- Email: [email protected]
Extreme value theory applied to police performance analysis
Joanna! of criminal Jsu&,
vol . S, pp . 287.300 (1977) . Persa Pms. Printed in U .S .A.
EXTREME VALUE THEORY APPLIED TO POLICE PERFORMANCE ANALYSIS
ROBERT P . ECKERT AND PETER M . KELLY
Kelly Scientific Corporation 4708 Wisconsin Avenue, N .W . Washington, D . C . 20016
ABSTRACT Extreme value theory is used to provide, according to one reviewer, a quantitative measure of the probability of a police dispatch operation going "haywire" under pressure . Conventional queuing theory that concentrates on measures of average performance is not 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 . Neverthetess, 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 would appear to have broad applications and great practical value . The subsequent analysis to identify causal factors is not a part of the discussion.
INTRODUCTION The value of operations research techniques can lie in their ability to provide a quantitative measure of the seriousness of well-recognized management and operational problems . This is in contrast to the more orthodox use of operations research analysis to detect and identify problems that
287
288
ROBERT P . ECKERT and PETER M . KELLY
may have not been suspected earlier . This article illustrates the use of extreme value theory to provide a measure of problem importance and so to supply the information required to break through normal organizational inertia to management and operational changes . The analytic techniques used to determine the seriousness of the problems are described ; the later, detailed analysis to identify causal factors is not discussed . Analysis and the development of recommendations for improvement of the Baltimore County Police dispatching system was the assigned task . An initial walk-through, some preliminary measurements, and applications of accepted but approximate rules of thumb provided significant START TRT (Telephone Ring Time) Estimated 1 .3% of total delay time
TIN (Time for Telephone Interview) Estimated 7 .6% of total delay Tcp (Time for Dispatch Card Preparation) Estimated 8 .5% of total delay (Transit Time for Transport of Dispatch Card) Estimated 3 .2% of total delay
TTR
TVT (Time for Vehicle Travel) Estimated 64.3% of total delay
T o (Time for Dispatch) Estimated 15 .1% of total delay
Figure 1 . Individual delay components and initial estimates of their relative magnitudes . (The estimates of the relative magnitude of the various delay components were developed during the initial weeks of measurements . It was presumed that these estimates would be useful in guiding the analytic activity . In the instances where extreme values of delay occurred these approximate relationships based on average values did not apply . Hence the relationships proved to be useful only as a general overview of the system operations .)
Extreme Value Theory Applied To Police Performance Analysis
28 9
information . The problem was not one of too few dispatchers or overtaxed telephone or radio channels . It was also clear that the average delay response was satisfactory . The distribution mode was on the order of four to five minutes . It was the extreme tail of the response-time distribution that was the problem . There were reports of infrequent delays of unacceptable length and such delays were not restricted to trivial requests but had, from time to time, involved serious incidents . In addition, simple observation revealed that neither the architectural features of the dispatch center nor the operational practices were conducive to a disciplined, repetitive operation . Clearly the orthodox analysis, with emphasis on modal values of delay components, would lead to an illumination of the problem only indirectly and after an extended study . With the cooperation and support of the police executives involved, the decision was made to take a somewhat unusual approach that concentrated attention on the extreme delay values and developed a measure of the frequency of their occurrence .
DESCRIPTION OF THE SYSTEM Figure I identifies the components of the delay response . The physical layout was such that the telephone answering operations were completely isolated, in terms of sight and sound, from the vehicle dispatch operation . (See figure 2 .) The radio dispatch room itself was in semidarkness to facilitate use of electronic displays . Each dispatcher, because of the low light level and the layout of the consoles, operated effectively in isolation receiving the dispatch assignments by moving belt from the telephone room . The police patrol personnel were organized into ten police districts . Patrol personnel reported for their duty shifts through the district headquarters and while on duty might receive assignments from the district headquarters as well as from the emergency communications center . The features just described-the physical layout of the communications center and the district organization of the patrol-interfered with providing clear-cut supervision of the emergency services delivery system . During the initial observation period, three separate but sometimes interrelated phenomena were noted as possibly contributing to the occasional extremely long delays in service delivery time that were the focus of concern . The county has many widely separated streets, lanes, avenues, and roads with similar and sometimes identical names . Because of this, the telephone complaint operator identifies the location while the citizen is on the telephone . This procedure can result in confusion . The civilian personnel employed at the communications center are often not familiar with all parts of the county . In addition, the telephone answering room is a large one with many positions and, at the time of the contractual activity, it was used informally as the lunchroom for police department employees in that area of the building . Further, it was observed that the dispatchers often had difficulty getting acknowledgements from patrol cars for periods of perhaps forty-five minutes around the time of shift change . If one or more of these phenomena=location ambiguities, telephone room distractions, or patrol shift change--occurred about the time of peak load, it would be reasonable to expect at least isolated instances of misplaced, mishandled, or delayed assignments .
PHILOSOPHICAL BASIS FOR THE ANALYTIC APPROACH Machol (1975) in an article entitled "The Titanic Coincidence" pointed out the need for the use of operations research to illuminate the possibility of catastrophic failure in complex systems . The predictability of recurrence of rare events was first studied by von Bortkewitsch (1898) in the late
29 0
ROBERT P. ECKERT and PETER M . KELLY
Approx 40' Approx 26'
f-
Dispatch Station (I of 4) Card Transport Apparatus Cards Receive Time Stamp and Incident Number
O Address Look-up ..0 (I of 2)
O
s
O
\-T
O
0 Approx 18'
O
DARKENED RADIO DISPATCH ROOM
O O Telephone Position (1 of 10) *-O
0 0
COMPUTER INQUIRY AND TELETYPE OPERATIONS
OFFICE OF SUPERVISORY SERGEANT
Approx 11'
I
i
TELEPHONE ROOM
Figure 2 . General layout of the communications center . (The above is an overview intended to illustrate general features only . Note, for example, that the telephone room cannot be seen from the supervisor's position and--because of low light levels-the dispatch stations are also not readily visible .) nineteenth century . He examined such unlikely events as suicides among young children in Prussia and the numbers of Prussian soldiers who were killed by the kick of a horse . Modem applications of extreme value theory include prediction of floods (Gumbel, 1951) and quality control in mass production (Epstein and Brooks, 1948) . The techniques apply in situations in which the likelihood in a single trial of the event of interest is vanishingly small . Because of the large number of repetitive trials, however, the overall probability of the rare event occurring is not negligible . Evidently, disastrous delays in public safety dispatching operations can be events of negligible probability in individual trials but of non-negligible overall probability . Hence, extreme value theory can be expected to apply readily to public safety . The Wilks tolerance limit (1941) is used to ensure, with a high degree of confidence, that sufficiently large samples have been secured so that almost the entire range of possible values of time delays is included . Extreme value theory is used to predict the future behavior of the system . There are, however, important philosophical questions to be considered before the use of the extreme value
Extreme Value Theory Applied To Police Performance Analysis
29 1
approaches can be accepted as valid in the present application . The Wilks model assumes that the process is in a state of statistical equilibrium . This is certainly not true of the police dispatch operation, since the load on it can vary widely over a period as short as fifteen minutes . One can moderate this difference between model and real world to an extent by limiting the periods of sampling to the busy hours . There are also reasons of economy for such a practice, since, during the busy hours, a single observer can record from ten to one hundred times the number of measurements that might be made during a slow period . An alternative to taking busy-hour observations would be to develop families of measurements of delay components for different conditions of arrival rate of requests for services . At low arrival rates, however, virtually all personnel observe or aid in responding to each individual request . Hence at these times an important additional element of reliability, not present during the busy hours, is introduced through redundancy . A second possible objection to the application of extreme value theory to police emergency delivery systems is that there is no clearly defined threshold for disaster . A five-minute delay in the arrival time of a patrol car might, under certain circumstances, be too long . A fifteen-minute delay in other circumstances might be satisfactory . To eliminate or at least minimize this possible objection, the decision was made to consider only requests identified as emergency in nature . In the Baltimore County Police Department these are Priority One and Priority Two requests, defined formally as crime in progress and serious crime requiring immediate attention . There do exist quantitative indicators of the limits of acceptable response time . The President's Crime Commission reports indicated that delays in excess of six minutes in response to emergency requests were generally unsatisfactory . The more recent National Advisory Commission on Criminal Justice Standards and Goals indicated a similar upper level. There are, at this time, no data on the relative acceptability of various frequencies of occurrence of extreme delays . These events will necessarily occur in the course of many repetitions of the phenomenon . Is one delay in a thousand in excess of ten minutes acceptable? Is one in ten thousand more nearly what should be expected in a well-run operation? For the present, the answers to such questions must be a matter of judgment . It is significant that the extreme value approach described here ignores the queuing aspects of the delivery system and these aspects are, of course, the principal features of the conventional analytic approach . The all-important relationship between the arrival rate of requests for service and the service time, for example, has not been considered . Evidently, in any serving system with a finite service time, there will be periods of unavoidable delay during periods of excessively high arrival rate . It was noted that the system capacity of the Baltimore County system in all its important features of equipment and personnel was more than sufficient to respond to peak loads without the development of queues . A conventional queuing analysis, therefore, could have been expected to reveal that the serving system was adequately designed in the conventional sense .
STATISTICAL BASIS FOR THE ANALYSIS In work of this nature, one must determine the probability that, for a given sample size, a certain proportion of the total population is represented within the range of observed values . Sufficient measurements are then taken to ensure that substantially the complete range of delay magnitudes was observed . Wilks derived the general result . In this study, it was preferable to consider only the case of interest since the mathematics were thereby considerably simplified, and insight into the process was more readily obtained . Consider the x-axis as a one-sided continuum from 0 to m and divided into k mutually exclusive intervalsl,,12, . . . Ik withp,, p2 , . . . p3 as the associated probabilities that an individual observation, x,
292
ROBERT P. ECKERT and PETER M. KELLY
will fall into one of these intervals . In a sample of size n, the probability that n 1 , n2 , . ., nk values of n will fall into the intervals 1r , 12, . . . Ik respectively is
n! pi nr p2 rh . . . p (nl, n2 , . . . nk) = n' in!! . . . nkl
pk nt
(1)
where k
k
C p, = I and S'c = n .
1 = 1
i = 1
To obtain the distribution for which the largest of the observed values isX„ we use the intervals!,,1 2 , and 13 , which cover the ranges, respectively, of (0, Xr), (X„ X, + dY,), (X, + dX„ w) with populations n-1, 1, and 0, respectively . The values of the individualp's are just the integrals of the probability distribution f (X) over the appropriate interval . By substituting these values, one obtains n p (X,) dX, = (n-1)!
Xr n-1 dX f (X,) dX, f f (X)
as the probability that the random variable x, lies between Xr and X, + dX, . This may be written p (X,) dX, = n [ F (Xr) ) n - I d[ F (X,) )
(2)
where F (X,) is the cumulative probability that, x < X, . X, f f (X) dX = F (X,) . 0 With no assumptions about distribution except appropriate continuity, F(X,) represents in probability the fraction of the population included to the left of the largest of n observations, and .(2) is the probability that this fraction lies between F (X,) and F (X,) + d [ F (X,) ] . The probability, then, that at least the fraction a of the total population is included between 0 and X, is
r
,,I n-1
Pa (a) = n f I F (Xr11
d [F (X,)j (3)
P,, (a) =1-a' . With these results, then, one may calculate the confidence that for n measurements, at least a fraction, a, of the total measurements was observed . The results of such calculations are shown in table 1 . In our study, it was determined that a minimum of one thousand observations had to be obtained to have a reasonably high confidence that all but 0 .5 percent of the total range had been observed . This goal was achieved and exceeded for all measurements except for those of phone call duration and dispatch time . For dispatch time, the most difficult of the observations, only 287 observations were taken . A total of 574 observations were recorded for phone call duration. The next probability value of interest is the probability, P,, that, in N future trials, the rnr" largest delay observed in a reference sample of n measurements will be equalled or exceeded at least once. Indicating the probability that this event will not take place by Q,,,, then Pa = 1 - Q,a.
293
Extreme Value Theory Applied To Police Performance Analysis TABLE 1 SAMPLE SIZES AND CONFIDENCE LIMITS
n
Pn (0 .950)
Pn (0 .970)
Pit (0.990)
Pn (0 .995)
100 200 500 1,000 1,200 1,500
0.994 0 .999+ 0.999+ 0.999+ 0 .999+ 0 .999+
0 .952 0 .997 0.999+ 0.999+ 0 .999+ 0 .999+
0 .634 0 .866 0.993 0.999+ 0 .999+ 0 .999+
0 .394 0 .633 0.918 0.993 0 .998 0 .999+
.997) Pit (0 0 .260 0 .452 0.777 0.950 0 .973 0 .989
NOTE : P (a) is the probability that at least the fraction a of the population of delay times is included within the range of the largest of n observations . To determine Q',,, one forms the ratio of the number of possibilities for non-extreme values to the total number of possibilities . The m extreme values can occur in the n reference measures in n!l(n-m)! ways . TheN measurements along with the (n-m) non-extreme values can occur in (N + n - m)! ways . The universe of all possibilities is (N + n)!, hence
Q. Q
n!
(n-m)!
(N + n-m)! (N + n)!
(4)
The distribution expressed by equation (3) was derived by Gumbel and von Schelling (1950) in terms of an underlying distribution function that was general in form except that the variate was assumed to be continuous . The simpler derivation given here was described by Sarkadi (1957) . The approximation of interest in the present application is that of m«n, m<
= [ nl(N+n) ]
m
and finally, P," = I - [ n/(N+n) ]m'
(S)
RESULTS OF THE MEASUREMENT PROGRAM Table 2 shows the number of observations made in each of the various activities identified in figure 1 . The objective, based on the statistical analysis presented earlier, was one thousand observations of each activity . Figure 3 is typical of the results obtained . It shows the distribution of observed delays for transport of the complaint card from the telephone operator's position to that of the dispatcher . This is designated as "transit time" to avoid any connotation of patrol vehicle travel . (The time for patrol vehicle travel is a separate and substantial portion of total delay .) Transit time was defined as beginning at the moment the complaint card was placed on the conveyor belt and continued while the card traveled to the stamping position, was inspected, stamped with a complaint number, and
ROBERT P. ECKERT and PETER M . KELLY
294
TABLE
2
SAMPLE SIZES OBTAINED
Number of Operation
Telephone Ring Time
Measurements
1,090
Comment
This number of measurements provides a confidence level of 99% that no more than 0 .5% of the range of ring time delay lies above the largest delay observed .
Phone Call Duration
574
Observations relatively difficult to make . Resulting confidence is 95% that no more than 0,5% of range escaped observation .
Complaint Card Preparation
961
Objective of 1,000 observations effectively obtained .
Delay for Card Transport
Dispatching Time Vehicle Travel Time
1,018
287
1,539
Many of these observations were made during same hourly periods as phone call duration permitting some predictions about coincidence of extremes . Difficult to observe . Confidence only about 80% for inclusion of all but 0 .5% of range . Data obtained from keypunched records of police department.
returned to the conveyor belt . Because of the division of the measurement tasks, transit time measurements excluded a final phase of fixed and almost negligible duration amounting to about three seconds for the transport of the complaint card from the stamping station to the dispatcher . If one were to assign a reasonably expected value to the transit time duration, a suitable choice might be fifteen to twenty seconds from consideration of the mechanical and manual activities involved . This assumes normal alertness on the part of the operator and reflects the simplicity of the operation . In fact, the distribution in figure 3 shows that a substantial 40 percent of the samples exceed twenty seconds . Further, the fact that there is an extremely long tail in the distribution indicates a tendency for system breakdown . Characteristically long tails were observed in the distributions of all the delay components of figure 1 . As might be expected in these cases of one-sided distributions involving measurements that algebraically are all positive, the mean values uniformly and substantially exceed the modal values . The general characteristics of the observed distributions are presented in table 3 . Of particular significance are the extreme values of the longest delays observed . Reflection on the data in table 3 produces the alarming generalization that the observed modes and means correspond approximately to the upper and lower bounds of what many law enforcement
~.i.t~on Room
••.•U
. ~~m ∎ • •o
monsoons on owns lnewommosommmon mosommon sommums onMINEEM NINE UMMINEEMME so an Mommummom :
:Co
∎• U •U=fm MIMEii
m~
~i.Riii=i
∎∎∎∎∎∎ ∎∎ i∎∎∎ _.∎∎__~/∎~ ._/ .
∎
w./∎ . . ∎ ./ ∎ff∎∎/∎=~f ∎.∎ . .∎
∎~ ..=. : nnu • u∎G Ease i ∎.
m.f∎ . / . .// m
as on n asomm=i/ tt3/w .mtttf
oo
`-' ---
-i
o'?
c
---
p t
; ;-
m a
-
∎∎/∎.∎.w∎
own
f.
w ∎ ∎ ∎
wf
i∎.
smammm / ∎ ∎ E..a ∎ Boom
No : of data points : 1 .018 Average transit time : 25 sec . • No . of observations of the specific transit time . ° Cumulative no . of observations for transit times equal to or less than the abscissa value .
i ~- -
u
T rn
c d
TRANSIT TIME (IN SECONDS)
mannounm~amom Eninloom OEM ani:~ am I MEMO ~a° w f/∎so mumm iii= ∎®Zi==iii=== iii=ii i a nowQ .nnw os as n on o/~owaa.n.o.w._ ./a>?t...∎/ff.w =
∎∎∎∎∎nfl7 .f.∎∎ ./f∎∎
Mommons low us ones zo
30
∎∎
. . . ∎ No n. /iiu. . i ∎. •a. ua loll i ugong
∎
Figure 3 . Observed distribution of transit times .
100
200
300
700
800
1001
296
ROBERT
P . ECKERT and PETER M . KELLY TABLE 3
COMPARISON OF MODE, MEAN, AND TAO. EXTENSION OF THE DISTRIBUTION OF OBSERVED DELAY COMPONENTS
COMPONENT ACTIVITY OF DISPATCHING OPERATION
CHARACTERISTIC OF OBSERVED DISTRIBUTION MODE MEAN LONGEST DELAY (Approx . Sec .) (Sec .) OBSERVED (Sec .)
Telephone Ring Time
2
10
193
Phone Call Duration
42
58
286
Card Preparation Time
50
65
960
Complaint Card Transport
15
25
282
Dispatching Time
35
115
1,142
180
490
3,420
Vehicle Travel Time
professionals consider to be acceptable performance . The observed mode and mean for ring time, for example, correspond roughly to response within three rings . For vehicle travel time, the mode and mean span the range between three and eight minutes . Indeed if, as is the conventional approach, the emphasis had been on the average values, the department's performance would have appeared to be satisfactory . The consideration of the extreme values reveals in quantitative terms the shortcomings of this particular operation . We may speculate about whether these long delays would have been recorded at all in a measurement program that, in the conventional fashion, concentrated on the observation of the typical dispatching performance and the development of steady-state queuing behavior .
SIGNIFICANCE OF THE
RESULTS OBTAINED
Table 4 provides the reccurrence predictions that were developed for all of the delay components . These results indicate that no one of the individual activities of the dispatching operation could be considered to be within acceptable performance limits in spite of acceptable average performance . Through use of extreme value theory, therefore, it was possible to demonstrate to the executives of the Baltimore County Police Department the operational implications of shortcomings both in the operational procedures and in the facilities of their existing police emergency service delivery system . In turn, the development in quantitative terms of the probable future effects of these shortcomings provided justification for a program of improvements that was readily acceptable to both the management and the operating personnel .
Extreme Value Theory Applied To Police Performance Analysis
297
TABLE 4 EXPECTED LANG DELAYS AND THEIR IMPLICATIONS
DELAY COMPONENT
EXTREME DELAY VALUE (Minutes)
OPERATIONAL IMPLICATIONS
Ring Time
1
Potential hang-up by caller that may occur in crime-in-progress situation .
Phone Call Duration
3
Telephone interviews lasting this long suggest confusion and high probability of misunderstandings in critical situations .
Card Preparation Time
5
Suggests the need for a complete analysis of the communication center operation since this relatively routine operation should not be a cause of delay .
Transit Time
3
Besides adding substantially to overall delay, three-minute delays indicate difficulties in basic procedures .
Dispatching Time
14
Adds substantially to overall delay in response and implies poor communications between dispatchers and field forces .
Vehicle Travel Time
50
A delay too large to be tolerated .
NOTE : On the basis of the observational data and the mathematical prediction techniques of extreme value theory, delay values greater than those above can be expected to occur at least once in every 2,000 emergency calls with probability exceeding 99 .5% .
Figure 4 presents recurrence predictions for the dispatching delay . The dispatch operation commences when the assignment card arrives at the radio dispatch desk and terminates when the patrol car acknowledges the receipt of assignment . The predictions take the form of the probability of delays equal to or exceeding various specific values in the course of two thousand future emergency calls . It was estimated that two thousand of these peak-period emergency requests for assistance occur in two weeks' time . The specific delay values in figure 4 are the largest values actually observed in the course of the measurement program, and these serve as benchmarks in the prediction technique employed here . The largest observed delay at the dispatcher's station was 1,142 seconds, and figure 4 indicates that a value this large or larger will occur with about 90 percent probability in the handling of two thousand future peak-period requests of an emergency nature . Figure 4 also
ROBERT P . ECKERT and PETER M . KELLY
298
Y\ •\ r w tN•\•MtSL LAOarM~\t~/Y tL•Y R•~LLL iiiiiii'o"MOM= e ....L.u.n . .L\:iLlla aA'YLLLLLm a \Y a""Y A LoW~L\At'Lww•a \w • L° •o
LLLML..w.. .. . aL.Mm ..ty
100
is ^
U a trl Q H~ 1 .1
Me
„e
p
95
V {rl wz
o <. a¢ o
,LL uu mLLMN.~N\\/MtL.m/A .\. .r ... .\ ..X. .W .. o n . IYtN/\ NLpW _ ML.p LLLSLLL ..LLL SN - . uI • L% .~.q.LS •m . .• N \ .\~ .N~YSY .LS m • tM . Y.m N •.M..Y .NN ..N.Nm. ..a..M . .lmrNm .M .. YW. .RR . .WY ../M . ..AYW.flfl.n NW ...\. .. .NLALu .•n . . .n .s s . . ...HON.\.GSM . ..t Y .\.. . .t Y Ym . .LL M M .mwNmmY a .N. L..~ .. t.Mt ... .W.. .~ . /. ... .L. L~~/t/MN. ..mtm....Y w.M\M .m MMtt/pWR . .I. .mR LL NYwW ..wR. . .WYMr.Am .mw.M Y NX. ..I M.LL • N .. .M L/ ../ .m.\ ..\Y . ..AWNNYN.m. . WN Y /.\ /\. ./t \ N\,.I~rt ...T'T.R.. ./ .Yta ...taM . ..Y.aM .WMr .6. .\ .. .Y. ..M/t .AtYwXNM .mMYMAY\, V . .N . .w/ •X N\MM. .NRMmN\/ . M. . • .MLOM\ ..M\..\ . .a. . .N.. .MMY~ .V .4LAG \A Y.m ^M .wY./ .w.. .yllNwY . .IyISLLL~ .l \. u R at Y \ l.N • .w . .LY Mma... L . \MY . .\N N 1,..\RM. X.yNWW..SLLLLLLL ... .Wa M S .Y ..mM . \ ..\.La{.. .\.. M.LL pA.w..Ynawo .WNMw. .Y .a.v+X .. .. N LN°LL Rma.a .m\ .w .. .N •.L M.M\SNiL.LLAmL: .Mw WMNtY4 .S..M .LM .YS SLoMYM.. • .. Nlw .rMMA • • . W .Y\.w WwA.YI..AYafm w.N. : 'LLt . .MN// \ \t\ .\pLMapp . . .tr LMr wm .a LSLAYRM.y.AMaY .YM.Y. .vA ...M.SL M~L wmmMY • .MM. .wt.YtNNa .Ym.YN\1 \ LLL • .Y .W. . .n. LLw.m.mmww S!. .\ ...LY .SY'u.Y. .. L..l pLLS • •• L •• SS•u5 LLLSSSSLLLSS .Y un. t ..ft • LLLS•Y. . .YSYSL eusn.nn pwgRYY. . . . Y RR ..O . N .m. .R ..WR..Wr i ° .M\ m\. M . .. LL ::: .:LLNMa. .Li..'LLLLLL' .SLL Y .' N.aL NpYMY/.\t ..YYNMYM.N .\t . ...l. . .MY .LN. .Y\..~,~ /..Nt/MYMlwmMMY .. . . . . N LLLLH+N°L°wtLm Ny~. .rrmp NM .LLL~\t/. .\t .t\N..I. .Y\ . ./ ... .NYYN. M . . .. .nu u. . .a..N .N.\n ./\ /LL t. .wYt .w . . ..\a/. N.ma . .. .• m. ./ .M .m.wm . ..q ./N YmM ..t .. . .Y R t.t .SY.\1.~ .NN ym ...M"t ..at..YMYS ..M YN .. .amwnM..„ . iu . ap. XM \ . .Y .l EM.MM EM L .NN MS Y .l .M.IM.M m. ..N M\. .\//.~ nflYnuu . . . . lt .YILSa ..a wM wo /LLSLL.. .MSLMYMLML/ta. .wM t.LL. M.tt.M.L . M. . .a..AWWY . ..ln fn w... .MyY . N. ... . .Y\.m.\ \MW ...MW. . . ..l. .Wnaanun •A w. . ..mM. ...Mmm. .N . . .... .m ..t\.N .. LL\N..m.YNMM. .r.. LM .L.°Y •. L •m w4Lm .. .AM.NNYMMAN oM.ML..wRW . ..... . .. ... .. . ... ..L MY ..Ym.m\ M' LLS YMmM. W.wW .. L\Y . . ./. .NWYY Y/. ...1. N\M N`t .M NWW • • Y NLm . Nm.. LN. .. . mm .. .. . .MMLr ./ .\. .mI.LL..WL m Nt / ../../Y~ N ..t..M• .MW./. . ..nwA ...\Mawaq .NmwM.X • A./\.lm • 1 Mm .NM.Y .w.u L. BLT. ~Y. . .• • N u u .u.n Y lu . W w=~ w" •. amwr ..A.wy w. w ..^!, • . .CM . .M. N . . Y .°.L.L T'• L° "Memo~~ \\A y ' .\mo .am.t S l\.lY. .=,L MLL1~ .lm .N... . INM .N Li ... M. •N.Li M\. .\ ...~m., .ltl SIM \.Y MUMUrM • .m .N.M ML . .. .' w~imNw.rMW\.m •~.1. .•. Na..l u~ .S M.Ia .l.w. .•A . , ... M. .~ .\INmY.RNYNmm L°nufM~ .m. ~ WY.m .lNa M .. .~..~ Y LLLL. . .ML .w..m\.Y/ •Y t\m .\..\w . .L rN\NY .. ..w. ./.~ L . ..O.L .mRm . Yw/l. ..w•Y .RRW N .M ./t. S~ \AY ..\tmr .0 .LLL°MOnL~ .. N N . .Woo .m\Mm an . \\ .Auaunun nun ML .L °lL M ~NA~O .L .NM .wM .MLm.N.N Nwm N .\m m .tLLM. .S ..iuioLmL .NL.LMu.NNN ...m .M..iLM° •. M .L Nmm a MN\. ~Nt.A. .LLStLLLLL ..I. . .at.XaNNI . . .LA MYI. .N .m~ N .~~ .YMLN . .~~ .. ..M.. ~. .N . .lypY .WM .YMNYy .Y .INy .RR ..~ Y. .. .M~L.\/AN .MN m .. . . L ..W'aMS\ .. .. . w\ M. . .! .l ..Y. .rm.. .NMMmt . ..\M • .S\ ..t . .. . • W9 . .. .LMY . . .N. =. ..mL r.N . M. . ml .Y.Mn n.n .Lw aA . . . .m~~//m. .L •N N .. wN . ..Y .mM.M .\.r r M ..L. . 'L ` . .t .. ' :::X.M.LLL . .. 7N .m. .°.L~ •FL°w .°.Lw L` .. u.uu.LN"" . ..n.'jL N .w..~m NN .+MAW.RN .\..L •\.nw"j i"UMM af. i:. Lw LrLLL LLMMmrwm L..a.m~ WLmas . . . .aer." . . .• .Y •al / . MN NM .. . .m.Mw.MM t/ N ..M.N .LMN YL ..Nm..m •l mN~~ 4L .Yl.YI :1\ . w MM .N .. . .. .m ..YM.Y .m•m YMMO.MLLWL\ ..\L.Nw. .mY.W .S .\ .m ..tjjw.y n..t M 0 Mr. n.nn r. . \.a . l . .MY ..M. \.m ..LN M. ..m.//N.. .. .. M .M MLN Nm..M\ : NMY •M NNY .mt\.. . : .\.l\ .NNw Y m.Mmm/.Mm • lNYmlw..mY NY . .t./\Nt.t .. . . Nm ..M.a .. . / tY.M .m ..M w m .YmLgt N r\wY..mY M~ SS ..L r.mlYpy.lR\Yt .. .Rt.NM.A. .MOM . .LLSLL.N.'r' /MMNN..nY Mm Y M .M ..~ ..NNtNmNa. ..../ ...N . .. XtYNm. .m. . NmY m .Y %. l IWn.M.NL •iil. . .\ .MMLLI. . .NM p ~LL . M MYMMMmuw..unuu YlM \. .N ..mmMY .•MMRMr ...\. ..~ •Y. .NmW.Y •Y.. L.N.\ ruL \ MM MLNM.. ..\ .Ywm .W.m~\M .MMYmN YM WY .m. l ...\..Y m p..mYN mM .nu• ..l Yw .NYM • ./mp m . ... . .Msu ~L LMMNu .N..Nmuu.MMN .MwMLL. .mNNX . m .. .M.MMS L u . . .LLuo . u . n .M N~nM~nL Mu .L u u.unuL.w . xML=~ .~ .=Yg .„u : •.uS:LL . . .u.L
90
800
900
1,000
1,100
1,200
DISPATCHING TIME (SECONDS) Figure 4 . Probability that in 2,000 emergency calls there will be at least one call in which dispatching time will equal or exceed the indicated value .
indicates that it is nearly certain that in two thousand of these trials, a dispatch duration time will occur that is at least as large as fourteen minutes . Figure 5 presents recurrence predictions for vehicle travel time . The extreme values are here measured in minutes, and it appears nearly certain that delays exceeding fifty minutes can be expected to occur every one or two weeks .
ACKNOWLEDGMENTS The civilian and uniformed personnel of the Baltimore County Police Department contributed in important ways in making it possible to obtain these results. The operational personnel cooperated wholeheartedly in the data collection . The department management, in addition, consistently supported the work ; both immediate and long-term improvement programs were initiated in response to the findings described here . The statistical development has benefited from a review by Dr . George R . Cooper of the Purdue University School of Engineering .
Extreme Value Theory Applied To Police Performance Analysis
ME
100
299
=a
unn
IIA s. 9n1 ∎ . slow wG=•Gi G '3w ~N/∎∎ ∎ GGG//M.A~JrN/ ∎∎i~rA∎ ra~rr∎∎G∎∎∎ ∎G∎∎∎∎∎∎∎∎/ ~MG∎∎pr pMEN∎G'YGC∎N/GG~k' MEN ME ~∎'GGG∎\pM∎%M ∎ iGGGNGG=G° nuu GGG= Y ∎= ∎o':000GG .N~GG=GGGrM /r • 90 MOON • .. MOM ∎ ∎ ME iG M ∎N∎∎∎pu∎u Y m mm m m r Now ®∎r∎ MO"' q ~∎r∎∎prNgN/ p/∎u∎∎∎NN∎YY∎∎\'r/ /∎N∎/∎∎∎∎r∎ /rY∎∎NN/Y∎∎∎YaGY∎∎∎∎r∎∎r/ ∎//∎ /NN∎∎∎p//∎N W .G∎N∎∎Y∎ ∎∎!~Y!Y f . 77\NM∎Ne N Y.Nw~MOON ...GGw CA Y∎ pASJJ:a'O/qu/Y so F GGquG°GGGG NNp∎/∎∎ ∎N∎N ∎/∎~Y/ ' ∎iGGI`G' W J Q ; ONE m udiM~GiiMCiii M '::: p~ GM ∎a m;G∎N N MN ∎~MEN=MG.a; Z • ; GGGp G:\GGGGGGNG N ∎∎/∎∎r∎∎Np/r//NrN//N/∎∎∎∎∎∎r∎∎pr/∎r∎'\∎∎∎∎/r∎∎// 4
~r .
'n
.u
• W 13.
ri 70 z
V
Np∎∎
MMMrNN0/∎YG ~'GM~'p \G1~000~G /∎N/∎~∎N∎~GG ∎NN∎GG∎∎∎∎∎∎/∎∎∎∎∎/N N∎∎/N∎/∎N∎ //∎\NN∎∎N/N∎∎r∎/∎/∎/\/∎∎∎∎///
u/NN
Z 00
mom MENEM . ' GGGGGG ∎C :000GGGGGGG N°~°GGGGI' 60M∎rr/YNY/N mass ∎pY∎∎//N∎∎∎∎/rrr∎~∎/N////,~///∎/∎ //r∎/∎ mYN/NNp/NN∎pY/∎/N/r/∎∎/r/Nr///∎r∎rrI 0 me • wM a '' •∎• G'~'°'∎∎∎NMN∎/aG ∎N∎/NN∎∎∎∎∎i∎G∎N∎∎∎iGG:∎GG∎NN∎ ~wu :000GGGGG.:GGGGGGGGGG GG∎a∎ ∎∎~IGGGKM∎NGG GGG :g.~000∎ sr∎∎/rN∎∎/∎∎∎∎∎YNN/ /∎∎NN∎N∎ /Yr∎/∎r∎∎r∎r∎∎r∎∎ a sp∎/∎∎p∎∎/∎NN=NN∎GG∎N∎∎N∎∎∎∎ a u∎NY∎∎∎Y∎∎/p∎ /NN/ 50 ENRON .GM.GGaG∎/' ::G:NGGGM000GG//GGGG/%GGG∎GGG NY∎NN∎N//∎∎u∎∎∎ ∎∎/∎∎/∎/∎∎∎NN∎/∎∎∎∎∎/∎∎∎∎∎/∎ uN ∎Y∎∎/∎/M∎GGp/NN∎NN∎NN∎Yr/rNN/Yr∎ N ~/~∎~∎N//∎NN/uN∎∎/pN∎p∎/∎Y∎∎∎//∎//∎//u/∎ ∎GGG∎∎∎%∎G//G// Nrqm~ ∎ pNG∎Nor/NNN/N NGGGGG∎O∎N∎ n∎∎∎∎mm∎ ∎∎r∎N∎r/uG∎NN/∎∎=N∎ G= G::000 ∎ .1∎ ∎'o'%GGGGGaG rp∎/N∎∎G∎∎∎∎∎∎∎∎/∎~N∎Y∎∎∎ N. . . .∎ •
w .∎∎.N .atN .∎ .sa aw 53
54
55
/∎//∎
56
57
TIME DELAY (IN MINUTES) Figure 5 . Probability that in 2,000 emergency runs there will be at least one tun in which travel delay will equal or exceed the indicated value .
REFERENCES Epstein, B ., and Brooks, H . (1948) . The theory of extreme values and its implication of the study of dielectric strength of paper capacitors . Journal of applied physics, 19.54450 . Gumbel, E. 1 . (1951). Statistical theory of extreme values and some practical examples . Applied Mathematics Series No. 33, National Bureau ds, Gaithersburg, Maryland . Gumbet, E . 7., and von Schelling, H . (1950) . The distribution of the number of exceedaaces . Annals of matheamrical statistics, 21 :247-62 .
300
ROBERT P . ECKERT and PETER M . KELLY
Machol, R . E . (1975) . The Titanic coincidence . Interfaces . 53 :53-54 . Sark" . K . (1957) . On the distribution of the number of exceedances . Annals of mathenatical statistics, 28 :1021-23 . von Bonkewitach, L . (1898) . Das Gesetz der Kleinen Zahlen . Leipzig : Teubner . Willis, S . C . (1941) . On the determination of sample sizes for setting tolerance limits. Annals of mathematical statistics, 12:91-96 .
vol . S, pp . 287.300 (1977) . Persa Pms. Printed in U .S .A.
EXTREME VALUE THEORY APPLIED TO POLICE PERFORMANCE ANALYSIS
ROBERT P . ECKERT AND PETER M . KELLY
Kelly Scientific Corporation 4708 Wisconsin Avenue, N .W . Washington, D . C . 20016
ABSTRACT Extreme value theory is used to provide, according to one reviewer, a quantitative measure of the probability of a police dispatch operation going "haywire" under pressure . Conventional queuing theory that concentrates on measures of average performance is not 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 . Neverthetess, 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 would appear to have broad applications and great practical value . The subsequent analysis to identify causal factors is not a part of the discussion.
INTRODUCTION The value of operations research techniques can lie in their ability to provide a quantitative measure of the seriousness of well-recognized management and operational problems . This is in contrast to the more orthodox use of operations research analysis to detect and identify problems that
287
288
ROBERT P . ECKERT and PETER M . KELLY
may have not been suspected earlier . This article illustrates the use of extreme value theory to provide a measure of problem importance and so to supply the information required to break through normal organizational inertia to management and operational changes . The analytic techniques used to determine the seriousness of the problems are described ; the later, detailed analysis to identify causal factors is not discussed . Analysis and the development of recommendations for improvement of the Baltimore County Police dispatching system was the assigned task . An initial walk-through, some preliminary measurements, and applications of accepted but approximate rules of thumb provided significant START TRT (Telephone Ring Time) Estimated 1 .3% of total delay time
TIN (Time for Telephone Interview) Estimated 7 .6% of total delay Tcp (Time for Dispatch Card Preparation) Estimated 8 .5% of total delay (Transit Time for Transport of Dispatch Card) Estimated 3 .2% of total delay
TTR
TVT (Time for Vehicle Travel) Estimated 64.3% of total delay
T o (Time for Dispatch) Estimated 15 .1% of total delay
Figure 1 . Individual delay components and initial estimates of their relative magnitudes . (The estimates of the relative magnitude of the various delay components were developed during the initial weeks of measurements . It was presumed that these estimates would be useful in guiding the analytic activity . In the instances where extreme values of delay occurred these approximate relationships based on average values did not apply . Hence the relationships proved to be useful only as a general overview of the system operations .)
Extreme Value Theory Applied To Police Performance Analysis
28 9
information . The problem was not one of too few dispatchers or overtaxed telephone or radio channels . It was also clear that the average delay response was satisfactory . The distribution mode was on the order of four to five minutes . It was the extreme tail of the response-time distribution that was the problem . There were reports of infrequent delays of unacceptable length and such delays were not restricted to trivial requests but had, from time to time, involved serious incidents . In addition, simple observation revealed that neither the architectural features of the dispatch center nor the operational practices were conducive to a disciplined, repetitive operation . Clearly the orthodox analysis, with emphasis on modal values of delay components, would lead to an illumination of the problem only indirectly and after an extended study . With the cooperation and support of the police executives involved, the decision was made to take a somewhat unusual approach that concentrated attention on the extreme delay values and developed a measure of the frequency of their occurrence .
DESCRIPTION OF THE SYSTEM Figure I identifies the components of the delay response . The physical layout was such that the telephone answering operations were completely isolated, in terms of sight and sound, from the vehicle dispatch operation . (See figure 2 .) The radio dispatch room itself was in semidarkness to facilitate use of electronic displays . Each dispatcher, because of the low light level and the layout of the consoles, operated effectively in isolation receiving the dispatch assignments by moving belt from the telephone room . The police patrol personnel were organized into ten police districts . Patrol personnel reported for their duty shifts through the district headquarters and while on duty might receive assignments from the district headquarters as well as from the emergency communications center . The features just described-the physical layout of the communications center and the district organization of the patrol-interfered with providing clear-cut supervision of the emergency services delivery system . During the initial observation period, three separate but sometimes interrelated phenomena were noted as possibly contributing to the occasional extremely long delays in service delivery time that were the focus of concern . The county has many widely separated streets, lanes, avenues, and roads with similar and sometimes identical names . Because of this, the telephone complaint operator identifies the location while the citizen is on the telephone . This procedure can result in confusion . The civilian personnel employed at the communications center are often not familiar with all parts of the county . In addition, the telephone answering room is a large one with many positions and, at the time of the contractual activity, it was used informally as the lunchroom for police department employees in that area of the building . Further, it was observed that the dispatchers often had difficulty getting acknowledgements from patrol cars for periods of perhaps forty-five minutes around the time of shift change . If one or more of these phenomena=location ambiguities, telephone room distractions, or patrol shift change--occurred about the time of peak load, it would be reasonable to expect at least isolated instances of misplaced, mishandled, or delayed assignments .
PHILOSOPHICAL BASIS FOR THE ANALYTIC APPROACH Machol (1975) in an article entitled "The Titanic Coincidence" pointed out the need for the use of operations research to illuminate the possibility of catastrophic failure in complex systems . The predictability of recurrence of rare events was first studied by von Bortkewitsch (1898) in the late
29 0
ROBERT P. ECKERT and PETER M . KELLY
Approx 40' Approx 26'
f-
Dispatch Station (I of 4) Card Transport Apparatus Cards Receive Time Stamp and Incident Number
O Address Look-up ..0 (I of 2)
O
s
O
\-T
O
0 Approx 18'
O
DARKENED RADIO DISPATCH ROOM
O O Telephone Position (1 of 10) *-O
0 0
COMPUTER INQUIRY AND TELETYPE OPERATIONS
OFFICE OF SUPERVISORY SERGEANT
Approx 11'
I
i
TELEPHONE ROOM
Figure 2 . General layout of the communications center . (The above is an overview intended to illustrate general features only . Note, for example, that the telephone room cannot be seen from the supervisor's position and--because of low light levels-the dispatch stations are also not readily visible .) nineteenth century . He examined such unlikely events as suicides among young children in Prussia and the numbers of Prussian soldiers who were killed by the kick of a horse . Modem applications of extreme value theory include prediction of floods (Gumbel, 1951) and quality control in mass production (Epstein and Brooks, 1948) . The techniques apply in situations in which the likelihood in a single trial of the event of interest is vanishingly small . Because of the large number of repetitive trials, however, the overall probability of the rare event occurring is not negligible . Evidently, disastrous delays in public safety dispatching operations can be events of negligible probability in individual trials but of non-negligible overall probability . Hence, extreme value theory can be expected to apply readily to public safety . The Wilks tolerance limit (1941) is used to ensure, with a high degree of confidence, that sufficiently large samples have been secured so that almost the entire range of possible values of time delays is included . Extreme value theory is used to predict the future behavior of the system . There are, however, important philosophical questions to be considered before the use of the extreme value
Extreme Value Theory Applied To Police Performance Analysis
29 1
approaches can be accepted as valid in the present application . The Wilks model assumes that the process is in a state of statistical equilibrium . This is certainly not true of the police dispatch operation, since the load on it can vary widely over a period as short as fifteen minutes . One can moderate this difference between model and real world to an extent by limiting the periods of sampling to the busy hours . There are also reasons of economy for such a practice, since, during the busy hours, a single observer can record from ten to one hundred times the number of measurements that might be made during a slow period . An alternative to taking busy-hour observations would be to develop families of measurements of delay components for different conditions of arrival rate of requests for services . At low arrival rates, however, virtually all personnel observe or aid in responding to each individual request . Hence at these times an important additional element of reliability, not present during the busy hours, is introduced through redundancy . A second possible objection to the application of extreme value theory to police emergency delivery systems is that there is no clearly defined threshold for disaster . A five-minute delay in the arrival time of a patrol car might, under certain circumstances, be too long . A fifteen-minute delay in other circumstances might be satisfactory . To eliminate or at least minimize this possible objection, the decision was made to consider only requests identified as emergency in nature . In the Baltimore County Police Department these are Priority One and Priority Two requests, defined formally as crime in progress and serious crime requiring immediate attention . There do exist quantitative indicators of the limits of acceptable response time . The President's Crime Commission reports indicated that delays in excess of six minutes in response to emergency requests were generally unsatisfactory . The more recent National Advisory Commission on Criminal Justice Standards and Goals indicated a similar upper level. There are, at this time, no data on the relative acceptability of various frequencies of occurrence of extreme delays . These events will necessarily occur in the course of many repetitions of the phenomenon . Is one delay in a thousand in excess of ten minutes acceptable? Is one in ten thousand more nearly what should be expected in a well-run operation? For the present, the answers to such questions must be a matter of judgment . It is significant that the extreme value approach described here ignores the queuing aspects of the delivery system and these aspects are, of course, the principal features of the conventional analytic approach . The all-important relationship between the arrival rate of requests for service and the service time, for example, has not been considered . Evidently, in any serving system with a finite service time, there will be periods of unavoidable delay during periods of excessively high arrival rate . It was noted that the system capacity of the Baltimore County system in all its important features of equipment and personnel was more than sufficient to respond to peak loads without the development of queues . A conventional queuing analysis, therefore, could have been expected to reveal that the serving system was adequately designed in the conventional sense .
STATISTICAL BASIS FOR THE ANALYSIS In work of this nature, one must determine the probability that, for a given sample size, a certain proportion of the total population is represented within the range of observed values . Sufficient measurements are then taken to ensure that substantially the complete range of delay magnitudes was observed . Wilks derived the general result . In this study, it was preferable to consider only the case of interest since the mathematics were thereby considerably simplified, and insight into the process was more readily obtained . Consider the x-axis as a one-sided continuum from 0 to m and divided into k mutually exclusive intervalsl,,12, . . . Ik withp,, p2 , . . . p3 as the associated probabilities that an individual observation, x,
292
ROBERT P. ECKERT and PETER M. KELLY
will fall into one of these intervals . In a sample of size n, the probability that n 1 , n2 , . ., nk values of n will fall into the intervals 1r , 12, . . . Ik respectively is
n! pi nr p2 rh . . . p (nl, n2 , . . . nk) = n' in!! . . . nkl
pk nt
(1)
where k
k
C p, = I and S'c = n .
1 = 1
i = 1
To obtain the distribution for which the largest of the observed values isX„ we use the intervals!,,1 2 , and 13 , which cover the ranges, respectively, of (0, Xr), (X„ X, + dY,), (X, + dX„ w) with populations n-1, 1, and 0, respectively . The values of the individualp's are just the integrals of the probability distribution f (X) over the appropriate interval . By substituting these values, one obtains n p (X,) dX, = (n-1)!
Xr n-1 dX f (X,) dX, f f (X)
as the probability that the random variable x, lies between Xr and X, + dX, . This may be written p (X,) dX, = n [ F (Xr) ) n - I d[ F (X,) )
(2)
where F (X,) is the cumulative probability that, x < X, . X, f f (X) dX = F (X,) . 0 With no assumptions about distribution except appropriate continuity, F(X,) represents in probability the fraction of the population included to the left of the largest of n observations, and .(2) is the probability that this fraction lies between F (X,) and F (X,) + d [ F (X,) ] . The probability, then, that at least the fraction a of the total population is included between 0 and X, is
r
,,I n-1
Pa (a) = n f I F (Xr11
d [F (X,)j (3)
P,, (a) =1-a' . With these results, then, one may calculate the confidence that for n measurements, at least a fraction, a, of the total measurements was observed . The results of such calculations are shown in table 1 . In our study, it was determined that a minimum of one thousand observations had to be obtained to have a reasonably high confidence that all but 0 .5 percent of the total range had been observed . This goal was achieved and exceeded for all measurements except for those of phone call duration and dispatch time . For dispatch time, the most difficult of the observations, only 287 observations were taken . A total of 574 observations were recorded for phone call duration. The next probability value of interest is the probability, P,, that, in N future trials, the rnr" largest delay observed in a reference sample of n measurements will be equalled or exceeded at least once. Indicating the probability that this event will not take place by Q,,,, then Pa = 1 - Q,a.
293
Extreme Value Theory Applied To Police Performance Analysis TABLE 1 SAMPLE SIZES AND CONFIDENCE LIMITS
n
Pn (0 .950)
Pn (0 .970)
Pit (0.990)
Pn (0 .995)
100 200 500 1,000 1,200 1,500
0.994 0 .999+ 0.999+ 0.999+ 0 .999+ 0 .999+
0 .952 0 .997 0.999+ 0.999+ 0 .999+ 0 .999+
0 .634 0 .866 0.993 0.999+ 0 .999+ 0 .999+
0 .394 0 .633 0.918 0.993 0 .998 0 .999+
.997) Pit (0 0 .260 0 .452 0.777 0.950 0 .973 0 .989
NOTE : P (a) is the probability that at least the fraction a of the population of delay times is included within the range of the largest of n observations . To determine Q',,, one forms the ratio of the number of possibilities for non-extreme values to the total number of possibilities . The m extreme values can occur in the n reference measures in n!l(n-m)! ways . TheN measurements along with the (n-m) non-extreme values can occur in (N + n - m)! ways . The universe of all possibilities is (N + n)!, hence
Q. Q
n!
(n-m)!
(N + n-m)! (N + n)!
(4)
The distribution expressed by equation (3) was derived by Gumbel and von Schelling (1950) in terms of an underlying distribution function that was general in form except that the variate was assumed to be continuous . The simpler derivation given here was described by Sarkadi (1957) . The approximation of interest in the present application is that of m«n, m<
= [ nl(N+n) ]
m
and finally, P," = I - [ n/(N+n) ]m'
(S)
RESULTS OF THE MEASUREMENT PROGRAM Table 2 shows the number of observations made in each of the various activities identified in figure 1 . The objective, based on the statistical analysis presented earlier, was one thousand observations of each activity . Figure 3 is typical of the results obtained . It shows the distribution of observed delays for transport of the complaint card from the telephone operator's position to that of the dispatcher . This is designated as "transit time" to avoid any connotation of patrol vehicle travel . (The time for patrol vehicle travel is a separate and substantial portion of total delay .) Transit time was defined as beginning at the moment the complaint card was placed on the conveyor belt and continued while the card traveled to the stamping position, was inspected, stamped with a complaint number, and
ROBERT P. ECKERT and PETER M . KELLY
294
TABLE
2
SAMPLE SIZES OBTAINED
Number of Operation
Telephone Ring Time
Measurements
1,090
Comment
This number of measurements provides a confidence level of 99% that no more than 0 .5% of the range of ring time delay lies above the largest delay observed .
Phone Call Duration
574
Observations relatively difficult to make . Resulting confidence is 95% that no more than 0,5% of range escaped observation .
Complaint Card Preparation
961
Objective of 1,000 observations effectively obtained .
Delay for Card Transport
Dispatching Time Vehicle Travel Time
1,018
287
1,539
Many of these observations were made during same hourly periods as phone call duration permitting some predictions about coincidence of extremes . Difficult to observe . Confidence only about 80% for inclusion of all but 0 .5% of range . Data obtained from keypunched records of police department.
returned to the conveyor belt . Because of the division of the measurement tasks, transit time measurements excluded a final phase of fixed and almost negligible duration amounting to about three seconds for the transport of the complaint card from the stamping station to the dispatcher . If one were to assign a reasonably expected value to the transit time duration, a suitable choice might be fifteen to twenty seconds from consideration of the mechanical and manual activities involved . This assumes normal alertness on the part of the operator and reflects the simplicity of the operation . In fact, the distribution in figure 3 shows that a substantial 40 percent of the samples exceed twenty seconds . Further, the fact that there is an extremely long tail in the distribution indicates a tendency for system breakdown . Characteristically long tails were observed in the distributions of all the delay components of figure 1 . As might be expected in these cases of one-sided distributions involving measurements that algebraically are all positive, the mean values uniformly and substantially exceed the modal values . The general characteristics of the observed distributions are presented in table 3 . Of particular significance are the extreme values of the longest delays observed . Reflection on the data in table 3 produces the alarming generalization that the observed modes and means correspond approximately to the upper and lower bounds of what many law enforcement
~.i.t~on Room
••.•U
. ~~m ∎ • •o
monsoons on owns lnewommosommmon mosommon sommums onMINEEM NINE UMMINEEMME so an Mommummom :
:Co
∎• U •U=fm MIMEii
m~
~i.Riii=i
∎∎∎∎∎∎ ∎∎ i∎∎∎ _.∎∎__~/∎~ ._/ .
∎
w./∎ . . ∎ ./ ∎ff∎∎/∎=~f ∎.∎ . .∎
∎~ ..=. : nnu • u∎G Ease i ∎.
m.f∎ . / . .// m
as on n asomm=i/ tt3/w .mtttf
oo
`-' ---
-i
o'?
c
---
p t
; ;-
m a
-
∎∎/∎.∎.w∎
own
f.
w ∎ ∎ ∎
wf
i∎.
smammm / ∎ ∎ E..a ∎ Boom
No : of data points : 1 .018 Average transit time : 25 sec . • No . of observations of the specific transit time . ° Cumulative no . of observations for transit times equal to or less than the abscissa value .
i ~- -
u
T rn
c d
TRANSIT TIME (IN SECONDS)
mannounm~amom Eninloom OEM ani:~ am I MEMO ~a° w f/∎so mumm iii= ∎®Zi==iii=== iii=ii i a nowQ .nnw os as n on o/~owaa.n.o.w._ ./a>?t...∎/ff.w =
∎∎∎∎∎nfl7 .f.∎∎ ./f∎∎
Mommons low us ones zo
30
∎∎
. . . ∎ No n. /iiu. . i ∎. •a. ua loll i ugong
∎
Figure 3 . Observed distribution of transit times .
100
200
300
700
800
1001
296
ROBERT
P . ECKERT and PETER M . KELLY TABLE 3
COMPARISON OF MODE, MEAN, AND TAO. EXTENSION OF THE DISTRIBUTION OF OBSERVED DELAY COMPONENTS
COMPONENT ACTIVITY OF DISPATCHING OPERATION
CHARACTERISTIC OF OBSERVED DISTRIBUTION MODE MEAN LONGEST DELAY (Approx . Sec .) (Sec .) OBSERVED (Sec .)
Telephone Ring Time
2
10
193
Phone Call Duration
42
58
286
Card Preparation Time
50
65
960
Complaint Card Transport
15
25
282
Dispatching Time
35
115
1,142
180
490
3,420
Vehicle Travel Time
professionals consider to be acceptable performance . The observed mode and mean for ring time, for example, correspond roughly to response within three rings . For vehicle travel time, the mode and mean span the range between three and eight minutes . Indeed if, as is the conventional approach, the emphasis had been on the average values, the department's performance would have appeared to be satisfactory . The consideration of the extreme values reveals in quantitative terms the shortcomings of this particular operation . We may speculate about whether these long delays would have been recorded at all in a measurement program that, in the conventional fashion, concentrated on the observation of the typical dispatching performance and the development of steady-state queuing behavior .
SIGNIFICANCE OF THE
RESULTS OBTAINED
Table 4 provides the reccurrence predictions that were developed for all of the delay components . These results indicate that no one of the individual activities of the dispatching operation could be considered to be within acceptable performance limits in spite of acceptable average performance . Through use of extreme value theory, therefore, it was possible to demonstrate to the executives of the Baltimore County Police Department the operational implications of shortcomings both in the operational procedures and in the facilities of their existing police emergency service delivery system . In turn, the development in quantitative terms of the probable future effects of these shortcomings provided justification for a program of improvements that was readily acceptable to both the management and the operating personnel .
Extreme Value Theory Applied To Police Performance Analysis
297
TABLE 4 EXPECTED LANG DELAYS AND THEIR IMPLICATIONS
DELAY COMPONENT
EXTREME DELAY VALUE (Minutes)
OPERATIONAL IMPLICATIONS
Ring Time
1
Potential hang-up by caller that may occur in crime-in-progress situation .
Phone Call Duration
3
Telephone interviews lasting this long suggest confusion and high probability of misunderstandings in critical situations .
Card Preparation Time
5
Suggests the need for a complete analysis of the communication center operation since this relatively routine operation should not be a cause of delay .
Transit Time
3
Besides adding substantially to overall delay, three-minute delays indicate difficulties in basic procedures .
Dispatching Time
14
Adds substantially to overall delay in response and implies poor communications between dispatchers and field forces .
Vehicle Travel Time
50
A delay too large to be tolerated .
NOTE : On the basis of the observational data and the mathematical prediction techniques of extreme value theory, delay values greater than those above can be expected to occur at least once in every 2,000 emergency calls with probability exceeding 99 .5% .
Figure 4 presents recurrence predictions for the dispatching delay . The dispatch operation commences when the assignment card arrives at the radio dispatch desk and terminates when the patrol car acknowledges the receipt of assignment . The predictions take the form of the probability of delays equal to or exceeding various specific values in the course of two thousand future emergency calls . It was estimated that two thousand of these peak-period emergency requests for assistance occur in two weeks' time . The specific delay values in figure 4 are the largest values actually observed in the course of the measurement program, and these serve as benchmarks in the prediction technique employed here . The largest observed delay at the dispatcher's station was 1,142 seconds, and figure 4 indicates that a value this large or larger will occur with about 90 percent probability in the handling of two thousand future peak-period requests of an emergency nature . Figure 4 also
ROBERT P . ECKERT and PETER M . KELLY
298
Y\ •\ r w tN•\•MtSL LAOarM~\t~/Y tL•Y R•~LLL iiiiiii'o"MOM= e ....L.u.n . .L\:iLlla aA'YLLLLLm a \Y a""Y A LoW~L\At'Lww•a \w • L° •o
LLLML..w.. .. . aL.Mm ..ty
100
is ^
U a trl Q H~ 1 .1
Me
„e
p
95
V {rl wz
o <. a¢ o
,LL uu mLLMN.~N\\/MtL.m/A .\. .r ... .\ ..X. .W .. o n . IYtN/\ NLpW _ ML.p LLLSLLL ..LLL SN - . uI • L% .~.q.LS •m . .• N \ .\~ .N~YSY .LS m • tM . Y.m N •.M..Y .NN ..N.Nm. ..a..M . .lmrNm .M .. YW. .RR . .WY ../M . ..AYW.flfl.n NW ...\. .. .NLALu .•n . . .n .s s . . ...HON.\.GSM . ..t Y .\.. . .t Y Ym . .LL M M .mwNmmY a .N. L..~ .. t.Mt ... .W.. .~ . /. ... .L. L~~/t/MN. ..mtm....Y w.M\M .m MMtt/pWR . .I. .mR LL NYwW ..wR. . .WYMr.Am .mw.M Y NX. ..I M.LL • N .. .M L/ ../ .m.\ ..\Y . ..AWNNYN.m. . WN Y /.\ /\. ./t \ N\,.I~rt ...T'T.R.. ./ .Yta ...taM . ..Y.aM .WMr .6. .\ .. .Y. ..M/t .AtYwXNM .mMYMAY\, V . .N . .w/ •X N\MM. .NRMmN\/ . M. . • .MLOM\ ..M\..\ . .a. . .N.. .MMY~ .V .4LAG \A Y.m ^M .wY./ .w.. .yllNwY . .IyISLLL~ .l \. u R at Y \ l.N • .w . .LY Mma... L . \MY . .\N N 1,..\RM. X.yNWW..SLLLLLLL ... .Wa M S .Y ..mM . \ ..\.La{.. .\.. M.LL pA.w..Ynawo .WNMw. .Y .a.v+X .. .. N LN°LL Rma.a .m\ .w .. .N •.L M.M\SNiL.LLAmL: .Mw WMNtY4 .S..M .LM .YS SLoMYM.. • .. Nlw .rMMA • • . W .Y\.w WwA.YI..AYafm w.N. : 'LLt . .MN// \ \t\ .\pLMapp . . .tr LMr wm .a LSLAYRM.y.AMaY .YM.Y. .vA ...M.SL M~L wmmMY • .MM. .wt.YtNNa .Ym.YN\1 \ LLL • .Y .W. . .n. LLw.m.mmww S!. .\ ...LY .SY'u.Y. .. L..l pLLS • •• L •• SS•u5 LLLSSSSLLLSS .Y un. t ..ft • LLLS•Y. . .YSYSL eusn.nn pwgRYY. . . . Y RR ..O . N .m. .R ..WR..Wr i ° .M\ m\. M . .. LL ::: .:LLNMa. .Li..'LLLLLL' .SLL Y .' N.aL NpYMY/.\t ..YYNMYM.N .\t . ...l. . .MY .LN. .Y\..~,~ /..Nt/MYMlwmMMY .. . . . . N LLLLH+N°L°wtLm Ny~. .rrmp NM .LLL~\t/. .\t .t\N..I. .Y\ . ./ ... .NYYN. M . . .. .nu u. . .a..N .N.\n ./\ /LL t. .wYt .w . . ..\a/. N.ma . .. .• m. ./ .M .m.wm . ..q ./N YmM ..t .. . .Y R t.t .SY.\1.~ .NN ym ...M"t ..at..YMYS ..M YN .. .amwnM..„ . iu . ap. XM \ . .Y .l EM.MM EM L .NN MS Y .l .M.IM.M m. ..N M\. .\//.~ nflYnuu . . . . lt .YILSa ..a wM wo /LLSLL.. .MSLMYMLML/ta. .wM t.LL. M.tt.M.L . M. . .a..AWWY . ..ln fn w... .MyY . N. ... . .Y\.m.\ \MW ...MW. . . ..l. .Wnaanun •A w. . ..mM. ...Mmm. .N . . .... .m ..t\.N .. LL\N..m.YNMM. .r.. LM .L.°Y •. L •m w4Lm .. .AM.NNYMMAN oM.ML..wRW . ..... . .. ... .. . ... ..L MY ..Ym.m\ M' LLS YMmM. W.wW .. L\Y . . ./. .NWYY Y/. ...1. N\M N`t .M NWW • • Y NLm . Nm.. LN. .. . mm .. .. . .MMLr ./ .\. .mI.LL..WL m Nt / ../../Y~ N ..t..M• .MW./. . ..nwA ...\Mawaq .NmwM.X • A./\.lm • 1 Mm .NM.Y .w.u L. BLT. ~Y. . .• • N u u .u.n Y lu . W w=~ w" •. amwr ..A.wy w. w ..^!, • . .CM . .M. N . . Y .°.L.L T'• L° "Memo~~ \\A y ' .\mo .am.t S l\.lY. .=,L MLL1~ .lm .N... . INM .N Li ... M. •N.Li M\. .\ ...~m., .ltl SIM \.Y MUMUrM • .m .N.M ML . .. .' w~imNw.rMW\.m •~.1. .•. Na..l u~ .S M.Ia .l.w. .•A . , ... M. .~ .\INmY.RNYNmm L°nufM~ .m. ~ WY.m .lNa M .. .~..~ Y LLLL. . .ML .w..m\.Y/ •Y t\m .\..\w . .L rN\NY .. ..w. ./.~ L . ..O.L .mRm . Yw/l. ..w•Y .RRW N .M ./t. S~ \AY ..\tmr .0 .LLL°MOnL~ .. N N . .Woo .m\Mm an . \\ .Auaunun nun ML .L °lL M ~NA~O .L .NM .wM .MLm.N.N Nwm N .\m m .tLLM. .S ..iuioLmL .NL.LMu.NNN ...m .M..iLM° •. M .L Nmm a MN\. ~Nt.A. .LLStLLLLL ..I. . .at.XaNNI . . .LA MYI. .N .m~ N .~~ .YMLN . .~~ .. ..M.. ~. .N . .lypY .WM .YMNYy .Y .INy .RR ..~ Y. .. .M~L.\/AN .MN m .. . . L ..W'aMS\ .. .. . w\ M. . .! .l ..Y. .rm.. .NMMmt . ..\M • .S\ ..t . .. . • W9 . .. .LMY . . .N. =. ..mL r.N . M. . ml .Y.Mn n.n .Lw aA . . . .m~~//m. .L •N N .. wN . ..Y .mM.M .\.r r M ..L. . 'L ` . .t .. ' :::X.M.LLL . .. 7N .m. .°.L~ •FL°w .°.Lw L` .. u.uu.LN"" . ..n.'jL N .w..~m NN .+MAW.RN .\..L •\.nw"j i"UMM af. i:. Lw LrLLL LLMMmrwm L..a.m~ WLmas . . . .aer." . . .• .Y •al / . MN NM .. . .m.Mw.MM t/ N ..M.N .LMN YL ..Nm..m •l mN~~ 4L .Yl.YI :1\ . w MM .N .. . .. .m ..YM.Y .m•m YMMO.MLLWL\ ..\L.Nw. .mY.W .S .\ .m ..tjjw.y n..t M 0 Mr. n.nn r. . \.a . l . .MY ..M. \.m ..LN M. ..m.//N.. .. .. M .M MLN Nm..M\ : NMY •M NNY .mt\.. . : .\.l\ .NNw Y m.Mmm/.Mm • lNYmlw..mY NY . .t./\Nt.t .. . . Nm ..M.a .. . / tY.M .m ..M w m .YmLgt N r\wY..mY M~ SS ..L r.mlYpy.lR\Yt .. .Rt.NM.A. .MOM . .LLSLL.N.'r' /MMNN..nY Mm Y M .M ..~ ..NNtNmNa. ..../ ...N . .. XtYNm. .m. . NmY m .Y %. l IWn.M.NL •iil. . .\ .MMLLI. . .NM p ~LL . M MYMMMmuw..unuu YlM \. .N ..mmMY .•MMRMr ...\. ..~ •Y. .NmW.Y •Y.. L.N.\ ruL \ MM MLNM.. ..\ .Ywm .W.m~\M .MMYmN YM WY .m. l ...\..Y m p..mYN mM .nu• ..l Yw .NYM • ./mp m . ... . .Msu ~L LMMNu .N..Nmuu.MMN .MwMLL. .mNNX . m .. .M.MMS L u . . .LLuo . u . n .M N~nM~nL Mu .L u u.unuL.w . xML=~ .~ .=Yg .„u : •.uS:LL . . .u.L
90
800
900
1,000
1,100
1,200
DISPATCHING TIME (SECONDS) Figure 4 . Probability that in 2,000 emergency calls there will be at least one call in which dispatching time will equal or exceed the indicated value .
indicates that it is nearly certain that in two thousand of these trials, a dispatch duration time will occur that is at least as large as fourteen minutes . Figure 5 presents recurrence predictions for vehicle travel time . The extreme values are here measured in minutes, and it appears nearly certain that delays exceeding fifty minutes can be expected to occur every one or two weeks .
ACKNOWLEDGMENTS The civilian and uniformed personnel of the Baltimore County Police Department contributed in important ways in making it possible to obtain these results. The operational personnel cooperated wholeheartedly in the data collection . The department management, in addition, consistently supported the work ; both immediate and long-term improvement programs were initiated in response to the findings described here . The statistical development has benefited from a review by Dr . George R . Cooper of the Purdue University School of Engineering .
Extreme Value Theory Applied To Police Performance Analysis
ME
100
299
=a
unn
IIA s. 9n1 ∎ . slow wG=•Gi G '3w ~N/∎∎ ∎ GGG//M.A~JrN/ ∎∎i~rA∎ ra~rr∎∎G∎∎∎ ∎G∎∎∎∎∎∎∎∎/ ~MG∎∎pr pMEN∎G'YGC∎N/GG~k' MEN ME ~∎'GGG∎\pM∎%M ∎ iGGGNGG=G° nuu GGG= Y ∎= ∎o':000GG .N~GG=GGGrM /r • 90 MOON • .. MOM ∎ ∎ ME iG M ∎N∎∎∎pu∎u Y m mm m m r Now ®∎r∎ MO"' q ~∎r∎∎prNgN/ p/∎u∎∎∎NN∎YY∎∎\'r/ /∎N∎/∎∎∎∎r∎ /rY∎∎NN/Y∎∎∎YaGY∎∎∎∎r∎∎r/ ∎//∎ /NN∎∎∎p//∎N W .G∎N∎∎Y∎ ∎∎!~Y!Y f . 77\NM∎Ne N Y.Nw~MOON ...GGw CA Y∎ pASJJ:a'O/qu/Y so F GGquG°GGGG NNp∎/∎∎ ∎N∎N ∎/∎~Y/ ' ∎iGGI`G' W J Q ; ONE m udiM~GiiMCiii M '::: p~ GM ∎a m;G∎N N MN ∎~MEN=MG.a; Z • ; GGGp G:\GGGGGGNG N ∎∎/∎∎r∎∎Np/r//NrN//N/∎∎∎∎∎∎r∎∎pr/∎r∎'\∎∎∎∎/r∎∎// 4
~r .
'n
.u
• W 13.
ri 70 z
V
Np∎∎
MMMrNN0/∎YG ~'GM~'p \G1~000~G /∎N/∎~∎N∎~GG ∎NN∎GG∎∎∎∎∎∎/∎∎∎∎∎/N N∎∎/N∎/∎N∎ //∎\NN∎∎N/N∎∎r∎/∎/∎/\/∎∎∎∎///
u/NN
Z 00
mom MENEM . ' GGGGGG ∎C :000GGGGGGG N°~°GGGGI' 60M∎rr/YNY/N mass ∎pY∎∎//N∎∎∎∎/rrr∎~∎/N////,~///∎/∎ //r∎/∎ mYN/NNp/NN∎pY/∎/N/r/∎∎/r/Nr///∎r∎rrI 0 me • wM a '' •∎• G'~'°'∎∎∎NMN∎/aG ∎N∎/NN∎∎∎∎∎i∎G∎N∎∎∎iGG:∎GG∎NN∎ ~wu :000GGGGG.:GGGGGGGGGG GG∎a∎ ∎∎~IGGGKM∎NGG GGG :g.~000∎ sr∎∎/rN∎∎/∎∎∎∎∎YNN/ /∎∎NN∎N∎ /Yr∎/∎r∎∎r∎r∎∎r∎∎ a sp∎/∎∎p∎∎/∎NN=NN∎GG∎N∎∎N∎∎∎∎ a u∎NY∎∎∎Y∎∎/p∎ /NN/ 50 ENRON .GM.GGaG∎/' ::G:NGGGM000GG//GGGG/%GGG∎GGG NY∎NN∎N//∎∎u∎∎∎ ∎∎/∎∎/∎/∎∎∎NN∎/∎∎∎∎∎/∎∎∎∎∎/∎ uN ∎Y∎∎/∎/M∎GGp/NN∎NN∎NN∎Yr/rNN/Yr∎ N ~/~∎~∎N//∎NN/uN∎∎/pN∎p∎/∎Y∎∎∎//∎//∎//u/∎ ∎GGG∎∎∎%∎G//G// Nrqm~ ∎ pNG∎Nor/NNN/N NGGGGG∎O∎N∎ n∎∎∎∎mm∎ ∎∎r∎N∎r/uG∎NN/∎∎=N∎ G= G::000 ∎ .1∎ ∎'o'%GGGGGaG rp∎/N∎∎G∎∎∎∎∎∎∎∎/∎~N∎Y∎∎∎ N. . . .∎ •
w .∎∎.N .atN .∎ .sa aw 53
54
55
/∎//∎
56
57
TIME DELAY (IN MINUTES) Figure 5 . Probability that in 2,000 emergency runs there will be at least one tun in which travel delay will equal or exceed the indicated value .
REFERENCES Epstein, B ., and Brooks, H . (1948) . The theory of extreme values and its implication of the study of dielectric strength of paper capacitors . Journal of applied physics, 19.54450 . Gumbel, E. 1 . (1951). Statistical theory of extreme values and some practical examples . Applied Mathematics Series No. 33, National Bureau ds, Gaithersburg, Maryland . Gumbet, E . 7., and von Schelling, H . (1950) . The distribution of the number of exceedaaces . Annals of matheamrical statistics, 21 :247-62 .
300
ROBERT P . ECKERT and PETER M . KELLY
Machol, R . E . (1975) . The Titanic coincidence . Interfaces . 53 :53-54 . Sark" . K . (1957) . On the distribution of the number of exceedances . Annals of mathenatical statistics, 28 :1021-23 . von Bonkewitach, L . (1898) . Das Gesetz der Kleinen Zahlen . Leipzig : Teubner . Willis, S . C . (1941) . On the determination of sample sizes for setting tolerance limits. Annals of mathematical statistics, 12:91-96 .