3). Sector allocation of patrol unit time.

given deployment. Further, the way in which such factorswould contribute to the results reported in the model wouldbe difficult to quantify.

It is assumed that all patrol units are composed of onepatrol car and one patrol officer. In some cities, two-officerpatrol units have been deployed. Given that one officer istypically insufficient for the safe service of dangerousincidents, the model assumes at least two patrol units are

assigned to high-priority calls. To adapt the model for theassumption that two officers are employed in each unitwould not be difficult.

If preemption is not allowed by the dispatching rule,certain limitations are imposed on a patrol unit’s availabil-ity for service. Patrol units are not considered available

during the travel times both to and from a CFS location.

Although this sounds like quite a strict assumption, itfollows actual experience rather well [1].

Preemption,when

allowed,is

capableof

suspendingservice of lesser-priority calls during the delivery of thatservice. The model is also capable of suspending servicewhile the patrol unit is still traveling to the location of alesser-priority call. As mentioned above, these assumptions and limitations

do not pose a threat to the model’s capability of supportingthe deployment decision process. Rather, the user mustrecognize their existence when interpreting the results

presented.

Verification

The simulation model was verified using two techniques.First, a special subroutine was constructed. The subroutine,when called, reports the contents of all files, all variables,and the event calendar. This subroutine was used in

conjunction with several sets of test data to insure thatevent logic executed as expected.

The second verification technique applied the simulatorto the task of modelling an M/M/S queueing system. Themodel was

providedwith

inputdata that described a set of

specific M/M/S systems. Each was run, and performancedata was collected. Measures of average wait in queue and

average length of queue were then statistically comparedwith the analytical values calculated using standard

queueing formulas.

Twenty replications were run at each interarrival rate.The lengths of the replications were computed usingFishman’s subrun method [7] ; a length of 61000 minuteswas used. A transient period was computed based on thestandard deviation of response time observed in prelimi-nary simulation runs. It was found that this standard

deviation tended to stabilize after 7200 minutes of simu-

lated time. Referring to Table 1, it can be seen that themodel was found to represent these queueing systems quitewell. For nine out of ten interarrival rates studied, thesimulated performance value were not significantlydifferent from the analytical results for an M/M/S system(confidence level = 98%).

Conclusion

This paper has presented a discrete-event computersimulation that can be used to support the deploymentdecision for police patrol operations. The model is veryflexible in terms of the types of deployments that can bestudied. The user also enjoys great flexibility in the patrolenvironments that can be defined.

A simulation model can present the effects of system

Table 1. Verification analysis for simulation.

*

average measure from 20 simulated replications*&dquo; the hypothesis (simulation value not significantly different from

analytical) was rejected at a confidence level of98%


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complexity that are oftentimes difficult to represent in staticmathematical formulations. Such is the case in police patroldeployment. The deployment is actually the interactiveresult of three decisions: sector design, initial allocation, anddispatching. Administrators can apply the simulation to studies of

their particular situation. Environmentally-specific factors,such as CFS frequency, high-priority call frequency, andtravel time distribution may have a significant impact onthe relative effectiveness of a

deploymentwhich includes

preemption or intersector dispatching. Studies like thesemay be undertakenthrough the use of the simulationmodel.

The model has proven to be an excellent vehicle for

developing experiments which address the design of

deployment. Through the use of simulation, variousdeployments can be routinely tested without actuallyaffecting patrol resources. The model should prove helpfulin the future to both researchers and patrol administrators.

References

Baker, M. Cops. 1985. Simon and Shuster.

Cahn, M. F. and J. M. Tien. 1981. An Alternative Approach to Police Response:Wilmington Management of Demand Program.. National Institute of JusticeReport.

Chaiken, J. M. and P. Dormont. 1975. Patrol Car Allocation Model: Executive

Summary. Rand CorporationReport number R-1786/1-HUD/DOJ,September.

Chaiken, J. M. and P. Dormont. 1975. Patrol Car Allocation Model: User’sManual. Rand CorporationReport number R-1786/2-HUD/DOJ,September.

Chaiken, J. M.; T. Crabill; L. Holliday; D. Jaquette; M. Lawless; and E.Quade. 1975. Criminal Justice Models: An Overview. Rand CorporationReport number R-1859-DOJ, October.

Chaiken, J. M. and P. Dormont. 1978. "Patrol Car Allocation Model:Capabilities and Algorithms." Management Science, August, 1978.

Fishman, G. S. 1978. "Grouping Observations in Digital Simulation."Management Science, January, 1978.

Green, L. 1984. "Multiple DispatchQueueingModel of Police Patrol

Operations." Management Science, June, 1984.

Green, L. and P. Kolesar. 1984. "Comparison of Multiple Dispatch and M/M/c Priority Queueing Models of Police Patrol." Management Science,June, 1984.

Kern, G. M. "The Effect of Intersector Dispatchingon Police PatrolPerformance." Omega, August, 1987.

Kolesar, P.and W. E. Walker. 1975. A Simulation Model of Police Patrol

Operations: Program Description. Rand Corporation Reportnumber R-1625/2-HUD/NYC.

Larson, R C. 1972. Urban Police Patrol Analysis. LexingtonBooks.

Larson, R C. and E. A. Franck. 1978. "Evaluating DispatchingConsequencesof Automatic Vehicle Location in Emergency Services." Computers andOperations Research 5: 11-30.

Larson, R C. 1978. Police Deployment. LexingtonBooks.

Larson, R C. and M. A. McKnew. 1982. ’’Police Patrol-Initiated ActivitiesWithin a Systems Queueing Model." Management Science, July, 1982.

McKnew, M. 1983. "An Approximation to the Hypercube Model With Patrol-Initiated Activities: An Application to Police." Decision Sciences 14: 408-418.

Parks, R P. 1976. "Police Response to Victimization: Effects on Citizen Attitudes and Perceptions." Report number R77-1, Workshop in PoliticalTheory and Policy Analysis, Indiana University.

Pritsker, A. A. B. 1986. Introduction to Simulation and SLAM II. Halstead Press,

John Wiley and Sons.

Gary M. Kem is an assistant professor ofmanagement for the College of Business Administration at the University of NotreDame. He received a PhD from Indiana

University in 1985, an MBA from CaseWestern Reserve University in 1979, and a BSfrom the University of Virginia in 1978. He hastaught at the University of Virginia, IndianaUniversity, Indiana State University, and the

Shanghai Institute of Mechanical Engineering as well as at NotreDame. His teaching and research interests are in the fields of

Operations Management, Management Science, and ManagementInformation Systems. He is a member of the Institute of Manage-ment Sciences, the Decision Sciences Institute, and the Society forInformation Management.

a computer simulation model for the study of police patrol deployment

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