Statistical Analysis with Little s Law. Supplementary Material: More on the Call Center Data. by Song-Hee Kim and Ward Whitt

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1 Saisical Analysis wih Lile s Law Supplemenary Maerial: More on he Call Cener Daa by Song-Hee Kim and Ward Whi Deparmen of Indusrial Engineering and Operaions Research Columbia Universiy, New York, NY {sk311,ww}@columbia.edu hp:// {sk311,ww} March 5, 1 1 Inroducion In his supplemen o he main paper [1], we provide more deails abou he call cener daa we used. As explained in he main paper, he daa are from a elephone call cener of a mediumsized American bank from he daa archive of Mandelbaum [], colleced from March, 1 o Ocober, 3. This banking call cener had sies in New York, Pennsylvania, Rhode Island, and Massachuses, which were inegraed o form a single virual call cener. The virual call cener had 9 1 agen posiions on weekdays and 5 agen posiions on weekends. The cener processed abou 3, calls per day during weekdays, wih abou, (%) handled by agens, wih he res being served by inegraed voice response (IVR) echnology. As in many modern call ceners, in his banking call cener here were muliple agen ypes and muliple call ypes, wih a form of skill-based rouing (SBR) used o assign calls o agens. Since we were only concerned wih esimaion relaed o he hree parameers L, λ and W, we did no ge involved wih he full complexiy of his sysem. Specifically, we used daa for only one class of cusomers, denoed by Summi. Furhermore, among hem, only he sub-calls ha had agen ineracions during weekdays in May 1 were considered. The res of his supplemen is organized as follows. In we briefly describe he full daabase of Mandelbaum [] and how we exraced he required daa in order o produce he resuls in he main paper. In 3 we describe he saisics colleced for he 1 weekdays in May we used in our analysis and give an overview of he saisics. The Daa Available and Used The full daabase of [] provides nine pre-processed ACCESS ables for each day in he sudy period, from March, 1 o Ocober, 3. In he pre-processing, issues such as midnigh calls, incorrec ime samps and incorrec idenifiers (id s) are already aken care of. As shown on he lef pane of Figure 1, he nine daabase ables are iled: calls, cusomer sub-calls, server sub-calls, queue records, even deails, agen evens, agen profile, agen records, and agen shifs. The calls able includes general informaion on each call ha

2 eners he call cener on a paricular day. Each call hen consiss of sub-call(s) ha sar and end wih a paricular service such as IVR, agen ineracion and announcemen. We focus on he sub-calls of Summi cusomers ha involve agen ineracion, and hence use only he cusomer sub-calls able of each day. Figure 1: Example of ACCESS ables: call cener daa from May 5, 1. Figure 1 is an example of he cusomer sub-calls able. There are 3 fields in he able, which are: call id, cus subcall, server subcall, record id, node, cusomer id, cusomer ype, service group, service, f irs service, segmen sar, queue exi, service enry, segmen end, seg ype, oucome, seg paries, wai ime, queue ime, preservice wai, service ime, hold ime, and pary answered. More informaion abou he differen ables, including deailed descripions of each field, can be found a: hp://ie.echnion.ac.il/labs/serveng/files/ Model Descripion and Inroducion o User Inerface.pdf To creae he daa se we used in [1], we used he following seps: Each sub-call is served by a service group. There are five main service groups, which are IVR, Business line, non-business line, Announcemen and Message. We kep he sub-calls ha were handled by he Business line (service group = ). In he ACCESS cusomer sub-calls able, we filered ou hese sub-calls by selecing service group =, as illusraed in Figure 1. We kep he sub-calls ha are from Summi cusomers by keeping records wih service = 1 (The service field indicaes he ype of service received by he caller. For example, here are Reail = 1, P remier =, Business = 3 and P lainum = ). We dropped he records wih no agen ineracion, which involve he caller hanging up (abandoning) while waiing o speak o he nex agen. This was done by dropping records wih oucome = 11,1, or 13. The oucome field indicaes he cause of call erminaion such as wheher hey were handled, ransferred and abandoned. oucome = 11 indicaes he cusomer abandoned shor (he caller abandons wihin an abandon hreshold ime), oucome = 1 indicaes he cusomer abandoned (afer he abandon

3 hreshold ime) and oucome = 13 indicaes he call was no handled wih oher reason ha is no specified in he daa). To ensure ha each sub-call spen posiive amoun of ime wih an agen, we omied records wih service ime =, where service ime is defined as he sum of alk ime and hold ime. I can also be defined as he difference beween segmen end and service enry. Since we already dropped he records wih cusomer abandonmen, here were no many records wih service ime =, less han 5 for each day. In order o compue he hree parameers L, λ and W, we used he ime each subcall eners he queue, leaves he queue (hence eners he service) and leaves he service. Therefore, we kep only he fields call id, segmen sar (queue enry ime), queue exi (queue exi ime), service enry (service enry ime) and segmen end (service exi ime). (The ime samps are records in seconds, using he origin ime, :: on 1/1/197. Finally, we expored he able o an EXCEL file using he Expor o Excel spreadshee funcion. The seps above were carried ou by he auhors for 1 weekdays of May. (In he nex secion we explain how he 1 days were seleced.) The combined daa se for all 1 weekdays (lile xls) is available from he auhors web sies. 3 Saisics from Eigheen Weekdays in May, 1 There were 3 weekdays in May 1. (May 1, 1, was a Tuesday.) Four weekdays were no normal, and so were excluded, for he following reasons: May 9 (Wed): shudown from :53:1 AM unil 11::5 AM May 1 (Thurs): shudown from :59:1 PM unil 11:31: PM May (Mon): Memorial Day May 31 (Thurs): daa missing In addiion, he daa from May 3 were excluded because he number of arrivals was exraordinarily high. In paricular, he number of arrivals was 31 on May 3 wih abou 5% arriving before 9 AM, whereas for he oher 1 weekdays, he average number of arrivals was 51.5 arrivals, wih a sandard deviaion 1.5. For each day, here were daa over a 17-hour period, from AM o 11 PM, referred o as [,3]. (There were no arrivals before AM and afer 11 PM.) We primarily focused on he number of Summi cusomers in he sysem, bu we also considered wheher hey were in service or waiing (in queue). Thus we measured he numbers in he sysem, in service and in queue. Similarly, we measured he ime ha each cusomer spen in he sysem, in he queue and in service. Using he daa se lile xls, we colleced he following saisics: L sys : he number in sysem L ser : he number in service L q : he number in queue 3

4 A sys = A q : number of arrivals ino he sysem/queue A ser : number of arrivals ino service W sys : ime spen in he sysem W ser : ime spen in service W q : ime spen in queue These are undersood o be funcions of he measuremen inerval. For example, A sys A sys ([9,1]) is he number of arrivals ino he sysem during he inerval [9,1]. For he inerval [9,1], he average arrival rae per minue is λ() A sys([9,1]) m([9,1]) = A sys([9,1]), (1) where m([9, 1]) = is he number of minues in he inerval [9, 1]. Thus he saisics are consisen wih he definiions in equaion (1) of [1]. 3.1 The Hourly Arrival Raes Figure shows he overall average (over he full 17-hour day) and he hourly averages of he arrival raes per minue, as defined in (1), ogeher wih esimaes of he 95% confidence inerval (reaing he daily values as i.i.d. Gaussian variables) for he 1 Figure shows ha he arrival rae is nonsaionary over he day. Figure also shows ha he arrival rae is highly variable from day o day, because of he wide confidence inervals for he hourly averages. Par of his day-o-day variaion can be explained by day-of-week effec. Figure 3 shows ha he average call volume on Mondays is he larges, followed by ha of Tuesdays, and hen he ohers. Figures - furher illusrae day-o-day variaion in he same day of week.

5 1 1 Hourly Avg of λ arrival rae per minue Figure : Overall average and hourly average of λ and is 95% confidence inerval over 1 5

6 average arrival rae per minue Mon Tues Wed Thurs Fri average arrival rae per minue Monday Avg May 7 May 1 May Figure 3: Average arrival rae and he dayof-week effec in May. average arrival rae per minue Tuesday Avg May 1 May May 15 May May Figure 5: Average arrival rae of Mondays in May. average arrival rae per minue Thursday Avg May 17 May Figure 7: Average arrival rae of Thursdays in May Figure : Average arrival rae of Mondays in May. average arrival rae per minue Wednesday Avg May May 1 May 3 May Figure : Average arrival rae of Wednesdays in May. average arrival rae per minue Friday Avg May May 11 May 1 May Figure : Average arrival rae of Fridays in May.

7 3. The Hourly Average Waiing Times 7 ime spen in he sysem, min Hourly Avg of W sys Figure 9: Overall average and hourly average of W sys and is 95% confidence inerval over 1 ime spen in he service, min 3 Hourly Avg of W ser Figure 1: Overall average and hourly average of W ser and is 95% confidence inerval over 1 Hourly Avg of W q ime spen in he queue, min Figure 11: Overall average and hourly average of W q and is 95% confidence inerval over 1 7

8 Figures 9-11 show he overall average (average of he 1 daily averages) and hourly average of he waiing ime in he sysem, service and queue and heir 95% confidence inerval of 1 Figure 1 suggess ha he service imes are approximaely saionary over ime. However, by comparing Figure 1 o Figure 9, we can conclude ha i is hard o say ha he imes in sysem is approximaely equal o he service imes because he ime in queue is oo long in he inerval [17,], which migh be due o inadequae saffing during his inerval. Furhermore, Figure 11 suggess he waiing imes in sysem is no approximaely saionary over ime, again possibly due o inadequae saffing. Nex, Figure 1 shows he hisogram of all waiing imes in he inerval [1,1] of Friday, May 5, 1 in our call cener example. In addiion, Figure 13 and Figure 1 illusrae he hisograms of all waiing imes and service imes in he inerval [9,17] over 1 weekdays in May in our call cener example. As usual for call ceners, he disribuion is approximaely lognormal, bu he SCV very close o 1 indicaes ha an exponenial approximaion is reasonable. Figure 1: The hisogram (empirical disribuion) of he imes spen in he sysem of all arrivals during he inerval [1,1].

9 5 µ = 3. σ = 3.17 c = µ = 3.15 σ = 3.51 c = 1.99 Frequency 3 Frequency ime spen in he sysem, min ime spen in service, min Figure 13: The hisogram (empirical disribuion) of he imes spen in he sysem of all arrivals during he inerval [9,17] over 1 weekdays in May (n = 7,535 and 1 observaions ha had W sys > 35 are no represened). Figure 1: The hisogram (empirical disribuion) of he imes spen in service of all arrivals o he service during he inerval [9,17] over 1 weekdays in May (n = 7, 9 and 39 observaions ha had W ser > 35 are no represened). 3.3 The Hourly Average Number in Sysem) Figures show he overall average (average of he 1 daily averages) and hourly average number in he sysem, service and queue and heir 95% confidence inerval of 1 weekdays in May. We observe ha he number in sysem is approximaely equal o he number in service, excep a he imes when here are slighly longer queues, in he inervals [9,13] and [17,]. 9

10 35 Hourly Avg of L sys 3 number in sysem Figure 15: Overall average and hourly average of L sys and is 95% confidence inerval over Hourly Avg of L ser number in service Figure 1: Overall average and hourly average of L ser and is 95% confidence inerval over 1 7 Hourly Avg of L q number in queue Figure 17: Overall average and hourly average of L q and is 95% confidence inerval over 1 1

11 Acknowledgemens. We hank Avishai Mandelbaum, Gali Yom-Tov, Ella Nadjharov and he Cener for Service Enerprise Engineering (SEE) a he Technion for access o he call cener daa and advice abou is use. We hank he Samsung Foundaion and NSF for suppor (NSF gran CMMI 137). References [1] Kim, S., W. Whi. 1. Saisical analysis wih Lile s law, Available from: hp:// ww/allpapers.hm [] Mandelbaum, A. 1. Service Engineering of Sochasic Neworks web page: hp://iew3.echnion.ac.il/serveng/ 11