Optml Order Processn Polces or E- commerce Servers Yon Tn Vy S Mooeree Kmrn Monzde Unversty o Wsnton Busness Scool Box 353 Settle Wsnton 9895-3 USA Unversty o Texs t Dlls Scool o Mnement PO Box 83688 JO44 Rcrdson TX 7583-688 USA Unversty o Wsnton Busness Scool Box 353 Settle Wsnton 9895-3 USA ytn@usntonedu vym@utdllsedu mrn@usntonedu Te explosve rot n onlne soppn s provded onlne retlers mpressve opportuntes or revenue nd prot At te sme tme retlers my lose consderble onlne busness rom slo response tmes t electronc soppn stes Altou ncresn server cpcty my mprove response tme te resources nd cptl needed to do so re clerly not ree In ts study e propose sceme tt cn mprove server s perormnce under ts current cpcty Ts sceme s bsed on prorty order processn ere te prorty o n order depends on te potentl revenue tt ould be enerted rom te order Te results or snle-perod nlyss so tt te benet rom prorty processn ncreses s te server becomes buser We ve lso modeled mult-perod verson o te problem ere te demnd n perod depends on te Qulty o Servce QoS tt buyers receve n te prevous perod In multperod problem bot te server cpcty nd te order processn polcy n ec perod re determned optmlly ere t my be optml or te retler to scrce prot nd ncrese QoS n ntl perods n order to ncrese demnd nd revenue n lter perods Tus te order processn polcy o te server evolves rom n empss on QoS n ntl perods to one on prot n subsequent perods Electronc ommerce; Prorty Order Processn; Queues; Renen
Introducton Te lst e yers ve observed n explosve rot n onlne busness Ts rot s lely to sustn snce te number o Internet users round te orld s contnues to ro Te omputer Industry Almnc s reported tt by te yer 49 mllon people round te orld ll ve Internet ccess tt s 794 per people orldde nd 8 people per by yerend 5 More sncntly Jupter ommunctons estmtes tt onlne soppn ll ro to $78 bllon by Yer 3 Accordn to Jupter ommunctons retlers seen to benet rom te ue onlne mret must desn te onlne experence t ve to estbls lon term reltonsps t ne onlne buyers Ts s becuse meetn nd exceedn te expecttons o ne buyers s n eect r beyond te current seson; postve onlne experence t specc retler could o lon y tord securn uture strem o revenues In ts study e consder te problem o coneston n e-commerce servers Our concern s specclly to do t slo response tme A sncnt sttstc ere s one provded by Forrester Reserc Inc 66% o ll electronc soppn crts re bndoned beore sle s completed Wle te bndonn o crt my occur rom resons tt re not relted to response tme t s qute cler tt slo response tme s ndeed sncnt cuse One study enttled Te Web Is o Sopper s Prdse publsed n te ovember 999 ssue o Fortune mzne sttes tt Mrdesc 999: Amon te Web consumers surveyed or onttve s et strtey consultn rm bsed n Sn Frncsco qurterly Pulse o te ustomer report te o reson tt customers ot ed up nd too ter busness
elseere s tecncl problems ncludn uncceptbly slo response tmes On smlr lnes Green 999 conducted more detled survey usn onlne soppers to study coneston problems n e-commerce servers Te survey reveled tt 8% o te respondents ound stes too slo nd let never to return An obvous soluton to te bove coneston problems s to ncrese server cpcty An nterestn cse n pont ere s te e-mercnt 8com tt sells consumer electroncs Ater ts lunc n October 998 te compny tred lunced promoton tt oered tree moves or tree Ds or $ Te promoton ored perps too ell On ovember 7 998 te compny s ste s smped nd vrtully sut don s undreds o tousnds o customers loced to te ste to obtn cep moves nd Ds Bsed on ts experence 8com no reles on 5 rter tn 6 mn servers to ndle ste trc! oever le ddn more servers sould clerly llevte coneston problems; tere s no esy ormul to determne te optml number o computers Furtermore ddn cpcty s obvously not ree nd sould be done only te ddtonl cpcty cn be economclly usted A resource neutrl soluton to te server coneston problem s to mprove perormnce usn prorty order processn Tere could be mny dmensons o perormnce e vere response tme trouput mxmum response tme revenue enerted etc; our ocus ere s on to o tese mesures: prot per unt tme nd percente o buyers lost Wle prot s n obvous perormnce mesure te percente o buyers lost s used to mesure Qulty o Servce QoS A buyer s stscton t te electronc soppn process sould depend n prt on eter or not te purcse trnscton s successully completed n resonble mount o tme Tereore te percente o buyers lost cn be used s mesure o te qulty o servce ceved by certn electronc ste Wen prot s te ol o te processn polcy 3
te mount o processn poer ssned to n order depends on to consdertons: te potentl revenue tt ould be enerted rom te order t s successully executed nd b te probblty tt te order ll be lost becuse te customer s dely tolernce s exceeded On te oter nd en QoS s te ol te mount o processn poer ssned to n order depends only on te probblty tt te order ll be lost In dervn te optml polces our study spns tree reserc res: tme-sred systems renen n queun teory nd derentted or prortzed servces Te concept o tmesrn s been dely dopted n sceduln computer processn Klenroc 964 967 976; Stllns 997 In tme-srn processn poer cn be dvded n to ys Te rst metod oten clled Round-Robn s true tme-srn polcy Ec process ets slce quntum o tme durn c te ull cpcty o te processor s dedcted to te process Te second metod s to dvde processn poer on ull-tme but prt-cpcty bss Ts metod reerred to s Processor-Srn represents te contnuous lmt o tme-srn ere te tme slce pproces zero Klenroc 964 967 obtned te expected response tme tme spent n te system tn tme plus requred processn tme or exponentlly dstrbuted rrvl nd deprture tme-sred M/M/ Te tn tme s proportonl to te processn tme ttned; suestn tt round robn systems re more r tn sequentl queues snce te tn tme n sequentl systems does not depend on te ctul mount o processn ncurred on ob omn et l 97 derved te expresson or te tn tme dstrbuton n tme-sred systems tt s ound to devte rom te exponentl orm tt descrbes te sequentl M/M/ queun model Morrson 985 lter ound te response tme tn plus requred process tme dstrbuton or M/M/ processor-srn systems St ouc nd Ozum 97 nd O Donovn 974 ve extended te results o expected response tme to 4
M/G/ processor-srn models ustomer mptence s drect source or loss o sles n onlne retln; customers qut soppn te tn exceeds ter tolernce Ts penomenon termed s queun t renen nd bln s been studed or bout 4 yers nd s stll n ctve reserc re Brrer 957 obtned te results or te queun problem t mptent customers bsed on eurstcs Gnedeno nd Kovleno 989 provded more rorous nlyss tt veres Brrer s results In ter studes tey ssume M/M/ system t eter constnt or exponentlly dstrbuted renen tme More detled studes on M/M/ system t n exponentlly dstrbuted renen tme ve been presented by Ancer nd Grn 96 963 Ro 968 extended te results to te renen n M/G/ systems In bot studes n exponentlly dstrbuted renen tme s ssumed Recent studes re ocused on extendn to more enerl queun systems or exmple GI/G/ model Stnord 979 In ddton Wtt 999 studed te renen problem en customers re normed bout ntcpted delys Tere s been no reserc to our nolede tt ncludes te renen n te tme-sred queun systems Ass nd vv 99 ve studed te strtey o renen rom processor srn systems oever te mpct o renen on te queues s not explctly modeled Klenroc 967 ntroduced te noton o prorty processn or computer processors nd derved te expected tn tme or derent prorty clsses t exponentlly dstrbuted processn tme O Donovn 974 soed tt te result olds even or M/G/ processorsrn systems Smlrly n netorn te vson o provdn derentted servces s been round or ell over decde Turner 986 Future pcet netors ll lely support Qulty o Servce n order to provde ull rry o servces Aello et l Reserc n ts re s becomn ncresnly populr s ne tecnoloy mes mplementton possble Oter studes 5
ve been ocused on economc sde nmely prcn or provdn prorty servces n communcton netors s ell s n oter settns Mrcnd 974 In ts study e propose nd nlyze optml order processn polces tt cn be used by e-commerce servers to mprove perormnce We lso provde vrety o nlytcl nd numercl results tt prescrbe o E-retlers sould best run ter servers to ceve ter perormnce ols We lso study mult-perod verson o te server coneston problem ere bot te server cpcty nd te processn polcy n ec perod re optmlly determned In te mult-perod problem e consder QoS externltes; demnd n ven perod depends on te QoS ceved n te prevous perod Te mult-perod problem s solved usn dynmc prormmn Te results rom te mult-perod nlyss cn provde udelnes or e-retlers to dust te order processn polcy over tme For exmple te nlyss revels tt e-retlers sould ntlly py more ttenton to QoS oever s te customer bse o n e-retler mtures prot-orented order processn polces sould be pursued Te rest o te pper s ornzed s ollos Sectons nd 3 re dedcted to te nlyss o optml server opertn polces A snle perod model s presented n Secton eres Secton 3 extends te snle perod model to mult-perod one umercl results re presented nd prctcl mplctons re dscussed Secton 4 provdes summry o results nd oers drectons or uture reserc Te Model In ts secton e set up te bsc model ere te demnd s ssumed to be sttonry Ts s snle perod problem E-retlers rst determne nd cqure te optml cpcty tt remns uncned or te rest o te plnnn orzon 6
Assumptons nd Prelmnres We crcterze orders rrvn t n E-commerce server by ter vlues Te predcted nl purcse vlue o n order ollos vlue dstrbuton One y to predct te nl purcse vlue s to use te customer s storc dt or exmple n vere vlue or movn vere Ater ec purcse te dt cn be moded nd ne vere vlue cn be clculted or use or te next vst For rst tme buyer typcl vlue suc s te mret vere or ll ne buyers cn be used Ts study ssumes sttc polcy e ec customer s ssned n estmted vlue tt remns uncned trouout te course o soppn In more sopstcted model te predcted nl vlue cn vry dynmclly or exmple by utlzn normton bout te vlue o tems n te soppn crt Tese nd oter metods e dt mnn tecnques cn be ppled A dynmcl polcy oers more ccurte vlue predcton oever t te cost o extr computtonl overed A sttc polcy estmtes te vlue olne tereore does not consume onlne resources but my suer rom nccurcy We ll ttempt to ddress ts trdeo queston n subsequent studes Te totl rrvl rte o orders s ts s te number o orders per unt tme We ssume tt orders rrve ccordn to Posson process c represents rel stuton qute ell Moe nd Fder Ec order requres certn mount o tme to be processed A typcl onlne soppn process nvolves vrous ctvtes or exmple brosn sercn ddn tems to soppn crt nd cecout For sttc polcy e nore te detls o nvolved soppn ctvtes s tese normton re not used to updte te predcted vlue nd nsted dene te requred processn tme s te totl tme tt n order receves processn rom te server ncludn ll soppn ctvtes Te requred processn tme or orders t vlue s ssumed to be exponentlly dstrbuted t n vere tme c s uncton o te vlue 7
Tereore te overll processn tme dstrbuton or ll orders s descrbed by te yperexponentl dstrbuton or lner combnton o exponentl dstrbutons t derent mens Te ssumpton o exponentl processn tme smples polcy nlyss We ve not ttempted to enerlze ts ssumpton; oever t s ood pproxmton or tme-sred systems ere t s been son St ouc nd Ozum 97; O Donovn 974 tt severl mportnt results suc s men delys depend only on te men processn tme nd not on te dstrbuton In enerl vere requred processn tme s non-decresn uncton o te vlue In our model ny uncton orm o cn be used Imptent customers my qut soppn te tme spent exceeds ter tolernce or tn We dene te tolernce level o sopper n terms o vere tme n te server tt s buyers t te vlue re lln to t or rndom tme tt s exponentlly dstrbuted Ancer nd Grn 96 t men It s lely tt customer mptence vres durn te course o soppn Smlrly or sttc polcy e suppress ts vrton nd dopt n vere level over vrous ponts o te soppn process Te mptence o n on-lne sopper s modeled usn n nloy rom te express lne n trdtonl rocery store Typclly store express lnes provde ster servce to customers t reltvely eer tems Te mplct ssumpton ere s tt customers t eer tems my be less tolernt o delys; more enerlly te tolernce level o customer s ssumed to depend n some y on te number o tems tt te customer ntends to purcse In our model te dely tolernce cn be ny uncton o ; oever t s plusble tt s ncresn n Smlr to te sceduln o computer processor server cpcty s sred by orders n te server Te srn sceme n typcl e-commerce server s bsed upon Round-Robn tmesrn Specclly n order s ven slce o processn tme sy Q en t enters te 8
processn unt o te server It exts rom te system t nses te desred processn durn ts llotted tme Oterse t oes bc to te end o te queue nd ts or ts next turn Round-Robn sceduln s better tn sequentl sceduln n c customers re served one t tme becuse sequentl sceduln my ste cpcty le tn or clent s response We ntroduce te prortzed processn by ssnn et or orders n prorty clss- so tey receve processn tme Q Ts sceme s rst proposed by Klenroc 967 Te prorty o n order s determned by te potentl nl vlue o te purcse tt buyer mes In ts study e lmt ourselves to te cse ere tere re n nnte number o clsses Ts llos us to ocus on te propertes o prorty sceme tout nvolvn complcted problem o ssnn orders to clsses Te processn et becomes contnuous nd s uncton o vlue Oern dscrete clsses my be optml te cost o mplementn te polcy s consdered A model or dscrete clsses s been publsed elseere Tn As mentoned erler customers ll bndon ter soppn crt or renee tey ve been mde to t or too lon We ve solved queun problem tt ncorportes renen n prorty-bsed tme-srn system Te result s summrzed n te ollon proposton Proposton Te loss uncton densty dened s number o customers lost per unt tme per unt vlue l or orders t vlue s ere must stsy l d 9
nd s te set o ll possble vlues Te proo s presented n te Appendx ere te proos or ll te propostons corollres nd lemms cn be ound In te next secton e ntroduce mult-perod verson o ts model ere cpcty coces cn be mde t te bennn o ec perod In ts secton oever e ve ssumed tt cpcty cn be cosen once t te bennn o te perod Te eect o ncresn server cpcty s to reduce te requred processn tme o n order: / ere s te processn tme or stndrd unt o cpcty nd s te server cpcty A more poerul server s lrer vlue o nd tereore orders cn be processed ster Tere s cost ssocted t cqurn cpcty We ssume lner cost γ ere γ s te unt cost normlzed to per unt tme Te lner orm cn be usted s n most cses cpcty cn be ddtve or exmple more servers re dded We nore te xed cost s t does not ect our results Prot-ocused Polcy Te totl expected revenue per unt tme cn be rtten s l S d 3 A retler s obectve s to coose cpcty nd processn et suc tt te expected prot per unt tme s mxmzed Explctly e dene n E-retler s problem: mx π subect to d γ
d 4 Equton 4 s dentcl to Equton t cpcty explctly expressed Te soluton to ts problem s ven n te ollon proposton: Proposton Te optml processn et llocton s ; ; / / S S ere s te Lrne multpler ssocted t Equton 4 nd te servceble set S s dened s S : Te optml cpcty s d γ Proposton ndctes tt tere s revenue relzton tresold dened by ; only orders t revenue relzton bove ts tresold receve processn cpcty Te revenue relzton tresold s rte t c n order s revenue s relzed Te vlue o cn be obtned by substtutn te expressons o nd n Equton 4 Te problem s reduced to solvn or nd Te ollon corollry provdes bounds or numercl serc orollry Te optml cpcty stses te ollon nequltes: S γ S s te revenue en te optml polcy s dopted Te rst nequlty ndctes tt n
order ll et processed t cn recover te cost o cpcty Te system cn lso ord to dmt some orders ose vlues re belo te cost o cpcty Te second nequlty urntees non-netve prot; representn te ndvdul rtonlty condton or te retler We ve son te ollon comprtve sttc results old orollry For te optml processn et llocton / > ; / < Orders t er rte o revenue relzton e te / rto receve more processn tme Te rto / mesures customer s ptence level Orders t er vlue o ts rto more ptent buyers cn tolerte more dely nd ence receve less processn tme Fure sos te optml processn et or lner processn tme nd tn tolernce: 7 3 nd 5 ere e ssume tt cpcty s xed so only te tresold needs to be clculted Te vlue dstrbuton s unorm nmely or [ ] ncreses t vlue s te rto / s n ncresn uncton o le / decreses t oever n Fure t nonlner orms o 3 3 7 nd 5 8 s not monotonc It turns out tt te rto / pes round te vlue 6 Ts exmple sos te mportnce o tern normton on customer purcsn bevor e te dely tolernce; er vlue lone does not rrnt er prorty
8 4 6 4 3 7 3 4 6 8 Fure Optml processn et plotted s uncton o vlue or lner processn tme nd tn tolernce 4 6 8 Fure Optml processn et plotted s uncton o vlue or nonlner processn tme nd tn tolernce orollry 3 Wt te cne o demnd te ollon results old: π > ; > ; > > ere te expresson or te coecent s ven n te proo Wen te totl demnd ncreses te expected prot ll lso ncrese Ts cn be ceved by ddn more cpcty Te server becomes more dscrmntn by rsn te tresold en te tresold exceeds certn level Te tresold / trnsltes to vlue tresold c c c Fure 3 sos te vlue tresold c s uncton o demnd t te sme prmeter settn o Fure I te cpcty s xed n E-retler s to drop more lo vlue orders so tt er 3
prorty orders re more lely to complete Oterse te cpcty ll be dusted optmlly t demnd tout muc ncrese n te deree o derentton c 5 4 3 Optml pcty onstnt pcty 6 5 4 3 Prot-ocused QoS-ocused Unorm 4 6 8 4 6 8 Fure 3 Vlue tresold or servceble orders c plotted s uncton o demnd or xed cpcty nd cpcty determned optmlly Fure 4 Optml processn et plotted s uncton o vlue or protocused QoS-orented nd unorm nonderentted polces orollry 4 Wt te cne o cpcty cost γ te ollon results old: π < ; γ < ; γ > γ It s ntutve tt en te cost o cpcty ncreses te expected prot nd cpcty ll drop Wen te cost o cpcty ncreses te server becomes more dscrmntn by rsn te tresold 3 QoS-ocused Polcy In ts subsecton e consder polcy tt ocuses on te Qulty o Servce QoS We dene te QoS s te number o lost orders per unt tme rerdless o ter order vlue Te E- 4
retlers problem becomes mn L l d 5 ere l s te loss uncton densty ven n Proposton Te optml processn llocton cn be obtned smlrly c / / < c ; c Te processn tme tresold c cn be ound by substtutn te bove expresson n Equton It s obvous rom te bove equton tt orders more processn ll be ssned less processn tme More ptent buyers t er / rto lso receve less processn tme Te vlue tresold c s ven by Assumn ncreses t te c c vlue orders t vlue bove te tresold c ll not receve ny processn cpcty We next compre tree polces: prot-ocused QoS-ocused nd unorm Te unorm polcy s non-dscrmntn polcy ere s constnt ere lner orms o 7 3 nd 5 re used nd te cpcty s set to be Fure 4 clerly sos te derence beteen prot-ocused nd QoS-orented polces QoS polces vor lo vlue customers s orders rom suc buyers consume less processn tme Fure 5 plots te expected revenue or prot s te cpcty s xed Te QoS-ocused polcy s te orst perormer s t vors lo vlue orders t sorter processn tme For te Prot-ocused polcy te mprovement n revenue ncreses en server becomes buser er Fure 6 plots te loss rto o customers L/ mesure or te qulty o servce Te QoS-ocused polcy s most eectve n te mddle rne o demnd snce L/ converes to s nd s or ll polces 5
8 6 9 Revenue S 4 Prot-ocused QoS-ocused Unorm L/ 8 7 6 Prot-ocused QoS-ocused Unorm 4 6 8 Fure 5 Sles plotted nst demnd or prot-ocused QoS-orented nd unorm non-derentted polces 5 4 6 8 Fure 6 QoS percent lost customers L/ nst demnd or prot-ocused QoS-orented nd unorm polces 3 Mult-Perod Model In Secton e presented vrous polces tt ttempt to optmze n E-retler s prot n snle perod oever retlers oten vlue ter customer bse more tn mmedte prot Ts consderton s not tout mert snce lost customers rrely come bc Ts suests tt durn ntl perods te vlue o orders sould be pd less mportnce so s to buld sold customer bse In ts secton e propose mult-perod model tt ncludes eedbc on qulty o servce Specclly te qulty o servce tt customers receve n perod ects te demnd n te next perod E-retlers re lloed to vry cpcty to mtc te demnd We dd n ndex or te perod to te notton used n te prevous secton We strt t te loss uncton densty l n te -t perod Te expresson or l s te sme s te one son n Proposton t subscrpts or te perod erever necessry Te expected number o buyers lost per unt tme n te -t perod L s clculted usn Equton 5 nd l We model te demnd n te -t perod n te ollon y: L p L r 6 ere p s te probblty tt unstsed buyers return nd r s te rte o rot n demnd 6
due to successul purcses or exmple trou ord o mout Equton 6 cn lso be expressed n terms o te expected percente o buyers lost L / mesure o QoS E-retlers re lloed to dust cpcty bot uprds nd donrds; mplyn tt e ccommodte stutons n c cpcty cn be rented We ssume tt te cost o cqurn cpcty s proportonl to te cpcty dded nmely γ ere s te cpcty n te -t perod Smlrly s n Equton e ve / Snce rdre cost typclly decreses t tme te coecent γ s ssumed to decrese rom perod to perod Wtout loss o enerlty e ssume tt te demnd becomes sttonry n te -t perod E-retlers ve opportuntes to dust cpcty ter c te cpcty remns uncned rom te -t perod onrds Te dscounted prot s mx δ π δ δ S γ S δ ere δ s te dscount ctor; nd nd re sort nds or te processn et nd cpcty vectors Te prot π S or becuse te opertn polcy nd cpcty remn uncned rom te -t perod onrds Te dscounted prot cn be represented by nte orzon ormulton δ π ere π s redened s S π γ δ We cn solve ts mult-perod problem usn te metod o dynmc prormmn Dreyus nd L 977 more specclly bcrd nducton Let us dene Π l mx δ π l l l l l l ; l l ere l nd l or l re te decson vrbles l s te demnd or te l-t perod 7
8 c cn be obtned recursvely usn te demnd enerton model s speced by Equton 6 Π s te mxmum dscounted prot strtn rom te -t perod It depends on te demnd rrvn n ts perod nd te cpcty crred over Ts llos us to rte te recursve relton ; mx Π Π δ π 7 ere Π s te obectve uncton We rst ntroduce te ollon lemm Lemm Te eect o crry-over cpcty on te mxmum dscounted prot s descrbed s belo γ Π Ts result s ntutve s more exstn cpcty reduces te cost o cpcty n te current perod nd consequently ncreses te dscounted prot o e re n poston to crcterze te optml polces Proposton 3 Te optml processn et llocton n te -t perod < s ; ; / / S S κ ere te servceble vlue set S s dened s { } S : κ nd p r δ κ Π 8
Te optml cpcty s d γ δγ For perods te processn et nd cpcty re ven by Proposton t te eectve unt cpcty cost δγ As dsplyed n Equton 8 ts polcy seems to combne te prot-ocused nd QoSorented polces or te snle perod problem; te ctor κ s lner combnton o to rtos / nd / Te reltve et o ts combnton s determned by te dscount ctor δ nd to prmeters r nd p tt determne te demnd rot Ts controls te evoluton o te opertn polcy rom QoS-ocus t te bennn to prot ocus en te demnd s stedy In te ollon e present some numercl demonstrtons Gven te results descrbed n Proposton 3 n ec perod e numerclly evlute nd orollry 5 Te ollon recursve relton olds nd me use o orollry 5 Π d δp Π Te computton procedure s s ollos We strt rom te -t perod ere te polcy n ts perod s descrbed by Proposton Te recurson n orollry 5 s ten ppled n te prevous perod to obtn te optml polcy n te --t perod usn Proposton 3 9
8 / 6 4 Perod Perod Perod 3 / 8 6 4 Perod Perod Perod 3 4 6 8 4 6 8 Fure 7 Optml processn ets normlzed / plotted s uncton o vlue or dscountn ctor δ 3 Fure 8 Optml processn ets normlzed / plotted s uncton o vlue or dscountn ctor δ 6 Fures 7 nd 8 plot te optml processn lloctons or 3; r p ; 7 5 ; γ 4 γ 35 γ 3 3; nd unorm vlue dstrbuton 3 or Te processn llocton s purely prot-ocused rom te trd perod onrds E-retlers t er δ re less dscrmntn s s evdent rom loer vlue tresold c A er vlue o dscount ctor δ ndctes tt te E-retler s more concerned bout te lonterm Tere s ever cpcty nvestment n te ntl perod to ccommodte more customers As δ ncreses te polces n te ntl perods becomn less dscrmntn or more QoSorented due to te nluence o uture demnd rot Fure 9 sos te optml cpcty ncrements n tree perods We ssume te cpcty remns uncned rom te trd perod onrds E-retlers t lo vlue o δ nvest n cpcty n te ntl perod nd mntn more or less constnt cpcty level even t te decresn cost o cpcty Tey ocus on prot mn even n te ntl perods Ts results n poor QoS nd tereore tere s no rot n demnd E-retlers t er δ nvest evly n te ntl perod nd dopt QoS-orented polcy to buld uture demnd Te second perod
sees smll ncrese or even decrese n cpcty Prtly te demnd rot s yet to te ull eect so te sltly ncresed or even decresed cpcty cn sustn te sme level o QoS Also te cqurn o cpcty cn be postponed to te trd perod tt s loer cpcty cost Recll tt te eectve unt cpcty cost s δγ It decreses t dscount ctor δ tereore e expect te optml cpcty ncrement n te trd perod to ncrese t δ Fure sos tt te prots n te second trd nd ourt perods ncrese t dscount ctor Ts s becuse o te ncresed customer bse or demnd In te second perod relesn o cpcty ves sltly ster ncrese o prot t er δ Te derence beteen te trd perod nd perods rom te ourt onrds s te cost o cpcty It s pprent tt E-retlers t loner orzons er δ suer more n te ntl perods rom evy nvestments n cpcty tt re med t mprovn QoS 8 Perod Perod Perod 3 8 6 4 6 π -6 4 6 8 δ - -4-6 Perod Perod Perod 3 Perod 4 4 6 8 δ Fure 9 Optml cpcty ncrements plotted nst dscountn ctor δ Fure Prot n ec perod π plotted s uncton o dscountn ctor δ 4 onclusons nd Future Reserc In ts study e propose n order processn sceme tt mproves te perormnce o n E- commerce ste Ts sceme s bsed on prorty order processn ere te prorty o n order
depends on te potentl vlue o te order We present te model nd te results or snleperod It s son tt te benet rom prorty processn ncreses s te server becomes buser We ve lso modeled mult-perod problem ere te demnd n perod depends on te Qulty o Servce QoS tt buyers receve n te prevous perod It s observed tt te retler usully loses money n te rst perod n order to provde better servce nd rot n uture demnd Te opertn polcy o te server evolves rom QoS-ocused one n ntl perods to prot-ocused one n subsequent perods In mult-perod problem te server cpcty plnnn n ec perod s determned optmlly Te current study nvolves one server or trets multple servers s snle server It s o prctcl mportnce to study te lod-blncn problem Ts s to-level problem Ec server ll be optmzed usn prorty sceme condtonl on n optml mount o trc drected to te server Anoter nterestn extenson s to control or nluence te vlue dstrbuton Ts cn be ceved usn pproprte ncentve scemes suc oern on te spot dscount to buyers tt my oterse leve Fnlly e re orn on te nlyss o dynmc order processn polces tt llo te prorty o n order to vry durn te course o soppn especlly n response to more ccurte predcton o te nl vlue nd te dely tolernce Reerences Aello W A Mnsour Y Ropoln S nd Rosen A ompettve Queue Polces or Derentted Servces Proceedns o IFOOM Ancer J nd Grn A V Queun t Imptent ustomers Wo leve t Rndom Journl o Industrl Enneern 3 pp 84-9 96 Ancer J nd Grn A V Some Queun Problems t Bln nd Renen I
Opertons Reserc pp 88-963 Ancer J nd Grn A V Some Queun Problems t Bln nd Renen II Opertons Reserc pp 98-937 963 Ass D nd vv M Renen rom Processor Srn Systems nd Rndom Queues Mtemtcs o Opertons Reserc 5 9-38 99 Brrer D Y Queun t Imptent ustomers nd Inderent lers Opertons Reserc 5 644-649 957 Brrer D Y Queun t Imptent ustomers nd Ordered Servce Opertons Reserc 5 65-656 957 omn E G Muntz R R nd Trotter Wtn Tme Dstrbutons or Processor- Srn Systems Journl o te Assocton or omputn Mcnery 7 pp 3-3 97 Dreyus S E nd L A M Te Art And Teory O Dynmc Prormmn Acdemc Press e Yor 977 Gnedeno B V nd Kovleno I Introducton to Queun Teory Bräuser Boston 989 Green Te Gret Yuletde Seout Busness Wee pp EB 8-8 ovember 999 Klenroc L Anlyss o Tme-Sred Processor vl Reserc Lostcs Qurterly 59-73 964 Klenroc L Tme-sred Systems: A Teoretcl Tretment Journl o te Assocton or omputn Mcnery 4 pp 4-6 967 Klenroc L Queuen Systems: Volume omputer Applctons Wley e Yor 976 Mrcnd M G Prorty Prcn Mnement Scence 7 pp 3-4 July 974 Mrdesc J Te Web Is o Sopper s Prdse Fortune pp 88-98 ovember 8 999 Moe W W nd P S Fder Modeln Onlne Store Vst Ptterns s Mesure o ustomer Stscton Wrton Scool Worn Pper Morrson J A Response-Tme Dstrbuton or Processor-Srn System SIAM Journl o 3
Appled Mtemtcs 45 5-67 985 O Donovn T M Drect Solutons o M/G/ Processor-Srn Models Opertons Reserc pp 3-35 974 Ro S S Queun t Bln nd Renen n M / G / Systems Metr pp 73-88 968 Sn Introducton to te lculus o Vrtons McGr-ll e Yor 969 St M ouc S nd Ozum J An nlyss o te M/G/ Queue under Round-Robn Sceduln Opertons Reserc 9 37-385 97 Stllns W Opertn Systems: Internls nd Desn Prncples Prentce ll e Jersey 997 Stnord R Renen Penomen n Snle nnel Queues Mtemtcs o Opertons Reserc 4 6-78 979 Tn Y Vlue-bsed Desn o Electronc ommerce Servers Doctorl Dssertton Unversty o Wsnton Turner J S e Drectons n ommunctons IEEE ommunctons Mzne 986 Wenstoc R lculus o Vrtons McGr-ll e Yor 95 Wtt W Improvn Servce by Inormn ustomers bout Antcpted Delys Mnement Scence 45 pp 9-7 999 Appendx Proo o Proposton Dscrete lsses Let us rst solve te dscrete verson o te problem Assume tt tere re prorty clsses; clss- orders ve rrvl rte processn rte µ nd renen rte ν Let E be te expected number o clss- orders n te system n te stedy stte We ollo te metod used by Klenroc 964 Introduce ted order ssume t s one o te prorty clss- orders Ec order n clss- s ven tme slce quntum o Q ere 4
5 Q s nntesml Te cycle tme s te durton beteen to consecutve eects o te ted order rom te server nmely Q QE Wt probblty Q µ n order o clss- ll ns needed processn nd ext rom te system durn te processn tme Q llocted to ts order Ater cycle expected number o clss- orders ll be E Q QE Q QE E Q ν µ Te rst term s te expected number o clss- orders returnn to te system Te second term s te expected number o clss- orders rrvn durn cycle Te trd term s te expected number o clss- orders renen durn cycle In te stedy stte te bove expresson sould be te sme s E Tereore e et E E E E ν µ or G G E ν µ ere te constnt G G E G ν µ A Te bove constnt cn be solved sel-consstently Te expected loss uncton or clss- orders s ten G G E L ν µ ν ν
Ts s te expected number o clss- orders lost per unt tme Exstence o Unque Postve Vlue o G Let s rerte Equton A s G ϕ G G µ ν G Te uncton ϕg s netve poles t G µ / ν nd netve zeros roots n beteen tese poles For te reon ere G nd ϕ Its second dervtve ϕ > µ ν d ϕ G dg µ ν G µ ν 3 < Tereore ϕg s concve uncton over G t ϕ > nd ϕ cross zero once nd only once t postve vlue o G 6 It must Rescln nd ontnuous Lmts Te exstence o postve G llos us to rescle te et vector explctly G Te rescled must stsy Equton A ere G s set to Wt te contnuous vlue dstrbuton requred processn tme nverse te servce rte nd tn tme e cn te contnuous lmt t respect to vlue le retnn te dscreteness o prorty clsses For orders tt ve vlues n te ntervl [ d] belonn to prorty clss- te loss uncton becomes l d d ere l s loss uncton densty Proposton ollos e urter ve
7 Proo o Proposton We dopt te metod o clculus o vrtons Sn 969; Wenstoc 95 tt s dely used to nd te unctonl orm tt ll optmze ven obectve uncton Optml Processn Allocton We strt by rtn te Lrne o Problem d L L ere te nternd or Lrne densty s ξ L s te Lrne multpler or constrnt ven by Equton 4 ξ s te Lrne multpler densty or constrnts It s postve nd zero > Te rst vrton yelds ξ L A Settn te bove equton to zero e et > ; A3 nd ξ Obvously
8 ξ becomes zero tereore te correspondn constrnts re non-bndn Te bove results re summrzed n Proposton Te vlue o Lrne multpler cn be solved by substtutn te expresson o nto constrnt ven by Equton 4 Second Vrton nd oncvty Let s perorm te second vrton round Ts s done by derenttn Equton A t respect to nd substtutn n Equton A3 We ve or > L Ts s strctly netve s > Tereore te soluton yelds mxmum vlue or te prot uncton π For te soluton tt les on te boundry e te prot ll decrese t rte ξ s ncresed rom Optml pcty Te optml cpcty cn be obtned by derenttn te Lrne explctly γ d L Substtutn n Equton A3 te bove expresson cn be smpled to γ d A4 Solvn Equtons A3 A4 nd constrnt ven by Equton 4 e cn nd te optml polcy descrbed by nd In te ollon e so te cpcty solved bove s optml Frst tn dervtve o Equton 4 nd combned t Equton A3 e ve
Te second dervtve o Lrne t respect to s L A5 Te coecents re 3 d ; 4 3 3 4 3 3 4 4 3 We nd tt coecents nd ve te ollon propertes: > ; > > ; 3 < d ; d Ten t s strtorrd to so tt Equton A5 s netve I < t s obvous tt < ; le e ve < Proo o orollry Snce s nonnetve rom Equton A4 e ve d γ γ Equton A3 yelds n nequlty Substtute nto Equton A4 e et 9
3 γ γ S d ere S s te expected revenue Proo o orollry Prt s strtorrd nd / / < Proo o orollry 3 Prt Te dervtve o te expected prot t respect to te demnd s ccordn to envelope teorem d π π Substtutn Equton A3 e ve > d π Prts nd Tn dervtves t respect to o Equtons 4 nd A4 nd mn usn o Equton A3 e et ; γ ; Solvn tese equtons e obtn γ
Ts becomes postve > γ orollry nd propertes o coecents nd mmedtely render γ > Proo o orollry 4 Prt To so tt te expected prot decreses t te ncresed cpcty cost e nd tt π γ γ π < Prts nd Smlrly e te dervtves t respect to γ o Equtons 4 nd A4 ts yelds γ γ ; γ γ Solvn tese to equtons smultneously e ve γ > ; γ < Proo o Lemm Ts s obvous s ppers explctly n Π n orm γ 3
3 Proo o Proposton 3 We ollo Equton 7 nd rte te -t perod problem ; mx Π δ π subect to: d ; d p r ; Smlr to snle perod problem e cn dene Lrne ; Π d d p r d ξ χ δ π L ere χ nd ξ re Lrne multplers or densty Te rst vrton ves p r χ A6 ere Π δ χ Te results descrbed n Proposton 3 ollo Te optml cpcty or -t perod s ven by
33 Π d δγ γ δ γ Smlr to te snle-perod problem te dervton o ts result utlzes te soluton ven by Equton A6 nd Lemm ote tt one requres tt te cpcty s nondecresn nmely postve number ε te Lrne multpler sould be subtrcted rom te rt nd sde o te bove expresson Proo o orollry 5 Usn te denton o Π ven by Equton 7 nd te envelope teorem e ve Π d p r χ χ Substtutn n Equton A6 reduces to te recursve equton n orollry 5