Submed o Operaons Research manuscrp (Please, provde he manuscrp number!) Auhors are encouraged o subm new papers o INFORMS journals by means of a syle fle emplae, whch ncludes he journal le. However, use of a emplae does no cerfy ha he paper has been acceped for publcaon n he named journal. INFORMS journal emplaes are for he exclusve purpose of submng o an INFORMS journal and should no be used o dsrbue he papers n prn or onlne or o subm he papers o anoher publcaon. Mul-prory Onlne Schedulng wh Cancellaons Xnshang Wang, Van-Anh Truong Deparmen of Indusral Engneerng and Operaons Research, Columba Unversy, New York, NY 10027, USA, xw2230@columba.edu, varuong@eor.columba.edu We sudy a fundamenal model of resource allocaon n whch a fne amoun of servce capacy mus be allocaed o a sream of jobs of dfferen prores arrvng randomly over me. Jobs ncur coss and may also cancel whle wang for servce. To ncrease he rae of servce, overme capacy can be used a a cos. Ths model has applcaon n healhcare schedulng, server applcaons, make-o-order manufacurng sysems, general servce sysems, and green compung. We presen an onlne algorhm ha mnmzes he oal cos due o wang, cancellaons and overme capacy usage. We prove ha our schedulng algorhm has cos a mos wce of an opmal offlne algorhm. Ths compeve rao s he bes possble for hs class of problems. We also provde exensve numercal expermens o es he performance of our algorhm and s varans. Key words : Analyss of algorhms, Approxmaons/heursc, Cos analyss 1. Inroducon In many applcaons, a fne amoun of a servce resource mus be allocaed o a sream of jobs arrvng randomly over me. Jobs are prorzed based on ceran crera such as profably or urgency. When mmedae servce s no avalable, arrvng jobs jon a prory queue o be served a a laer me. Whle wang, jobs may cancel her requess and leave he queue randomly. A cancellaon s any job ha expres or leaves he sysem whou beng processed. To ncrease he rae of servce, overme resource can be used a a hgher cos. The sysem mus dynamcally 1
Wang and Truong: Mul-prory Onlne Schedulng wh Cancellaons 2 Arcle submed o Operaons Research; manuscrp no. (Please, provde he manuscrp number!) deermne he servce rae ha mnmzes he oal cos due o wang, cancellaons and overme resource usage. The above problem s cenral o many applcaons n Operaons Research. For example, n healhcare facles, jobs correspond o paen requess for resources such as dagnosc devces and operang rooms. Paens are ofen prorzed based on her urgency and served n order of prory (Mn and Yh 2010). Longer wa mes, whch resul n lower qualy of care, are represened as a cos on he sysem. Tme s ofen sloed. Wh a lmed number of me slos avalable each day, only a ceran number of paens can be served on each day; he remanng paens mus jon a wals (Denon e al. 2010, Ayvaz and Huh 2010, Gerchak e al. 1996). Paens n he wals may randomly cancel her requess, hus leavng he sysem. Ofen, paens can be served usng surge capacy or overme (Parck e al. 2008) a an addonal cos. The scheduler mus selec he number of paens o serve each day, usng surge capacy or overme as needed, o mnmze he oal cos, ncludng wang coss, los revenue due o cancellaons, and he cos of overme work. Anoher applcaon s nework roung for server applcaons. In hs seng, jobs correspond o daa requess sen o server applcaons by local clens. In order o process jobs, servers make use of compung resources such as CPU and dsk I/O. These fne resources lm he rae of servce. Arrvng daa requess ha canno be mmedaely processed are sored n he memory. If he wa me s oo long, some daa requess mgh expre or lose he value of beng processed, hus leavng he queue. For nsance, n some applcaons, daa requess are sen wh me-ous and need no be processed afer her delay exceeds he me-ou (Xong e al. 2008); n ohers, daa conan nformaon ha gradually loses value over me such as he locaon of a moble devce. The prory of a daa reques s deermned by s expraon dae or he probably of exprng. In he case of server congeson, daa packages can ofen be roued o remoe (exernal) dle servers a a cos of propagaon delay (Ln e al. 2012); such roung ncreases he servce rae emporarly. The roung decson needs o be dynamcally made o reduce he overall cos generaed from daa expraon, local congeson and processng delays due o daa roung.
Wang and Truong: Mul-prory Onlne Schedulng wh Cancellaons Arcle submed o Operaons Research; manuscrp no. (Please, provde he manuscrp number!) 3 In many servce sysems, make-o-order manufacurng sysems, real sores and call ceners, jobs correspond o cusomers arrvng randomly over me. Dependng on he applcaon, cusomers may be served n order of her prores. Backlogged cusomers may cancel her orders (Rubno and Aa 2009, Blackburn 1972), resulng n los sales and even reshelvng coss (Marn e al. 1992). In hese sengs, he servce rae can ofen be ncreased by usng overme work (Dellaer and Melo 1998, Özdamar and Yazgaç 1997), on-call workers (Greenhouse 2012), or expeded procuremen of pars. These sraeges have he effec of emporarly ncreasng he servce rae a an ncreased varable cos. The manager needs o dynamcally deermne he servce polcy so as o conrol he oal sysem cos. In he scalng of compuer processng speed for mnmzng energy usage (Bansal e al. 2009a, Yao e al. 1995), jobs correspond o sequences of CPU nsrucons ha arrve randomly. Jobs are ofen prorzed and processed n order of prory. Wh recen echnologes, he processng speeds of CPUs can be dynamcally rased a he cos of a hgher rae of power usage. Such a speed-scalng echnque ofen helps o save more power han he smple sraegy of urnng off a devce durng dle perods. The goal s o mnmze he sum of some measure of qualy of servce, such as job compleon me and oal energy consumpon (Bansal e al. 2009a). Our model capures mos, f no all, of hese applcaons. Specfcally, we consder a dscree-me plannng horzon of T perods, where T s possbly nfne. Jobs are caegorzed no n prory classes. Each class s assocaed wh a wang cos, a cancellaon probably and a cancellaon cos. Jobs are eher processed n he curren perod or are added o a prory queue. In each perod, a number C of jobs of any prory can be processed. Addonal jobs can be processed a an exra varable cos. The above schedulng problem s especally dffcul o analyze n real applcaons due o he dffculy n forecasng fuure nformaon. On he demand sde, fuure arrvals are ofen classdependen and me-dependen (Huh e al. 2013), whch requres an enormous amoun of daa o esmae he jon dsrbuon of demand for mulple classes. For nsance, a paen reques
Wang and Truong: Mul-prory Onlne Schedulng wh Cancellaons 4 Arcle submed o Operaons Research; manuscrp no. (Please, provde he manuscrp number!) ofen leads o subsequen perodc requess, resulng n he me correlaon of demand. Also, n markes of new producs or servces, demand s ofen drven by nensve promoon campagns, n whch case fuure demand depends on promoonal and socal facors and s hghly unceran. On he supply sde, processng capaces are ofen subjec o occasonal falures such as saff abseneesm, machne breakdown (Federgruen and So 1990) and server crashes, whch can be very hard o predc. Even wh access o accurae jon dsrbuons of fuure demand and supply, he compuaon of an opmal schedulng polcy s ofen nracable. When here are jobs n he sysem, akes O( n ) space o sore all possble saes n a gven perod. Ths dffculy s referred o as he curse of dmensonaly. In vew of hese dffcules, we am o develop near-opmal schedulng polces ha are robus o fuure nformaon and are easy o compue. In hs paper, we make no assumpons abou he jon dsrbuon of fuure arrvals and capaces. Insead, we sudy an onlne verson of he problem. A problem s onlne f a all pons n me, exogenous fuure nformaon s compleely unknown and he algorhm has o make adapve decsons based on pas and curren nformaon. In conras, an offlne algorhm knows all fuure nformaon up-fron. Compeve Analyss s he mos wdely used mehod for evaluang onlne algorhms (Borodn and El-Yanv 1998). I consders he relave performance beween an onlne algorhm and an opmal offlne algorhm under he wors npu nsance. The maxmum rao beween he cos acheved under he onlne algorhm and ha under he opmal offlne algorhm s called he compeve rao for ha onlne algorhm. An algorhm wh a compeve rao of α s sad o be α-compeve. For he schedulng problem whou cancellaons, we propose 2-compeve randomzed and deermnsc onlne algorhms. For he schedulng problem wh cancellaons, we relax he assumpon of he onlne problem by makng he offlne polcy unaware of whch jobs wll cancel,.e., he random cancellaon evens are exogenous o boh he onlne and offlne polces. Under hs defnon, we propose 2-compeve onlne algorhms for he model wh cancellaons. Furher, we show ha he compeve rao of our deermnsc algorhm s he bes ha can be acheved.
Wang and Truong: Mul-prory Onlne Schedulng wh Cancellaons Arcle submed o Operaons Research; manuscrp no. (Please, provde he manuscrp number!) 5 Our proofs of he compeve raos use a cos-balancng approach n conjuncon wh he followng new deas. We consruc a novel dsance funcon whch summarzes n a sngle number he dfference beween he hsory of he onlne algorhm OLN and he opmal offlne algorhm OFF. The dsance funcon φ (OLN, OFF) has a nce physcal nerpreaon. A any me, f we mmedaely servce φ (OLN, OFF) addonal jobs under he onlne algorhm, he remanng jobs wll have lower prores han he curren remanng jobs under he offlne algorhm. The dsance funcon dynamcally accouns for he dfference n he number of scheduled and cancelled jobs beween he wo algorhms. Dependng on he sgn of he dsance funcon n each perod, we paron he perods n he plannng horzon no wo ses. We show ha n each ype of he perods, one cos componen of he onlne algorhm s domnaed by he correspondng componen of he offlne algorhm. Ths resul naurally leads o he proof of he compeve raos. For he model wh cancellaons, we use sochasc couplng o compare he exogenous cancellaon evens under he onlne and offlne algorhms. When exended o he model wh cancellaons, our dsance funcon ncorporaes he dfference n he number of coupled cancellaon evens beween he wo algorhms. For he model wh cancellaons, we propose a new cos-accounng scheme whch ransforms cancellaon coss no new wang and overme coss. Ths ransformaon allows he algorhms for he model whou cancellaons o be easly exended o capure cancellaon behavours. 2. Leraure Revew Our work s relaed o he leraure on Apponmen Schedulng, whch has been suded nensvely. For comprehensve revews of he broader area, see Guerrero and Gudo (2011), May e al. (2011), Cardoen e al. (2010) and Gupa (2007). A large par of he leraure consders nra-day schedulng. In hese problems, he number of paens o be served on each day s gven or s exogenous, and he ask s o se he sequence and he sar me of each apponmen so as o conrol
Wang and Truong: Mul-prory Onlne Schedulng wh Cancellaons 6 Arcle submed o Operaons Research; manuscrp no. (Please, provde he manuscrp number!) paen wa me and provder dle me. Anoher par of he leraure models mul-day schedulng. In hese problems, he allocaon of paens o days s dynamcally conrolled. Some of hs leraure allows paens o be scheduled no fuure days a he me of arrval. Ths paradgm s called advance schedulng. See, for example, Truong (2014b), Gocgun and Ghae (2012) and Parck e al. (2008). In he res of he mul-day leraure, only he number of paens o be scheduled o he curren perod s deermned. The res of he paens are added o a wals. Ths paradgm s called allocaon schedulng. See, for example, Huh e al. (2013), Mn and Yh (2010), Ayvaz and Huh (2010) and Gerchak e al. (1996). So far, very few works have suded he opmal advanceschedulng polcy. Recenly, Truong (2014b) lnked he soluons for he advance and allocaon schedulng problems by showng ha for a wo-class model, her opmal schedulng polces are equvalen. Ths resul pons o he mporance of allocaon schedulng as a fundamenal model. Our model s an allocaon-schedulng model. In allocaon schedulng, pas works have used dynamc programmng o explore srucural properes of he opmal schedulng polcy. When here are one or wo paen classes, he problem s easy o solve. For mul-class problems, some srucural resuls are known bu here s no polcy wh performance guaranees. Gerchak e al. (1996) and Huh e al. (2013) sudy schedulng problems wh wo paen classes. Paens n he emergen class requre same-day servce; paens n he elecve class can wa. Gerchak e al. (1996) show ha he opmal schedulng polcy s no a cu-off polcy; he opmal number of admssons ncreases n he sze of wals. Huh e al. (2013) develop heurscs for a correlaed and dynamc envronmen. Mn and Yh (2010) and Ayvaz and Huh (2010) sudy he allocaonschedulng problem wh mulple elecve paen classes. Mn and Yh (2010) develop bounds on he opmal number of admssons. They show ha prory-based dscrmnaon resuls n as much as a 30% dfference n he opmal number of admssons compared o an undscrmnaed scheme. Ayvaz and Huh (2010) analyze he srucural properes of an opmal schedulng polcy and sudy numercal performance of a proec-consan heursc. The heurscs presened n hese works do no come wh any performance guaranees. Moreover, for he sac polces ha hey propose,
Wang and Truong: Mul-prory Onlne Schedulng wh Cancellaons Arcle submed o Operaons Research; manuscrp no. (Please, provde he manuscrp number!) 7 such as he proec-consan polces, s easy o search for he bes proec-consan levels only when he number of demand classes s small. When he number of demand classes s large, s much harder o search for he bes se of proec-consan levels whou addonal srucural properes. Thus, n mul-class sengs, even heurscs wh good emprcal performance are hard o fnd. The schedulng sysem we consder s relaed o make-o-order manufacurng sysems n ha processng capacy s used o servce realzed demand. These make-o-order sysems are usually modeled as queung sysems. In he framework of queung sysems, servce mes and ner-arrval mes mus be saonary, ndependen and mos ofen, exponenally dsrbued o ensure ha he model s racable. Our approach dffers from hs leraure n ha we do no assume any jon dsrbuon on fuure arrvals and servce capaces. For revews on admsson conrol for make-o-order queues, see Sdham (1985) and more recenly, Carr and Duenyas (2000). Blackburn (1972) sudes he opmal sraeges for urnng on or off a server subjec o renegng cusomers. Ther work s relaed o ours n ha hey consder he dynamc expanson of he servce rae. Whle hey only consder one ype of jobs, we allow jobs o have mulple prores, each wh a dfferen cancellaon probably. Rubno and Aa (2009) consder a relaed problem n whch cusomers can be ousourced and have chances o renege. They propose a heursc based on he soluon o he problem n he heavy-raffc regme. Our model s relaed o he work of Kesknocak e al. (2001), who sudy sngle-server onlne schedulng problems wh lead-me quoaon, wh applcaon o make-o-order manufacurng sysems. In her model, jobs can be rejeced upon arrval, and wang coss are ncurred n each perod before he jobs are fnshed. The rejecon of jobs s smlar o he use of overme resource n our model. Our work can be seen as a mul-prory, mul-server exenson of her model, and wh furher consderaons for job renegng. We noe ha n her model, a job may span mulple perods, whle n our model, every job can be fnshed n a sngle perod. However, our model easly accommodaes bached arrvals. A job ha akes mulple perods o fnsh can be modelled as a bached arrval.
Wang and Truong: Mul-prory Onlne Schedulng wh Cancellaons 8 Arcle submed o Operaons Research; manuscrp no. (Please, provde he manuscrp number!) The class of machne and mulprocessor schedulng problems share some characerscs wh our work. In a ypcal machne-schedulng problem, jobs mus be assgned o one or more machnes so as o mnmze a chosen objecve such as he makespan, he oal compleon me or he oal wang me. Our model resembles a machne-schedulng problem n whch (1) jobs have un processng mes, (2) jobs can be rejeced or dvered afer beng released, (3) each job has a specfc release me, whch s used o defne a wang cos, and (4) jobs may cancel randomly. However, an onlne verson of hs model has no been consdered. Overme usage and job cancellaons are no common n he machne-schedulng leraure. We refer he reader o Chen e al. (1998) for a dealed survey of machne schedulng. Among he exsng leraure, he mos relevan works nclude Noga and Seden (2001) and Zhang e al. (2009). Noga and Seden (2001) consder an onlne machne-schedulng problem where jobs have release mes and he objecve s o mnmze he oal wang cos, bu he servce rae canno be dynamcally conrolled. Zhang e al. (2009) sudy a deermnsc offlne schedulng problem where jobs can be rejeced. Our work s relaed o speed scalng problems n he managemen of power for a sngle processor. In hese problems, CPU processng speeds can be rased by supplyng more power. One group of works consders he opmzaon problem of some energy relaed objecve, subjec o deadlnes for job compleon (Yao e al. 1995, Chan e al. 2007, Bansal e al. 2007a, 2009b, 2011). The frs heorecal sudy of such model s gven by Yao e al. (1995). They show ha an opmal offlne algorhm for any convex power funcon can be compued by a greedy mehod. They also gve an onlne algorhm wh consan compeve rao when he power funcon s polynomal. Anoher group of works consders energy usage and job wang me (Albers and Fujwara 2007, Bansal e al. 2007b, 2009a). Bansal e al. (2009a) propose an onlne algorhm ha mnmzes he sum of fraconal wang coss and energy usage for arbrary power funcons. When here are no cancellaons, our model capures he radeoff n Bansal e al. (2009a). In he leraure of speed scalng, mos works consder connuous-me models. Our model capures a dscree-me speed scalng problem n whch processng speeds can only be changed n dscree perods. Cancellaon behavours are generally no consdered n he speed-scalng leraure.
Wang and Truong: Mul-prory Onlne Schedulng wh Cancellaons Arcle submed o Operaons Research; manuscrp no. (Please, provde he manuscrp number!) 9 Our onlne algorhms and her performance guaranees are relaed o many oher approxmaon algorhms developed n Operaons Managemen. Approxmaon algorhms have performance guaranees ha are relave o he opmal sochasc dynamc polcy, whle onlne algorhms have performance guaranees ha are relave o an opmal offlne algorhm. The laer ype of guaranee s much sronger. Moreover, n order o use approxmaon algorhms s sll necessary o esmae he jon dsrbuon of fuure arrvals. These esmaes can be very hard o make. In conras, he onlne algorhms we sudy do no have hs requremen. Lev e al. (2005) propose a cos-balancng echnque for nvenory conrol problems. They prove ha hs cos-balancng algorhm s a 2-approxmaon. The cos-balancng echnque s found o be very adapable and s appled n approxmaon algorhms for many oher supply-chan problems (see, for example, Lev e al. (2008a) and Lev e al. (2008b)). Recenly, Truong (2014a) develops an approxmaon algorhm for he sochasc nvenory conrol problem by usng a look-ahead opmzaon approach. Many onlne algorhms have been developed recenly for problems n Operaons Managemen. Ball and Queyranne (2009) consder an onlne verson of a revenue managemen problem. They show ha he smple proecon-level polcy gves he bes possble compeve rao. The rao depends on he level of prce dscouns. Wagner (2010) consders he onlne economc lo-szng problem. They model he onlne prof-maxmzng problem as a mn-max game, and provde condons under whch he compeve rao s bounded. Buchbnder e al. (2013) sudy an onlne algorhm for a make-o-order varan of he jon-replenshmen problem for whch hey proved a compeve rao of hree. Elmachoub and Lev (2014) sudy a general class of cusomer-selecon problems where decsons are made n wo phases: In he frs phase, arrvng cusomers wh dfferen confguraons are seleced n an onlne manner. Then n he second phase, he cos of he servce sysem s generaed based on he se of seleced cusomers. They develop a framework of analyss for hs class of problems and apply o varous models. The remander of hs paper s organzed as follows. In Secon 3, we presen our onlne algorhm for he schedulng model whou cancellaons. In Secon 3.3, we generalze our resuls o he
Wang and Truong: Mul-prory Onlne Schedulng wh Cancellaons 10 Arcle submed o Operaons Research; manuscrp no. (Please, provde he manuscrp number!) case ha fuure coss are dscouned. In Secon 3.4, we dscuss lower bounds on he compeve rao and prove ha our algorhm s opmal. In Secon 4, we exend he model and algorhm o capure cancellaons. Fnally, n Secon 5, we repor he numercal performance of our schedulng polces. 3. Model of Allocaon Schedulng whou Cancellaons The plannng horzon has T perods, ndexed from 1 o T, where T may be nfne. There are n prory classes. Each class s assocaed wh a wang cos w 0, whch s ncurred when a class job says n he wals for one perod. Le he n classes be ordered n decreasng order of prory. We assume ha he wang coss sasfy w 1 w 2 w n. The schedulng polces we presen n he paper do no depend on he oal number n of classes, so n can be arbrarly large and he collecon of wang coss can even approach a connuous dsrbuon. A he begnnng of each perod, we observe he vecor s = (s 1, s 2,..., s n ) represenng he oal number of jobs currenly n he wals, where s s he number of jobs n class. Then, we observe he regular capacy C, whch s he number of jobs, regardless of prory, ha can be processed by regular resource n perod. Nex we observe he number of new arrvals δ = (δ 1, δ 2,..., δ n ), where δ sands for he number of arrvals of class jobs. We have s, δ Z n + and C Z +, where Z + s he se of all non-negave negers. For an onlne algorhm, C and δ are compleely unknown unl perod, whle for an offlne algorhm he enre sample pah {(C, δ )} =1,2,...,T s known a he begnnng of perod 1. Afer he new arrvals have occurred, he number of jobs n sysem s represened by he vecor s + δ. From among he s + δ 1 jobs n sysem, a schedulng polcy deermnes he number a Z + of jobs o servce n perod. I s nuve ha once he number a s decded, s opmal o serve he a jobs wh he hghes prores. Ths propery s proved n Ayvaz and Huh (2010) and Mn and Yh (2010) for opmal sochasc polces. The same resul holds here. However, we do no repea he proof. We resrc our aenon o he class of polces ha follow hs servce scheme.
Wang and Truong: Mul-prory Onlne Schedulng wh Cancellaons Arcle submed o Operaons Research; manuscrp no. (Please, provde he manuscrp number!) 11 If a > C, we assume ha he addonal a C jobs wll be served by overme resource ncurrng a oal overme cos of (a C )p, where p s he cos of usng an overme slo. If a C, no overme cos wll be ncurred. Defne d (a C ) + as he number of overme slos used n perod. We normalze all cos values such ha p = 1. Then he oal overme cos n perod s jus d. Because we only consder polces ha schedule some number of hghes prory jobs n each perod, wo schedulng polces dffer only n he mng and number of jobs drawn from he wals. We nroduce he followng operaor ha exracs a ceran number of jobs wh he hghes prores from a gven sysem sae s Z n +. Defnon 1. For a vecor x Z n + and a non-negave neger k, we defne h(x, k) Z n + as he vecor ha conans he k jobs wh he hghes prores n x. Le h (x, k) be he h elemen of h(x, k). Le h(x, k) = 0 for k < 0, and h(x, k) = x for k > x 1. Snce he w s are decreasng n, we have for 0 k x 1, where h (x, k) = x, h (x, k) = 0, for < for > h (x, k) = x k, oherwse, = mn{j j x k}. Usng hs operaor, we can wre he number of jobs remanng n he wals a he end of perod as f = s + δ h(s + δ, d + C ). Nex, he wang cos ncurred n perod can be wren as W = f τ w. In he nex perod, he nal sae of he sysem s s +1 = f. For each polcy Π, we add a superscrp Π o all he sae and decson varables ha resul from Π. If Π s an onlne algorhm, he decson d Π does no depend on any nformaon o be
Wang and Truong: Mul-prory Onlne Schedulng wh Cancellaons 12 Arcle submed o Operaons Research; manuscrp no. (Please, provde he manuscrp number!) realzed laer han perod. When we presen our onlne algorhm n Secon 3.2, he objecve s he undscouned oal cos over a fne number T of perods, namely, V Π T = T =1 (dπ + W Π ). We wll show ha our onlne algorhm gves a oal cos whch s a mos 2 mes he oal cos under an opmal offlne algorhm for any sample pah {(C, δ )}. In Secon 3.3 we furher show ha he same resul holds n dscouned, fne and nfne-horzon sengs. 3.1. Comparson of Schedulng Polces In hs secon we presen a mehod for comparng sysem saes generaed by dfferen schedulng polces. Specfcally, we consruc a dsance funcon whch, on he one hand, capures he dfference n he cumulave overme usage beween wo schedulng polces, and on he oher hand, ndcaes wheher he jobs under one polcy have lower prores han hose under he oher polcy. We wll use he dsance funcon o prove he compeve rao of our onlne algorhms. For wo schedulng polces Π and Θ, he dsance funcon φ (Π, Θ) s defned recursvely as φ 0 (Π, Θ) = 0, (1) φ (Π, Θ) = max{φ 1 (Π, Θ) d Π + d Θ, 0} for 1, where recall ha d Π and d Θ are he numbers of overme slos used under Π and Θ n perod, respecvely. Inuvely, he dsance funcon sores he cumulave dfference beween he number of overme slos used under Π and Θ. However, s value s always kep non-negave. Table 1 provdes an llusrave example. I lss he number of overme slos used n perods from 1 o 9 on a sample pah. The correspondng values of he dsance funcon are shown n he boom row of he able. Nex, we wll show n Theorem 1 ha he dsance funcon has a drec physcal nerpreaon. If we remove he number of jobs equal o he value of he funcon φ (Π, Θ) from sae f Π, he res of he jobs n f Π wll be domnaed by he jobs n f Θ n erms of prores. Fgure 1 llusraes hs nerpreaon. Before sang Theorem 1, we frs formalze he followng defnon of a domnance relaonshp and some of s mplcaons.
Wang and Truong: Mul-prory Onlne Schedulng wh Cancellaons Arcle submed o Operaons Research; manuscrp no. (Please, provde he manuscrp number!) 13 Table 1 Example of he dsance funcon. Based on he numbers of scheduled overme slos d Π and d Θ of wo schedulng polces Π and Θ, respecvely, he values of he dsance funcon φ (Π, Θ) are compued and lsed n he boom row. Perod 1 2 3 4 5 6 7 8 9 d Θ 0 2 0 0 0 1 0 0 1 d Π 1 0 1 2 0 0 0 1 2 φ (Π, Θ) 0 2 1 0 0 1 1 0 0 Fgure 1 Illusraon of he dsance funcon. There are 4 prory classes wh wang coss w = (4, 3, 2, 1). By he end of perod, f Π = (1, 1, 2, 1) and f Θ = (0, 2, 0, 2). The fgure dsplays all he jobs wh her wang coss marked. Assume ha φ (Π, Θ) = 2. Afer φ (Π, Θ) = 2 jobs wh he hghes prores are removed from f Π, he remanng jobs, marked by he black box, have lower prores han he jobs n f Θ. Noe ha f we only removed 1 job wh un wang cos of 4 from f Π, he remanng jobs would no be domnaed by he jobs n f Θ, as here would be 3 jobs wh wang coss of a leas 2 remanng n f Π, bu only 2 such jobs n f Θ. Defnon 2. For wo vecors x, x Z n +, we say x s domnaed by x and wre x x f l l x x l = 1, 2,.., n. A resul mmedaely followng hs defnon s ha, f x and x represen dfferen sysem saes a he end of perod, and x x, hen he oal wang cos ncurred by he jobs n x a s no greaer han ha ncurred by he jobs n x. Lemma 1. For wo vecors x, x Z n +, f x x, hen x τ w x τ w. Proof. l x l x l = 1, 2,..., n
Wang and Truong: Mul-prory Onlne Schedulng wh Cancellaons 14 Arcle submed o Operaons Research; manuscrp no. (Please, provde he manuscrp number!) = l x α n 1 = = l=1 l x α l = 1, 2,..., n and α > 0 l x (w l w l+1 ) + n x w n x w. n n 1 x w n l=1 l x (w l w l+1 ) + n x w n The followng lemma saes wo smple operaons ha preserve a domnance relaon: Lemma 2. Fx an neger l 0 and wo vecors x, x Z n + sasfyng x h(x, l) x. 1. For any neger l 0, x h(x, l + l ) x h(x, l ). 2. For any vecor δ N n 0, x + δ h(x + δ, l) x + δ. Proof. Ths lemma s easly proved by drecly checkng he defnon of domnance relaon. Nex, we prove an nvarance beween wo schedulng polces ha can be saed n erms of he dsance funcon and he domnance relaon. Ths nvarance wll help us laer o compare he cos of he polces. Theorem 1. For any wo schedulng polces Π and Θ, we have f Π h(f Π, φ (Π, Θ)) f Θ, = 1, 2,..., T. (2) Proof. Recall ha s = f 1, so equaon (2) s equvalen o s Π h(s Π, φ 1 (Π, Θ)) s Θ. (3)
Wang and Truong: Mul-prory Onlne Schedulng wh Cancellaons Arcle submed o Operaons Research; manuscrp no. (Please, provde he manuscrp number!) 15 Ths equaon (3) s clearly rue for = 1, as s Π 1 = s Θ 1 s he nal sae. Suppose ha (3) holds up o perod, we nex prove ha s also rue for perod + 1. In perod, afer δ new jobs arrve, accordng o Lemma 2 we have s Π + δ h(s Π + δ, φ 1 (Π, Θ)) s Θ + δ. Then afer Θ removes C + d Θ jobs, Lemma 2 gves us s Π + δ h(s Π + δ, φ 1 (Π, Θ) + C + d Θ ) s Θ + δ h(s Θ + δ, C + d Θ ). Now we le l = φ 1 (Π, Θ) + d Θ d Π and rewre he above equaon as s Π + δ h(s Π + δ, l + C + d Π ) f Θ. Dependng on he value of l, here are wo cases: 1. If l < 0, we have s Π + δ h(s Π + δ, C + d Π ) s Π + δ h(s Π + δ, l + C + d Π ) because he lef hand sde has more jobs removed from he vecor s Θ + δ. I s easy o check ha he bnary relaon s ransve, so he above equaon leads o s Π + δ h(s Π + δ, C + d Π ) f Θ = f Π f Θ = f Π h(f Π, 0) f Θ. 2. If l 0, we have s Π + δ h(s Π + δ, l + C + d Π ) = s Π + δ h(s Π + δ, C + d Π ) h(s Π + δ h(s Π + δ, C + d Π ), l) = f Π h(f Π, l) = f Π h(f Π, l) f Θ.
Wang and Truong: Mul-prory Onlne Schedulng wh Cancellaons 16 Arcle submed o Operaons Research; manuscrp no. (Please, provde he manuscrp number!) In sum, we have f Π h(f Π, max(l, 0)) f Θ = f Π h(f Π, φ (Π, Θ)) f Θ = s Π +1 h(s Π +1, φ (Π, Θ)) s Θ +1. Thus he heorem s proved. The nex heorem shows ha he dsance funcon separaes all perods no wo ypes, dependng on he sgn of he dsance funcon. In one case, he curren wang cos ncurred under polcy Π s bounded by ha under Θ. In he oher case, he cumulave overme cos ncurred under Π s bounded by ha under Θ. Theorem 2. In any perod, 1. f φ (Π, Θ) = 0, hen W Π W Θ ; 2. f φ (Π, Θ) > 0, le 0 = max{k : φ k (Π, Θ) = 0, k < }. Then k= 0 +1 d Π k < k= 0 +1 Proof. If φ (Π, Θ) = 0, we know from Theorem 1 ha f Π f Θ. Then Lemma 1 gves W Π d Θ k. W Θ. The case φ (Π, Θ) > 0 can be proved by drecly checkng he defnon of he dsance funcon. Usng Table 1, we llusrae he wo ypes of perods dsngushed n Theorem 2. 1. Perods n whch φ (Π, Θ) = 0. These are perods = 1, 4, 5, 8, 9 n Table 1. From he frs saemen of Theorem 2 we know ha for hs ype of perods, he wang coss under Π s bounded by he wang coss under Θ. 2. Perods n whch φ (Π, Θ) > 0. We can dvde hese perods no nervals of consecuve perods, e.g., nerval [2, 3] and nerval [6, 7] n Table 1. Durng each of hese nervals, he oal number of overme slos used n Π s no greaer han he number of overme slos used n Θ. Hence, n hese nervals he oal overme cos under Π s bounded by ha under Θ.
Wang and Truong: Mul-prory Onlne Schedulng wh Cancellaons Arcle submed o Operaons Research; manuscrp no. (Please, provde he manuscrp number!) 17 In sum, n any ype of perods, one cos componen of Π, eher he wang cos or he overme cos, s bounded by he correspondng cos of Θ. Snce Θ can be any schedulng polcy ncludng he opmal offlne polcy, Π wll have a compeve rao of 2 f can balance he wo cos componens evenly n an onlne manner. We nex show ha a smple mehod of balancng coss leads o a 2-compeve algorhm. 3.2. Onlne Algorhm In hs secon we presen a 2-compeve onlne algorhm for he allocaon-schedulng problem whou cancellaons. Defne an onlne algorhm OLN as follows. In each perod, OLN balances he oal cumulave wang cos and oal cumulave overme cos by mnmzng he maxmum of he wo. Mahemacally, le W (d) = (s OLN + δ h(s OLN perod f d overme slos are used n. Then d OLN + δ, d + C )) τ w be he wang cos o be ncurred n s deermned by (recall ha he un overme cos s p = 1) d OLN = arg mn d 1 max( 1 + d, W OLN + W (d)). (4) d OLN The dea of OLN s o keep hese wo cumulave coss as closely mached o each oher as possble. Theorem 3. For any polcy Π and any sample pah, max( d OLN, W OLN ) (d Π + W Π ), = 1, 2,..., T. (5) Proof. When = 0 he condon (5) s rvally rue. Suppose ha (5) s rue up o perod 1. We nex prove ha also holds for perod.. Le g = max( doln, W OLN Case 1: φ 1 (OLN, Π) + d Π d OLN ) be he maxmum of he wo cumulave coss up o perod < 0. We mmedaely have d OLN > 0 and φ 1 (OLN, Π) + d Π (d OLN 1) 0.
Wang and Truong: Mul-prory Onlne Schedulng wh Cancellaons 18 Arcle submed o Operaons Research; manuscrp no. (Please, provde he manuscrp number!) Then from Theorem 2 we know ha W (d OLN 1) W Π. In oher words, even f we schedule one fewer job n perod under OLN, he resulng wang cos for hs perod s sll less han or equal o W Π. The decson creron for OLN n (4) gves us g g 1 + W (d OLN 1) because oherwse usng d OLN 1 overme slos nsead of d OLN n perod would reduce he maxmum componen of cumulave coss. Connecng he above wo equaons we ge g g 1 + W (d OLN 1) g 1 + W Π 1 (d Π + W Π ) + W Π where he hrd nequaly follows from nducon on he ( 1)-h perod. Case 2: φ 1 (OLN, Π) + d Π d OLN > 0. Agan le (d Π + W Π ), 0 = max{k : φ k (OLN, Π) = 0, k < } (6) be he las perod n whch he dsance funcon was equal o 0. Snce n hs case φ (OLN, Π) = φ 1 (OLN, Π) + d Π d OLN > 0, from Theorem 2 we know ha = = 0 +1 = 0 +1 On he oher hand, defnon (4) gves us d OLN < d OLN g g 0 + ( = 0 +1 + 1 = 0 +1 d Π = 0 +1 d OLN + 1) because oherwse we could use one more overme slo o reduce g. Combnng he above wo equaons we ge d Π. g g 0 + ( = 0 +1 d OLN + 1) g 0 + = 0 +1 d Π 0 (d Π + W Π ) + = 0 +1 d Π (d Π + W Π ), where he hrd nequaly comes from nducon on he 0 -h perod.
Wang and Truong: Mul-prory Onlne Schedulng wh Cancellaons Arcle submed o Operaons Research; manuscrp no. (Please, provde he manuscrp number!) 19 Case 3a: φ 1 (OLN, Π) + d Π d OLN = 0, g = defnon of he dsance funcon we know ha doln. Le 0 be defned as n (6). From he = 0 +1 d OLN = = 0 +1 d Π. Then we have g g 0 + = 0 +1 d OLN 0 (d Π + W Π ) + = 0 +1 d Π (d Π + W Π ), where he frs nequaly comes from he condon for hs case, namely ha g = doln. Case 3b: φ 1 (OLN, Π) + d Π d OLN = 0, g = W OLN. From Theorem 2 we have W OLN W Π = g g 1 + W OLN g 1 + W Π 1 (d Π + W Π ) + W Π (d Π + W Π ), where he frs nequaly comes from he condon for hs case, namely ha g = W OLN. Fnally usng Theorem 3 we can show ha OLN s 2-compeve, by leng Π be he opmal offlne algorhm OFF. Corollary 1. On every sample pah, (d OLN + W OLN ) 2 (d OFF + W OFF ). Proof. (d OLN + W OLN ) 2 max( d OLN, W OLN ) 2 (d OFF + W OFF ).
Wang and Truong: Mul-prory Onlne Schedulng wh Cancellaons 20 Arcle submed o Operaons Research; manuscrp no. (Please, provde he manuscrp number!) 3.3. Generalzaon o Dscouned Coss Now we generalze our prevous resuls o he case of dscouned fuure coss. Gven a dscoun facor γ (0, 1), le he oal dscouned cos from perod 1 o T be V Π T (γ), V Π T (γ) = (d Π p + φ Π )γ 1. =1 The followng heorem ensures ha he compeve rao of our onlne algorhm s sll 2 n he dscouned-cos case. Theorem 4. For any polcy Π and any horzon T, where T s possbly nfne, we have V OLN T (γ) 2V Π T (γ). Proof. We already know from Corollary 1 ha for any lengh of he horzon and any sample pah we have V OLN 2V Π, where V Π s he undscouned cos from perods 1 o. Then for any polcy Π, V OLN T (γ) = = =1 =1 T 1 = =1 T 1 =1 (d OLN (V OLN V OLN 2V Π = 2V Π T (γ). p + φ OLN )γ 1 V OLN 1 )γ 1 (γ 1 γ ) + V OLN T γ T 1 (γ 1 γ ) + 2V Π T γ T 1 3.4. Lower Bounds We prove ha our onlne algorhm acheves he opmal compeve rao by reducng our schedulng problem no a sk-renal problem, and concludng ha he compeve raos for he sk-renal problem apply o our model.
Wang and Truong: Mul-prory Onlne Schedulng wh Cancellaons Arcle submed o Operaons Research; manuscrp no. (Please, provde he manuscrp number!) 21 The classcal sk-renal problem, whch s frs suded by Karln e al. (1988), s a smplfed verson of our allocaon-schedulng problem. In he sk-renal problem, a sngle job was o be processed some me n he fuure, bu he exac dae ha he job wll be processed s unknown. A wang cos of $1 s ncurred n each perod ha he job has o wa. The job can also be mmedaely processed a an addonal cos of $B a any me. If we know ha he job has o wa a leas B perods, hen s opmal o mmedaely process he job n he curren perod. If he job needs o wa no more han B perods, hen s opmal o le wa. Ths sk-renal problem s onlne f he exac me ha he job wll be processed s unknown and s chosen by an adversary. I s well known ha he opmal compeve rao of he sk-renal problem s 2 for deermnsc algorhms (Karln e al. 1988) and e/(e 1) for randomzed algorhms (Karln e al. 1990). Theorem 5. OLN s an opmal onlne algorhm for he allocaon-schedulng model. Proof. In our allocaon schedulng model, f here s only one job n he sysem and we always le C = 0 unl some fuure perod chosen by an adversary, hen he problem reduces o he skrenal problem. Thus, he sk-renal problem s a subclass of he allocaon-schedulng problem. Therefore, s lower bounds on he compeve rao also apply o he algorhms for he allocaonschedulng problem. From hs, we can conclude ha our 2-compeve deermnsc algorhm has he lowes possble compeve rao. 4. Model of Allocaon Schedulng wh Cancellaons In hs secon, we consder he allocaon-schedulng problem wh cancellaons. The onlne algorhm we propose n hs secon s adaped from he cos-balancng algorhm of he prevous secon. The algorhm n hs secon s a deermnsc one. We wll only prove he compeve rao over an undscouned and fne horzon, bu smlar o he resuls n Secon 3.3, our compeve analyss can be easly generalzed o a dscouned and nfne horzon. Sarng from he model whou cancellaons, we assume ha a class job has a cancellaon probably of q [0, 1], and a cancellaon cos of r 0. We assume ha he cancellaon cos domnaes he overme cos for each class,.e., r p = 1 for all. We furher assume ha he
Wang and Truong: Mul-prory Onlne Schedulng wh Cancellaons 22 Arcle submed o Operaons Research; manuscrp no. (Please, provde he manuscrp number!) cancellaon probables and coss are hgher for hgher-prory classes,.e., r 1 r 2 r n, and q 1 q 2 q n. Ths assumpon makes sense n mos applcaons. In healhcare, hgher prory paens have a hgher need o be seen quckly, less wllngness o wa, and hgher endency o leave for oher care arrangemens f hey are made o wa for oo long. In server applcaons, hgher prory jobs have shorer deadlnes. In servce sysems, hgher prory cusomers are more mpaen o wa, and ofen brng hgher profs o he sysem whch would be los f hey leave he queue. In each perod, he followng evens happen n sequence 1. A he begnnng of perod, s = (s 1, s 2,..., s n ) s he oal number of jobs n sysem, where s s he number of jobs of class. 2. Each job n class ndependenly leaves he sysem wh probably q. The remanng jobs form a sae m, m s. The oal cancellaon cos ncurred n perod s R = (s m ) τ r. 3. The capacy C and new arrvals δ are observed. The sysem sae becomes m + δ. 4. The schedulng decson d for perod s made. The number of jobs remanng n he queue s f, f = m + δ h(m + δ, d + C ). The overme cos ncurred n perod s d, and he wang cos ncurred s W = f τ w. 5. In he nex perod we have s +1 = f. For he compeve analyss of he onlne algorhm wh cancellaons, we assume ha an offlne algorhm sees fuure arrvals and capaces, δ, C, = 1, 2,..., T, bu does no see whch jobs wll cancel. Le F = σ(δ 1, δ 2,..., δ T, C 1, C 2,..., C T ) conan he nformaon ha an offlne algorhm can see. The objecve s E[V T F] = E[ (R + d + W ) F], where he expecaon s aken over he random cancellaon evens. =1
Wang and Truong: Mul-prory Onlne Schedulng wh Cancellaons Arcle submed o Operaons Research; manuscrp no. (Please, provde he manuscrp number!) 23 Before presenng he onlne algorhm, s necessary o reexamne he queson of, n he presence of job cancellaons, wheher s sll opmal for he offlne algorhm o serve jobs wh he hghes prores frs,.e., wheher we can sll use he h(, ) operaor o represen an opmal offlne schedulng decson. The followng heorem ensures ha hs servce rule s sll opmal. Theorem 6. The opmal offlne algorhm OFF always schedules jobs wh he hghes prores n each perod. Proof. As an offlne algorhm, OFF knows all he arrvals and capaces upfron. However, snce he cancellaon evens are exogenous o offlne algorhms, OFF faces a sochasc seng n whch jobs cancel randomly n each perod. In hs sochasc decson process, le u 1 (s) be he expeced cos of OFF from o T when he sysem sae a s s. Tha s, le u 1 (s) = E[ = (R OFF + d OFF + W OFF ) F, s = s]. Le u 2 (s) be he cos of OFF from o T mmedaely afer cancellaons have occurred n perod, and when he sysem sae a s s, u 2 (s) = E[d OFF + W OFF + =+1 (R OFF We nex show by nducon ha for any s 1 s 2, + d OFF + W OFF ) F, m = s]. u 1 (s 1 ) u 1 (s 2 ), and (7) u 2 (s 1 ) u 2 (s 2 ). (8) These wo resuls wll naurally lead o he proof of hs heorem. Frs, s clear ha (8) holds n he las perod T, as no cancellaon wll ever happen sarng a ha me, and hence he resul reduces o he case whou cancellaons. Suppose ha (8) holds sarng from perod. We nex prove ha (7) also holds for perod and ha (8) holds for perod 1.
Wang and Truong: Mul-prory Onlne Schedulng wh Cancellaons 24 Arcle submed o Operaons Research; manuscrp no. (Please, provde he manuscrp number!) Le e be he un vecor wh 1 for he h elemen and 0 for all oher elemens. Snce addng more jobs o he sysem only mposes a larger cos, we mus have u 1 (s) u 1 (s + e ) for any = 1, 2,..., n. Then o prove (7) suffces o prove ha for any > j, u 1 (s + e j ) u 1 (s + e ). For any s s, le P (s, s) be he probably ha all he jobs n s reman whle all he jobs n s s cancel. Then he offlne cos value can be wren as u 1 (s + e ) = s s P (s, s) [ (s s) τ r + q (r + u 2 ( s)) + (1 q )u 2 ( s + e ) ], where r + u 2 ( s) s he oal cos value under he condon ha he addonal job e cancels, and u 2 ( s + e ) s he cos value under he condon ha he addonal job does no cancel. By nducon we know ha u 2 ( s + e j ) u 2 ( s + e ) f > j. Moreover, he margnal cos of u 2 ( ) mus be bounded by he overme cos, namely, u 2 ( s + e ) u 2 ( s) p r because oherwse he offlne polcy would servce he addonal job e by overme and reduce he margnal cos o p. Then for any > j, u 1 (s + e j ) = s s P (s, s) [ (s s) τ r + q j (r j + u 2 ( s)) + (1 q j )u 2 ( s + e j ) ] s s s s P (s, s) [ (s s) τ r + q j (r + u 2 ( s)) + (1 q j )u 2 ( s + e ) ] P (s, s) [ (s s) τ r + q (r + u 2 ( s)) + (1 q )u 2 ( s + e ) ] = u 1 (s + e ). where he las nequaly follows from he fac ha q > q j and ha r + u 2 ( s) u 2 ( s + e ).
Wang and Truong: Mul-prory Onlne Schedulng wh Cancellaons Arcle submed o Operaons Research; manuscrp no. (Please, provde he manuscrp number!) 25 Thus we have proved (7) for perod. Now s mmedaely clear ha he opmal offlne schedulng rule n perod 1 always servces he jobs wh he hghes prores, because (7) saes ha s beer o have lower prory jobs n he sysem a he begnnng of perod, and ha he coss o serve any wo jobs are he same. I also follows ha (8) holds for perod 1, as havng lower-prory jobs n sysem leads o lower wang coss and, a he same me, lower-prory jobs a he begnnng of he nex perod. 4.1. Couplng of wo schedulng polces Agan we wan o compare he sysem saes under wo polces Π and Θ. Suppose hey sar wh he same nal sae and experence he same capacy C and arrvals δ for each perod. Snce boh onlne and offlne algorhms do no know whch jobs wll cancel, we can couple he cancellaon evens under Π and Θ. We show ha a new dsance funcon can be defned based on he couplng of cancellaons. Le o Π = s Π m Π 1 be he oal number of cancelled jobs n perod for polcy Π. We defne a new dsance funcon φ(π, Θ) for any wo polces Π and Θ as follows φ 0 (Π, Θ) = 0, φ (Π, Θ) = max{ φ 1 (Π, Θ) d Π o Π + d Θ + o Θ, 0} for 1. (9) Ths new dsance funcon akes boh he number of overme slos and he number of cancelled jobs no accoun. Suppose ha a he begnnng of perod we have s Π h(s Π, φ 1 (Π, Θ)) s Θ. (10) Then we can always smulae he cancellaons n perod n hree phases as follows (see Fgure 2 for an llusraon): 1. Le he φ 1 (Π, Θ) jobs wh he hghes prores n sae s Π,.e., hose couned n h(s Π, φ 1 (Π, Θ)), make her cancellaon decsons.
Wang and Truong: Mul-prory Onlne Schedulng wh Cancellaons 26 Arcle submed o Operaons Research; manuscrp no. (Please, provde he manuscrp number!) 2. Le l = ( s Π 1 φ 1 (Π, Θ)) + be he number of remanng jobs n sae s Π ha have no made her cancellaon decsons ye. Le U 1, U 2,..., U l be..d. [0, 1] unform random varables. For each of he l jobs, gong from he hghes prory o he lowes prory, f he h job s n class j, le he h job cancel f and only f q j U. Then, for he l jobs wh he hghes prores n sae s Θ, le hem cancel smlarly, by usng he same sequence of unform random varables U 1, U 2,..., U l (bu usng possbly dfferen cancellaon probables). In hs way we have coupled he cancellaon evens beween he l jobs wh he lowes prores under Π and he l jobs wh he hghes prores under Θ. 3. Le he oher jobs n s Θ make her cancellaon decsons. Fgure 2 Sochasc couplng of cancellaon evens. Afer removng φ 1(Π, Θ) jobs wh he hghes prores from he sae under Π, he remanng l jobs are domnaed by he jobs under Θ, n ha he prory of each remanng job under Π s a mos ha of he job wh he same prory rankng under Θ. In Phase 2, he cancellaon evens of each par of jobs havng he same prory rankng under he wo polces are coupled ogeher. The followng heorem shows ha, under he above couplng of cancellaon evens, he dsance funcon sll enables us o se up a domnance relaonshp beween Π and Θ. Theorem 7. Suppose (10) holds n perod. Afer he above coupled cancellaon process, we have on every sample pah, m Π h(m Π, φ 1 (Π, Θ) o Π + o Θ ) m Θ. (11) In parcular, φ 1 (Π, Θ) o Π + o Θ 0. (12)
Wang and Truong: Mul-prory Onlne Schedulng wh Cancellaons Arcle submed o Operaons Research; manuscrp no. (Please, provde he manuscrp number!) 27 By he end of perod, f Π h(f Π, φ 1 (Π, Θ) d Π o Π + d Θ + o Θ ) f Θ. (13) Proof. Recall ha l = ( s Π 1 φ 1 (Π, Θ)) +. Le x Π and x Θ be he vecors of jobs consdered n he h couplng phase under Π and Θ, respecvely, for = 1, 2, 3 (see Fgure 2). In parcular, we have n Phase 1, x Π 1 = h(s Π, φ 1 (Π, Θ)), n Phase 2, x Π 2 = s Π x Π 1 and x Θ 2 = h(s Θ, l), and n Phase 3, x Θ 3 = s Θ x Θ 2. Le x Π and x Θ be he vecors of remanng jobs n x Π and x Θ, respecvely, afer cancellaons have occurred. Le y Π = x Π 1 x Π 1 and y Θ = x Θ 1 x Θ 1 be he number of cancelled jobs n phase under Π and Θ, respecvely. Under couplng, he h job n x Π 2, ranked by prory, s coupled wh he h job n x Θ 2. Accordng o he nal condon (10), we have x Π 2 x Θ 2,.e., he h job n x Π 2 has equal or lower prory han he h job n x Θ 2. Accordng o he couplng process, f he h job n x Π 2 cancels, hen he h job n x Θ 2 cancels. So we mus have y Π 2 y Θ 2. Snce x Π 2 x Θ 2, by removng y Θ 2 y Π 2 jobs wh he hghes prores from x Π 2, he resulng sae mus be domnaed by x Θ 2,.e., x Π 2 h( x Π 2, y Θ 2 y Π 2 ) x Θ 2. In phase 1, here are φ 1 (Π, Θ) y Π 1 jobs n x Π 1. Pluggng hese jobs no he domnance relaon, we ge x Π 1 + x Π 2 h( x Π 1 + x Π 2, φ 1 (Π, Θ) y Π 1 + y Θ 2 y Π 2 ) x Θ 2.
Wang and Truong: Mul-prory Onlne Schedulng wh Cancellaons 28 Arcle submed o Operaons Research; manuscrp no. (Please, provde he manuscrp number!) By furher addng he jobs n phase 3, we ge x Π 1 + x Π 2 h( x Π 1 + x Π 2, φ 1 (Π, Θ) y Π 1 + y Θ 2 y Π 2 + y Θ 3 ) x Θ 2 + x Θ 3, whch s jus (11). To prove (12), noe ha φ 1 (Π, Θ) y Π 1 because no more han φ 1 (Π, Θ) jobs can cancel n phase 1, and y Θ 2 y Π 2 due o he couplng process. Hence φ 1 (Π, Θ) o Π + o Θ = φ 1 (Π, Θ) y Π 1 + y Θ 2 y Π 2 + y Θ 3 φ 1 (Π, Θ) y Π 1 + y Θ 2 y Π 2 0, provng (12). To see ha (11) and (12) lead o (13), we rea m Π and m Θ as saes n an nermedae perod, and rea φ 1 (Π, Θ) o Π + o Θ 0 as he dsance funcon value for he nermedae perod. Then (13) follows accordng o Theorem 1. Corollary 2. For any wo schedulng polces Π and Θ, we have on every sample pah, f Π h(f Π, φ (Π, Θ)) f Θ, = 1, 2,..., T. Proof. Ths saemen s equvalen o equaon (13), whch s also he same as s Π +1 h(s Π +1, φ (Π, Θ)) s Θ +1. Ths fnshes he proof ha (10) holds for all perod by nducon. Therefore, (13) holds for all perod. In he remander of hs paper, we use he couplng of cancellaons whenever we compare wo schedulng polces. We wll drecly use Theorem 7 and Corollary 2 whou each me specfyng ha cancellaons are coupled.
Wang and Truong: Mul-prory Onlne Schedulng wh Cancellaons Arcle submed o Operaons Research; manuscrp no. (Please, provde he manuscrp number!) 29 4.2. New Cos-Accounng Scheme Nex we presen a new cos-accounng scheme ha separaes each cancellaon cos no wo pars: r p and p (recall ha r > p for each ). The onlne algorhm ncorporaes he wo pars of he cancellaon cos no he orgnal wang cos and overme cos respecvely. I acheves a compeve rao of wo by rebalancng he wo componens. In he new cos-accounng scheme, le he new cancellaon cos be r = p = 1 for all class, and le he new wang cos n perod for class jobs be w + γ(r 1)q for perod < T, w, = w for perod = T <, where γ s he dscoun facor. Noe ha he new wang cos n he las perod s dfferen from ha n oher perods. I s also easy o check ha w,1 w,2 w,n, and hus f Π f Θ mples w τ f Π w τ f Θ for all. The dea of he new cos-accounng scheme s ha, by seng he new cancellaon cos equal o he overme cos, we can rea a cancellaon as a job ha s forced o be served n overme. Wh hs change, we can apply he proof of he performance bound n Theorem 3 wh a few changes. Now he oal wang cos n perod s W = f τ w. And he oal cos n perod can be wren as Ω = (o + d ) + W. one. The followng heorem saes ha he new cos-accounng scheme s equvalen o he orgnal Theorem 8. For any horzon T, where T can be nfne, he oal cos for any schedulng polcy dffers only by a consan beween he orgnal and new cos-accounng scheme.
Wang and Truong: Mul-prory Onlne Schedulng wh Cancellaons 30 Arcle submed o Operaons Research; manuscrp no. (Please, provde he manuscrp number!) Proof. Le Ω and Ω be he cos ncurred n perod under he orgnal and new cos-accounng schemes, respecvely. We have for any schedulng polcy, E[ γ 1 Ω F] =1 = E[ = E[ = E[ = E[ γ 1 ((s m ) τ r + d + f τ w) F] =1 γ 1 ((s m ) τ (r 1) + s m 1 + d + f τ w) F] =1 γ 1 ((s m ) τ (r 1) + o + d + f τ w) F] =1 γ 1 (E[(s m ) τ (r 1) F, s ] + o + d + f τ w) F] =1 n = E[ γ 1 ( s q (r 1) + o + d + f τ w) F] = = =1 n s 1, q (r 1) + E[ n =2 1 s 1, q (r 1) + E[ =1 =1 γ 1 γ 1 n s q (r 1) + γ 1 (o + d + f τ w) F] =1 n f, γq (r 1) + n = s 1, q (r 1) + E[ γ 1 (o + d + f τ w ) F] = n s 1 q (r 1) + E[ γ 1 Ω F]. =1 γ 1 (o + d + f τ w) F] =1 Noe ha he second erm on he las lne s he oal cos value under he new cos-accounng scheme, and he frs erm s a consan ha depends only on he nal sae. Ths heorem mples ha he opmal polcy remans he same under he new cos-accounng scheme. Moreover, snce he oal cos value decreases by a consan n s 1q (r 1) when new coss are appled, he onlne algorhms under he new cos-accounng scheme are also onlne algorhms for he orgnal coss, wh he same compeve raos. We nex consruc he onlne algorhms under he new cos-accounng scheme.