Congestion-based leadtime quotation and pricing for revenue maximization with heterogeneous customers

Transcription

1 Queueig Sys 13 73:35 78 DOI 1.17/s Cogesio-based leadime quoaio ad pricig for reveue maximizaio wih heerogeeous cusomers BarışAa Tava Leo Olse Received: 14 April 11 / Published olie: 1 March 1 Spriger Sciece+Busiess Media, LLC 1 Absrac This paper sudies a queuig model where wo cusomer classes compee for a give resource ad each cusomer is dyamically quoed a meu of price ad leadime pairs upo arrival. Cusomers selec heir preferred pairs from he meu ad he server is obligaed o mee he quoed leadime. Cusomers have covex cocave delay coss. The firm does o have iformaio o a give cusomer s ype, so he offered meus mus be iceive compaible. A meu quoaio policy is give ad prove o be asympoically opimal uder radiioal large-capaciy heavy-raffic scalig. Keywords Asympoic opimaliy Dyamic schedulig Leadime quoaio Dyamic pricig Mahemaics Subjec Classificaio 6K3 6K5 1 Iroducio We cosider he problem of dyamic leadime quoaio ad pricig i he coex of amae-o-orderfirmservigwocusomerclasses.thecusomershavehesameservice requiremes bu differig oliear leadime sesiiviies. The firm is a reveue maximizig moopolis ad mus decide he price/leadime meu o offer each arrivig cusomer; a meu is ecessary o ideify he rue class desigaios of cusomers because he firm cao observe a give cusomer s ype. B. Aa Kellogg School of Maageme, Norhweser Uiversiy, Evaso, IL 68, USA b-aa@ellogg.orhweser.edu T.L. Olse Uiversiy of Auclad Busiess School, Uiversiy of Auclad, Auclad, New Zealad .olse@auclad.ac.z

2 36 Queueig Sys 13 73:35 78 We assume a covex cocave shape for he cusomers delay cos curves. Acovex cocave,or S-shaped,coscurvemodelshesiuaiowherecusomers have a paricular deadlie i mid bu oce ha deadlie has passed, hey are icreasigly isesiive o he addiioal leadime. Leclerc e al. [3] arguehrough various behavioral experimes ha he shape of he delay cos fucio depeds o he coex effecs. I follows from heir experimes ha he delay cos fucio is cocave i he rage where delay is relaively log, while i ca be covex oherwise. We assume ha our wo cusomer classes have similarly shaped curves bu oe class, he impaie class, has a shorer deadlie ha he oher paie class. I order o solve he leadime quoaio ad pricig problem we mus also address how o schedule cusomers. If cusomer delay coss are covex, he Firs-Come- Firs-Serve FCFS wihi a cusomer class is opimal e.g., [54], ex bu his is o he case for covex-cocave cos curves, as we wish o sudy here. Eve wih covex coss he schedulig problem across classes is challegig ad was firs addressed from a asympoic opimaliy objecive i [49]. Va Mieghem [49] cosidersasympoicallyopimalscheduligfor cusomer classes each wih covex icreasig delay coss. He shows he geeralized cµ, or Gcµ, ruleobeasympoicallyopimal,whereheexcusomerclassservedisha wih he larges value of margial cos imes service rae. Our paper exeds va Mieghem s wor i four sigifica ways. Firs, we prove he asympoic opimaliy of a discree ime policy remiisce of he Gcµ rule. Secod, we allow he cos fucios o be covex cocave. Third, we allow for leadime quoaio a he ime of arrival ad require ha all quoed leadimes are respeced. Fially, we require iceive compaibiliy ad hece quoe meus raher ha jus leadimes. Each of hese exesios is explaied i more deail below. We cosider a shif-based approach o leadime quoaio. Leadimes are quoed o he eares shif raher ha i arbirarily small uis ad oce a leadime is quoed i mus be me. I a shif-based approach, he miimum pracically achievable producio delay is oe period. However, for ease of exposiio, i wha follows we will refer o his miimal delay as quoig zero leadimes. Producio quaiies for a shif are assumed o be ow ad always me i.e., service imes are i effec deermiisic. I his way we ca deermie feasible leadimes as cusomers arrive. We cosider alarge-capaciyasympoicregimewhereboharrivalraesadcapaciygrowproporioally i a radiioal heavy-raffic sese ad allow he shif legh o shri i a appropriae maer o be described laer. As discussed above, we assume a covex-cocave shape for he cusomers delay cos curves. We use he covex hull of he delay cos fucio o lower boud he cos wih is value uder he Gcµ rule ad he provide a quoaio ad schedulig rule ha asympoically achieves his lower boud. A impora feaure of our model is ha we are o resriced o FCFS service, eve wihi a cusomer class, ad ca isead schedule o maximize reveue. Despie his, our asympoically opimal schedulig rule will be see o be boh simple ad highly iuiive. The hird ey feaure of our model is ha we quoe cusomers leadimes as hey arrive. Cusomers who are less paie will receive shorer, ad ofe zero, leadimes. However, due o variabiliy, here will be imes ha eve hough oal sysem cogesio implies arrivig impaie cusomers are supposed o receive zero delay, say, hey

3 Queueig Sys 13 73: acually receive leadimes of muliple shifs. The fracio of such deviaios will be asympoically egligible bu i he real sysem his may occur. Because our policies quoe leadimes based o acual wor observed ad oly quoe achievable leadimes, hese imes cause o implemeaio issues. I oher words, we do o blihely quoe delays ad he argue ha hese delays are asympoically achievable; isead, we quoe wha is achievable ad show ha deviaios from prescribed delays will be asympoically egligible. The fial sigifica characerisic of our model is ha we cosider iceive compaible meus. I a rece impora wor i his area, Afèche [1]cosidersasysem wih wo cusomer classes where he server cao observe cusomer ype ad he cusomer cao observe queue legh. Cusomers mae heir choice based o he price ad expeced queue legh; opimal saic price ad schedulig mechaisms are cosidered. Lie Afèche we cosider iceive compaible mechaism desig for wo cusomer classes, bu our meus are dyamic raher ha saic. As oulied above, our coribuio is he provisio of asympoically opimal ad readily-implemeable iceive-compaible policies for dyamic leadime quoaio ad pricig for reveue maximizaio uder covex cocave cos curves for wo cusomer classes. The paper is orgaized as follows. Secio coais a brief review of he releva lieraure. Secio 3 preses our model ad iiial formulaio. Our large capaciy asympoic regime is preseed i Sec. 4.Ourproposedpolicyishe give i Sec. 5 ad Sec. 6 proves is asympoic opimaliy. Fially, Sec. 7 cocludes he paper. A umber of he more echical proofs are relegaed o he Appedix. Lieraurereview Research o leadime quoaio has a log hisory ad has accumulaed a sigifica body of lieraure see, for example, [8], for a review; mos of his wor does o cosider reveue maximizaio, especially i he presece of uobservable cusomer ypes, as we cosider here. Lieraure ha has cosidered reveue maximizaio bu o iceive compaibiliy i he coex of dyamic leadime quoaio icludes [7, 1 1, 14 18, 6, 7, 3, 39, 4, 48, 51, 5], oe of which cosider he covex cocave delay cos curves cosidered here. Lieraure ha has cosider iceive compaibiliy i he coex of saic leadime quoaio icludes [1, 1,, 5, 31, 33 35, 37, 38, 43, 5, 55]. We also refer he reader o [3]forabroaderreviewofreveueopimizaioiqueuigsysems. Wor ha has cosidered dyamic leadime quoaio wih a meu of opios icludes [, 9]. Çeli ad Maglaras [9] cosideramodelwhereademadfucio maps he dyamic meu of prices ad fixed leadimes o a vecor of demad raes; wih such a arrival process here are o issues of iceive-compaibiliy. Dyamic leadime quoaio is also cosidered i [], where a fluid model is used o aalyze asocialwelfaremaximizaiomodel.thapaperprovidesfurhermoivaioforhe covex cocave leadime cos curve cosidered here. We cosider a large-capaciy asympoic regime where boh arrival raes ad capaciy grow proporioally i a radiioal heavy-raffic sese. I order o preve he

4 38 Queueig Sys 13 73:35 78 delay cos fucios from becomig eiher egligible or overwhelmig, we also scale he cos fucio o maiai a cosise order of magiude as he sysem scales. Such scalig of coss ad/or delays experieced have bee doe by may researchers i he field icludig [6, 8, 9,, 36, 4 4, 47, 49]. As described i he iroducio, we assume a covex cocave shape for he cusomers delay cos curves where cusomers have a paricular deadlie i mid bu oce ha deadlie has passed, hey are icreasigly isesiive o he addiioal leadime. Kahema ad Tversy s [4] prospecheoryspeasoheexiseceofsuch se performace expecaios ad he wor of Leclerc e al. [3] specializeshiso aseigofdelays.oherlieraurealsocosiderswaiigcossaduiliiesfroma cusomer s perspecive ad some releva refereces iclude Aoides e al. [4]who perform a experime o evaluae peoples percepios of waiig ime ad Frederic e al. [19] whogiveareviewofieremporaldiscouig. Aa ad Olse [6] cosiderahomogeeouscusomeraalogofheproblemcosidered here. I ha paper he pricig problem was elimiaed by he assumpio of asigleclass;pricigwasahecusomer smargialcos,adherewereoissues of muli-class schedulig or of iceive compaibiliy. Moreover, he sae space collapse resul esablished i Aa ad Olse is much simpler ha he oe here as i oly cocers sae space collapse wihi a class so as o achieve he covex hull of he delay cos fucio, whereas here we mus esablish sae-space collapse boh across classes ad wihi each class. 3 Themodel We cosider a mae-o-order moopolisic firm servig wo classes of delay sesiive cusomers, who differ i heir persoal coss of delay. The firm is modeled as a sigle-server queue, where a sysem maager quoes leadimes ad prices o arrivig cusomers dyamically over ime. Paymes are colleced upo arrival. The firm s objecive is o maximize he log-ru average paymes received per ui ime. I order o do his, he sysem maager chooses he sequece i which he cusomer orders are processed so ha he quoed leadimes are respeced. A class i cusomer has a delay cos of c i τ associaed wih receivig a leadime quoaio of τ.shorerleadimes are more aracive o cusomers, i.e., he delay cos fucio c i τ is icreasig i he quoed leadime τ for each ype i. Allcusomersreceivehesamepersoalreward R from service herefore he uiliy hey receive from he produce equals R c i τ mius he price hey pay. The sysem is iiially empy ad class i cusomers arrive a he sysem a rae λ i accordig o a Poisso process {A i : }. Adopighe ermiology ha is sadard i queuig heory, cusomer orders o be processed are referred o as jobs. I order o guaraee ha he quoed leadimes will always be me, we resric aeio o deermiisic producio, where he producio decisios are made a discree pois i ime, say a imes,κ,κ,... ;adcusomersarequoedleadimes ha are ieger muliples of κ. Eachimeiervalbeweewocosecuivereview pois may be hough of as a producio shif. We assume ha ew arrivals ca be served immediaely if here is ucommied capaciy. We also allow early delivery,

5 Queueig Sys 13 73: should here be exra capaciy, bu i pracice his may mea sorig he iem off lie for a o-ime delivery. The sequecig decisios ae he form of cumulaive corol processes. I paricular, le T i be he cumulaive amou of service effor ha he server devoes o servig class i jobs durig [,]. Thehevecorprocess{T: }, where T= T 1, T,deoeshesysemmaager ssequecigpolicy.clearly,t is odecreasig, ad saisfies [ T 1 + T ] [ T 1 s + T s ] µ s for s <, where µ is he service rae. The, defiig S = for, ST i deoes he cumulaive umber of class i jobs processed up o ime. Recallhabohclassesof jobs have he same service requireme. Also, defiig L = µ T 1 T for as he cumulaive uused capaciy up o ime, werequireha L is odecreasig wih L =. As meioed earlier, we will assume ha he delay coss c 1, c are boh covex cocave, which correspods o c i beig covex o a ierval [,d i ] ad cocave o [d i, for i = 1,. The basic idea is ha d i represes he cusomer s deadlie ad he is icreasigly impaie leadig up o his deadlie ad icreasigly more olera oce he deadlie has passed. I our model, class 1 is he impaie class while class is he paie class. I paricular, he deadlie of a class 1 cusomer is sooer ha ha of a class cusomer, i.e., d 1 <d,adaclass1cusomer s delay cos icreases a a faser rae uil is deadlie. For simpliciy, we model his siuaio as follows. Le c deoe he delay cos of class, where c is covex o [,d ] ad cocave o [d,. Thegived 1 <d,le c 1 x = c x + d d 1 c d d 1 for x. 1 I is easy o chec ha 1 implieshac 1 is also covex cocave. The uderlyig assumpio i 1ishaiisasifbohclasseshavehesameuderlyigcosfucio, bu class 1 cusomers place heir orders closer o he deadlie. Tha is, we assume ha impaie cusomers have a shor deadlie, whereas paie cusomers pla furher i advace ad have a loger ime frame uil heir iolerace poi he poi where he curve swiches from covex o cocave is reached. To faciliae our aalysis, le h i deoe he covex hull of c i for i = 1,, i.e., h i is he maximal covex fucio wih h i c i.defiig x i = if { x : c i x c} for i = 1,, i is easy o see ha h i x = { c i x for x x i, x x i c + c i x i oherwise.

6 4 Queueig Sys 13 73:35 78 Fig. 1 Illusraive delay cos fucios ad heir covex hulls Also defie x i = if { x>d i : c i x = c} for i = 1,. Illusraive delay cos fucios ad heir covex hulls are displayed i Fig. 1. I addiio o 1, which defies c 1 i erms of c, wemaehefollowig assumpios o he cos curves. Assumpio 1 i c is sricly covex icreasig o,d which implies ha c 1 is sricly covex icreasig o,d 1 ; ii c is cocave icreasig o [d, which implies ha c 1 is cocave icreasig o [d 1, ; iii lim x c i x = c>fori = 1, ; iv c <c; v x i < for i = 1, ; ad vi c 1 x c x for x. Assumpios 1i ad 1ii have bee discussed previously. Assumpio 1iii is ecessary as he magiude of he limi poi c will be impora i deermiig he appropriae schedulig rule. If isead he derivaive of he coss eded asympoically o zero, he schedulig rule would pu oo much emphasis o very log wais, which appears o be urealisic. Assumpio 1iv rules ou he possibiliy of he covex hull of c beig liear. This assumpio esures ha boh he covex ad he cocave pars of he covex fucio c are releva for decisio maig. Assumpio 1v is made for echical reasos o esure ha overly log delays are o quoed. I pracice, i is a modes assumpio as x i may be allowed o be very large. Fially, Assumpio 1vi is cosise wih our ierpreaio of paie ad impaie cusomers. This assumpio esures ha a impaie cusomer icurs a higher delay cos ha a paie cusomer whe hey are quoed he same leadime. I is easy o chec ha Assumpio 1vi is saisfied for ay covex cocave delay cos fucio c ad c 1 which is defied by 1 provided c 1 = c d d 1 c, hais,wheeverheimpaiecusomersareoooimpaie, which, of course, is oly a sufficie codiio. However, our aalysis allows arbirary

7 Queueig Sys 13 73: Fig. A illusraive fucio c 1 c d 1 such ha d 1 d.thus,oemaycosrucpahologicaldelaycosfucios c 1, c such ha vi is violaed, which we rule ou o esure ha he delay cos fucios c 1, c are cosise wih our ierpreaio of impaie ad paie cusomers. As a aside, oe ha a weaer sufficie codiio for vi o hold for a arbirary covex cocave delay cos fucio c ishad d 1 c c d d 1. The followig lemma summarizes useful properies of he fucio c 1 c, which will be used i subseque secios. A illusraive fucio c 1 c is porrayed i Fig.. Lemma 1 There exis a uique maximizer d x d 1,d of c 1 c such ha c 1 c x is icreasig o [,d ], decreasig o [d, x ], ad is cosa for x x. Moreover, c 1 c x 1 c 1 c x for all x 1 x 1 x. The sysem maager cao observe he ypes of he arrivig cusomers ad he cusomers may misreprese hemselves. Therefore, he sysem maager offers a meu of prices ad leadimes o disiguish he wo ypes of cusomers. The meu will be updaed dyamically over ime depedig o he sysem saus o maximize profis. Namely, he sysem maager offers a meu of prices ad leadimes {p i, τ i : i = 1, } a ime, wherep i, τ i is ieded for a class i cusomer arrivig a ime. Lep = p 1, p ad τ = τ 1, τ,ad deoe he sysem maager s policy by PT ; p; τ:,whichwerequire o be o-aicipaig i he usual sese. For admissibiliy of a policy PT ; p ; τ, heassociaedmeuofprice leadime pairs mus saisfy he followig idividual raioaliy IR ad iceive compaibiliy IC cosrais: R p i c i τi, i = 1,, 3 R p 1 c 1 τ1 R p c 1 τ, 4 R p c τ R p 1 c τ1, 5 where he idividual raioaliy cosrai 3 esureshaeachcusomerreceivesa oegaive surplus. To esure ha all cusomers ideed receive a oegaive sur-

8 4 Queueig Sys 13 73:35 78 plus, we allow he possibiliy of egaive prices alhough i pracice his is uliely o occur. This would ae he form of a discou give o he cusomer o he base price of he iem which is o icluded i he leadime/price meu for he excessive delay received. Ideed, i he large capaciy asympoic regime, oe ca show ha he cogesio cocers are of secod order relaive o he cusomer rewards ad hece, he cusomers are always quoed posiive prices. The iceive compaibiliy cosrais 4 5 esurehaeachclassi cusomer reveals his ype ruhfully by choosig he price, leadime pair p i, τ i ieded for him. I oher words, he iceive compaibiliy cosrais guaraee ha choosig p i, τ i is i he bes ieres of a class i cusomer arrivig a ime. I wha follows, moivaed by he IR cosrais, we will express he prices as follows: p i = R c i τi i for i = 1, ad, where i deoeshepoeialpricediscouforclassi cusomers a ime i.e., he discou he firm mus give because i does o ow he cusomer s ype ad herefore cao jus charge R c i τ i. The, he IR cosrais ca be wrie as follows: i for i = 1, ad. Similarly, he iceive compaibiliy cosrais 4 5areequivaleohefollowig: c 1 c τ 1 c 1 c τ 1. The sysem maager srives o maximize he expeced log-ru average reveues per ui ime give by [ ] 1 lim E p i s λ i ds i=1 [ 1 = lim E R ci τi s i s ] λ i ds, i=1 which ca equivalely be saed as miimizig [ 1 lim E ci τi s + i s ] λ i ds. i=1 We deoe he umber of class i jobs i he sysem a ime by Q i ; alsoleα i deoe he delay experieced by he class i cusomer arrivig a ime. Noeha α i is a raher complicaed fucio of he policy employed. The sysem is empy iiially. To faciliae fuure aalysis, also defie Q as he oal umber of jobs i he sysem a ime. Thais, Q = Q 1 + Q for.

9 Queueig Sys 13 73: Also, deoe he worload remaiig i he sysem a ime by W ad oe ha W= A µ + L,, where A is defied as A 1 + A for. Noe ha for ay wor-coservig policy he processes W,L joily saisfy 1 {W>} dl =. 6 Ideed, i wha follows we propose a wor-coservig policy which, of course, saisfies 6. The sysem maager s problem ca be saed as follows: Choose a policy PT ; ; τ so as o miimize lim subjec o [ 1 E ci τi s + i s ] λ i ds i=1 Q i = A i S T i, i = 1,, 8 L = µ [ T 1 + T ], 9 T,Lare odecreasig wih T = L =, 1 α i τ i, i = 1,, 11 i, i = 1,, 1 c 1 c τ 1 c 1 c τ 1, 13 where he cosrais 8 9 describehedyamicsofhebaclogadhecumulaive idleess processes, while he cosrai 1 assersheauralrequiremeha he cumulaive allocaio ad he cumulaive idleess processes are odecreasig. Cosrai 11 esureshaallquoeddelaysarerespecedbyrequirighaheacual delays are less ha or equal o he quoed delays. Fially, cosrais 1 13 are he IR ad IC cosrais, respecively. The followig proposiio is immediae ad simplifies he problem. Proposiio 1 For ay feasible policy PT ; ; τ for he problem formulaio 7 13, we ca fid a feasible policy P T ; ; τ such ha 1 = ad = c 1 c τ 1 for ad he objecive 7 is a leas as small as uder PT ; ; τ. The, defiig c 1 x = c 1 x + λ λ 1 c 1 c x for x, 7

10 44 Queueig Sys 13 73:35 78 ad subsiuig 1 =, = c 1 c τ 1, heformulaio7 13 reduces o he followig: [ 1 miimize lim E λ 1 c 1 τ1 s ds + λ c τ s ] ds 14 subjec o 8 11, ad 15 c 1 c τ c 1 c τ We ca equivalely express he problem as follows: miimize lim 1 E[ C P ] subjec o 15 16, 17 where he cumulaive cos 1 uder policy PT ; τ up o ime is give by C P = c 1 τ1 s da 1 s + c τ s da s. Also deoe he cumulaive profi up o ime uder policy PT ; τ by P, where P = Rλ 1 + λ C P. Uforuaely, he problem formulaio 17 doesoseemoberacableaalyically. Therefore, we cosider a closely relaed sequece of problems i he heavy raffic asympoic regime, ad provide a asympoically opimal policy. As a auxiliary sep, we ex iroduce a lower boudig problem. Alowerboudigproblem To derive a lower boud o he objecive 17 wefirs relax he cosrai 16. The puig τ i = α i,wearriveahefollowigiermediae lower boudig problem: Choose a schedulig policy T so as o [ 1 miimize lim E c 1 α1 s da 1 s + c α s ] da s subjec o 8 1. To faciliae fuure aalysis, le h 1 be he covex hull of c 1 i.e., h i is he maximal covex fucio wih h i c i so ha h 1 is a lower boud o he cos c 1. Defiig x 1 = if { x : c 1 x c}, i is easy o see ha x 1 x 1,adha { c 1 x if x x h 1 x = 1, x x 1 c + c 1 x 1 oherwise. 1 We adop he equivale defiiio of cos o faciliae compariso wih he lower boud advaced by va Mieghem [49].

11 Queueig Sys 13 73: Moreover, h 1 x > h 1 x for x<x 1,providedc 1 <c. Fially, replacig he delay cos fucios c 1 ad c by heir covex hulls h 1 ad h,wearriveahelowerboudigproblem:chooseascheduligpolicyt so as o where H= miimize lim 1 E[ H ] subjec o 8 1, 18 h 1 α1 s da 1 s + h α s da s,. This problem is sudied i [49], which shows ha he geeralized cµ Gcµ rule miimizes he oal delay cos experieced by cusomers asympoically i he heavy raffic limi provided ha he cusomers have covex icreasig delay coss. This resul provides a asympoic lower boud for he formulaio 7 13 iheheavy raffic limi. I wha follows, we propose a policy for he formulaio 7 13, which achieves his lower boud i he heavy raffic limi. To ha ed, we iroduce he large capaciy asympoic regime ad some auxiliary defiiios i he ex secio. 4 Largecapaciyasympoicregime Cosider a sequece of sysems idexed by = 1,,...Asuperscrip will be aached o he quaiies correspodig o he h sysem i his sequece. The asympoic regime we are ieresed i is he oe where he arrival raes ad he processig capaciy grow wih as specified i he ex assumpio. Heavy raffic assumpio We assume for all ad some θ<ha λ i = λ i for i = 1, ad µ = λ 1 + λ θ. The heavy raffic assumpio correspods o havig a large, balaced-flow sysem for large. For such sysems, he worload i he sysem is expeced o be of order,whileweexpecdelaysobeoforder1/.thus,wescalehecosofdelayas follows: c i = c i for i = 1, ad 1. I is easy o chec ha he covex hull h i of c i is give by he followig: h i = h i for i = 1, ad 1. Similarly, he covex hull h 1 of c 1 is give by h 1 = h 1 for 1.

12 46 Queueig Sys 13 73:35 78 As is cusomary i he heavy raffic lieraure, we ex iroduce he diffusioscale processes. Defie he scaled queue legh, worload, ad cumulaive idleess processes as follows: Q = Q, Q i = Q i, Ŵ = W, ad L = L for 1,. Also defie he scaled prices p i = pi R for adi = 1,. Fially, defie he diffusio scaled cumulaive delay coss ad profis as follows: Ĉ P = C P, Ĥ P = H, ad P = 1 [ P Rλ 1 + λ ]. Noe ha P = Ĉ P for all,,reflecighefachamiimizigdiffusioscaled coss is equivale o maximizig diffusio-scaled profis. The followig auxiliary resul is immediae from Theorem.4 of Csorgo ad Horvah [13]. This is a powerful resul sice i cosrucs a srog approximaio of he arrival process, i.e., a approximaio o he same probabiliy space where he arrival process lives, which i ur allows us o esablish all sochasic process covergece resuls i he almos sure sese. Therefore, i wha follows all covergece resuls are i he almos sure sese uless saed oherwise. Proposiio There exis idepede Browia moios B i ={B i : } for i = 1, such ha A i = λ i + λ i B i + εi for, where sup T εi =OlogT for T>. Havig iroduced he diffusio scaled quaiies, we give a precise meaig o he erm asympoic opimaliy ex. Recall ha he sysem maager srives o maximize expeced log-ru average profi. A eve more ambiious objecive would be o maximize he cumulaive profi up o ime almos surely. Ideed, we adop his more ambiious objecive as give by he followig defiiio of asympoic opimaliy, which is also i lie wih he objecive crierio used i [49]. Defiiio 1 Asympoic Opimaliy A sequece of policies {P T ; τ : 1} is called asympoically opimal if for ay oher sequece of policies {P T ; τ : 1},isaisfies lim P lim P a.s. for. To faciliae he aalysis o follow, we ex derive a upper boud.

13 Queueig Sys 13 73: A asympoic lower boud As a prelimiary o iroducig he asympoic performace boud, he followig auxiliary defiiios are eeded. Le B = λ 1 B 1 + λ B,, so ha B is a,λ 1 + λ Browia moio. The defie L = if s Bs ad W = B + L so ha W,L isaoe-dimesioalreflecedbrowiamoiorbm.alsole { } q1 q qw arg mi λ 1 h 1 + λ h : q 1 + q = w, q i fori = 1,, q=q 1,q λ 1 λ for w suchhaq 1 w is miimal wheever here are muliple miimizers. The followig proposiio summarizes releva properies of q i for i = 1, adis proved i Appedix A. Proposiio 3 For i = 1,, q i is odecreasig ad Lipschiz coiuous wih Lipschiz cosa L i = 1. Moreover, for w we have q 1 w λ 1 q w λ ad q 1 w λ 1 x 1, q w λ λ 1 + λ w. Noe ha q i will be applied o boh he worload ad queue legh processes, which will be show o be close i he limi. The followig proposiio follows immediaely from Proposiio 6 of va Mieghem [49]. Proposiio 4 For ay sequece of admissible policies {P T ; τ : 1}, we have [ lim q1 W P λ 1 h q W ] 1 d + λ h d. λ 1 λ 5 Proposedpolicyadisasympoicopimaliy The idea behid he proposed leadime quoaio ad sequecig policies is o esure ha whe he worload is w, adelaycosraeofλ 1 h 1 q 1 w + λ h q w is icurred. Whe worload is small boh classes are ep i he covex cos regio ad a modified versio of he Gcµ rule is i operaio. Whe worload is large, class 1 is ep i is covex regio bu class wo is operaig i is cocave regio. To esure asympoic opimaliy, class is divided io wo arificial subclasses o

14 48 Queueig Sys 13 73:35 78 esure ha he cos rae of he covex hull is icurred. I paricular, we arificially segme class io wo subclasses a ad b, ad will give prioriy o subclass a over subclass b while providig a small bu posiive amou of service capaciy o subclass b. Small asympoically egligible modificaios are he made o he origial policy o esure iceive compaibiliy. Recall ha we resric aeio o deermiisic producio wih a discree-review framewor. Here we choose a review-period legh or a producio shif legh κ for each sysem such ha deermiisic producio is a reasoable assumpio ad a quoa of µ κ jobs may be processed durig each review period. I paricular, we le κ = z 1 1 α 1λ 1 + λ µ for 1, 19 where 1/ <α 1 < 1adz 1 > isafixediegerhaservesasascaligcosa. I he h sysem, he sysem maager reviews he sysem saus a imes = κ for =, 1,...Ahebegiigofeachperiod,shedecideshowoallocaeresources durig ha period, which will be described below. The leadime quoaio ad pricig decisios are made as cusomers arrive o he sysem. To be more specific, as he cusomers arrive, he sysem maager quoes hem leadimes ad correspodig prices ad decides he order bu o he exac imig i which hey will be processed i he upcomig periods so ha he quoed leadimes are respeced. This resuls i a deailed producio pla for he upcomig producio shif so ha he server ca jus process jobs as prescribed by his pla ad ay ew arrivals should here be sufficie capaciy. Give our choice of he review-period legh κ,he server ca process z 1 1 α 1λ 1 + λ jobs i each shif, ad κ is of order 1/ α 1, which esures ha he sysem maager reviews he sysem saus sufficiely frequely provided α 1 1/, 1,whichiscosisewih[5]. To faciliae he descripio of he proposed sequecig ad leadime quoaio policies, i is helpful o imagie ha we have four server pools, where server pool i has capaciy µ i for i = 1, a,b, whilehefourhserverpoolisa flexible server wih capaciy µ f ad will serve eiher class 1 or class. I pracice, his is jus a allocaio of he review period producio quoa o he differe classes. We se µ f = z α, µ b = z 3 α 3, µ 1 = λ1 µ κ µ f λ 1 + λ, µ a = µ κ µ 1 µ f µ b, where 1 α 1 >α 3 >α ad α 1 α 1 /, 1 α 1,adz,z 3 > arefixediegers ha ca be used o scale he capaciies. To mae his more cocree, cosider he siuaio where α 1 = 3/4, α = 3/16, α 3 = 7/3, ad z 1 = z = z 3 = 1; hese saisfy he above requiremes ad could be used as he values of hese parameers for he policy. However, here are a rage of oher values possible for hese parameers ha are asympoically ideical bu may be used i pracice o fie ue he policy s performace. Coiuig wih he umerical example, suppose ha = 16 = 65536, λ 1 = λ = 5, so ha µ κ = 16. The we have µ f = 4 = 8, µ b = 11.3,

15 Queueig Sys 13 73: µ 1 = 8 4 = 796, ad µ a = = As discussed above, µ 1 ad µ a receive he bul of he capaciy, bu oice how seig he scalig parameers z = z 3 = woulddoublehecapaciyallocaedoµ f ad µ b,respecively. I each period he flexible server is allocaed o eiher class 1 or class, depedig o he delay cos rae associaed wih each class. To be more specific, he flexible server gives prioriy o class 1 durig period if h 1 Q 1 λ 1 h Q oherwise i gives prioriy o class. Whe servig class, he flexible server gives sric prioriy o class b ad serves class a wheever class b is empy. O he corary, server i gives prioriy o class i for i = 1, a,b. Thespecificprioriyrule for he flexible server ca be viewed as a discree-ime versio of he Gcµ rule. Moreover, he service policy is collecively o-idlig i he sese ha oe of he servers idle uless he sysem is empy; wheever a server fids is ow prioriy class empy i gives prioriy o he class wih he highes idex ha is o-empy, where b>a >1. The sysem maager updaes her rouig decisios a he review pois. To be more specific, a each review poi for = 1,,... he sysem maager observes he sysem saus. Le Q a be he queue-legh of subclass a ad Q b he queue legh of subclass b so ha Q = Q a + Q b. Sheroueshe firs µ a class jobs o subclass a ad roues he ex µ b jobs o subclass b. The he ex [ λ x Q a ] + jobs are agai roued o class a. Ay furher class arrivals are roued o subclass b. Noe ha his rouig policy esures ha Q a λ x + µ a a all imes. Moreover, for large eough, i.e., λ x >µ a,uderheproposedpolicyheumberofclassjobsq a a each review poi is less ha or equal o λ x. Noe ha whe baclogs are small, class operaes i he covex regio of is cos curve. Furher, mos of he worload is ep i class a; ad class b receives jus eough wor o eep is server occupied. However, as he baclog i class icreases, i is class b ha absorbs he added baclog, possibly receivig quie log delays, while class a is ep a a moderae legh. Cusomers are quoed leadimes ad prices whe hey arrive o he sysem as follows. For adi = 1, a,b,le Q τ i = i µ 1periods. i The se λ ; { τ = τ a if he class job arrivig a ime is o be roued o subclass a, τ b oherwise. For large, hevariousprocessesofheorigialsysemarewellapproximaedby heir couerpars i he limiig Browia model. Thus, we expec wih high lielihood ha he leadimes quoed i he origial sysem will iheri he propery

16 5 Queueig Sys 13 73:35 78 of iceive compaibiliy from heir couerpars i he limiig Browia model. Noeheless, here is a vaishigly small bu posiive probabiliy ha he IC cosrai may be violaed. Moreover, he segmeaio of class o achieve he boud give by he covex hull approach requires us o ae addiioal cauio o mae sure ha he quoed leadimes are ideed iceive compaible. To isure iceive compaibiliy, we chec o see if τ 1 x 1 /κ periods. If so, we se Noe i his case ha which implies by Lemma 1 ha τ 1 = τ 1 ad τ = τ 1 τ. τ 1 κ x 1 / τ κ, c 1 c τ κ c1 c τ 1 κ. Thus, he quoed leadimes are iceive compaible. Ideed, as will be see i Sec. 6.4, hiscasewillarisewihprobabiliyoeiheheavyrafficlimi.ohe oher had, if τ 1 > x 1 /κ periods, he cosider he followig wo cases: Case i. ICcosraiisoviolaed.Thais, c 1 c τ κ c1 c τ 1 κ. The, we se τ = τ. Case ii. ICcosraiisviolaedby τ. These τ 1 = τ 1 x κ ad τ = τ x κ, which saisfy he IC cosrai. Fially, recall from Proposiio 1 ha 1 = ad = c 1 c τ 1 κ for. The, he prices are give by p 1 = R c 1 τ 1 κ ad p 1 = R c τ κ. The followig heorem saes he mai resul of he paper ad is prove i Sec Theorem 1 The sequece of proposed policies is asympoically opimal. Namely, we have [ lim P = q1 W λ 1 h q W ] 1 d + λ h d λ 1 λ for all a.s.

17 Queueig Sys 13 73: Proofofhemairesul The proof of asympoic opimaliy proceeds i hree major seps. Firs, we esablish he covergece of he worload process hrough a coiuous mappig argume usig he coiuiy of he oe dimesioal reflecio map; cf. Sec Thesecod sep is o esablish a sae-space collapse resul. Secio 6. provides some auxiliary resuls eeded for ha. I Sec. 6.3 we esablish wo ids of sae-space collapse resuls: Firs, we esablish ha Q i q i W i a appropriae sese, cf. Theorem, from which i follows, i paricular, ha he umber of class 1 jobs i he sysem are sufficiely low ha oly he covex par of he delay cos fucio c 1 is releva. Secod, we esablish a sae-space collapse resul wihi class ; cf. Theorem 3. Tha is, class jobs are roued o arificial subclasses a ad b i such a way ha he covex hull h of c is achieved. The las sep is he proof of asympoic opimaliy iself buildig o he earlier resuls. Secio 6.4 provides furher auxiliary resuls for he proof of asympoic opimaliy. These are o he regulariy of delays quoed o each class, ad he regulariy of he rouig paers of class jobs o subclasses a ad b. Fially, Sec. 6.5 esablishes he asympoic opimaliy of he proposed policy by showig ha is asympoic performace coicides wih he lower boud provided i Proposiio 4. As a prelimiary o esablishig he covergece resuls for he queue-legh processes uder he proposed policy, we firs esablish a covergece resul for he worload process i he ex secio. 6.1 Covergece of scaled worload process To esablish he covergece of he scaled worload process, le ψ, ϕ deoe he oe-dimesioal reflecio map defied o he space D[, of r.c.l.l. fucios, cf. [53]. Tha is, for a real-valued fucio x : D[, D[,, ϕx = if xs ad ψx= x + ϕx. s The ex proposiio esablishes he covergece of he scaled worload process. Proposiio 5 Uder he proposed policy, Ŵ, L W,L u.o.c. a.s. as, where W = ψb, L = ϕb ad B = θ + B for. Proof Recall ha i he h sysem we have he followig: For W = A µ + L, 1 {W s>} dl s =. 1

18 5 Queueig Sys 13 73:35 78 I follows from 1ha W = ψ A µ ad L = ϕ A µ. By scalig hese, we have A Ŵ µ = ψ ad A L µ = ϕ. Noe ha A µ / B u.o.c. a.s. as,byproposiio.the, by coiuiy of he reflecio map ψ, ϕ, cf.[53], we coclude ha Ŵ, L W,L u.o.c. a.s. as. As a immediae corollary, we ex show he covergece of he oal queue legh process. Corollary 1 Uder he proposed policy, Q W u.o.c. a.s. as. Proof The resul follows from he fac ha W Q = Q 1 + Q a + Q b W + 3for adfromproposiio5. 6. Auxiliary resuls for esablishig sae space collapse This secio preses auxiliary resuls ad defiiios which will be used o faciliae he proofs of he sae-space collapse resuls of Sec For T,ζ >, β 1 α 1, 1 α 1, 1, ad = 1,..., T/κ,defie A = { A i A i λ i ζ β,i= 1, }. The followig lemma provides a boud o he probabiliy of A ;isproofisgive i Appedix B. Lemma For T>, = 1,..., T/κ ad sufficiely large, we have P A 1 C1 exp { C ζ β 1+α } 1, where C 1 ad C are posiive cosas idepede of,. For T>, 1, defie M T = { A i A i λ i 1 α,i= 1,, = 1,..., T/κ }. Proposiio 6 For T>, PMT 1 as.

19 Queueig Sys 13 73: Proof I suffices o show ha PM T c as.noeha P M T T/κ c =1 T/κ =1 { A P i } i λ i 1 c α,i = 1, { C 1 exp C } 4 α 1+α 1 T κ + 1 C 1 exp { C } 4 α 1+α 1. Sice κ = O 1 α 1,herigh-hadsideedsozero,cocludigheproof. I he ex subsecio we esablish a sae space collapse resul. 6.3 Sae space collapse As a prelimiary, we firs provide some probabiliy bouds uder he proposed policy. Give δ>ad<ε<mi{.5 α,α 1.5},defie { B = Q q Q δ } ε for, 1. The followig lemma provides a useful probabiliy boud ad is prove i Appedix B. Lemma 3 For T>, ad = 1,..., T/κ, uder he proposed policy, P [ 1 C 3 exp { C 4 α }] 1+α 1, j=1 B j where C 3 ad C 4 are posiive cosas idepede of,. Defiig NT probabiliy. = T/κ =1 B,hefollowigproposiiocharacerizesislimiig Proposiio 7 For T>, PNT 1 as uder he proposed policy. Proof Noe by Lemma 3 ha P NT 1 C3 exp { C 4 α } 1+α 1 T/κ where he righ-had side coverges o 1 as because T/κ C 3 exp { C 4 α } 1+α 1 as. Hece he resul follows.

20 54 Queueig Sys 13 73:35 78 Defiig he processes Q i = Q i ξ i = sup s for [ +1,,=, 1,..., ad i = 1,, Q i s Q i s for adi = 1,, he followig proposiio shows ha Q i well approximaes Q i. Proposiio 8 For T> ad i = 1,, uder he proposed policy we have ha ξ i T / as. Proof Firs, oe ha for i = 1,, sup Q i Q i T Also oe ha for =, 1,..., T/κ The by Proposiio, we have max sup =,..., T/κ [, +1 ] Q i Q i. Q i Q i A i A i + µ κ. A i A i λi + sup ε i s + λi sup Bi B i. s T [, +1 ] Thus, for i = 1,, sup Q i Q i λ i κ + µ κ + sup ε i s [, +1 ] s T + λ i sup Bi B i, [, +1 ] which i ur gives he followig: 1 sup Q i Q i λ i κ + µ κ + sup s T εi s T + λ i max sup Bi B i, =,1,..., T/κ [, +1 ] where he firs erm o he righ eds o zero as by Proposiio ad he choice of κ,cf.19, while he secod erm goes o zero as by Levy s modulus of coiuiy heorem for Browia moio; cf. [44]. The followig heorem saes he firs par of he sae-space collapse resul uder he proposed policy.

21 Queueig Sys 13 73: Theorem For T>ad i = 1,, we have uder he proposed policy ha sup Q i q i W as. T Proof Noe ha sup q Q Q sup T + sup T q T q Q 1 + Q Q + sup L ξ 1 T + ξ T + 1 ξ T + sup L + 1 ξ 1 T + ξ T + max Q Q 1 q + Q =1,..., T/κ L + 1 ξ 1 T + ξ T + δ ε 1 {NT } + T T q q Q sup T Q Q Q 1 + Q Q Q Q 1 {N T c }. Leig,wecocludefromProposiios5,6,adLemma3 ha sup q Q Q as a.s. T Combiig his wih he facs ha q is Lipschiz coiuous ad sup T Q W as gives sup Q q W T as. The usig Q 1 = Q Q ad q 1W = W q W for W, he resul follows. To esablish he secod par of he sae-space collapse resul, he followig lemma is eeded. Is proof is give i Appedix C. Lemma 4 Uder he proposed policy, for T>, = 1,..., T/κ, ad sufficiely large, we have he followig o he se M T : If Q a < λ x, he Q b 3µ b. The followig saes he secod par of he sae-space collapse resul. Theorem 3 Uder he proposed policy Q a, Q b Q a,q b u.o.c. as a.s.,

22 56 Queueig Sys 13 73:35 78 where Proof Noe ha Q a = Q λ x = q W λ x,, Q b = Q Q a = [ q W λ x ] +,. sup T Q a Q λ x max =1,..., T/κ + max =1,..., T/κ + max =1,..., T/κ Q a Q λ x sup Q [, ] a Q a sup Q [, ] λ x Q λ x. I ca be argued as i he proof of Proposiio 6 ha he las wo erms o he righhad side ed o zero as.thus,wecosiderhefirsermoherigh-had side. Nex we show ha { Q a Q 3µ b λ x, o M T, λ x, o MT c. I is easy o see ha Q a Q λ x λ x everywhere, bu specifically o MT c. Cosider MT.SiceQ a λ x, isufficesochec ha Q λ x Q a 3µ b. 3 Noe ha 3isimmediaeifQ a = λ x. Oherwise, i.e., if Q a < λ x,hebylemma4,weobservehaq b 3µ b.thus, Q λ x Q a = Q a + Q b λ x Q a = Q b λ x Q a Q b Tha is, 3holds.The,clearly, max =1,..., T/κ 3µ b. Q a Q 3µ λ x b 1 {M T } + λ x 1 {M T c }, where he righ-had side eds o zero sice PMT 1. Thus, we coclude ha Q a Q λ x as a.s. sup T

23 Queueig Sys 13 73: Combiig his wih he fac ha Q q W u.o.c. as cocludes he proof. Combiig Theorems ad 3, wehavehefollowigsae-saecollapseresul. Corollary Uder he proposed policy, Q 1, Q a, Q b Q 1,Q a,q b u.o.c. as. 6.4 Auxiliary resuls for he proof of asympoic opimaliy Defie C T = { τ 1 κ x 1 /, τ 1 κ τ a κ for all [,T] }. Proposiio 9 For T>, we have uder he proposed policy ha PC T 1 as. Proof Fix T>. The i suffices o show ha wih probabiliy oe Noe ha lim τ 1 κ = Q 1 lim sup T sup T µ 1 τ a κ = Q a µ a The he resul follows sice q 1w λ 1 τ 1 κ x 1, τ 1 τ a. 1 κ q 1W Noe ha o he se CT,wehaveτ 1 = τ 1, ad λ 1 u.o.c. as, 1 κ q W x λ u.o.c. as. x 1 ad q 1w λ 1 q w x λ λ for all w. { τ = τ a if he class job arrivig a ime is roued o class a, τ b τ 1 oherwise. Le A a ad A b deoe he cumulaive umber of jobs roued o classes a ad b,respecively,upoime.thefollowiglemmawillbeisrumealiprovig he asympoic opimaliy of he proposed policy. Lemma 5 Uder he proposed policy, for T>, sup A λ as, 4 T

24 58 Queueig Sys 13 73:35 78 A sup a T λ as, 5 A sup b z 3 z 1 α 1+α 3 as. 6 T Proof Recall ha A = λ + λ B + ε. The A λ B λ + ε λ. Taig he sup of boh sides, sup T A λ sup T B λ + sup T ε, λ where he righ-had side eds o zero as which proves 4. Noe ha 5 follows from 4 ad6. Thus, i suffices o prove 6. Recall ha PMT 1as.Thus,isufficesoproveha6 holdso MT.OheseM T,for = 1,,..., λ κ 1 α µ a µ b A b b λ κ + 1 α µ a. I is easy o see ha for sufficiely large µ b λ κ 1 α µ a ad λ κ + 1 α µ a z + 1 α + z 3 α 3. Therefore, for = 1,,..., z 3 α 3 A b A b z + 1 α + z 3 α 3. The for T,wehave κ z 3 α 3 A z b κ + 1 α + z 3 α 3. Thus, we coclude ha A sup b z 3 T z 1 α 1+α 3 { max sup T/κ z + 1 α + z 3 α 3 z 3 T z 1 α 1+α 3, sup T/κ z 3 α 3 z 3 T z 1 α 1+α 3 }, where he righ-had side eds o zero as.hece,heresulfollows.

25 Queueig Sys 13 73: Proof of asympoic opimaliy Proof of Theorem 1 Fix T>, ad recall ha Also recall ha Ĉ P T = 1 T Clearly, we ca wrie P CT 1 as. c 1 τ 1 κ da T c τ κ da. Ĉ P T = Ĉ P T 1 {C T } + Ĉ P T 1 {C T c }. 7 Cosider he firs erm o he righ-had side: { 1 T ĈP T 1 {CT } = + 1 T { 1 T + 1 T + 1 T { T = + T T + c 1 τ 1 κ da T c τ κ da a c τ b τ 1 κ } da b 1 {C T } c 1 τ 1 κ da T c τ a κ da a c τ b κ da b c τ 1 κ } da b 1 {C T } c 1 Q 1 µ 1 c Q a z 3 /z 1 1 α 1 α 3 c µ a 1 κ d 1 κ d Q b µ b T z 3 /z 1 Q + 1 α c 1 1 α 3 µ 1 A } d b z 3 /z 1 α 1 1+α 3 {C T } A c 1 Q 1 κ µ d 1 1 { T A 1 A a 1 1 κ A κ d b z 3 /z 1 α 1+α 3

26 6 Queueig Sys 13 73:35 78 T + c T + T + A Q a κ d a µ a z 3 /z 1 1 α c Q 1 α 3 z 3 /z 1 1 α 1 α 3 c Q b κ µ b 1 κ µ 1 A d b z 3 /z 1 α 1+α 3 d A b z 3 /z 1 α 1+α 3 } 1 {C T }. The passig o he limi o boh sides as,adusigcorollary1saespace collapse resul, Lemma 5, he geeralized Lebesgue covergece heorem, cf. Proposiio 18 o p. 7 of [45], he facs ha c x/ cx ad c x/ as,adhapct 1as,wecocludeha T lim Ĉ P T 1 {CT } T + c 1 q1 W T + c [q W λ x ] + d. c q W λ x d Noe ha c 1 q 1 w = h 1 q 1 w ad h w = c q w λ x + c[q w λ x ] + for all w. Subsiuighisioherigh-hadsidegives T lim Ĉ P T 1 {CT } h 1 q1 W T d + h q W d. 8 Nex, cosider he secod erm o righ-had side of 7, ad oe ha o C T c we have τ i κ τ i κ x / for i = 1,, ad 1. Thus, we wrie Ĉ P T 1 {C T c } { 1 T = { 1 T { 1 T + c 1 τ 1 κ da T c τ κ } da 1 {C T c } } c 1 x da T c x da c 1 τ 1 κ da T { = T c 1 x A 1 T + Tc x A T + 1 T } 1 {C T c } + 1 {C T c } c τ κ } da 1 {C T c } { 1 T c 1 τ 1 κ da 1 c τ a κ da a + 1 T c τ b κ } da b 1 {C T c }

27 Queueig Sys 13 73: { = T c 1 x A 1 T { T + + Tc x A T } 1 {C T c } Q c 1 A 1 µ 1 κ d 1 1 A 1 κ d a T Q + c T + { T z 3 /z 1 1 α 1 α 3 c c 1 x A 1 T T + c T + µ a Q a κ z 3 /z 1 1 α 1 α 3 c Q b µ a lim Ĉ P T 1 {C T c } µ b + c x A T A d a Q b κ µ b 1 T + A κ } d b z 3 /z 1 α 1 1+α 3 {C T c } A c 1 Q 1 κ µ d 1 1 A } d b z 3 /z 1 α 1 1+α 3 {C T c }. 9 The passig o he limi as,wecaargueasbeforeha { T c 1 x λ 1 + c x λ [ T + h 1 q1 W T d + h q W ]} d =, 3 where he firs iequaliy follows sice he firs muliplier of he righ-had side of 9 covergesoafiielimiad1 {C T c } as,a.s.thecombiig 7, 8, ad 3 wihproposiio3adhefacha π P = Ĉ P for all cocludesheproof. 7 Coclusios This paper sudies a wo-class model of dyamic leadime quoaio ad pricig, providig policies ha are boh iuiive ad asympoically opimal. To he bes of our owledge, his paper is he firs o sudy he quesio of how o desig dyamic meus of iceive-compaibleprices ad leadime quoaios for reveue maximizaio. We assumed hroughou our aalysis ha class cusomers are paie, i.e., c <c.thisassumpioiseasyorelax,burelaxigidoesoleadoayew isighs. Ideed, relaxig his assumpio esseially reduces he problem o a sigle class problem. To be specific, whe c c he covex hull of boh c 1 ad c are give by he liear fucio wih slope c. Thus,byquoigasmallfracioof

28 6 Queueig Sys 13 73:35 78 class cusomers very log leadimes, while quoig all oher cusomers zero delays, ad chargig prices equal o he margial coss, he sysem maager achieves a iceive compaible asympoically opimal soluio. Afuureresearchdirecioisocosiderhecasewherehewoclassesofcusomers have differe rewards. If paie cusomers have a higher reward ha impaie cusomers he here is o esio bewee he orderig of rewards ad he delay cos fucios ad he soluio is similar o he oe derived i his paper, excep wih he paie cusomers receivig more value. The opposie case where impaie cusomers have he higher reward is more complex sice boh classes of cusomers may have a iceive o preed o be a cusomer of he oher class. A iiial sep owards udersad differig rewards was made i [] buhaworcosideredwelfare maximizig; dyamic reveue maximizaio uder heerogeeous rewards is a ieresig ad ope problem. I he ieres of simpliciy, we did o cosider admissio corol. However, such a exesio ca easily be added o our model. I ha case, we would expec o ge ahresholdbasedadmissiopolicysimilarohai[6]. We also did o cosider capaciy opimizaio. Ideed, we believe ha i maes sese o firs pursue he operaioal problem of leadime quoaio, pricig, ad schedulig before aswerig he more sraegic quesios of demad ad capaciy opimizaio. Asympoically opimal schedulig ad capaciy seig i he presece of oliear delay coss has bee cosidered by Kumar ad Radhawa [9], who sudy he effecs of mare size o capaciy opimizaio. I coras o heir wor, for our curre model we i o way claim ha he sysem should be operaed i heavyraffic. Raher we solve a hard problem by a sesible ad racable approximaio. We leave he problem of opimal capaciy sizig, ad i paricular he ivesigaio of he appropriae raffic iesiy, as a opic for possible fuure research. Afialassumpiowehavemadeishacusomersmaeheirpurchasigdecisios upo arrival. However, uder he assumpios of our model, here is o reaso for hem o defer heir orders. To see his, oe ha he worload i he sysem evolves very slowly across periods due o he separaio of ime scales pheomeo ihere i large scale sysems. Thus, he cusomers will face similar leadime quoaios across several periods. Because waiig is cosly, o cusomer will chose o do so. To be more specific, he cusomers are presumably sesiive o delays ha are i he order of days i mos seigs, while he baclog of wor i he sysem may evolve i he ime scale of wees for large scale sysems. Therefore, o see a desirable chage i he worload ad hece i he quoed leadimes ad prices a cusomer may have o wai for several wees, which is very cosly for him relaive o he sligh gai he may have from waiig. Thus, we do o cosider he case ha cusomers may be sraegic or arrival-imig see, for example, [46] ad[3]. Acowledgemes We would lie o ha Mary Lariviere ad Musafa Aa for useful discussios ad Tiglog Dai for his echical assisace. This research was suppored i par by he Boeig Ceer for Techology, Iformaio, ad Maufacurig.

29 Queueig Sys 13 73: Appedix A: Techical proofs i Secs. 3 ad 4 Proof of Lemma 1 Firs, observe ha c 1 c is icreasig o,x d 1.To see his, oe ha c 1 c x = c x + d d 1 c x. For x d 1,wehavex + d d 1 d.thus,c 1 c x > forx d 1.Nex,suppose wihou loss of geeraliy ha x >d 1 ad x d 1,x.The,sicec x < c for x x,adc x + d d 1 c, wecocludehac 1 c x >. Thus, c 1 c is icreasig o [,x d 1.Theobservehac 1 c x = c x + d d 1 c x is decreasig over x d 1,d sice he firs erm o he righ had side is decreasig while he secod oe is icreasig. Also oe ha c 1 c x d 1 whereas c 1 c d <. Thus, here exiss a uique d x d 1,d such ha c 1 c d =. To coclude ha d is he uique maximizer oe ha c 1 c x < forx [d, x ],whilec 1 c x = forx>x,whichalso proves ha c 1 c is cosa o [x,. Now if x d,he followsimmediaelybecausec 1 c is icreasig o [,d ].Ifx >d he o prove, i suffices o show ha c 1 c x c 1 c x This is because c 1 c x c 1 c x by he firs par of he lemma. Moreover, because c 1 c x 1 c 1 c x 1 agai by he firs par of he lemma ad because x 1 <d isufficesoshow31. To his ed, we cosider he followig wo cases: Case i c 1 c; Caseiic 1 <c.icasei,wehavex 1 =. Thus, provig reducesochecigc 1 c x, which follows from Assumpio 1vi. I Case ii, we have c 1 c x c 1 c x 1 = c x + d d 1 c x [ c x 1 + d d 1 c x 1 ] x +d d 1 = c y dy x = d d 1 c x1 +d d 1 x 1 x1 +d d 1 x 1 >d d 1 c d d 1 c =, c y dy c y dy where we use he fac ha <x 1 o coclude ha c 1 x = c x + d 1 d <cfor x<x 1. Proof of Proposiio 1 Give a feasible policy for which here exiss a ierval 1, such ha 1 >, we ca improve he objecive by modifyig 1 o 1, such ha 1 =. This also relaxes he cosrai 1. Thus, wihou

30 64 Queueig Sys 13 73:35 78 loss of opimaliy, we have 1 = forall. Similarly,ifollowsha = c 1 c τ 1 for all. Proof of Proposiio 3 Firs, we verify ha q i is moooe. To ha ed, oe ha if W>λ 1 x 1 + λ x,heq 1 W = x 1,adq W = W x 1,adheresulfollows. Oherwise, i.e., W λ 1 x 1 + λ x,heaecessarycodiioforopimaliyisha h q1 1 h q 3 λ 1 is miimized over q 1,q suchhaq 1 + q = W. Le W >W 1 ad suppose ha q i W <q i W 1 for some i {1, }, hewe mus have q j W >q j W 1 + W W 1 for j i. Noicehaseigq i W = q i W 1 ad q j W = q j W 1 +W W 1 decreases he differece i 3byhesric covexiy of h 1 o, x 1 ad h o,x which follows from sric covexiy of c i o [,α i ]. Thus, q i is moooe. Tha is, for W >W 1, Puig q 1 W = W q W i 34 gives λ q W q W 1, 33 q 1 W q 1 W q W q W 1 W W The combiig 33 ad35 giveshelipschizcoiuiyofq : q W q W 1 W W 1. The Lipschiz coiuiy of q 1 follows similarly. Fially, we oe by he miimaliy of q 1 amog muliple opimal soluios ha q 1 W λ 1 q W λ ad q 1 W λ 1 x 1, from which we coclude ha q W Wλ /λ 1 + λ. Appedix B: Proofs of auxiliary resuls i Sec. 6. Proof of Lemma I suffices o show ha for i = 1, P A i i λ i > ζ β C 1 e C ζ β 1+α1. By Theorem.1 of Csorgo ad Horvah [13], here exiss a sadard Browia moio B i ad a error process ε i for each such ha A i = λ i + λ i B i + ε i for,

31 Queueig Sys 13 73: where for posiive cosas C 1, C P sup T ε > C 1 logλ i T + x Usig his resul, we wrie for sufficiely large ha C 1 e C x. P A i i 1 λi > ζ β = P λi B i B i + ε i ε i ζ > λi P κ B i 1 ζ > 4 β P B i 1 > ζ 4 β 1 α i λ i + P sup + P T sup T ε i > ζ 4 β β ε i > ζ 8 β. Noe by a sraighforward applicaio of Marov s iequaliy ha P B i 1 >x exp { x / } The for i = 1,, A P i λi ζ > i for x>. β { ζ } { exp β 1+α ζ 1 + C 1 exp C 3λ i { ζ + C } 1 exp 3λ 1 + λ β 1+α 1 16 β } for i = 1,, where he las iequaliy follows for sufficiely large because 1 α 1 <β<1 α 1. The leig he resul follows. C 1 = + C 1 ad C = 1 3λ 1 + λ, Appedix C: Proofs of echical resuls i Sec. 6.3 Proof of Lemma 3 Fix T>, ad oe ha PB = 1, which provides he iducio basis. As he iducio hypohesis assume ha P B [,...,B j 1 C3 exp { C 4 α }] 1+α 1 j

32 66 Queueig Sys 13 73:35 78 for j =, 1,..., 1. The i suffices o show ha P B c, B,...,B C3 exp { C 4 α 1+α 1 } P B,...,B. 36 To his ed, oe ha P B c, B,...,B B = P c, B,...,B ; Ŵ δ 3 ε Cosider he firs erm o he righ-had side: B c, P B,...,B, Ŵ δ 3 ε P B,...,B, Ŵ δ B c, + P B,...,B ; Ŵ δ > 3 ε >q Q δ + ε 3 ε, Q + P B,...,B, Ŵ δ 3 ε, Q <q Q δ ε Nex, we cosider each erm o he righ-had side: P B,...,B, Q >q Q δ + ε, Ŵ δ 3 ε P B,...,B, Q δ > ε, Q δ 3 ε P B,...,B, Q Q δ > 3 ε P B,...,B,A δ > 3 ε P B,...,B P A δ > P B δ,...,b C1 exp { C } 36 α 1 ε for sufficiely large,wherehefirsiequaliyfollowssiceq Q, ad 3 ε Q Q W δ + ε 3 ε for sufficiely large.thehirdiequaliyfollowsfromhefacha Q Q A A