Are Call Center and Hospital Arrivals Well Modeled by ...ww2040/Kim_W_MSOM_2014.pdf · Kim and Whitt: Are Call Center and Hospital Arrivals Well Modeled by NHPPs? 466 Manufacturing

MANUFACTURING & SERVICEOPERATIONS MANAGEMENT

Vol. 16, No. 3, Summer 2014, pp. 464–480ISSN 1523-4614 (print) � ISSN 1526-5498 (online) http://dx.doi.org/10.1287/msom.2014.0490

© 2014 INFORMS

Are Call Center and Hospital Arrivals Well Modeled byNonhomogeneous Poisson Processes?

Song-Hee Kim, Ward WhittDepartment of Industrial Engineering and Operations Research, Columbia University, New York, New York 10027

{[email protected], [email protected]}

Service systems such as call centers and hospitals typically have strongly time-varying arrivals. A naturalmodel for such an arrival process is a nonhomogeneous Poisson process (NHPP), but that should be tested by

applying appropriate statistical tests to arrival data. Assuming that the NHPP has a rate that can be regardedas approximately piecewise-constant, a Kolmogorov–Smirnov (KS) statistical test of a Poisson process (PP)can be applied to test for a NHPP by combining data from separate subintervals, exploiting the classicalconditional-uniform property. In this paper, we apply KS tests to banking call center and hospital emergencydepartment arrival data and show that they are consistent with the NHPP property, but only if that data isanalyzed carefully. Initial testing rejected the NHPP null hypothesis because it failed to account for three commonfeatures of arrival data: (i) data rounding, e.g., to seconds; (ii) choosing subintervals over which the rate varies toomuch; and (iii) overdispersion caused by combining data from fixed hours on a fixed day of the week overmultiple weeks that do not have the same arrival rate. In this paper, we investigate how to address each of thesethree problems.

Keywords: arrival processes; nonhomogeneous Poisson process; Kolmogorov–Smirnov statistical test;data rounding; overdispersion

History : Received: June 17, 2013; accepted: March 12, 2014. Published online in Articles in Advance June 2, 2014.

1. IntroductionSignificant effort is being made to apply operationsmanagement approaches to improve the performance ofservice systems such as call centers and hospitals (Aksinet al. 2007, Armony et al. 2011). Because call centersand hospitals typically have strongly time-varyingarrivals, when analyzing the performance to allocateresources (e.g., staffing), it is natural to model thearrival process as a nonhomogeneous Poisson process(NHPP). We usually expect these arrival processesto be well modeled by NHPP’s, because the arrivalstypically arise from the independent decisions of manydifferent people, each of whom uses the service systeminfrequently. Mathematical support is provided by thePoisson superposition theorem; e.g., see Barbour et al.(1992), §11.2 of Daley and Vere-Jones (2008) and §9.8 ofWhitt (2002).

Nevertheless, there are phenomena that can pre-vent the Poisson property from occurring. For exam-ple, scheduled appointments at a doctor’s office andenforced separation in airplane landings at airportstend to make the arrival processes less variable orless bursty than Poisson. On the other hand, arrivalprocesses tend to be more variable or more bursty thanPoisson if they involve overflows from other finite-capacity systems, as occurs in hospitals (Asaduzzamanet al. 2010, Litvak et al. 2008) and with requests for

reservations at hotels, because the overflows tend tooccur in clumps during the intervals when the first sys-tem is full. Indeed, there is a long history of studyingoverflow systems in teletraffic engineering (Cooper1982, Wilkinson 1956). Bursty arrival processes alsooccur if the arrivals occur in batches, as in arrivals tohospitals after accidents. Restaurant arrivals occur ingroups, but these groups usually can be regarded assingle customers. In contrast, batch arrivals in hospi-tals typically use resources as individuals. From theextensive experience in teletraffic engineering, it isknown that departures from the Poisson property canstrongly impact performance; that is supported byrecent work in Li and Whitt (2013), Pang and Whitt(2012). We emphasize this key point by showing theresults of simulation experiments in §3 of the onlinesupplement (available at http://dx.doi.org/10.1287/msom.2014.0490).

1.1. Exploiting the Conditional Uniform PropertyHence, to study the performance of any given servicesystem, it is appropriate to look closely at arrival dataand see if an NHPP is appropriate. A statistical testof an NHPP was suggested by Brown et al. (2005).Assuming that the arrival rate can be regarded asapproximately piecewise-constant (PC), they proposedapplying the classical conditional uniform (CU) propertyover each interval where the rate is approximately

464

Dow

nloa

ded

from

info

rms.

org

by [

128.

59.2

22.1

2] o

n 31

Jul

y 20

14, a

t 16:

40 .

For

pers

onal

use

onl

y, a

ll ri

ghts

res

erve

d.

Kim and Whitt: Are Call Center and Hospital Arrivals Well Modeled by NHPPs?Manufacturing & Service Operations Management 16(3), pp. 464–480, © 2014 INFORMS 465

constant. For a Poisson process (PP), the CU propertystates that, conditional on the number n of arrivals inany interval 601T 7, the n ordered arrival times, eachdivided by T , are distributed as the order statisticsof n independent and identically distributed (i.i.d.)random variables, each uniformly distributed on theinterval 60117. Thus, under the NHPP hypothesis, ifwe condition in that way, the arrival data over severalintervals of each day and over multiple days can all becombined into one collection of i.i.d. random variablesuniformly distributed over 60117.

Brown et al. (2005) suggested applying theKolmogorov–Smirnov (KS) statistical test to see if theresulting data is consistent with an i.i.d. sequence ofuniform random variables. To test for n i.i.d. randomvariables Xj with cumulative distribution function (cdf) F ,the KS statistic is the uniform distance between theempirical cdf (ecdf)

Fn4t5≡1n

n∑

j=1

184Xj/T 5≤t91 0 ≤ t ≤ 11 (1)

and the cdf F , i.e., the KS test statistic is

Dn ≡ sup0≤t≤1

�Fn4t5− F 4t5�0 (2)

We call the KS test of a PP directly after applying theCU property to a PC NHPP the CU KS test; it uses (2)with the uniform cdf F 4t5= t. The KS test compares theobserved value of Dn with the critical value, �4n1�5; thePP null hypothesis H0 is rejected at significance level �if Dn >�4n1�5, where P4Dn >�4n1�5 �H05= �. In thispaper, we always take � to be 0005, in which case it isknown that �4n1�5≈ 1036/

√n for n> 35; see Simard

and L’Ecuyer (2011) and the references therein.

1.2. The Possibility of a Random Rate FunctionIt is significant that the CU property eliminates allnuisance parameters; the final representation is inde-pendent of the rate of the PP on each subinterval. Thathelps for testing a PC NHPP because it allows us tocombine data from separate intervals with differentrates on each interval. The CU KS test is thus thesame as if it were for a PP. However, it is important torecognize that the constant rate on each subintervalcould be random; a good test result does not supportany candidate rate or imply that the rate on eachsubinterval is deterministic. Thus, those issues remainto be addressed. For dynamic time-varying estimationneeded for staffing, that can present a challengingforecasting problem, as reviewed in Ibrahim et al. (2012)and references therein.

By applying the CU transformation to differentdays separately, as well as to different subintervalswithin each day as needed to warrant the PC rateapproximation, this method accommodates the com-monly occurring phenomenon of day-to-day variation

in which the rate of the Poisson process randomlyvaries over different days; see, e.g., Avramidis et al.(2004), Ibrahim et al. (2012), Jongbloed and Koole(2001). Indeed, if the CU transformation is applied inthat way (by combining the data over multiple daystreated separately), then the statistical test should beregarded as a test of a Cox process, i.e., for a doublystochastic PP, where the rate is random over the daysbut is constant over each subinterval over which theCU transformation is applied.

Indeed, even though we will not address that issuehere, there is statistical evidence that the rate functionoften should be regarded as random over successivedays, even for the same day of the week. It is importantto note that, where these more complex models withrandom rate function are used, as in Bassamboo andZeevi (2009) and Ibrahim et al. (2012), it invariablyis assumed that the arrival process is a Cox process,i.e., that it has the Poisson property with time-varyingstochastic rate function. The statistical tests we considercan be used to test if that assumed model is appropriate.

1.3. An Additional Data TransformationIn fact, Brown et al. (2005) did not actually applythe CU KS test. Instead, they suggested applying theKS test based on the CU property after performingan additional logarithmic data transformation. Kimand Whitt (2014) investigated why an additional datatransformation is needed and what form it shouldtake. They showed through large-sample asymptoticanalysis and extensive simulation experiments that theCU KS test of a PP has remarkably little power againstalternative processes with nonexponential interarrival-time distributions. They showed that low power occursbecause the CU property focuses on the arrival timesinstead of the interarrival times; i.e., it converts thearrival times into i.i.d. uniform random variables.

The experiments in Kim and Whitt (2014) showedthat the KS test used by Brown et al. (2005) has muchgreater power against alternative processes with nonex-ponential interarrival-time distributions. Kim and Whitt(2014) also found that Lewis (1965) had discovered adifferent data transformation in Durbin (1961) to useafter the CU transformation and that the Lewis KS testconsistently has more power than the log KS test fromBrown et al. (2005) (although the difference is smallcompared with the improvement over the CU KS test).Evidently, the Lewis test is effective because it bringsthe focus back to the interarrival times. Indeed, the firststep of the Durbin (1961) transformation is to reorderthe interarrival times of the uniform random variablesin ascending order. We display the full transformationin the online supplement.

Kim and Whitt (2014) also found that the CU KStest of a PP should not be dismissed out of hand.Even though the CU KS test of a PP has remark-ably little power against alternative processes with

Dow

nloa

ded

from

info

rms.

org

by [

128.

59.2

22.1

2] o

n 31

Jul

y 20

14, a

t 16:

40 .

For

pers

onal

use

onl

y, a

ll ri

ghts

res

erve

d.

Kim and Whitt: Are Call Center and Hospital Arrivals Well Modeled by NHPPs?466 Manufacturing & Service Operations Management 16(3), pp. 464–480, © 2014 INFORMS

nonexponential interarrival-time distributions, the sim-ulation experiments show that the CU KS test of a PPturns out to be relatively more effective against alterna-tives with dependent exponential interarrival times.The data transformations evidently make the othermethods less effective in detecting dependence becausethe reordering of the interarrival times weakens thedependence. Hence, here we concentrate on the Lewisand CU KS tests. For applications, we recommendapplying both of these KS tests.

1.4. Remaining Issues in ApplicationsUnfortunately, it does not suffice to simply performthese KS tests on arrival data because there are othercomplications with the data. Indeed, when we firstapplied the Lewis KS test to call center and hospitalarrival data, we found that the Lewis KS test inap-propriately rejected the NHPP property. In this paper,we address three further problems associated withapplying the CU KS test and the Lewis refinement fromKim and Whitt (2014) to service system arrival data.After applying these additional steps, we conclude thatthe arrival data we looked at are consistent with theNHPP property, but we would not draw any blanketconclusions. We think that it is appropriate to conductstatistical tests in each setting. Our analysis shows thatthis should be done with care.

First, we might inappropriately reject the NHPPhypothesis because of data rounding. Our experienceindicates that it is common for arrival data to berounded, e.g., to the nearest second. This often producesmany 0-length interarrival times, which do not occurin an NHPP and thus cause the Lewis KS test toreject the PP hypothesis. As in Brown et al. (2005),we find that inappropriate rejection can be avoidedby unrounding, which involves adding i.i.d. smalluniform random variables to the rounded data. In §2 weconduct simulation experiments showing that roundinga PP leads to rejecting the PP hypothesis and thatunrounding avoids it. We also conduct experiments toverify that unrounding does not change a non-PP into aPP, provided that the rounding is not too coarse. If theKS test rejects the PP hypothesis before the roundingand unrounding, and if the rounding is not too coarse,then we conclude that the same will be true after therounding and unrounding.

Second, we might inappropriately reject the NHPPhypothesis because we use inappropriate subintervalsover which the arrival rate function is to be regardedas approximately constant. In §3, we study how tochoose these subintervals. As a first step, we makethe assumption that the arrival-rate function can bereasonably well approximated by a piecewise-linearfunction. In service systems, nonconstant linear arrivalrates are often realistic because they can capture arapidly rising arrival rate at the beginning of the day

and a sharply decreasing arrival rate at the end of theday, as we illustrate in our call center examples. (It isimportant to note that some fundamental smoothnessin the arrival rate function is being assumed; see §3.7for more discussion.) Indeed, ways to fit linear arrivalrate functions have been studied in Massey et al. (1996).However, we do not make use of this estimated arrivalrate function in our final statistical test; we use it onlyas a means to construct an appropriate PC rate functionto use in the KS test. We develop simple practicalguidelines for selecting the subintervals.

Third, we might inappropriately reject the NHPPhypothesis because, in an effort to obtain a largersample size, we might inappropriately combine data frommultiple days. We might avoid the time-of-day effectand the day-of-the-week effect by collecting data frommultiple weeks, but only from the same time of day onthe same day of the week. Nevertheless, as discussedin §1.2, the arrival rate may vary substantially overthese time intervals over multiple weeks. We mayfail to recognize that data from multiple weeks mayin fact have variable arrival rates even though welook at the same time of day and the same day ofthe week. That is, there may be overdispersion in thearrival data. It is often not difficult to test for suchoverdispersion, using standard methods, provided thatwe remember to do so. Even better is to use moreelaborate methods, as in Ibrahim et al. (2012) andthe references therein. If these tests do indeed findthat there is such overdispersion, then we should notsimply reject the NHPP hypothesis. Instead, the datamay be consistent with i.i.d. samples of a Poissonprocess, but one for which the rate function should beregarded as random over different days (and thus astochastic process). In §4 we review how to test foroverdispersion in the arrival data.

After investigating those three causes for inappro-priately rejecting the NHPP hypothesis in §§5 and 6,we illustrate these methods with call center and hos-pital emergency department arrival data. We drawconclusions in §7. Supporting material is provided inthe online supplement (available at http://dx.doi.org/10.1287/msom.2014.0490) and in the online appendix(available at http://www.columbia.edu/∼ww2040/allpapers.html).

2. Data RoundingA common feature of arrival data is that arrival timesare rounded to the nearest second or even the near-est minute. For example, a customer who arrivesat 11:15:25.04 and another customer who arrives at11:15:25.55 may both be given the same arrival timestamp of 11:15:25 (rounding to seconds). That pro-duces batch arrivals or, equivalently, interarrival timesof length 0, which do not occur in an NHPP. If we

Dow

nloa

ded

from

info

rms.

org

by [

128.

59.2

22.1

2] o

n 31

Jul

y 20

14, a

t 16:

40 .

For

pers

onal

use

onl

y, a

ll ri

ghts

res

erve

d.


do not take account of this feature, the KS test mayinappropriately reject the NHPP null hypothesis.

The rounding problem can be addressed by havingaccurate arrival data without rounding, but often thatis not possible, e.g., because the rounding is done inthe measurement process. Nevertheless, as observedby Brown et al. (2005), it is not difficult to addressthe rounding problem in a reasonable practical wayby appropriately unrounding the rounded data. If thedata has been rounded by truncating, then we canunround by adding a random value to each observation.If the rounding truncated the fractional componentof a second, then we add a random value uniformlydistributed on the interval 60117 seconds. We let theserandom values be i.i.d. It usually is straightforward tocheck if rounding has been done, and we would onlyunround to undo the rounding that we see.

2.1. The Need for UnroudingTo study rounding and unrounding, we conductedsimulation experiments. We first simulated 1,000 repli-cations of an NHPP with constant rate �= 11000 (anordinary PP) on the interval 60167, with time mea-sured in hours, so that a mean interarrival time is306 seconds. We then apply the CU KS test and theLewis KS test, as described in Kim and Whitt (2014), tothree versions of the simulated arrival data: (i) raw, asthey are; (ii) rounded, rounded to the nearest second;and (iii) unrounded, in which we first round to thenearest second and then afterward unround by addinguniform random variables on 60117 divided by 3,600(since the units are hours and the rounding is to thenearest second) to the arrival times from (ii), as wassuggested in Brown et al. (2005).

Table 1 summarizes the results of the 1,000 experi-ments. For each of the three forms of the data (raw,rounded, and unrounded) and two KS tests (CU andLewis), we display the number of the 1,000 KS teststhat fail to reject the PP hypothesis at significance level�= 0005 (#Pass), the average p-values (Ave p-value),and the average percentage of 0 values (Ave %0). TheLewis test consistently rejects the PP null hypothesiswhen the arrival data are rounded, but the CU KS testfails to do so. In fact, it is clearly appropriate to rejectthe PP null hypothesis when the data are roundedbecause the rounding produces too many 0-valued

Table 1 Results of the Two KS Tests with Rounding and Unrounding:Poisson Data

CU Lewis

Type #Pass Ave p-value Ave %0 #Pass Ave p-value Ave %0

Raw 944 0.50 0.0 955 0.50 000Rounded 945 0.50 0.0 0 0.00 1207Unrounded 945 0.50 0.0 961 0.50 000

interarrival times. Table 1 shows that the roundingturns 1207% of the interarrival times into 0.

These test results illustrate the advantage of theLewis KS test over the CU KS test. Because the Lewistest looks at the ordered interarrival times, all these0-valued interarrival times are grouped together at theleft end of the interval. As a consequence, the Lewistest strongly rejects the Poisson hypothesis when thedata are rounded. The CU test looks at the data inorder of the initial arrival times, so the 0 interarrivaltimes are spread out throughout the data and are notdetected by the CU KS test. Fortunately, the problemof data rounding is well addressed by unrounding.After the rounding, the Lewis KS test of a PP fails toreject the Poisson hypothesis when applied to a PP.

As in Kim and Whitt (2014), we find that plots ofthe empirical cdf’s used in the KS tests are revealing.Figure 1 compares the average ecdf based on 100replications of a rate-1,000 Poisson process on thetime interval 60167 with the cdf of the null hypothesis.We do not show the 95% confidence intervals for theaverage ecdf’s as they overlap with the average ecdf’s.From these plots, we clearly see that the Lewis test iseffective, whereas the CU KS test fails to detect anyproblem at all.

2.2. The Possible Loss of PowerEvidently, the rounding and subsequent unroundingmakes an arrival process more like an NHPP than itwas before the rounding was performed. It is thusnatural to wonder if the rounding and subsequentunrounding causes a serious loss of power. To exam-ine that issue, we performed additional experiments.First, we simulated 1,000 replications of a renewalarrival process with a constant rate �= 11000 and i.i.d.hyperexponential (H2, a mixture of two exponentials,and hence more variable than exponential) interarrivaltimes Xj with the squared coefficient of variation c2

X = 2on the interval 60167. The cdf of H2 is P4X ≤ x5 ≡

1−p1e−�1x −p2e

−�2x. We further assume balanced meansfor (p1�

−11 = p2�

−12 ) as in (3.7) of Whitt (1982) so that

pi = 61 ±√

4c2X − 15/4c2

X + 157/2 and �i = 2pi. Table 1 ofKim and Whitt (2014) shows that the Lewis test isusually able to detect this departure from the Poissonproperty and to reject the Poisson hypothesis.

Table 2 here shows the results of applying the CUtest and the Lewis test to the renewal arrival processwith H2 interarrival times. We see that the Lewis KS testconsistently rejects the Poisson hypothesis for the rawdata, as it should, but the CU KS test fails to reject in70% of the cases. Moreover, we observe that roundingand unrounding does not eliminate the non-Poissonproperty. This non-Poisson property of the H2 renewalprocess is detected by the Lewis test after the roundingand unrounding. Figure 2 again provides dramaticvisual support as well. (The ecdf’s from the CU testlook similar to the ones provided in Figure 1.)

Dow

nloa

ded

from

info

rms.

org

by [

128.

59.2

22.1

2] o

n 31

Jul

y 20

14, a

t 16:

40 .

For

pers

onal

use

onl

y, a

ll ri

ghts

res

erve

d.


Figure 1 Comparison of the Average ecdf for a Rate-1,000 Poisson Process

0 0.2 0.4 0.6 0.8 1.00

0.2

0.4

0.6

0.8

1.0

x

F(x

)

0 0.2 0.4 0.6 0.8 1.00

0.2

0.4

0.6

0.8

1.0

x

F(x

)

0 0.2 0.4 0.6 0.8 1.00

0.2

0.4

0.6

0.8

1.0

x

0 0.2 0.4 0.6 0.8 1.00

0.2

0.4

0.6

0.8

1.0

x

0 0.2 0.4 0.6 0.8 1.00

0.2

0.4

0.6

0.8

1.0

x

0 0.2 0.4 0.6 0.8 1.00

0.2

0.4

0.6

0.8

1.0

x

Notes. From top to bottom: CU, Lewis test. From left to right: raw, rounded, unrounded.

Table 2 Results of the Two KS Tests with Rounding and Unrounding: H2

Interarrival Times

CU Lewis

Type #Pass Ave p-value Ave %0 #Pass Ave p-value Ave %0

Raw 705 0.21 0.0 0 0.00 000Rounded 706 0.21 0.0 0 0.00 1602Unrounded 706 0.21 0.0 0 0.00 000

We also conducted other experiments of the samekind to show that the unrounding does not inappropri-ately cause the Lewis KS test to fail to reject a non-PP,provided that the unrounding is not done too coarsely.

Figure 2 Comparison of the Average ecdf of a Rate-1,000 Arrival Process with H2 Interarrival Times

0 0.2 0.4 0.6 0.8 1.00

0.2

0.4

0.6

0.8

1.0

x

F(x

)

0 0.2 0.4 0.6 0.8 1.00

0.2

0.4

0.6

0.8

1.0

x0 0.2 0.4 0.6 0.8 1.0

0

0.2

0.4

0.6

0.8

1.0

x

Notes. Lewis test only. From left to right: raw, rounded, unrounded.

Among the more interesting cases are two forms ofbatch Poisson processes. The first is a rate-1,000 renewalprocess in which the interarrival times are 0 with aprobability p and an exponential random variable witha probability 1 − p. The second is a modification ofa PP in which every kth arrival occurs in batches ofsize 2; the arrival rate is reduced to 11000k/4k + 15,so the overall arrival rate is again 1,000. Assumingthat the rounding is done to the nearest seconds asin the PP and H2 examples above, the unroundingconsistently detects the deviation from the PP when pis not too small (e.g., when p ≥ 0005) in the renewalprocess example and when k is not too large (e.g., whenk ≤ 9) in the second modification of a PP with batches.

Dow

nloa

ded

from

info

rms.

org

by [

128.

59.2

22.1

2] o

n 31

Jul

y 20

14, a

t 16:

40 .

For

pers

onal

use

onl

y, a

ll ri

ghts

res

erve

d.


As long as the rounding is not too coarse, the story forthese examples is just like the H2 renewal process wehave already considered. The unrounding removes all0 interarrival times, but it still leaves too many veryshort interarrival times, so the PP hypothesis is stillrejected. The details appear in the online supplement.

On the other hand, if the rounding is too coarse, as inthe batch-Poisson examples above when the rounding isto the nearest minute instead of the nearest second, thenthe unrounding can hide the non-PP character of theoriginal process and thus reduce the power of the KStest. We also illustrate this phenomenon in the onlinesupplement. Overall, rounding should not matter, sounrounding is unnecessary if the rounding is fine, e.g.,to less than 0001 mean service time, whereas there is adanger of a loss of power if the rounding is too coarse,e.g., to more than a mean service time. We recommendusing a simulation to investigate in specific instances,as we have done here. See the online supplement andthe online appendix for more discussion.

3. Choosing Subintervals with a NearlyConstant Rate

To exploit the CU property to conduct KS tests of anNHPP, we assume that the rate function is approxi-mately PC. Because the arrival rate evidently changesrelatively slowly in applications, the PC assumptionshould be reasonable, provided that the subintervals arechosen appropriately. However, some care is needed,as we show in this section. Before starting, we shouldnote that there are competing interests. Using shorterintervals makes the PC approximation more likely to bevalid, but interarrival times are necessarily truncated atboundary points and any dependence in the processfrom one interval to the next is lost when combiningdata from subintervals, so we would prefer longersubintervals unless the PC approximation ceases to beappropriate.

As a reasonable practical first step, we proposeapproximating any given arrival rate function by apiecewise-linear arrival rate function with a finite num-ber of linear pieces. Ways to fit linear arrival ratefunctions were studied in Massey et al. (1996), whichcan be extended to piecewise-linear arrival rate func-tions (e.g., by choosing roughly appropriate boundarytimes and applying the least-squares methods thereover each subinterval with the endpoint values con-strained). However, it usually should not be necessaryto have a formal estimation procedure to obtain asuitable rough approximation. In particular, we do notassume that we should necessarily consider the arrivalrate function as fully known after this step; instead, weassume it is sufficiently well known to determine howto construct an appropriate PC approximation.

In this section, we develop theory to support choos-ing subintervals for any given linear arrival rate func-tion, which we do take as fully known. This theoryleads to simple practical guidelines for evaluatingwhether (1) a constant approximation is appropriatefor any given subinterval with a linear rate and (2) aPC approximation is appropriate for any candidatepartition of such a subinterval into further equallyspaced subintervals; see §§3.4 and 3.6, respectively.Equally spaced subintervals are only one choice, butthe constant length is convenient to roughly judge thedependence among successive intervals.

3.1. A Call Center ExampleWe start by considering an example motivated by thebanking call center data used in Kim and Whitt (2013a,b). For one 17-hour day, represented as 661237 in hours,they produced the fitted arrival rate function

�4t5=

1404t − 65 on [6,10],560 on [10,16],560 − 2304t − 165 on [16,18],100 − 204t − 185 on [18,23],

(3)

as shown in Figure 3 (taken from Kim and Whitt 2013a, b).This fitted arrival rate function is actually constant inthe subinterval 6101167, which of course presents nodifficulty. However, as in many service systems, thearrival rate is increasing at the beginning of the day, asin the subinterval 661107, and decreasing at the end ofthe day, as in the two intervals 6161187 and 6181237.

We start by considering an example motivated byFigure 3. The first interval 661107 in Figure 3 with linearincreasing rate is evidently challenging. To capturethe spirit of that case, we consider an NHPP witha linear arrival rate function �4t5 = 11000t/3 on theinterval 60167. The expected total number of arrivals

Figure 3 Fitted Piecewise-Linear Arrival Rate Function for the Arrivalsat a Banking Call Center

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 230

100

200

300

400

500

600

700

t

Arr

ival

rat

e pe

r ho

ur

Dow

nloa

ded

from

info

rms.

org

by [

128.

59.2

22.1

2] o

n 31

Jul

y 20

14, a

t 16:

40 .

For

pers

onal

use

onl

y, a

ll ri

ghts

res

erve

d.


Table 3 Performance of the Alternative KS Test of an NHPP as aFunction of the Subinterval Length L

CU Lewis

Ave Ave Ave AveL Type #Pass p-value %0 #Pass p-value %0

6 Raw 0 0.00 0.0 0 0.00 000Rounded 0 0.00 0.0 0 0.00 1602Unrounded 0 0.00 0.0 0 0.00 000



0.5 Raw 62 0.01 0.0 946 0.47 000Rounded 69 0.01 0.1 0 0.00 1602Unrounded 66 0.01 0.0 932 0.47 000

0.25 Raw 570 0.19 0.0 953 0.48 000Rounded 578 0.19 0.1 0 0.00 1603Unrounded 563 0.19 0.0 953 0.49 000

over this interval is 6,000. We apply simulation tostudy what happens when we divide the interval 60167into 6/L equally spaced disjoint subintervals, each oflength L; apply the CU construction to each subintervalseparately; and then afterward combine all the datafrom the subintervals.

Table 3 and Figure 4 show the performance of theLewis and CU KS tests as a function of the subintervallength. As before, #Pass is the number of KS testspassed at significance level �= 0005 out of 1,000 replica-tions. It shows the average p-values under Ave p-value

Figure 4 Comparison of the Average ecdf of an NHPP with Different Subinterval Lengths

0 0.2 0.4 0.6 0.8 1.00

0.2

0.4

0.6

0.8

1.0

x

F(x

)

0 0.2 0.4 0.6 0.8 1.00

0.2

0.4

0.6

0.8

1.0

x

F(x

)

0 0.2 0.4 0.6 0.8 1.00

0.2

0.4

0.6

0.8

1.0

x

0 0.2 0.4 0.6 0.8 1.00

0.2

0.4

0.6

0.8

1.0

x

0 0.2 0.4 0.6 0.8 1.00

0.2

0.4

0.6

0.8

1.0

x

0 0.2 0.4 0.6 0.8 1.00

0.2

0.4

0.6

0.8

1.0

x

0 0.2 0.4 0.6 0.8 1.00

0.2

0.4

0.6

0.8

1.0

x

0 0.2 0.4 0.6 0.8 1.00

0.2

0.4

0.6

0.8

1.0

x

Notes. From top to bottom: CU, Lewis test. From left to right: L= 6, 3, 1, 0.5.

and the average percentage of 0 values in the trans-formed sequence under Ave %0. First, we see, justas in §2, that the Lewis test sees the rounding, butthe CU test misses it completely. Second, we concludethat both KS tests will consistently detect this strongnonconstant rate and reject the PP hypothesis with ahigh probability if we use L= 6 (the full interval 60167)or even if L= 1 or 005. However, the Lewis KS testswill tend not to reject the PP hypothesis if we dividethe interval into appropriately many equally spacedsubintervals.

Because we are simulating an NHPP, the actualmodel differs from the PP null hypothesis only throughtime dependence. Consistent with the observations inKim and Whitt (2014), we see that the CU KS test isactually more effective in detecting this nonconstant ratethan the Lewis test. The nonconstant rate produces aform of dependence, for which the CU test is relativelygood. However, for our tests of the actual arrivaldata, we will wish to test departures from the NHPPassumption. Hence, we are primarily interested in theLewis KS test. The results in §§3.3–3.6 below indicatethat we could use L= 005 for the Lewis test.

In the remainder of this section, we develop a theorythat shows how to construct PC approximations of therate function. We then derive explicit formulas for theconditional cdf in three cases: (1) in general (whichis complicated), (2) when the arrival rate is linear(which is relatively simple), and (3) when the datais obtained by combining data from equally spacedsubintervals of a single interval with linear rate (which

Dow

nloa

ded

from

info

rms.

org

by [

128.

59.2

22.1

2] o

n 31

Jul

y 20

14, a

t 16:

40 .

For

pers

onal

use

onl

y, a

ll ri

ghts

res

erve

d.


remains tractable). We then apply these results todetermine when a PC approximation can be consideredappropriate for KS tests.

3.2. The Conditioning PropertyWe first observe that a generalization of the CU methodapplies to show that the scaled arrival times of a generalNHPP, conditional on the number observed within anyinterval, can be regarded as i.i.d. random variables, butwith a nonuniform cdf, which we call the conditional cdf,depending on the rate of the NHPP over that interval.That conditional cdf then becomes the asymptoticvalue of the conditional-uniform Kolmogorov–Smirnovtest statistic applied to the arrival data as the samplesize increases, where the sample size increases bymultiplying the arrival rate function by a constant.

Let N ≡ 8N 4t52 t ≥ 09 be an NHPP with arrival ratefunction � over a time interval 601 T 7. We assume that� is integrable over the finite interval of interest andstrictly positive except at a finite number of points. Letå be the associated cumulative arrival rate function,defined by

å4t5≡

∫ t

0�4s5 ds1 0 ≤ t ≤ T 0 (4)

We will exploit a basic conditioning property of theNHPP, which follows the same reasoning as for thehomogeneous special case. It is significant that thisconditioning property is independent of scale, i.e., it isunchanged if the arrival rate function � is multipliedby a constant. We thus later consider asymptotics inwhich the sample size increases in that way.

Theorem 1 (NHPP Conditioning Property). Let Nbe an NHPP with the arrival rate function c�, where c is anarbitrary positive constant. Conditional upon N4T 5= n forthe NHPP N with arrival rate function c�, the n orderedarrival times Xj , 1 ≤ j ≤ n, when each is divided by theinterval length T , are distributed as the order statisticsassociated with n i.i.d. random variables on the unit interval60117, each with cumulative distribution function (cdf ) Fand probability density function 4pdf5 f , where

F 4t5≡å4tT 5/å4T 5 and f 4t5≡ T�4tT 5/å4T 51

0 ≤ t ≤ 10 (5)

In particular, the cdf F is independent of c.

We call the cdf F in (5) the conditional cdf associatedwith N ≡N4c�1T 5. Let Xj be the jth ordered arrivaltime in N over 601 T 7, 1 ≤ j ≤ n, assuming that we haveobserved n≥ 1 points in the interval 601 T 7. Let Fn4x5 bethe empirical cdf (ecdf) after scaling by dividing by T,defined by

Fn4t5≡1n

n∑

k=1

184Xj/T 5≤t91 0 ≤ t ≤ 10 (6)

We naturally are more likely to obtain larger and largervalues of n if we increase the scaling constant c.

Observe that the ecdf 8Fn4t52 0 ≤ t ≤ 19 is a stochasticprocess with

E6Fn4t57= F 4t5 for all t1 0 ≤ t ≤ 11 (7)

where F is the conditional cdf in (5). As a consequenceof Lemma 1 below and the Glivenko-Cantelli theorem,we immediately obtain the following asymptotic result.

Theorem 2 (Limit for Empirical cdf). Assuming aNHPP with an arrival rate function c�, where c is a scalingconstant, the empirical cdf of the scaled order statisticsin (6), obtained after conditioning on observing n points inthe interval 601 T 7 and dividing by T , converges uniformlyw.p.1 as n→ � 4which may be obtained by increasing thescaling constant c5 to the conditional cdf F in (5), i.e.,

sup0≤t≤1

�Fn4t5− F 4t5� → 0 as n→ �0 (8)

We will usually omit the scaling constant in ourdiscussion but do so with the understanding that italways can be introduced. Becasue we want to see howthe NHPP fares in a KS test of a PP, it is natural tomeasure the degree of nonhomogeneity in the NHPP by

�4NHPP5≡ �4�1T 5=D ≡ sup0≤t≤1

�F 4t5− t�1 (9)

where F is the conditional cdf in (5). The degree ofnonhomogeneity is closely related to the CU KS teststatistic for the test of a PP, which is the absolutedifference between the ecdf and the uniform cdf, i.e.,

Dn ≡ sup0≤t≤1

�Fn4t5− t�3 (10)

see Marsaglia et al. (2003), Massey (1951), Miller (1956),Simard and L’Ecuyer (2011).

As a consequence of Theorem 2, we can describe thebehavior of the conditional-uniform (CU) KS test ofa Poisson process applied to a NHPP with a generalarrival rate function �.

Theorem 3 (Limit of the KS Test of a PoissonProcess Applied to an NHPP). As n→ � in a NHPPwith rate function � over 601T 7,

Dn →D ≡ sup0≤t≤1

�F 4t5− t�1 (11)

where Dn is the CU KS test statistic in (10) and D is thedegree of nonhomogeneity in (9) involving the conditionalcdf F in (5).

Corollary 1 (Asymptotic Rejection of the Pois-son Process Hypothesis If NHPP Is Not a PoissonProcess). The probability that an NHPP with rate functionn� will be rejected by the CU KS test for a PP converges to1 as the scaling parameter n→ � if and only if the � is notconstant w.p.1, i.e., if and only if the NHPP is not actuallya PP.

Proof. It is easy to see that the cdf F in (5) coincideswith the uniform cdf t if and only if �4t5 is constant. �

Dow

nloa

ded

from

info

rms.

org

by [

128.

59.2

22.1

2] o

n 31

Jul

y 20

14, a

t 16:

40 .

For

pers

onal

use

onl

y, a

ll ri

ghts

res

erve

d.


Corollary 1 suggests that a PC approximation of anon-PP NHPP never makes sense with enough data,but we develop a positive result exploiting appropriatesubintervals, where the number of subintervals growswith the sample size n; see Theorem 6.

3.3. An NHPP with Linear Arrival Rate FunctionWe now consider the special case of an NHPP with alinear arrival rate function

�4t5= a+ bt1 0 ≤ t ≤ T 1 (12)

The analysis is essentially the same for increasing anddecreasing arrival rate functions, so we will assume thatthe arrival rate function is increasing, i.e., b ≥ 0. Thereare two cases, a > 0 and a= 0, and we consider themboth. If a > 0, then cumulative arrival rate function canbe expressed as

å4t5≡

∫ t

0�4s5 ds = at +

bt2

2= a

(

t +rt2

2

)

(13)

where r ≡ b/a is the relative slope. If a= 0, then å4t5=

bt2/2.

Theorem 4 (Aymptotic Maximum Absolute Dif-ference in the Linear Case). Consider an NHPP with alinear arrival rate function in (12) observed over the interval601 T 7. If a > 0, then the conditional cdf in (5) assumes theform

F 4t5=tT + 4r4tT 52/25T + 4rT 2/25

1 0 ≤ t ≤ 13 (14)

if a= 0, thenF 4t5= t21 0 ≤ t ≤ 10 (15)

Thus, if a > 0, then the degree of nonhomogeneity of theNHPP can be expressed explicitly as

D ≡ D4rT 5≡ sup0≤t≤1

8�F 4t5− t�9= �F 41/25− 1/2�

=12

−4T /2 + 4rT 25/854T + 4rT 25/25

=rT

8 + 4rT0 (16)

If a= 0, then D = 1/4 4which agrees with (16) when r = �5.

Proof. For (16), observe that �F 4t5− t� is maximizedwhere f 4t5= 1, so that it is maximized at t = 1/2. �

3.4. Practical Guidelines for a Single IntervalWe can apply formula (16) in Theorem 4 to judgewhether an NHPP with a linear rate over an intervalshould be close enough to a PP with a constant rate.(We see that should never be the case for a singleinterval with a= 0 because then D = 1/4.) In particular,the rate function can be regarded as approximatelyconstant if the ratio D/�4n1�5 is sufficiently small,where D is the degree of homogeneity in (16) and�4n1�5 is the critical value of the KS test statistic Dn

with sample size n and significance level �, which we

always take to be �= 0005. Before looking at data, wecan estimate n by the expected total number of arrivalsover the interval.

We have conducted simulation experiments to deter-mine when the ratio D/�4n1�5 is sufficiently small thatthe KS test of a PP applied to an NHPP with that ratefunction consistently rejects the PP null hypothesis withprobability approximately �= 0005. Our simulationexperiments indicate that a ratio of 0010 (0050) shouldbe sufficiently small for the CU (Lewis) KS test with asignificance level of �= 0005.

Table 4 shows the values of D, �4n1�5 and D/�4n1�5along with the test results for selected subintervals ofthe initial example with �4t5= 11000t/3 on the timeinterval 60167. (The full table with all intervals andother examples appear in the online appendix.)

3.5. Subintervals for an NHPP witha Linear Arrival Rate

In this section, we see the consequence of dividing theinterval 601 T 7 into k equal subintervals when the arrivalrate function is linear over 601T7 as in §3.3. As in theCU KS test discussed in Kim and Whitt (2014), we treateach interval separately and combine all the data. Animportant initial observations is that the final cdf F canbe expressed in terms of the cdf’s Fj associated withthe k subintervals. In particular, we have the followinglemma.

Lemma 1 (Combining Data from Equally SpacedSubintervals). If we start with a general arrival ratefunction and divide the interval 601 T 7 into k subintervals oflength T /k, then we obtain i.i.d. random variables with aconditional cdf that is a convex combination of the conditionalcdf’s for the individual intervals, i.e.,

F 4t5 =

k∑

j=1

pjFj4t51 0 ≤ t ≤ 11 where

Fj4t5 =åj4tT /k5

åj4T /k51 0 ≤ t ≤ 11 1 ≤ j ≤ k1

åj4t5 = å444j − 15T /k5+ t5−å44j − 15T /k51

0 ≤ t ≤ T /k1 1 ≤ j ≤ k1

pj =å4jT /k5−å44j − 15T /k5

å4T 51 1 ≤ j ≤ k0 (17)

For the special case of a linear arrival rate function as in(12) with a > 0,

åj4t5 =at4k42+rt5+24j−15rT 5

2k1 0≤ t≤T /k11≤ j≤k1

Fj4t5 =t42k+42j−2+t5rT 5

2k+42j−15rT1 0≤ t≤111≤ j≤k1

pj =2k+42j−15rTk242+rT 5

and

rj =b

�44j−15T /k5=

bk

a4k+4j−15rT 50 (18)

Dow

nloa

ded

from

info

rms.

org

by [

128.

59.2

22.1

2] o

n 31

Jul

y 20

14, a

t 16:

40 .

For

pers

onal

use

onl

y, a

ll ri

ghts

res

erve

d.


Table 4 Judging When the Rate Is Approximately Constant: The Ratio D/�4n1�5 for Single Subintervals with �= 0005

CU Lewis

L Interval Ave[n] r D Ave[�4n1 �5] D/Ave6�4n1 �57 #Pass Ave p-value #Pass Ave p-value

6 60167 5199703 � 00250 00018 14028 0 0.00 0 0.003 60137 1149808 � 00250 00035 7015 0 0.00 0 0.00

63167 4149805 0.33 00083 00020 4012 0 0.00 481 0.151 60117 16608 � 00250 00104 2040 0 0.00 46 0.01

61127 49907 1.00 00083 00060 1038 22 0.01 896 0.4362137 83204 0.50 00050 00047 1007 145 0.03 928 0.4863147 1116609 0.33 00036 00040 0090 300 0.08 931 0.4964157 1150100 0.25 00028 00035 0079 358 0.09 949 0.4965167 1183006 0.20 00023 00032 0072 453 0.13 948 0.49

0.5 6010057 4200 � 00250 00207 1021 46 0.01 562 0.186005117 12408 2.00 00083 00121 0069 479 0.14 918 0.486105127 29200 0.67 00036 00079 0045 766 0.29 945 0.506205137 45609 0.40 00023 00063 0036 833 0.35 960 0.516305147 62303 0.29 00017 00054 0031 865 0.38 938 0.516405157 79201 0.22 00013 00048 0027 882 0.41 936 0.506505167 95606 0.18 00011 00044 0025 893 0.42 951 0.50

0.25 60100257 1004 � 00250 00418 0060 588 0.17 888 0.426002510057 3106 4.00 00083 00239 0035 841 0.37 946 0.496005100757 5108 2.00 00050 00187 0027 885 0.41 943 0.4960075117 7300 1.33 00036 00157 0023 907 0.44 947 0.5061075127 15605 0.57 00017 00108 0015 924 0.48 940 0.4962075137 23807 0.36 00011 00087 0012 931 0.48 956 0.5063075147 32200 0.27 00008 00075 0011 941 0.47 946 0.5064075157 40607 0.21 00006 00067 0010 937 0.48 953 0.5065075167 48902 0.17 00005 00061 0009 941 0.50 943 0.50

For the special case of a linear arrival rate function as in(12) with a= 0,

åj4t5 =bt4kt + 24j − 15T 5

2k1 0 ≤ t ≤ T /k1 1 ≤ j ≤ k1

Fj4t5 =t42j − 2 + t5

2j − 11 0 ≤ t ≤ 11 1 ≤ j ≤ k1

pj =2j − 1k2

and rj =k

4j − 15T1 1 ≤ j ≤ k0 (19)

We now apply Lemma 1 to obtain a simple charac-terization of the maximum difference from the uniformcdf when we combine the data from all the equallyspaced subintervals.

Theorem 5 (Combining Data from EquallySpaced Subintervals). If we start with the linear arrivalrate function in (12), divide the interval 601 T 7 into k subin-tervals of length T /k, and combine all the data, then weobtain

D ≡ sup0≤t≤1

8�F 4t5− t�9=

k∑

j=1

pjDj

=

k∑

j=1

pj sup0≤t≤1

8�Fj4t5− t�90 (20)

If a > 0, then

D =

k∑

j=1

pjrjT /k

8 + 4rjT /k≤C/k for all k ≥ 1 (21)

for a constant C. If a= 0, then

D =p1

4+

k∑

j=2

pj/4j − 15

8 + 4/4j − 15≤C/k for all k ≥ 1 (22)

for a constant C.

Proof. By Theorem 4, by virtue of the linearity, foreach j ≥ 1, �Fj4t5− t� is maximized at t = 1/2. Hence,the same is true for �F 4t5− t�, where F 4t5=

∑kj=1 pjFj4t5,

which gives us (20). For the final bound in (21), userj ≤ 1 + 4T /ka5 for all j . For the final bound in (22), userj = 4j/4j − 1552 ≤ 4 for all j ≥ 2 with p1 = 1/k2. �

3.6. Practical Guidelines for Dividing an IntervalInto Equal Subintervals

Paralleling §3.4, if the rate is strictly positive on theinterval (or if it is 0 at one endpoint), then we canapply formula (21) (respectively, (22)) in Theorem 5 tojudge whether the partition of a given interval withlinear rate into equally spaced subintervals producesan appropriate PC approximation. As before, we lookat the ratio D/�4n1�5, requiring that it be less than 0010(0050) for the CU (Lewis) KS test with significance level�= 0005, where now D is given by (21) or (22) and�4n1�5 is again the critical value to the KS test, butnow applied to all the data, combining the data afterthe CU transformation is applied in each subinterval.In particular, n should be the total observed sample

Dow

nloa

ded

from

info

rms.

org

by [

128.

59.2

22.1

2] o

n 31

Jul

y 20

14, a

t 16:

40 .

For

pers

onal

use

onl

y, a

ll ri

ghts

res

erve

d.


Table 5 Judging If a PC Approximation Is Good for an Interval DividedInto Equal Subintervals: The Ratio D/Ave6�4n1 �57

CU Lewis

L D D/Ave6�4n1 �57 #Pass Ave p-value #Pass Ave p-value

6 002500 140278 0 0.00 0 0.003 001250 70139 0 0.00 0 0.001 000417 20380 0 0.00 797 0.330.5 000208 10190 62 0.01 946 0.470.25 000104 00595 570 0.19 953 0.480.1 000042 00238 896 0.43 955 0.480.09 000038 00214 902 0.43 954 0.480.08 000033 00190 914 0.45 948 0.480.07 000029 00167 923 0.47 960 0.490.06 000025 00143 927 0.47 941 0.490.05 000021 00119 941 0.50 958 0.490.01 000004 00024 953 0.50 948 0.480.005 000002 00012 944 0.49 943 0.480.001 0000004 00002 952 0.50 959 0.49

size or the total expected number of arrivals, addingover all subintervals.

We illustrate in Table 5 by showing the values ofD and D/�4n1�5 along with the test results for eachsubinterval of the initial example with �4t5= 11000t/3on the time interval 60167, just as in Table 3. In all cases,Ave6n7 is 5,997.33, and hence the ave6�4n1�57 valuesare the same and are approximately 0.0175. (Again,more examples appear in the online appendix.)

In summary, we present the following algorithm forchoosing an appropriate subinterval length to enable aPC approximation of a linear arrival rate over someinterval.

1. Given an interval T with a fitted arrival ratefunction that is linear (�4t5= a+bt), let n be the numberof arrivals in that interval.

2. Compute the critical value of the KS test, �4n1�5.It can be approximated as �4n1�5≈ 1036/

√n if n> 35

and when we choose �= 0005. (See Simard and L’Ecuyer2011 and references therein for other values of n and �.)

3. Start with subinterval length L= T /2. Given L,compute the degree of nonhomogeneity of the NHPP Dusing (21) if a > 0 and using (22) if a= 0.

4. Compute D/�4n1�5. Use bisection method to findthe value of L that gives the ratio D/�4n1�5 less than0.10 (0.50) for the CU (Lewis) KS test.

3.7. Asymptotic Justification ofPiecewise-Constant Approximation

We now present a limit theorem that provides usefulinsight into the performance of the CU KS test of aNHPP with a linear rate. We start with a nonconstantlinear arrival rate function � as in (12) and then scale itby multiplying it by n and letting n→ �. We show thatas the scale increases, with the number of subintervalsincreasing as the scale increases appropriately, the KStest results will behave the same as if the NHPP had aconstant rate. We will reject if it should and fail to reject

otherwise (with probability equal to the significancelevel). In particular, it suffices to use kn equally spacedsubintervals, where

kn√n

→ � as n→ �0 (23)

To enable the sample size in each interval to also growwithout bound, we also require that

n

kn→ � as n→ �0 (24)

For example, kn = np satisfies both (23) and (24) if1/2 < p < 1.

Theorem 6 (Asymptotic Justification of the PCApproximation of Linear Arrival Rate Functions).Suppose that we consider a nonconstant linear arrivalrate function over the fixed interval 601T 7 as above scaledby n. Suppose that we use the CU KS test with anyspecified significance level � based on combining data overkn subintervals, each of width T /kn. If conditions (23) and(24) hold, then the probability that the CU KS test ofthe hypothesis of a Poisson process will reject the NHPPconverges to � as n→ �. On the other hand, if

kn√n

→ 0 as n→ �1 (25)

then the probability that the CU KS test of a Poisson processwill reject the NHPP converges to 1 as n→ �.

Proof. Recall that the critical value �4n1�5 of the CUKS test statistic Dn has the form c�/

√n as n→ �, where

n is the sample size (see Simard and L’Ecuyer 2011),and here the sample size is Kn for all n, where K issome constant. Let D4n5 be D above as a function of theparameter n. Hence, we can compare the asymptoticbehavior of �4n1�5 with the asymptotic behavior of D4n5,which has been determined above. Theorem 5 showsthat D4n5 is asymptotically of the form C/kn. Hence, itsuffices to compare kn with

√n as in (23) and (25). �

In §5 of the online supplement, we conduct a sim-ulation experiment to illustrate Theorem 6. In §6 ofthe online supplement, we also obtain an asymptoticresult paralleling Theorem 6 for a piecewise-continuousarrival rate function where each piece is Lipschitzcontinuous.

4. Combining Data from MultipleDays: Possible Overdispersion

When the sample size is too small, it is natural tocombine data from multiple days. For example, wemay have hospital emergency department arrival data,and we want to test whether the arrivals from 9 a.m. to10 a.m. can be modeled as an NHPP. However, if thereare only 10 arrivals in 691107 on average, then data

Dow

nloa

ded

from

info

rms.

org

by [

128.

59.2

22.1

2] o

n 31

Jul

y 20

14, a

t 16:

40 .

For

pers

onal

use

onl

y, a

ll ri

ghts

res

erve

d.


from one day alone will not be sufficient to test the PPproperty. A common way to address this problem is tocombine data from multiple days; e.g., we can use allinterarrival times in 691107 from 20 weekdays, whichwill give us a sample size of about 200 interarrivaltimes. From Kim and Whitt (2014), we know that thatsample size is sufficient.

In call centers, as in many other service systems,there is typically significant variation in the arrival rateover the hours of each day and even over different daysof the week. It is thus common to estimate the arrivalrate for each hour of the day and day of the week bylooking at arrival data for specified hours and days ofthe week, using data from several successive weeks.The natural null hypothesis is that those counts oversuccessive weeks are i.i.d. Poisson random variables.However, that null hypothesis should not be takenfor granted. Indeed, experience indicates that thereis often excessive variability over successive weeks.When that is found, we say that there is overdispersionin the arrival data.

In some cases, overdispersion can be explained byspecial holidays and/or seasonal trends in the arrivalrate. The seasonal trends often can be identified byapplying time-series methods. However, it can be thecase that the observed overdispersion is far greaterthan can be explained in those systematic ways, as wewill illustrate for the arrival data from a banking callcenter in §5.

4.1. Directly Testing for OverdispersionIn the spirit of the rest of this paper, we recommenddirectly testing whether or not there is overdispersionin arrival data. The null hypothesis is that the hourlyarrival counts at fixed hours on fixed days of week overa succession of weeks constitute independent Poissonrandom variables with the same mean. A commonlyused way to test if n observations x11 0 0 0 1 xn can beregarded as a sample from n i.i.d. Poisson randomvariables is the dispersion test, involving the statistic

D≡Dn≡4n−15�2

n

xn=

∑ni=14xi− xn5

2

xn1 where

�2≡ �2

n ≡

∑ni=14xi− xn5

2

n−1and x≡ xn≡

∑ni=1xin

3 (26)

e.g., see Kathirgamatamby (1953). Because we areconcerned with excessive variability, we consider theone-sided test and reject if Dn >�4n1�5 where P4Dn >�4n1�5 �H05= �, again using �= 0005. Under the nullhypothesis, Dn is distributed as �2

n−1, a chi-squaredrandom variable with n− 1 degrees of freedom, whichin turn is distributed as the sum of squares of n− 1standard normal random variables. Thus, under the nullhypothesis, E6Dn �H07= n− 1, Var4Dn �H05= 24n− 15and 4�2

n −n5/√

2n converges to the standard normal

as n increases. Thus, �4n100055 = �2n−110095, the 95th

percentile of the �2n−1 distribution.

See Brown and Zhao (2002) for a discussion andcomparison of several tests of the Poisson hypothesis.The dispersion test above is called the conditionalchi-squared test in §3.3 there; it is shown to performwell along with a new test that they introduce, whichis based on the statistic

Dbz≡Dbz

n ≡4n∑

i=1

4yi− yn52 where yi ≡

√

xi+43/851 (27)

where yn = n−1∑ni=1 yi. Under the null hypothesis, D4bz5

n

is distributed as �2n−1 as well. We used both tests for

overdispersion and found that the results were similar,so we only discuss Dn in (26); see the online appendixfor details.

4.2. Avoiding Overdispersion andTesting for It with KS Tests

An attractive feature of the KS tests based on theCU property is that we can avoid the overdispersionproblem while testing for an NHPP. We can avoid theoverdispersion problem by applying the CU propertyseparately to intervals from different days and thenafterwards combining all the data. When the CU prop-erty is applied in this way, the observations becomei.i.d. uniform random variables, even if the rates ofthe NHPP’s are different on different days, becausethe CU property is independent of the rate of eachinterval. Of course, when we apply KS tests based onthe CU property in that way and conclude that thedata is consistent with an NHPP, we have not yet ruledout different rates on different days, which might bemodeled as a random arrival rate over any given day.

One way to test for such overdispersion is to conductthe KS test based on the CU property by combiningdata from multiple days, by both (1) combining all thedata before applying the CU property and (2) applyingthe CU property to each day separately and thencombining the data afterward. If the data are consistentwith an NHPP with fixed rate, then these two methodswill give similar results. On the other hand, if there issignificant overdispersion, then the KS test will rejectthe NHPP hypothesis if all the data is combined beforeapplying the CU property. By conducting both KS testsof a PP, we can distinguish among three alternatives:(1) PP with a fixed rate, (2) PP with a random rate,and (3) neither of those.

5. Banking Call Center Arrival DataWe now consider arrival data from service systems,first a banking call center and then, in the next section,a hospital emergency department. We use the same callcenter data used in Kim and Whitt (2013a, b) from atelephone call center of a medium-sized American bankfrom the data archive of Mandelbaum (2012), collected

Dow

nloa

ded

from

info

rms.

org

by [

128.

59.2

22.1

2] o

n 31

Jul

y 20

14, a

t 16:

40 .

For

pers

onal

use

onl

y, a

ll ri

ghts

res

erve

d.


from March 26, 2001 to October 26, 2003. This bankingcall center had sites in New York, Pennsylvania, RhodeIsland, and Massachusetts, which were integrated toform a single virtual call center. The virtual call cen-ter had 900–1,200 agent positions on weekdays and200–500 agent positions on weekends. The center pro-cessed about 300,000 calls per day during weekdays,with about 60,000 (20%) handled by agents and withthe rest being served by voice response unit (VRU)technology. In this study, we focus on arrival data dur-ing April 2001. There are four significant entry points tothe system: through VRU ∼92%, announcement ∼6%,message ∼1% and direct group (callers that directlyconnect to an agent) ∼1%. There are a very smallnumber of outgoing and internal calls, and we do notinclude them. Furthermore, among the customers thatarrive to the VRU, there are five customer types: Retail∼91.4%, Premier ∼1.9%, Business ∼4.4%, CustomerLoan ∼0.3%, and Summit ∼2.0%.

5.1. Variation in the Arrival Rate FunctionFigure 5 shows average hourly arrival rate and indi-vidual hourly arrival rate for each arrival type onMondays. As usual, Figure 5 shows strong within-dayvariation. The variation over days of the week andover successive weeks in this call center data can bevisualized by looking at four plots shown in Figure 6.The first plot on the left shows the average hourlyarrival rate (per minute) over 18 weekdays (solid line)along with the daily average (horizontal dashed line).

Figure 5 Call Center Arrivals: Average and Hourly Arrival Rates for Five Mondays

7 10 13 16 19 220

0.5

1.0

1.5

2.0× 104

Ave

rage

hou

rly a

rriv

al r

ate

7 10 13 16 19 220

100

200

300

400

7 10 13 16 19 220

500

1,000

1,500

7 10 13 16 19 220

20

40

60

80

100

7 10 13 16 19 220

200

400

600

800

1,000

Ave

rage

hou

rly a

rriv

al r

ate

t7 10 13 16 19 22

0

50

100

150

200

250

300DirectVRU–summit

VRU–retail VRU–premier VRU–business VRU–customer loan

t7 10 13 16 19 22

0

500

1,000

1,500

2,000Announcement

t7 10 13 16 19 22

0

100

200

300

400

500

600Message

t

The average for each hour is the average of the arrivalcounts over that hour divided by 60 to get the averagearrival rate per minute. The first plot also shows 95%confidence intervals about the hourly averages, whichare wide in the middle of the day.

Part of the variability seen in the first plot can beattributed to the day-of-the-week effect. This is shownby the second plot, which displays the hourly averagesfor the five weekdays. From this second plot, we seethat the arrival rates on Mondays are the highest,followed by Tuesdays and then the others. Finally, thethird plot focuses directly on the overdispersion bydisplaying the hourly average rates by a specific day ofthe week, three Mondays. Even when restricting atten-tion to a single day of the week, we see considerablevariation.

However, we have yet to quantify the variation.When we applied the dispersion test to the call centerdata, we found overwhelming evidence of overdisper-sion. That is illustrated by the test results for 16 hourson 16 Fridays. Because the sample size for each hourwas n= 16, E6Dn �H07= 15, and Var4Dn �H05= 30. The95th and 99th percentiles of the �2

15 distribution are,respectively, 2500 and 3006. However, the 16 observedvalues of Dn corresponding to the 16 hours on theseFridays ranged from 16304 to 1,068.7, with an averageof 356. The average value of Dn exceeds the 99th per-centile of the chi-squared distribution by a factor of 10.Moreover, the sample sizes were not small; the averagehourly counts ranged from 29 to 503.

Dow

nloa

ded

from

info

rms.

org

by [

128.

59.2

22.1

2] o

n 31

Jul

y 20

14, a

t 16:

40 .

For

pers

onal

use

onl

y, a

ll ri

ghts

res

erve

d.


Figure 6 Average Arrival Rates

6 9 12 15 18 210

4

8

12

t

Arr

ival

rat

e pe

r m

inut

eOverall avgHourly avg

6 9 12 15 18 210

4

8

12

16

t

MonTues

6 9 12 15 18 210

4

8

12

16

t

Mon avg

Note. From left to right: Overall and hourly average arrival rates for 18 weekdays, average arrival rates by each day of week, and arrival rates on three Mondayswith its average.

5.2. One Interval with a NearlyConstant Arrival Rate

Figure 5 shows that the VRU-summit arrival rate atthe call center is nearly constant in the interval 6141157(i.e., from 2 p.m. to 3 p.m.). We want to test whetherthe arrival process in 6141157 can be regarded as aPP. Consistent with the observations above, we seethat there is a strong day-of-the-week effect. Whenwe applied the dispersion test to all 30 days, weobtained Dn = 21320, whereas �430100055= 4308. Whenwe considered individual days of the week, we hadfour samples for each weekday and five for eachweekend day. For Saturday we had Dn = 1303, while�45100055 = 905 and �45100015 = 1303, showing thatthe p-value is 0001, but in the other cases the Dn valuesranged from 3208 to 9007, so the arrival data for the timeinterval 6141157 on a fixed day of the week exhibitsstrong overdispersion.

Because the arrival rate is approximately constantover 6141157, we do not need to consider subintervals tohave a PC rate approximation. We can directly test forthe PP, treating the data from separate days separately(and thus avoiding the day-of-the-week effect andthe overdispersion over successive weeks). First, wenote that the arrival data were rounded to the nearestsecond. The results of the CU and Lewis KS tests, withand without unrounding, are shown in Table 6. Thistable shows that the Lewis test fails to reject the PPhypothesis in 29 of 30 cases after unrounding, but inonly 19 before unrounding. Just as in §2, the CU KStest fails to detect any problem caused by the rounding.Except for the overdispersion, this analysis supportsthe PP hypothesis for the arrival data in the singleinterval 6141157.

5.3. One Interval with Increasing Arrival RateFigure 5 shows that the VRU-summit arrival rate atthe call center is nearly linear and increasing in the

Table 6 Results of KS Tests of PP for the Interval 6141157

Before unrounding Unrounded

Test Ave p-value #Pass Ave p-value #Pass

CU 0054± 0012 28 0054± 0012 28Lewis 0020± 0008 19 0049± 0009 29

interval 671107. We want to test whether the arrivalprocess in 671107 can be regarded as an NHPP.

Just as in the previous example, we see that thereis a strong day-of-the-week effect. When we appliedto dispersion test to all 30 days, we obtained Dn =

41257, whereas �430100055= 4308. When we consideredindividual days of the week, we again had four samplesfor each weekday and five for each weekend day. ForWednesday we had Dn = 703, while �44100055= 708 and�44100015= 1103, and the p-value is 0006, but for theother days of the week the Dn values ranged from 6405to 418, so the arrival data for the time interval 671107 ona fixed day of the week exhibits strong overdispersion.

The arrival rate is nearly linear and increasing over671107, so we need to use subintervals, as discussedin §3. Table 7 shows the result of using differentsubinterval lengths, L= 3, 1.5, 1, and 0.5 hours. Theaverage number of arrivals over 30 days was 67707 ±

11101. We observe that more days pass the Lewis testas we decrease the subinterval lengths (and hencemake the PC approximation more appropriate in eachsubinterval). When we use L= 005, all 30 days in Aprilpass the Lewis test. We also see the importance ofunrounding; with L = 005, only 18 days instead of30 days pass the Lewis test when the arrival data arenot unrounded.

5.4. The KS Test of All the Call CenterArrival Data

We now consider all the call center arrival data. Table 8shows the result of applying the Lewis test to all the

Dow

nloa

ded

from

info

rms.

org

by [

128.

59.2

22.1

2] o

n 31

Jul

y 20

14, a

t 16:

40 .

For

pers

onal

use

onl

y, a

ll ri

ghts

res

erve

d.


Table 7 Results of KS Tests of NHPP for the Interval 671107


L (hours) Test Ave p-value #Pass Ave p-value #Pass

3 CU 0000± 0000 0 0000± 0000 0Lewis 0000± 0001 1 0004± 0005 4

1.5 CU 0002± 0003 1 0002± 0003 1Lewis 0009± 0008 7 0026± 0011 18

1 CU 0008± 0004 12 0008± 0004 12Lewis 0016± 0008 15 0048± 0010 29

0.5 CU 0023± 0009 21 0023± 0010 21Lewis 0020± 0009 18 0051± 0010 30

Table 8 Lewis KS Test Applied to the Call Center Data by Type withL= 1 and Unrounding

Average no. of obs. Ave p-value #Pass

VRU-retail 104× 105 ± 105× 104 0015± 0009 11VRU-premier 209× 103 ± 206× 102 0049± 0010 30VRU-business 608× 103 ± 102× 103 0049± 0012 24VRU-CL 403× 102 ± 507× 101 0044± 0012 25VRU-summit 303× 103 ± 507× 102 0046± 0010 28Business 105× 103 ± 206× 102 0044± 0012 25Announcement 907× 103 ± 104× 103 0042± 0013 22Message 206× 103 ± 501× 102 0050± 0011 30VRU-retail 671107 305× 104 ± 406× 103 0037± 0012 22VRU-retail 6101137 307× 104 ± 309× 103 0015± 0008 13VRU-retail 6131167 208× 104 ± 303× 103 0027± 0011 20VRU-retail 6161197 201× 104 ± 202× 103 0045± 0011 27VRU-retail 6191227 104× 104 ± 102× 103 0043± 0011 27

call center data by call type using subinterval length Lequal to one hour to unrounded arrival times. (Detailedresults as well as CU test results can be found in theonline appendix.) We avoid the overdispersion byapplying the CU transformation to all hours separatelyand then combining the data. The average numberof observations, average p-value with associated 95%confidence intervals, and the number of days (out of

Figure 7 Hospital ED Arrivals: Average and Hourly Arrival Rates for 10 Mondays

0 3 6 9 12 15 18 21 240

5

10

15

20

t

Ave

rage

hou

rly a

rriv

al r

ate

0 3 6 9 12 15 18 21 240

1

2

3

4

5AmbulanceWalk-in

t

30 days) that passed each test at significance level�= 0005 are shown.

The results of the tests lead us to conclude that thearrival data from all these groups of customers areconsistent with the NHPP hypothesis, with the possibleexception of the VRU-retail group. We conjecture thatthe greater tendency to reject the NHPP hypothesis forthe VRU-retail group is due to its much larger samplesize. To test that conjecture, we reduce the sample size.We do so by further dividing the time intervals into3-hour long subintervals. Table 8 shows that we aremuch less likely to reject the NHPP null hypothesiswhen we do this.

In conclusion, we find significant overdispersionin the call center data—i.e., variation over successiveweeks in counts for the same hour on the same dayof the week. Otherwise, we conclude that the arrivaldata is consistent with the NHPP hypothesis. However,failure to reject the NHPP null hypothesis dependscritically on (1) unrounding, (2) properly choosingsubintervals over which the rate can be regarded asapproximately constant, and (3) avoiding the overdis-persion by applying the CU transformation to differenthours separately.

6. Hospital Emergency DepartmentArrival Data

The emergency department (ED) arrival data are fromone of the major teaching hospitals in South Korea,collected from September 1, 2012 to November 15,2012. We focus on 70 days, from September 1, 2012 toNovember 9, 2012. There are two major entry groups:walk-ins and ambulance arrivals. On average, there are138.5 arrivals each day with ∼88% walk-ins and ∼12%ambulance arrivals. Figure 7 shows the average hourlyarrival rates for each arrival type on 10 Mondays.We observe less within-day variation among the EDarrivals than among the call center arrivals.

Dow

nloa

ded

from

info

rms.

org

by [

128.

59.2

22.1

2] o

n 31

Jul

y 20

14, a

t 16:

40 .

For

pers

onal

use

onl

y, a

ll ri

ghts

res

erve

d.


Table 9 KS Tests of NHPP for the Hospital ED Data


L= 24 L= 1 L= 24 L= 1

Type Day of week n CU Lewis CU Lewis CU Lewis CU Lewis

Walk-in Mon 11599 0.00 0.00 0.34 0.00 0.00 0.00 0.97 0.62Tues 11278 0.00 0.00 0.32 0.00 0.00 0.00 0.92 0.63Wed 11085 0.00 0.00 0.15 0.00 0.00 0.04 0.13 0.94Thurs 11063 0.00 0.00 0.58 0.00 0.00 0.02 0.68 0.36Fri 11122 0.00 0.00 0.03 0.00 0.00 0.00 0.03 0.54Sat 968 0.00 0.00 0.10 0.00 0.00 0.00 0.07 0.95Sun 11298 0.00 0.00 0.26 0.00 0.00 0.00 0.90 0.93Average 1120109 0.00 0.00 0.25 0.00 0.00 0.01 0.53 0.71#Pass (�= 0005) 0/7 0/7 6/7 0/7 0/7 0/7 6/7 7/7

Ambulance Mon 160 0.94 0.00 0.16 0.00 0.94 0.34 0.34 0.24Tues 162 0.08 0.00 0.19 0.00 0.08 0.69 0.16 0.88Wed 152 0.01 0.00 0.85 0.00 0.01 0.93 0.95 0.32Thurs 171 0.00 0.00 0.61 0.00 0.00 0.22 0.50 0.34Fri 169 0.03 0.00 0.00 0.00 0.03 0.71 0.01 0.28Sat 139 0.07 0.00 0.78 0.00 0.07 0.34 0.69 0.75Sun 192 0.15 0.00 0.35 0.00 0.15 0.46 0.48 0.08Average 16306 0.18 0.00 0.42 0.00 0.18 0.53 0.45 0.41#Pass (�= 0005) 4/7 0/7 6/7 0/7 4/7 7/7 6/7 7/7

We first apply the dispersion test to test the Poissonhypothesis for daily counts for all days (n= 70) and allweekdays (n= 50), for all arrivals and by the two types.We can compare the dispersion statistic D values to�2n−111−� values for each 4n1�5 pair: (70, 0.05): 89.4, and

(50, 0.05): 66.3. The dispersion test rejects the Poissonhypothesis for the walk-in arrivals and the daily totals,with 448 ≤ Dn ≤ 520 in the 4 cases, but it does not rejectfor the ambulance arrivals, with Dn = 7908 for n= 70(p-value 0.17) and Dn = 5005 for n= 50 (p-value 0.42).

However, in the analysis above we have not yet con-sidered the day-of-the-week effect. When we analyzethe walk-in arrivals by day of the week, we obtainn= 10 and �2

n−111−� = 1609 for 4n1�5= 410100055. Theobserved daily values of Dn on the 7 days of the week,starting with Sunday, were 1409, 904, 1508, 1007, 300,2503, and 1404. Hence, we would reject the Poissonhypothesis only on the single day Friday. The asso-ciated p-values were 0009, 0040, 0007, 0030, 0097, 0000,and 0011. While we might want to examine Fridaysmore closely, we tentatively conclude that there is nooverdispersion in the ED arrival data. We do not rejectthe Poisson hypothesis for ambulance arrivals on alldays and walk-in arrivals by day of the week.

Next, we apply the CU and Lewis KS tests of anNHPP to the ED arrival data. First, based on thedispersion test results, we combine the data over the 10weeks for each type and day of the week. We considertwo cases for L: L= 24 (the entire day) and L= 1 usingsingle hours as subintervals. Table 9 shows that, withunrounding and subintervals of length L= 1, the Lewistest never rejects the PP hypothesis, while the CU test

rejects only once (Fridays). As before, using unroundingand subintervals is critical to these conclusions.

7. ConclusionsWe examined call center and hospital arrival dataand found that they are consistent with the NHPPhypothesis—i.e., that the KS tests of an NHPP appliedto the data fail to reject that hypothesis, except thatsignificant overdispersion was found in the call centerdata. In particular, (1) variation in the arrival rate overthe hours of each day was strong for the call centerdata and significant for the ED data; (2) variationin the arrival rate over the days of the week wassignificant for both the call center and ED data, exceptfor ambulance arrivals; and (3) variation in the arrivalrate over successive weeks for the same time of dayand day of week (overdispersion) was significant forthe call center data but not the ED data.

The analysis was not entirely straightforward. Themajority of the paper was devoted to three issuesthat need to be addressed and showing how to do so.Section 2 discussed data rounding, showing that itsimpact can be successfully removed by unrounding.Consistent with Kim and Whitt (2014), the Lewis testis highly sensitive to the rounding, while the CUKS test is not. Section 3 discussed the problem ofchoosing subintervals so that the PC rate functionapproximation is justified. Simple practical guidelineswere given for (1) evaluating any given subinterval in§3.4 and (2) choosing an appropriate number of equallyspaced subintervals in §3.6. Again consistent with Kimand Whitt (2014), the CU KS test is more sensitive

Dow

nloa

ded

from

info

rms.

org

by [

128.

59.2

22.1

2] o

n 31

Jul

y 20

14, a

t 16:

40 .

For

pers

onal

use

onl

y, a

ll ri

ghts

res

erve

d.


to the deviation from a constant rate function thanthe Lewis KS test. Finally, §4 discussed the problemof overdispersion caused by combining data frommultiple days that do not have the same arrival rate.These three issues played an important role for bothsets of data.

Supplemental MaterialSupplemental material to this paper is available at http://dx.doi.org/10.1287/msom.2014.0490.

AcknowledgmentsThe authors thank Avishai Mandelbaum, Galit Yom-Tov, EllaNadjharov, and the Center for Service Enterprise Engineering(SEE) at Technion–Israel Institute of Technology for access tothe SEE call center data and advice about its use. They alsothank Won-Chul Cha and the Samsung Medical Center inSouth Korea for providing the hospital emergency departmentdata and advice about its use. Lastly, they thank the SamsungFoundation and the National Science Foundation for support[Grants CMMI 1066372 and 1265070].

ReferencesAksin OZ, Armony M, Mehrotra V (2007) The modern call center:

A multi-disciplinary perspective on operations managementresearch. Production Oper. Management 16(6):665–688.

Armony M, Israelit S, Mandelbaum A, Marmor Y, Tseytlin Y, Yom-TovG (2011) Patient flow in hospitals: A data-based queueing-scienceperspective. Working paper, New York University, New York.http://www.stern.nyu.edu/om/faculty/armony/.

Asaduzzaman M, Chaussalet TJ, Tobertson NJ (2010) A loss networkmodel with overflow for capacity planning of a neonatal unit.Ann. Oper. Res. 178:67–76.

Avramidis AN, Deslauriers A, L’Ecuyer P (2004) Modeling dailyarrivals to a telephone call center. Management Sci. 50(7):896–908.

Barbour AD, Holst L, Janson S (1992) Poisson Approximation (OxfordUniversity Press, Oxford, UK).

Bassamboo A, Zeevi A (2009) On a data-driven method for staffinglarge call centers. Oper. Res. 57(3):714–726.

Brown L, Gans N, Mandelbaum A, Sakov A, Shen H, Zeltyn S, ZhaoL (2005) Statistical analysis of a telephone call center: A queueing-science perspective. J. Amer. Statist. Assoc. 100(469):36–50.

Brown L, Zhao LH (2002) A test for the Poisson distribution. Sankhya:Indian J. Statist. 64(A-3):611–625.

Cooper RB (1982) Introduction to Queueing Theory, 2nd ed. (North-Holland, Amsterdam).

Daley DJ, Vere-Jones D (2008) An Introduction to the Theory of PointProcesses, Vol. II, 2nd ed. (Springer, Oxford, UK).

Durbin J (1961) Some methods for constructing exact tests. Biometrika48(1):41–55.

Ibrahim R, L’Ecuyer P, Regnard N, Shen H (2012) On the modelingand forecasting of call center arrivals. Proc. 2012 Winter SimulationConf., Berlin, 256–267.

Jongbloed G, Koole G (2001) Managing uncertainty in call centersusing Poisson mixtures. Appl. Stochastic Models Bus. Indust.17(4):307–318.

Kathirgamatamby N (1953) Note on the Poisson index of dispersion.Biometrika 40(1):225–228.

Kim S-H, Whitt W (2013a) Statistical analysis with Little’s law. Oper.Res. 61(4):1030–1045.

Kim S-H, Whitt W (2013b) Statistical analysis with Little’s law,supplementary material: More on call center data. ColumbiaUniversity, New York. http://www.columbia.edu/~ww2040/allpapers.html.

Kim S-H, Whitt W (2014) Choosing arrival process models for servicesystems: Tests of a nonhomogeneous Poisson process. Naval Res.Logist. 61(1):66–90.

Lewis PAW (1965) Some results on tests for Poisson processes.Biometrika 52(1):67–77.

Li AA, Whitt W (2013) Approximate blocking probabilities forloss models with independence and distribution assumptionsrelaxed. Performance Evaluation, ePub ahead of print August 13,http://dx.doi.org/10.1016/j.peva.2013.08.004.

Litvak N, van Rijsbergen M, Boucherie RJ, van Houdenhoven M(2008) Managing the overflow of intensive care patients. Eur. J.Oper. Res. 185(3):998–1010.

Mandelbaum A (2012) Service engineering of stochastic networkswebpage. Last accessed May 27, 2014, http://iew3.technion.ac.il/serveng/.

Marsaglia G, Tsang WW, Wang J (2003) Evaluating Kolmogorov’sdistribution. J. Statist. Software 8(18):1–4.

Massey FJ Jr (1951) The Kolmgorov-Smirnov test for goodness of fit.J. Amer. Statist. Assoc. 46(253):68–78.

Massey WA, Parker GA, Whitt W (1996) Estimating the parametersof a nonhomogeneous Poisson process with linear rate. Telecomm.Systems 5(253):361–388.

Miller LH (1956) Table of percentage points of Kolmogorov statistics.J. Amer. Statist. Assoc. 51(273):111–121.

Pang G, Whitt W (2012) The impact of dependent service times onlarge-scale service systems. Manufacturing Service Oper. Manage-ment 14(2):262–278.

Simard R, L’Ecuyer P (2011) Computing the two-sided Kolmogorov-Smirnov distribution. J. Statist. Software 39(11):1–18.

Whitt W (1982) Approximating a point process by a renewal process,I: Two basic methods. Oper. Res. 30(1):125–147.

Whitt W (2002) Stochastic-Process Limits (Springer, New York).Wilkinson R (1956) Theories of toll traffic engineering in the U.S.A.

Bell System Technical J. 35(2):421–514.

Dow

nloa

ded

from

info

rms.

org

by [

128.

59.2

22.1

2] o

n 31

Jul

y 20

14, a

t 16:

40 .

For

pers

onal

use

onl

y, a

ll ri

ghts

res

erve

d.

Documents

Are Call Center and Hospital Arrivals Well Modeled by ...ww2040/Kim_W_MSOM_2014.pdf · Kim and Whitt: Are Call Center and Hospital Arrivals Well Modeled by NHPPs? 466 Manufacturing