Density Forecasting of Intraday Call Center Arrivals. using Models Based on Exponential Smoothing

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1 Dnsiy Forcasing of Inraday Call Cnr Arrivals using Modls Basd on Exponnial Soohing Jas W. Taylor Saïd Businss School Univrsiy of Oxford Managn Scinc, 0, Vol. 58, pp Addrss for Corrspondnc: Jas W. Taylor Saïd Businss School Univrsiy of Oxford Park End Sr Oxford OX HP, UK Tl: +44 (0) Fax: +44 (0) Eail:

2 Dnsiy Forcasing of Inraday Call Cnr Arrivals using Modls Basd on Exponnial Soohing Absrac A ky inpu o h call cnr saffing procss is a forcas for h nubr of calls arriving. Dnsiy forcass of arrival ras ar ndd for analyical call cnr odls, which assu Poisson arrivals wih a sochasic arrival ra. Dnsiy forcass of call volus can b usd in siulaion odls, and ar also iporan for h analysis of ousourcing conracs. A forcasing hod, ha has prviously shown srong ponial, is Hol-Winrs xponnial soohing adapd for odling h inraday and inrawk cycls in inraday daa. To nabl dnsiy forcasing of h arrival volu and ra, w dvlop a Poisson coun odl, wih gaa disribud arrival ra, which capurs h ssnial faurs of his xponnial soohing hod. Th apparn saionary lvl in our daa lads us o dvlop vrsions of h nw odl for sris wih saionary lvl. W valua forcas accuracy up o wo wks ahad using daa fro hr organizaions. W find ha h saionary lvl odls iprov prdicion byond approxialy wo days ahad, and ha hs odls prfor wll in coparison wih sophisicad bncharks. This is confird by h rsuls of a call cnr siulaion odl, which donsras h us of arrival ra dnsiy forcasing o suppor saffing dcisions. Ky words: call cnrs; arrival ra; dnsiy forcasing; xponnial soohing; sasonaliy

3 . Inroducion Th nubr and siz of call cnrs has incrasd in rcn yars o h xn ha hy ploy approxialy 3% of h working populaions in h US and UK. For any organizaions, call cnrs provid h ain channl of counicaion wih hir cusors. As h huan rsourc coss for call cnrs ypically copris 60 o 80% of hir ovrall opraing budg, i is of gra iporanc o sablish an opial lvl of saffing. Ovrsaffing obviously incurs unncssary opraing coss, whil undrsaffing causs quuing is considrd by cusors as unaccpabl, which can lad o h abandonn of calls. Indd, saff dployn is a balanc bwn srvic qualiy and opraing coss (Akşin al. 007). A ky inpu o h schduling procss is a forcas for h nubr of calls arriving a h call cnr. Long-r prdicions ar ndd for rcruin purposs. Forcass wih lad is of svral wks ar rquird for saff schduling, and hs forcass and schduls nd o b rgularly rvisd unil h arg day islf (Gans al. 003). This can b coninud wihin h day, wih forcass for svral hours ahad usd o nabl dynaic updaing of h agns dployn (Mhrora al. 00). Th usfulnss of inraday forcas updaing props Shn and Huang (008a) and Winbrg al. (007) o xnd hir forcasing hods for his purpos. Ti sris of inraday call cnr arrivals consis of variabiliy around a sasonal parn ha is ad up of boh inrawk and inraday sasonal cycls. Anicipaing his variabiliy is iporan, as highr uncrainy rquirs highr saffing lvls in ordr o spcifid srvic lvl objcivs. A prdicion of h probabiliy dnsiy funcion (i.. a dnsiy forcas) of h call volu can b usd for h arrival procss in siulaion odls, which hav bn growing in populariy du, a las in par, o h incrasd coplxiy of call cnr opraions and dvlopns in copuaional powr (Gans al. 003; Mhrora and Faa 003). Dnsiy forcass of call volus can also b usd o analyz call cnr ousourcing conracs (s, for xapl, Akşin al. 008). Conracs ay b dfind so ha h call volu abov a spcifid bas lvl is ousourcd, or so ha only a bas load is ousourcd. For analyical call cnr odls, and for any siulaion odls, h arrival volu is odld as a Poisson procss, and so a forcas is ndd for h arrival ra. Epirical vidnc has shown ha

4 call cnr arrivals show significan ovrdisprsion rlaiv o a Poisson procss (i.. h varianc is grar han h an), and his has ld o h arrival ra bing rad as sochasic (s, for xapl, Jongblod and Kool 00; Avraidis al. 004; Akşin al. 007, Scion.4; Bassaboo and Zvi 009). In viw of his, i is a dnsiy forcas of h arrival ra ha is ndd for analyical odls. Jongblod and Kool (00) dscrib how o us such a dnsiy forcas o assss h ipac of h arrival ra uncrainy on h srvic lvl, dfind as h fracion of calls whos dlay falls blow a spcifid arg. Thy show how h % quanil of h arrival ra disribuion can b pluggd ino h Erlang C forula o g a srvic lvl ha will b xcdd wih % probabiliy. Jongblod and Kool also dscrib how, for a givn srvic lvl, h bounds of an arrival ra prdicion inrval can b usd wih h Erlang C forula o dlivr bounds for h rcondd saffing lvl. Thy no ha, if a flxibl workforc is availabl, h lowr bound for h saffing lvl can b usd as h fixd nubr of agns, and h inrval widh can b usd as h nubr of agns ndd wih flxibl conracs. In his papr, w dvlop odls for h dnsiy forcasing of boh h call volu and h arrival ra.. Univaria Modls for Inraday Arrivals On approach o forcasing call arrivals is o us a causal odl (s, for xapl, Soyr and Taricilar 008). Th alrnaiv is a hod basd on only h arrivals sris, and i is hs univaria hods ha nd o b usd for inraday call arrivals, which is our focus in his papr. Of h univaria approachs, hos involving a saisical odl hav h advanag ha hy nabl convnin gnraion of dnsiy forcass. To h bs of our knowldg, h only sudis of inraday call cnr arrivals ha focus on hods for dnsiy forcasing ar hos of Winbrg al. (007) and Shn and Huang (008a, 008b). Winbrg al. xnd h rando ffcs saisical odl of Brown al. (005), which involvs h produc of h daily oal and h proporion of h oal ha occurs in a givn priod of h day. Winbrg al. odl boh his proporion and h daily oal using yps of auorgrssiv odls. Thy us a Markov chain Mon Carlo algorih o sia h lan sas and odl parars, and hnc poin and dnsiy forcass for h arrival volu and ra.

5 Th novl hod of Shn and Huang (005, 008a) involvs h us of singular valu dcoposiion (SVD), and i procds by arranging h inraday daa as a (d ) arix, whr d is h nubr of days in h siaion sapl and is h nubr of priods in ach day. Each colun of his arix consius a i sris of daily obsrvaions for a paricular priod of h day. SVD is usd o xrac h ain undrlying coponns in hs coluns, and hus rduc h probl fro having o forcas daily sris o forcasing daily sris for jus h ain coponns. A sipl i sris forcasing hod is applid o ach of hs ain coponns. Th basic hod uss inforaion up o h nd of a givn day o produc forcass for all priods of a fuur day. To nabl inraday forcas updaing, an addiional sag is incorporad. A boosrap procdur is usd o produc dnsiy forcass for h call volus. To gnra dnsiy forcass for h arrival ra, Shn and Huang (008b) dvlop a coun daa vrsion of h hod, which involvs Poisson odling wih a sochasic arrival ra odld using a siilar dinsion rducion approach o ha usd in hir ohr paprs. If a i sris consiss of high couns ha ar Poisson-disribud, a squar roo ransforaion sabilizs h varianc in h sris and allows h fiing of Gaussian odls. This ransforaion is ployd by Brown al. (005), Winbrg al. (007) and Shn and Huang (005, 008a). For h yp of odls ha w considr in his papr, if such a ransforaion is usd, i is no clar how a dnsiy forcas for h arrival ra could b producd. In viw of his, w do no ransfor h daa, and insad follow Shn and Huang (008b) by fiing Poisson coun daa odls o our daa. Thy xplain ha, if a Poisson assupion is rasonabl, hn fficincy should iprov by odling h coun daa dircly, rahr han odling h ransford sris. A furhr oivaion for dvloping coun daa odls is ha hy ar ndd for call arrivals sris ha consis of rlaivly low volus. An xapl of his is h quarr-hourly sris of oal call arrivals a an Israli bank, analyzd by Brown al. (005, Scion 6), which has 0% of priods wih lss han 0 calls. Low volu arrivals ofn occur a uliskill call cnrs, which involv diffrn yps of calls bing srvd by diffrn groups wih spcializd skills (s, for xapl, Czik and L Ecuyr 008; Po al. 008). On of h sris considrd in his papr consiss of rlaivly low volu arrivals channld o a spcializd group of agns a a uliskill call cnr. 3

6 In h pirical analysis of Taylor (008), a hod ha prford wll agains ARMA and sa spac alrnaivs, for lad is up o abou four days ahad, was Hol-Winrs xponnial soohing adapd by Taylor (003) for odling boh h inraday and inrawk cycls in inraday daa. W r his hod HWT xponnial soohing. Exponnial soohing hods ar widly usd in a variy of businss applicaions (s Gardnr 006). Thy ar basd on xponnially wighd avraging, and hav h appal of inuiivnss and ransparncy, which is iporan if adopion by praciionrs is valud. In his papr, w dvlop h HWT hod by providing h following wo ain conribuions: - To nabl dnsiy forcasing of h arrival volu and ra, w dvlop a Poisson coun daa odl ha capurs h ssnial faurs of h HWT hod. W ipln a gaa disribud sochasic arrival ra, as in h work of Jongblod and Kool (00). W ar no awar of prvious sudis ha hav considrd a univaria sasonal i sris coun daa odl for inraday call cnr arrivals. - Lik all of h sandard xponnial soohing hods, h nw HWT coun daa odl posssss a nonsaionary lvl. As his can b inappropria for call arrivals daa, w dvlop nw vrsions of h odl ha hav a saionary lvl. In rs of concpual sipliciy and as of iplnaion, h HWT odls copar wll wih h xising call arrivals dnsiy forcasing hods, which ar prsnd by Winbrg al. (007) and Shn and Huang (008a, 008b). Ths auhors phasiz h iporanc of inraday forcas updaing, and for h HWT odl his is sraighforward bcaus h on odl is usd dircly o produc forcass for all lad is and fro all forcas origins. A liiaion of h hods of Winbrg al. and Shn and Huang (008a) ar ha hy ar suiabl only for high volu arrivals daa. 3. Inraday Call Cnr Arrivals Daa In his papr, our pirical analysis ainly focuss on on high volu and on low volu sris of half-hourly arrivals. Th high volu sris corrsponds o oal arrivals a h call cnrs of NHS Dirc, which is h 4-hour lphon hlplin providd by h Naional Halh Srvic in England and Wals. Th NHS Dirc srucur is siilar o ha dscribd by Shusky and Pinkr (003), wih 4

7 calls iniially handld by a gakpr who provids sipl inforaion or passs h call on o a nurs or halh inforaion saff. Th NHS Dirc sris consiss of 35 wks of obsrvaions, which vary fro 8 o,03 calls. This daa is also usd in h sudy of Taylor (00), which dvlops nw xponnially wighd hods for sris wih ulipl sasonal cycls ha igh occur in a variy of applicaions. In conras o our currn papr, Taylor (00) focuss on only poin forcasing and dos no considr coun daa odls. I is no clar how on would produc an arrival ra dnsiy forcas fro h hods prsnd in ha sudy. In ha papr, Figurs o 3 show ha h flucuaions in h NHS Dirc sris ar doinad by sasonaliy. Th firs of hs figurs shows an inraday sasonal cycl of duraion =48 priods, and an inrawk cycl of lngh =336 priods. Rpaing inraday and inrawk sasonal cycls is ypical of inraday call arrivals daa. For all sris in his papr, xponnial soohing and ARMA coun daa odl parars wr siad onc using an iniial sapl, and h forcas origin was hn rolld forward hrough a possapl valuaion priod o produc a collcion of forcass for ach horizon. W usd h firs 5 wks of NHS Dirc daa for siaion and h raining 0 for valuaion. Th hods considrd in his papr ar no of us for days on which h parn of calls is unusual, such as public holidays, and so w rplacd obsrvaions for hs days using sipl avraging procdurs. Ths days wr no includd in our pos-sapl forcas valuaion. Our scond sris cos fro h uliskill call cnr of a UK crdi card copany. I consiss of half-hourly arrivals channld o a group of agns spcializing in a paricular yp of nquiry. This channl is opn fro 9 a o 8 p on Mondays o Saurdays, and is closd on Sundays. Th 70-wk sris is shown in Figur. Abou 0% of h priods had calls or fwr, and abou % had no calls a all. For such low volu daa, coun daa odls ar appropria. W usd h firs 50 wks for odl siaion, and h final 0 for pos-sapl valuaion. Figur shows a four-wk priod of h daa. In spi of subsanial volailiy, hr is vidnc of a rpaing wkly cycl. Givn h opning hours, an inraday cycl would consis of = half-hourly priods, and an inrawk cycl would b of lngh =3 priods. Figur 3 shows h avrag daily call profils, calculad using h in-sapl daa. 5

8 Figurs HWT Exponnial Soohing for Coninuous Daa In his scion, w inroduc saisical odls basd on Taylor s (003, 008) HWT hod. Ths odls ar appropria for obsrvaions fro coninuous rando variabls. Alhough i can b rasonabl o ra high volu arrivals daa as obsrvaions fro a coninuous rando variabl, w do no apply h odls of his scion o arrivals daa in his papr. Th purpos of h odls in his scion is siply o provid h foundaion for Scion 5, whr w dvlop nw coun daa odls. Dnsiy forcass can b gnrad fro an xponnial soohing hod by xprssing i as a singl sourc of rror sa spac odl and hn using Mon Carlo siulaion (Hyndan al. 008). Th following singl sourc of rror sa spac HWT odl corrsponds o h addiiv HWT hod. HWT odl addiiv vrsion: y l d w (a) l y d w (b) l l (c) d d (d) w w () y is h arg variabl; l is a lvl r; d is h sasonal facor for h inraday cycl; w is h facor for h inrawk cycl raining afr h inraday cycl is rovd; and ar h rspciv lnghs of hs cycls;,, and ar parars; and ~ N(0, ). Th r involving is an iporan iid adjusn for auocorrlaion in h rsiduals,. No ha using xprssion (b) o subsiu for lswhr in h odl rsuls in a odl wih jus a singl rando rror,. Having wo rs wih lags of on in h obsrvaion quaion of xprssion (a) ans ha h lvl is odld by a cobinaion of l and, and so h updaing of h lvl is dicad by boh and. W rurn o his issu in Scion 5.3. Th HWT odl of xprssions (a)-() has addiiv sasonaliy and an addiiv rror,. If h sasonaliy dpnds on h lvl of h sris, hn uliplicaiv sasonaliy is appropria. A uliplicaiv rror r, which is also known as a rlaiv rror, is appropria if h varianc of h 6

9 randonss dpnds on h lvl and sasonaliy. On approach o daling wih uliplicaiv sasonaliy and rror is o apply a varianc sabilizing ransforaion, such as a squar roo or logarihic ransforaion, and hn us an addiiv odl. A or dirc approach is o apply a odl wih uliplicaiv srucur o h original unransford daa. For all of h sandard xponnial soohing odls, Hyndan al. (008, Scion.5) considr forulaions wih addiiv and uliplicaiv rror, and addiiv and uliplicaiv sasonaliy. In xprssions (a)-(), w prsn h HWT odl wih uliplicaiv sasonaliy and uliplicaiv rror. l is a lvl r; d and w ar uliplicaiv sasonal iid facors; and ~ N(0, ). To b consisn wih h for of h sandard uliplicaiv rror r (+ ) in xprssion (a), w hav incorporad h rsidual auocorrlaion as a uliplicaiv r (+ - ). HWT odl uliplicaiv vrsion: y l d w (a) y / l d w (b) l l d (c) d (d) w w () In our wo call arrivals sris, plod in Figur of Taylor (00) and in Figur of his papr, h flucuaions s o dpnd on h lvl of h daa. This is consisn wih Poisson arrivals, alhough h prcis rlaionship bwn h an and varianc will b affcd by h sochasic naur of h arrival ra. As h flucuaions dpnd on h lvl, i ss appropria o us a odl wih uliplicaiv sasonaliy and uliplicaiv rror. Howvr, h odl of xprssions (a)-() is unsuiabl bcaus, as w xplaind in Scions and, w rquir Poisson-basd coun daa odls. W dvlop such odls in h nx scion. 5. Coun Daa Modls Basd on Exponnial Soohing 5.. Rviw of Modls and Disribuions for Coun Ti Sris In his scion, w brifly rviw odls for coun i sris and dscrib h disribuions ha w us in h nw odls ha w inroduc in Scions 5. and 5.3. Th Poisson disribuion is ofn usd 7

10 in coun daa odls. Such odls involv h spcificaion of a suiabl srucur for h arrival ra. An xapl is Poisson rgrssion, which involvs odling h ra as hx xponnial funcion, x is a vcor of rgrssors, and is a vcor of parars., whr h is an Univaria odls for coun i sris ar ihr obsrvaion-drivn or parar-drivn (Jung al. 006). Th dynaics of obsrvaion-drivn odls ar capurd by laggd valus of h couns. Davis al. (003) considr a Poisson for of gnralizd linar ARMA odl, which is obsrvaion-drivn and rlas an xponnial funcion of ARMA rs o h ra of h Poisson disribuion. Wih parar-drivn odls, auocorrlaion in h couns is odld using a lan dynaic procss for h parars of h condiional disribuion. An xapl is h odl of Zgr (988), which allows for an auocorrlad r, l, wihin Poisson rgrssion, and is wrin as: y ~ Poiss (3a) l h x (3b) l l h (3c) whr y is h coun for h priod nding a i ; and ar parars; h is h xponnial funcion; iid and ~ N(0, ). Figin al. (008) xplain ha parar-drivn coun daa odls can b viwd as ulipl sourc of rror sa spac odls, whil obsrvaion-drivn coun daa odls hav a singl sourc of rror. Hyndan al. (008) xplain ha dnsiy forcasing and nonlinar odling is paricularly convnin wih singl sourc of rror odls. In viw of his, w lc o dvlop in his papr nw HWT coun daa odls of his yp. Thr is lil liraur on h i sris odling of sasonal coun daa, and w ar no awar of any xponnial soohing odls for such daa. Call cnr arrivals nd o b odld as a Poisson procss wih a i-varying arrival ra. Howvr, as discussd in Scion, h arrivals ofn xhibi ovrdisprsion, and his has propd h assupion of a sochasic arrival ra. Jongblod and Kool (00) considr h us of a Poisson disribuion wih gaa disribud arrival ra, which iplis an assupion of a ngaiv binoial disribuion for h arrival volu. In his papr, w us his disribuion wihin i sris odls. In our odls, h an and varianc of h ngaiv binoial disribuion ar i-varying, and can b wrin 8

11 as and, rspcivly, whr 0<. Th parar conrols h dgr of ovrdisprsion. (Th corrsponding an and varianc of h gaa disribud arrival ra ar and, rspcivly.) Alhough no considrd by Jongblod and Kool, sing (whr iplis ha h varianc of h ngaiv binoial disribuion is, a quadraic funcion of h an (s Hinn 003). (This varianc spcificaion is naural whn h ngaiv binoial is viwd as dscribing h nubr of failurs in a squnc of rials bfor h rh succss. In his conx, =/r.) 5.. An HWT Coun Daa Modl Exprssions (4a)-(4g) prsn a nw ngaiv binoial coun daa odl basd on h uliplicaiv HWT odl of Scion 4. W us a uliplicaiv srucur bcaus i avoids h possibiliy of a ngaiv arrival ra, and bcaus, as nod in Scion 4, a uliplicaiv forulaion is or suiabl for our daa. HWT coun daa odl: y NgBin, (4a) ~ l d w h (4b) y / l d w (4c) l lh d d h w w h whr h x (4d) (4) (4f) xp xp x y is h coun for h half-hour priod nding a i ; is h an and is h ovrdisprsion parar of h ngaiv binoial disribuion, as dfind in Scion 5.; and h is a logisic funcion wih parar. No ha wih h(x)=(+x), his nw odl would hav siilar for o h uliplicaiv HWT odl of Scion 4. Howvr, his choic for h funcion h canno nsur a non-ngaiv arrival ra in h scond of our xnsions of h odl for saionary lvl, which w considr in Scion 5.3. An alrnaiv for h is an xponnial funcion, which, as w discussd in Scion 5. has bn usd in svral coun daa odls. Indd, wih his choic of h, h sa quaions of xprssions (4d)-(4f) ar siilar in (4g) 9

12 srucur o xprssion (3c) of Zgr s (988) forulaion. Howvr, wih our daa, using an xponnial funcion for h ld o occasional xrly larg valus for h lvl and sasonal coponns whn prforing Mon Carlo siulaion o gnra dnsiy forcass. Th caus of his was low couns y, which ld o low valus for h lvl r l, which in urn rsuld in h rlaiv rsidual rror in xprssion (4c) bcoing larg, spcially whn h sasonal facors, d and w, wr a hir lows. Th cobinaion of a larg valu for and h us of an xponnial funcion for h causd xprssions (4d)- (4f) o dlivr occasional xrly larg valus for h lvl and sasonal coponns. This ld o us slcing h o b a funcion wih an uppr bound, as wll as a lowr bound of zro. A naural choic was h logisic funcion of xprssion (4g), which has h araciv propris ha i is non-ngaiv, i has an uppr bound (qual o (+xp())), i is onoonic, and i is qual o whn x=0. Th valu of dicas and liis h ipac of h prvious priod s rsidual on h currn priod s sias of h lvl and sasonal coponns. W no ha uppr and lowr bounds hav prviously bn considrd wihin a coun daa odl by Fokianos (00), and ha a logisic funcion is usd in logisic rgrssion, which is a for of gnralizd linar odl (s McCullagh and Nldr 989). Th HWT coun daa odl is an obsrvaion-drivn (singl-sourc of rror) odl. This can b sn by using xprssion (4c) o subsiu for in h ohr xprssions o giv a odl wrin in rs of y. For h HWT coun daa odls in his papr, analyical xprssions for poin and dnsiy forcass xis for only on sp-ahad prdicion. For longr lad is, Mon Carlo siulaion us b usd. Th us of hurisics is coon whn iniializing xponnial soohing odls (s Hyndan al. 008, Scion 5.). Our hurisic approach was basd on h firs hr wks of daa. W usd h an of hs obsrvaions o iniializ l. Our iniializaion of d procdd by calculaing, for a givn priod, h raio of h obsrvd valu, y, o h -poin oving avrag. Th iniial valu of d, for ach priod of h day, was s as h goric an of h firs svn of hs raios corrsponding o h sa priod of h day. To iniializ w, w firs calculad, for a givn priod, h raio of h obsrvd valu, y, o h -poin oving avrag. Th iniial valu of w, for ach priod of h wk, was s as 0

13 h goric an of h firs wo of hs raios, corrsponding o h sa priod of h wk, dividd by h iniial valu of d for h sa priod of h day. Maxiu liklihood was usd for parar siaion. Using h ngaiv binoial dnsiy funcion, w consrucd h liklihood funcion as: N 3 y y! whr N is h nubr of priods in h siaion sapl; h s P conains h priods corrsponding o public holidays (which wr oid fro h parar opiizaion); is h gaa funcion; is givn by xprssion (4b); and is h ovrdisprsion parar. To opiiz his objciv funcion, w followd a procdur siilar o ha usd by Engl and Manganlli (004) for a diffrn yp of odl. Maxiizaion of h liklihood procdd by sapling 0 4 vcors of parars using a unifor rando nubr gnraor wih bounds u L and u U. Exprinaion ld o h us of u L =0 and u U = for,,, and ; u L =-5 and u U =5 for ; and u L =0 and u U = for. For ach of h 0 4 randoly sapld vcors, w valuad h log liklihood funcion. Th hr vcors ha producd h highs log liklihood valus wr hn usd, in urn, as h iniial vcor in a quasi-nwon algorih. Of h hr rsuling vcors, h on producing h highs log liklihood was chosn as h final parar vcor. In h row labld HWT of Tabl, for h NHS Dirc daa, w prsn h siad parars for h HWT coun daa odl of xprssions (4a)-(4g) wih varianc odld as. Th valu of iplis ha h funcion, h, has an uppr bound of (+xp(-0.306))=.736. Subsanial ovrdisprsion is indicad by h low valu of. In Scion 5., w dscribd how h varianc can b odld as, a quadraic funcion of h an. In h row labld HWT of Tabl, for h NHS Dirc daa, w prsn h siad parars for h odl wih varianc odld in his way. Th posiiv valu of indicas ovrdisprsion Tabls and y

14 5.3. HWT Coun Daa Modls wih Saionary Lvl Fro Figur of Taylor (00), h NHS Dirc i sris appars o b saionary in h lvl. Alhough hr is or chang in h lvl of h crdi card copany sris in Figur, i could b argud ha h lvl of ha sris is also an-rvring. If a sris has a saionary lvl, and a odl wih nonsaionary lvl is usd, hr would b a isspcificaion probl and h accuracy of long-r forcasing is likly o b paricularly poor. Howvr, in ach of h HWT forulaions in Scions 4 and 5., unlss =0, h lvl, l, is odld as nonsaionary. Indd, all of h sandard fors of xponnial soohing, prsnd in Gardnr s (006) rviw, possss a nonsaionary srucur for h odling of h lvl. In his scion, w dvlop nw vrsions of h HWT odl wih saionary lvl. L us considr sing =0 in h HWT coun daa odl of xprssions (4a)-(4g). This lads o l bing qual o a consan, which can b inrprd as a consan long-run an lvl, l. W can hn rwri xprssions (4b) and (4c) as xprssions (5) and (6), rspcivly. L us dfin a nw lvl as in xprssion (7). l d w h (5) y / l d w (6) l l h (7) Using his, w can rwri xprssion (5) as xprssion (8b). L us dfin a nw rlaiv rror in rs of h nw lvl, as in xprssion (8c). Using xprssions (6) and (8c), w obain ( l l ) / l ( l / l ). W can us his o subsiu for in xprssions (7), (4) and (4f) o dlivr xprssions (8d)-(8f), rspcivly. In suary, h rsul of sing =0 in h HWT odl of xprssions (4a)-(4g) is h nw coun daa odl of xprssions (8a)-(8g). Th inrsing faur of his odl is ha xprssion (8d) rprsns a saionary odling of h lvl. Th lvl and wo sasonal rs ar influncd by dviaions of h prvious priod s lvl fro h long-run an, l. W opiizd l in h sa procdur as h odl parars. Opiizd odl parars for h NHS Dirc daa ar givn in Tabls and, in h rows nild HWT wih Saionary Lvl. In ach abl,

15 h valus of, and for his odl ar gnrally qui diffrn o h corrsponding valus for h HWT odl of Scion 5.. HWT coun daa odl wih saionary lvl: y NgBin, (8a) ~ l d w (8b) y / l d w l l h l l / l l / l d d h l l / l l / l w w h l l l l / l (8c) (8d) whr (8) (8f) / h x xp xp x (8g) Alhough foral saisical sing is no h nor whn slcing fro a s of xponnial soohing hods, w no ha, in principl, a uni roo s could b prford o sablish whhr or no a odl wih saionary lvl should b usd. Such a s would nd o b suiabl for sing for a uni roo in high frquncy daa in h prsnc of srong sasonaliy. If h odl of xprssions (8a)-(8g) was inroducd wih no rfrnc o h arlir HWT coun daa odl, on igh ask why, in ach sa quaion, h cofficins of ( l l ) / l and ( l / l ) ar spcifid o b h sa. This props an unconsraind vrsion of h odl, which involvs hr addiional parars,, and, and which w prsn in xprssions (9a)-(9g). W rfr o his as h gnralizd for of h odl. Opiizd parar valus for his odl ar prsnd in h boo rows of Tabls and. In ach abl, h parars and hav qui diffrn valus, and so also do and. This givs so jusificaion for considring h gnralizd vrsion of h odl. 3

16 HWT coun daa odl wih saionary lvl and gnralizd: y NgBin, (9a) ~ l d w (9b) y / l d w l l h l l / l l / l d d h l l / l l / l w w h l l l l / l (9c) (9d) whr (9) (9f) / h x xp xp x An HWT odl can b usd dircly o produc forcass for all lad is and fro all forcas origins. This rlas o inraday updaing, and also o how an HWT odl would b usd wihin a call cnr s capaciy planning cycl (s Gans al. 003, Scion 3.4). A forcas would b producd fro h HWT odl so wks in advanc, and a any poin up o and including h schduld day islf, h saffing plan would b updad basd on nw forcass gnrad fro h sa HWT odl. Th only hods pu forward so far in h liraur for prdicing dnsiis of boh call arrival volus and ras ar h hods of Winbrg al. (007) and Shn and Huang (008b). Th choic as o whhr o us on of hs hods or an HWT odl will ulialy b dcidd by forcas accuracy, and by how asily h hods can b undrsood and iplnd. (9g) 6. Epirical Analysis Scion 6. liss h forcasing hods includd in our pirical sudy. Scion 6. dscribs h saisical asurs usd o valua accuracy. Scions rpor h rsuls for h wo sris dscribd in Scion 3, and h US bank daa of Winbrg al. (007). Scion 6.6 uss a call cnr siulaion odl o valua h us of h arrival ra dnsiy forcass in sing saffing lvls. 4

17 5 6.. Forcasing Mhods HWT This is h odl prsnd in xprssions (4a)-(4g). W rfr o his as h basic HWT odl. For all h HWT odls and h ARMA odl dscribd blow, w considrd h hr disribuions discussd in Scion 5.: Poisson, ngaiv binoial, and ngaiv binoial wih varianc a quadraic funcion of h an. For ach forcas origin, w usd Mon Carlo siulaion wih,000 iraions o gnra a dnsiy forcas. A poin forcas was consrucd as h an of h,000 iraions. HWT wih saionary lvl This is h odl prsnd in xprssions (8a)-(8g). HWT wih saionary lvl and gnralizd This is h odl prsnd in xprssions (9a)-(9g). ARMA This is an ARMA coun daa odl of h yp considrd by Davis al. (003) wih h inclusion of lags corrsponding o h inraday and inrawk cycls:, ~ gbin N y z xp z z y xp xp z z z z z z z z z z whr,, h i and h i ar consan parars. W basd odl slcion on h Schwarz Baysian Cririon wih h rquirn ha all parars wr significan (a h 5% lvl). SVD-yp approach wih AR facor odls This is h hod of Shn and Huang (008b), dscribd brifly in Scion. Wihin a Poisson odling frawork, an SVD-yp procdur is usd o rval a daily sris for ach of h ain undrlying facors in h inraday cycl. Our iplnaion closly followd Shn and Huang, wih us of hir alrnaing axiu liklihood algorih o xrac h facors; iplnaion of hir varying inrcp AR() odls o forcas h daily sris of ach facor; us of hir pnalizd axiu liklihood o upda h wihin-day forcass of h facors, as h forcas origin ovd hrough ach day; and boosrapping wih,000 rplicas o gnra forcass. Forcass of h facors for a fuur day wr gnrad fro h varying inrcp AR() odls using, as

18 facors for h currn day, h updad wihin-day forcass. W usd cross-validaion o slc: h nubr of facors; h nubr of iraions o ploy wihin ach liklihood axiizaion; and h wigh in h pnalizd axiu liklihood. Th cross-validaion usd a hold-ou sapl of h final fiv wks of h siaion sapl for h NHS Dirc daa and h US bank daa, and for h longr crdi card copany sris, w usd h final 0 wks. This ld o h us of fiv facors for h NHS Dirc daa, wo for h crdi card copany daa, and four for h US bank daa. SVD-yp approach wih xponnial soohing facor odls In his vrsion of h hod of Shn and Huang (008b), w rplac h varying inrcp AR() odls by sandard Hol-Winrs xponnial soohing odls. Th inroducion of his hod was propd by h SVD-yp approach wih AR facor odls prforing rlaivly poorly for h NHS Dirc and crdi card copany sris. Taylor (00) prsns an SVD-basd approach ha uss jus on xponnial soohing odl o sooh and upda all h facors siulanously, wih h arrival of ach nw inraday obsrvaion. Howvr, i is no a coun daa odl, and so i is unclar how i could b usd o dlivr dnsiy forcass of h arrival ra. Sasonal an In his poin forcasing bnchark hod, h forcas for ach lad i is h an of a oving window of in-sapl arrivals for h sa half-hour of h wk as h priod o b prdicd. W s h oving window o b qual in lngh o h iniial siaion sapl. Siplisic hods, such as his, ar dscribd by Mhrora and Faa (003) as coonly usd in pracic. 6.. Forcas Evaluaion Masurs This papr is focusd on h dnsiy forcasing of h arrival ra and of call volus. Howvr, bcaus h ra is unobsrvabl, w follow Winbrg al. (007) by valuaing pos-sapl accuracy of only h dnsiy forcass of call volus. Givn h ordinal and discr naur of coun daa, o valua h dnsiy forcass w ployd h rankd probabiliy scor (RPS) (s Wilks 995, Scion 7.4.8): J RPS(F,y ) = F j Iy j j0 6

19 F is h forcas of h cuulaiv disribuion funcion (cdf), y is h acual call volu, I is h indicaor funcion, and J is h axiu possibl ouco for y. (W s J o b h largr of h following wo valus: y and h largs ouco gnrad by h Mon Carlo siulaion fro h odl ha is bing valuad.) Th RPS has bn widly usd, paricularly in h aosphric scincs, o valua discr dnsiy forcass. I is a squard forcas rror asur, whr h prdicd cdf is valuad a ach possibl ouco j. Lowr valus of h RPS ar prfrabl. Th asur capurs h wo iporan characrisics of a disribuional forcas: is locaion rlaiv o h obsrvd valu and is sharpnss around ha valu. For ach lad i, w calculad a an RPS valu by avraging ovr h RPS valus obaind for h diffrn forcas origins. W also valuad h 5%, 5%, 75% and 95% quanils of h dnsiy forcass by calculaing h hi prcnag, which is h prcnag of obsrvaions falling blow h quanil forcas. Idally, his prcnag should b. W xaind significan diffrnc fro his idal using a s basd on h binoial disribuion. W no ha h hi prcnag asurs h uncondiional covrag in a quanil forcas, and ha a horough valuaion rquirs a asur of condiional covrag (Chrisoffrsn 998). To valua poin forcas accuracy, w calculad h an absolu rror and roo an squard rror. For h NHS Dirc and US bank daa, w also calculad h an absolu prcnag rror, bu his could no b calculad for h crdi card copany sris bcaus i conaind zros. Th rankings of h hods, for ach sris, wr siilar for ach rror asur usd. For ach of h ARMA and saionary HWT odls, byond h vry arly lad is, h RPS rsuls wr siilar for h hr disribuions. Evaluaing h uppr quanils using h hi prcnag, w found ha h rsuls for hs odls ndd o b br for h wo ngaiv binoial disribuions han for h Poisson. W show his for h NHS Dirc daa in Scion 6.3. For h basic HWT odl, a Poisson assupion was slighly prfrabl, bu his is no paricularly noworhy bcaus his odl was gnrally uncopiiv. In h rs of h papr, unlss sad o h conrary, w rpor h rsuls for ach odl wih ngaiv binoial disribuion and varianc odld as a quadraic funcion of h an. 7

20 6.3. Forcasing Rsuls for h NHS Dirc Daa Figur 4 prsns h an RPS rsuls for h NHS Dirc sris. Th rsuls ar disappoining for h SVD-yp approach wih AR facor odls, alhough i was or accura han h ARMA odl for h vry arly lad is. Boh of hs hods wr coforably ouprford a all lad is by h SVD-yp approach wih xponnial soohing facor odls. This hod was also noicably or accura han h basic HWT odl. Inrsingly, boh of h HWT odls wih saionary lvl dlivrd a clar iprovn on h basic HWT odl byond abou on day ahad, and hs wo hods wr, ovrall, h os accura. Turning o poin forcas valuaion, h rankings of hods wr vry siilar o hos shown in Figur 4 for h an RPS. This indicas ha dnsiy forcas accuracy is srongly rlad o h qualiy of h siaion of h locaion of h dnsiy. Th sasonal an poin forcasing hod prford vry poorly rlaiv o h ohr hods Figurs In Figur 5, w show h uncondiional covrag rsuls for h 95% quanils. For h ohr quanils, h rlaiv prforancs of h hods wr siilar o hos for h 95% quanil, xcp ha nihr of h SVD-yp approachs was copiiv for h 5% and 5% quanils. For h HWT odls and h SVD-yp approach wih AR facor odls, h rsuls in Figur 5 ar rasonably consisn wih h an RPS rsuls of Figur 4. Th bs rsuls in Figur 5 ar for h ARMA odl, which conrass wih h an RPS rsuls in Figur 4. An xplanaion for his is ha uncondiional covrag only convys h avrag covrag, and dos no valua i-variaion in h quanil and in h covrag. This has propd h dvlopn of asurs of condiional covrag for quanil sias (s, for xapl, Engl and Manganlli 004). An appal of h RPS asur is ha i capurs h abiliy of h dnsiy forcas o vary ovr i wih h daa, and i asssss h qualiy of h whol dnsiy forcas, rahr han jus an individual quanil. In Figur 6, w valua forcass of h 95% quanil fro h ARMA odl and on of h HWT odls using h hr diffrn disribuions. Th figur shows ha using a Poisson disribuion ld o poor rsuls, and ha h rsuls for h ngaiv binoial wr slighly iprovd wih varianc odld as a quadraic funcion of h an. 8