Submd o Managmn Scnc manuscrp Srvc Capacy Compon wh Pak Arrvals and Dlay Snsv Cusomrs Hayan Wang Oln Busnss School, Washngon Unvrsy n S. Lous, S. Lous, MO 6330,USA wangha@oln.wusl.du Tava Lnnon Olsn Oln Busnss School, Washngon Unvrsy n S. Lous, S. Lous, MO 6330,USA olsn@wusl.du W sudy capacy dcsons n a srvc nvronmn whr h arrval ras ar hghly sasonal (.g., lunch m rushs) and cusomrs ar m snsv, so may dpar whou rcvng srvc f h wang m s oo long. W bgn by sudyng a monopols s capacy dcson, whr h ky rad-off s bwn h cos of xra capacy for low dmand prods and h loss of rvnu for hgh dmand prods. W hn sudy a duopoly modl, whr los dmand for on frm may bcom ncrasd dmand for h compor. In boh modls w us a flud modl for h analyss, whch allows us boh o provd xplc nsghs no h rad-offs whn sng capacy and o solv for h Nash qulbrum (whn xss) n h duopoly. Th canoncal nvronmn w hav n mnd for our modlng s a foodcour, bu any srvc nvronmn whr h pak arrval ra wll lkly xcd avalabl capacy s smlarly appropra.. Inroducon and Rlad Lraur In many srvc nvronmns arrval ras ar hghly sasonal (.g., lunch m rushs) bu capacy canno b fnly und o h arrval parn du o h nd o nsall fxd capacy and h nd for human capacy o b hrd for som mnmum shf lngh. Ths ofn lads o an undr-supply of capacy a pak arrval ms, whch n urn lads o long dlays and sgnfcan cusomr balkng, parcularly f h srvc s no crcal o h cusomr or f hr ar ohr compng provdrs of h srvc. Bcaus w consdr srvc nvronmns, hr s no possbly of nvnoryng for hs pak prod. Our prmary nrs s h sudy of capacy compon n h prsnc of xcss cusomrs causd by non-saonary arrval parns. Dos compon mak h ovrflow suaon br or wors for h cusomr? How should frms s capacy n h prsnc of compon? In ordr o answr hs qusons w frs provd a smpl modl for capacy sng for a sngl monopols srvng mpan cusomrs who may balk. Alhough lkly oo crud a modl o us o dsgn a srvc sysm, dos provd som nrsng nsghs no h rlvan rad-offs ha nd o b consdrd. Ths modl s hn xndd o h duopoly sng. W consdr a non-saonary arrval parn ha rss o a pak and hn dclns. Ths yp
Wang and Olsn: Srvc Capacy Compon wh Pak Arrvals Arcl submd o Managmn Scnc; manuscrp no. of arrval parn s qu common n a vary of srvc nvronmns. For xampl, around rush hour n h mornng, a oll booh may fac such a parn of raffc (s,.g., Ed, 954). Smlar parns occur a banks (say, durng h lunch brak prod) or pos offcs. In hs lar cas, Olvr and Samul (96) clam ha h rs-pak-fall-off parn of h avrag arrval ras for lrs whn a fw hours s h man conrbuor o procssng dlays n a pos offc. Th spcfc arrval parn w sudy s on whr h cusomr arrval ra grows lnarly and hn dsspas lnarly durng a cran prod. As sad n May and Kllr (967), basd on raffc suds, Thr s vdnc o suppor hs assumpon of rangular- or rapzodal-shapd dmand parns ; s nod ha h rapzod shap may com from blockng. Anohr ypcal nsanc for our arrval parn s food cour, whch s h canoncal applcaon w us for our modls. Durng h lunch prod, h cusomr arrval ra wll ncras o a pak, somwhr around h noon hour, and hn dcras (s Marnch, 00, for a dmand profl for a ypcal fas-food rsauran durng h lunch prod). Thr s a sram of lraur ha suds prformanc analyss of non-saonary arrvals usng saonary modls (s,.g., Grn and Kolsar, 997, and rfrncs hr-n). W do no ak hs rou. Insad, w xplcly modl h non-saonary parn of arrvals. Our focus s on xplc approxmaons, gnraon of nsghs, and a modl ha s usful for duopoly compon. Thrfor, w us a rlavly crud flud modl whr all flows ar drmnsc, as ar h cusomr balkng and swchng dcsons. Cusomrs wll lav f hr prdcd dlay xcds hr olranc for wa (bcaus h flud sysm s drmnsc, dlays may b prfcly prdcd). Alhough hr ar mor accura modls of ransnc han hs (.g., Nwll, 968a,b, Koopman, 97, Eck al. 993a,b, Wh, 006), hy wll no yld h xplc xprssons ha w sk; vn undr our smpl assumpons h xprssons bcom surprsngly complx. Th us of flud modls for ovrloadd sysms has bn movad by Nwll (97, Chapr ), Hall (99, Chapr 6), Klnrock (976, Scon.7), Mandlbaum and Massy (995), and Wh (004, 005, 006). Th basc da s ha as h sysm scal bcoms larg h funconal srong law of larg numbrs wll apply and sochasc procsss wll sar o look drmnsc. Hr, w provd no formal lm horms, nsad w bgn wh h assumpon of a flud modl and procd wh h analyss from hr. No ha h sandardzaon of srvc procdurs n fas food rsaurans may mak h assumpon of drmnsc procssng ms parcularly applcabl for our food cour xampl. Thr has bn sgnfcan work on saffng modls for non-saonary arrval parns (s Fldman al., 008, Grn, Kolsar, and Wh, 007, and Wh, 007 for rcn rvws). Much
Wang and Olsn: Srvc Capacy Compon wh Pak Arrvals Arcl submd o Managmn Scnc; manuscrp no. 3 of hs work spcfcally consdrs call-cnr saffng. Du o hs applcaon, mos of hs work assums ha saffng can also b adusd (usually n nrvals,.g., vry half hour) as h dmand rqurmns chang, prhaps wh som rqurmns on shfs (.g., Alason, Eplman and Hndrson, 004). Howvr, somms h xpns of changng capacy frqunly may b vry hgh (s,.g., Hol al., 960, for h xpnss assocad wh adusng work forc and ohr aspcs of producon capacy). Whn capacy s fxd durng h srvc prod, h srvc provdr mus rad off h ponal rvnu los durng h pak dmand prod wh h cos of dl capacy durng h low dmand prod. W consdr fxd capacy hr. In h food cour xampl, h lunch m prod s rlavly shor (probably wo o four hours). I ofn dos no mak sns o chang h capacy (say, saff numbrs) durng h shf bcaus h m wndow may b shorr han a work shf. Wh (999) consdrs h suaon whr m-varyng capacy s hr dffcul or vry cosly for a srvc sysm wh m-varyng cusomr arrval ras and hnc a fxd capacy mus b usd o m all dmand n a day. H proposs classfyng cusomrs no dffrn classs accordng o hr rspons-m rqurmns and prorzng accordngly. Harrson and Zv (005) sudy opmal saffng n a larg call cnr wh h goal of balancng h radoff bwn prsonnl coss and abandonmn pnals. Lk us, hy assum ha capacy s fxd ovr h prod of nrs. Bsds h m-varyng characrsc of h cusomr arrval ras, hy ncorpora h randomnss of arrval ra a any m pon by usng a sochasc flud modl. Chang al. (004) us a sochasc flud modl (flows ar drmnsc bu h arrval ra s sochasc) o sudy opmal schdulng of wo classs whr on of h classs has prods of ovrload. Broadr modls of h nracon of capacy and dlays n srvc nvronmns nclud Ig (994), Corsn and Suhlmann (998), and Pullman and Moor (999). Our sngl frm modl gnras nsghs no how capacy should dpnd on basc sysm prmvs and s also usd as a buldng block for our duopoly modl. I provds a back-ofh-nvlop soluon ha may b usd f n fac capacy chocs ar lmd and dscr (.g., hr on or wo srvrs?). In only modlng prm-m saffng, w ar, n ffc, assumng ha hr s som basln lvl of capacy ha s prsn for non-prm m (whch s no modld). Ths basln capacy (say a sngl srvr) wll also modra h undrlyng varably durng h prm m. Howvr, for a mor fn und answr o how capacy should b s w would rcommnd smulaon, whr a sarch ovr a sngl paramr (.., capacy) s vry fasbl and many mor subls of cusomr bhavor (.g., sochasc arrvals, abandonmn, c.) may b ncludd. Ths s h approach akn by Marnch (00), who uss smulaon o sudy whn o
Wang and Olsn: Srvc Capacy Compon wh Pak Arrvals 4 Arcl submd o Managmn Scnc; manuscrp no. add addonal capacy n a fas food rsauran n ordr o dal wh h pak dmand. Th smulaon rsuls dmonsra ha schdulng addonal srvrs a ll arlr can hav dramac mpacs on cusomr wang ms for an xndd prod n a non-saonary quung sysm. W fnd ha hr ar wo ky facors n drmnng h srvr s opmal capacy lvl, namly, cusomr panc and h margnal cos o rvnu rao. Unsurprsngly, whn facng mpan cusomrs and rlavly hgh cos o rvnu rao, w fnd may b opmal for h srvr o forgo som cusomrs durng h pak dmand prod. W quanfy h rgons whr hs s h cas. W show ha whl h opmal capacy s monooncally dcrasng n h cos o rvnu rao, s no always monoon n cusomr panc. Insad, s unmodal whr h hghs opmal capacy s a modra lvls of panc. Th nuon bhnd hs rsul s gvn n Scon 3. In our duopoly sng, cusomrs ar assumd o hav an nal srvr prfrnc (.g., pzza ovr Chns food) bu may swch from hr orgnal choc o hr scond choc f h wang m a hr favor srvr s oo long. Thrfor h wo srvrs comp wh ach ohr on m n ordr o arac cusomrs by sragcally choosng an appropra capacy lvl. W do no consdr prcng compon. Insad, w assum h frms ar prc-akrs or h prcs ar s by som cnral organzaon (.g., h franchs) whou consdraon of local compon. Our modl s rlad o h lraur ha suds h qulbrum bhavor of cusomrs and srvrs n quung sysms. Followng Naor s (969) smnal work, hr has bn rapdly growng rsarch n hs fld. Spcfcally, hs sram of rsarch suds h mpac of congson on h cusomrs and srvc provdr s dcsons by xplcly modlng cusomrs onng bhavor and hr sragc nracon wh ach ohr. Hassn and Havv (003) provd an xclln rvw on hs rsarch sram. Usng h rms n Hassn and Havv (003), w sudy srvr(s) capacy dcson(s) wh obsrvabl quu(s),.., cusomrs mak hr dcsons afr hy obsrv h quu lngh(s). Th lraur on srvr compon undr obsrvabl quus s sgnfcanly lss han ha wh non-obsrvabl quus. In par, hs s bcaus an qulbrum ofn fals o xs n hs cas (s p. 45 of Hassn and Havv for nuon). L and L (994) provd som rsrcons whr qulbra do xs n sysms wh obsrvabl quus. Thy sudy prc compon bwn wo srvrs wh dffrnad xognous procssng ras. Cusomrs arrv accordng o a Posson procss and hav homognous valuaons on h wo srvcs. No balkng s allowd and cusomrs may ocky from h rar of on quu o h ohr. Undr h condons of dncal cusomrs and obsrvabl quus, hr dos no xs a pur sragy Nash qulbrum. Thrfor, hy
Wang and Olsn: Srvc Capacy Compon wh Pak Arrvals Arcl submd o Managmn Scnc; manuscrp no. 5 analyz a rsrcd modl and show ha h frm wh a hghr srvc ra always noys a prc prmum. Kala, Kamn, and Rubnovch (99) sudy a wo-srvr gam horc modl wh a sngl quu whr cusomrs wll on h frs srvr o bcom mpy (rgardlss of srvc ra). Chrs and Av-Izhak (00) xnd hs work o allow balkng and mor gnral capacy cos funcons. In som sns, such a modl may b hough of as a sysm ha allows unlmd and coslss ockyng, whr a prson s plac n ln s prsrvd vn whn swchng quus. Dmand allocaon o wo compng srvrs s consdrd n Glbr and Wng (998) and Cachon and Zhang (007). Chapr 8 of Hassn and Havv (003) survys rsarch on srvc ra dcsons, prdomnanly for unobsrvabl quung sysms wh saonary arrvals. Ohr rcn work on capacy compon (wh unobsrvabl quus) ncluds Chn and Wan (005), Allon and Fdrgrun (007, 008), Allon and Gurvch (007), Chay and Hopp (007), and Dobson and Savrulak (007). W ar awar of no lraur ha consdrs capacy compon n h prsnc of non-saonary arrvals. W frs analyz cusomrs swchng and balkng bhavor and h mpac on srvrs profs. Bcaus of h non-saonary of h cusomr arrval procss, h cusomrs swchng and balkng parn s qu complcad. Ths poss a sgnfcan challng n characrzng h Nash qulbrum n h srvrs capacy compon gam bcaus s almos mpossbl o xplcly xprss h srvrs prof funcons for a gvn par of capacy chocs n a gnral sng. Thrfor, w dvlop ss of suffcn condons ha susan h xsnc of a unqu Nash qulbrum n h symmrc gam. Inuvly spakng, as long as cusomrs hav a srong prfrnc for hr orgnal choc, a symmrc Nash qulbrum xss a whch ach srvr works as f h wr a monopols, ha s, no swchng happns n h qulbrum. Howvr, whn cusomrs do no hav srong prfrnc (manng ha cusomrs hav smlar valuaons on wo srvcs), hr ofn dos no xs a pur-sragy Nash qulbrum. Th raonal s smlar o ha for h modl of prc compon ovr dncal cusomrs wh obsrvabl quus. W sudy asymmrc compon numrcally and fnd suaons wh no pur-sragy Nash qulbra, suaons wh mulpl pur-sragy Nash qulbra, and suaons wh a unqu pur-sragy Nash qulbrum; nuon bhnd hs rsuls s gvn. As oulnd abov, our conrbuons ar fourfold. Frs, w provd a back-of-h-nvlop approach for capacy sng undr sasonal dmand. Scond, w us ha approxmaon o provd nsghs no h rad-offs o b consdrd whn sng capacy. Thrd, w xnd h (rlavly lmd) lraur on compon wh obsrvabl quus o a modl wh capacy
Wang and Olsn: Srvc Capacy Compon wh Pak Arrvals 6 Arcl submd o Managmn Scnc; manuscrp no. compon. Fnally, w provd wha w blv o b h frs rsuls on capacy compon wh non-saonary arrvals. Th rs of h papr s organzd as follows. In Scon, w ouln our assumpons rgardng cusomrs, srvrs, coss and rvnus and provd som nal rsuls. Scon 3 consdrs h monopols s capacy dcson, provdng boh som gudanc on capacy sng and a buldng block for h duopoly compon n Scon 4. Scon 5 concluds h papr. All proofs may b found n h onln appndx.. Flud Modl of Quung Bhavor In hs scon w ouln our spcfc assumpons on h arrval, srvc, and balkng procsss and xplan hr mplcaons for cusomr quung. Ths modl s usd boh n h monopoly and duopoly scons. To capur h cos of dl capacy, w assum ha h capacy cos ha a srvc provdr ncurs s proporonal o boh h full capacy and h oal m h capacy s usd for. L dno h lngh of m prod durng whch arrvals ar sgnfcan (prm m). Assum h cusomr arrval ra ncrass lnarly n m unl a pak m pon, 0, and h arrval ra a hs m s λ 0. I hn dcrass lnarly n m o zro whn h prm m prod nds. Spcfcally, h cusomr arrval ra Λ( ) has h followng form: λ f 0 0, Λ() = λ 0 ( 0 )λ 0 /( 0 ) f 0 <, () 0 ohrws. Fgur shows h arrval ra parn. Fgur Cusomr Arrval Ras
Wang and Olsn: Srvc Capacy Compon wh Pak Arrvals Arcl submd o Managmn Scnc; manuscrp no. 7 Cusomrs ar srvd connuously a ra µ (whr µ wll b a dcson varabl). For a cran capacy lvl µ, dfn h pak prod as h m nrval durng whch h cusomr arrval ra s hghr han h capacy lvl. In hs flud sysm, h quu bulds up only durng h pak prod. Th pak prod sars a m µ/λ and nds a m µ( 0 )/(λ 0 ). If cusomrs ar no allowd o balk, hn h quu lngh rachs s maxmum a h nd of h pak prod (.., a m µ( 0 )/(λ 0 )). Dfn Q N max(µ) as h maxmum quu lngh n h cas of no balkng (h suprscrp N rprsns No balkng) whn h capacy lvl s s a µ. Thn, µ( 0 )/(λ 0 ) Q N max (µ) = (Λ() µ)d= (λ 0 µ). λ 0 µ/λ Afr m pon µ( 0 )/(λ 0 ), h arrval ra s lowr han h capacy and h quu dmnshs (s Fgur for an llusraon). No ha, as s wll known n h quung lraur (.g., Nwll, 97), h longs dlays occur sgnfcanly afr h pak of h arrval ra. Fgur Flud modl (no balkng opon) Th prod followng h pak prod unl s calld h pos pak prod; durng hs m h quu dmnshs. Howvr, h quu may no b fully clard by m. W l T dno h m h sysm s clard of prm m cusomrs. Bcaus of h cusomr arrval procss, T. Whn T >, w say h srvr works ovrm. Dfn µ T o b h mnmum lvl of µ for whch hr s no ovrm, undr h assumpon of no cusomr balkng. Th frs fram n Fgur 3 llusras ha a µ T (wh no balkng opon),
Wang and Olsn: Srvc Capacy Compon wh Pak Arrvals 8 Arcl submd o Managmn Scnc; manuscrp no. h quu s clard xacly a m. In h scond fram n Fgur 3, h capacy µ s lowr han µ T and hn quu s no clard a and h srvr works ovrm. In h hrd fram n Fgur 3, h capacy s hghr han µ T and hn h quu s clard bfor and h srvr works wh som xcss capacy durng h pos pak prod. Fgur 3 Quu clarng possbls (no balkng opon) Undr h assumpon of no balkng, h maxmum quu lngh s Q N max (µ). Furhr, durng h pos pak prod, h quu lngh can b rducd by µ ( 0 )/(λ 0 ). Thus, h quu s clard xacly a m whn Q N max(µ) = µ ( 0 )/(λ 0 ). Th followng lmma provds h xplc xprsson for µ T. Lmma. L µ T = λ ( 0 / ). Undr h assumpon of no cusomr balkng, f µ µ T, all cusomrs ar srvd bfor or a. Ohrws, hr wll b cusomrs lf a m. No ha µ T s lnar n λ (as would b xpcd) and s ncrasng n h rao of h pak arrval m o h nd of h horzon ( 0 / ). Unl hs pon w hav assumd ha hr s no balkng. Suppos, nsad, ha cusomrs balk whn hr xpcd (or n hs cas, acual) dlay s grar han w. Thn, h longs quu lngh ha an arrvng cusomr would on quals Q(µ) = wµ. Whn h quu lngh rachs Q(µ), balkng occurs, and h ffcv cusomr arrval ra rducs o h srvc ra µ so ha h maxmum wang m rmans a w unl h pak prod nds. A h nd of h pak prod h arrval ra s lowr han µ and h quung lngh dcrass. No balkng occurs ousd of h pak prod. Fgur 4 llusras a suaon wh balkng. I s asy o s ha Q N max(µ) s dcrasng n µ (for µ λ 0 ) whl Q(µ) s ncrasng n µ. Thus hr xss a unqu valu µ N, whch s h mnmum lvl of µ whr no balkng occurs. Ths lvl s whr Q N max(µ) = Q(µ). Furhr, f balkng occurs, hn h numbr of cusomrs los mus b h dffrnc bwn h quu lngh a h nd of h pak prod whou balkng, Q N max(µ),
Wang and Olsn: Srvc Capacy Compon wh Pak Arrvals Arcl submd o Managmn Scnc; manuscrp no. 9 Fgur 4 Flud Modl wh Balkng and wha h quu lngh acually s a h nd of h pak prod, Q(µ) (snc no balkng occurs afr h pak prod). Ths s summarzd n h followng lmma, whch also gvs an xplc xprsson for µ N, by solvng Q N max(µ) = Q(µ). Lmma. L µ N = λ 0 ( + w w ) + w. If µ µ N, hn no cusomr wll balk. Ohrws, som cusomrs balk and h numbr of cusomrs los s Q N max(µ) Q(µ). As mgh b xpcd, µ N s lnar n λ 0. I s asy o show ha µ N s dcrasng n w/. Tha s, as cusomrs ar wllng o wang longr rlav o h m horzon (hr bcaus hy hav hghr valuaon on h srvc or bcaus hy hav a lowr wang cos ra), h srvr nds a lowr capacy lvl o prvn balkng. W hav dfnd wo crcal capacy lvls. Namly, µ N, whch nvolvs cusomrs balkng bhavor bu no h srvr s workng m, and µ T whch drmns h srvr s workng m whou consdrng cusomr balkng. In fac, h rlaonshp bwn hs wo spcal capacy lvls s drmnd by cusomrs longs wang m w, as ndcad n h followng lmma, whch follows by smpl algbra. Lmma 3. If w > ( 0 ) ( 0 / )/( 0 ), hn µ N < µ T. Ohrws, µ N µ T
Wang and Olsn: Srvc Capacy Compon wh Pak Arrvals 0 Arcl submd o Managmn Scnc; manuscrp no. W rfr o h rgon w > ( 0 ) ( 0 / )/( 0 ) as h rgon whr cusomrs ar rlavly pan and o h rgon w ( 0 ) ( 0 / )/( 0 ) as h rgon whr cusomrs ar mpan. Impld by h abov lmma, n hs lar cas, n ordr o accommoda all cusomrs, h srvr mus hav som xcss capacy durng h pos pak prod and h backlog wll clar bfor m (h hrd fram n Fgur 3). Th capacy a whch all cusomrs ar clard a m whn hr s balkng s gvn n h nx lmma. Lmma 4. Suppos w ( 0 ) ( 0 / )/( 0 ) so ha cusomrs ar mpan. L µ C = (λ 0 w)/( 0 ). If µ µ C, hn (undr cusomr balkng) h quu wll mpy of all cusomrs a or bfor m. Ohrws, som cusomrs balk bu h quu sll has cusomrs a m. Clarly, µ C (C sands for clarng ) s lnar n boh λ 0 and w/( 0 ). In our flud modl, dos no mak sns for cusomrs o ocky or rng bcaus h srvrs procssng m s assumd o b consan so ha h n valu dffrnc of h wo srvcs dos no chang wh m. 3. A Monopols s Capacy Dcson W bgn by sudyng a monopols s capacy dcson. As n h prvous scon, cusomrs wll balk f hr dlay wll b abov w. W assum h frm arns r pr un of cusomr hy srv and ncurs a srvc cos c pr un of capacy pr un of m. Th radoff for h srvr s clar: h hghr h capacy, h hghr h rvnu bu wh hghr srvc cos. No ha, bcaus λ 0 s h largs cusomr arrval ra, suffcs o only consdr µ λ 0. L n dno h numbr of cusomrs srvd (whch mplcly dpnds on µ). Thn, n 0 λ()d = λ 0. L V (µ) dno h frms s prof ovr h prod whn h capacy lvl s µ. Thn, Dfn v a (µ) = (r c) + c 0 λ 0 V (µ) = rn cµt. () ( µ λ ) 0(r c)( + w) + λ 0(r c) ( + w). (r c) + c 0 [(r c) + c 0 ] Ths s h xprsson for V (µ) corrspondng o h scnaro whn hr s boh balkng and ovrm; s maxmzd a µ a = λ 0(r c)( + w) (r c) + c 0.
Wang and Olsn: Srvc Capacy Compon wh Pak Arrvals Arcl submd o Managmn Scnc; manuscrp no. Dfn v b (µ) = r ( µ λ 0 ( c λ 0 r + w ) ) + λ 0( c + w r ) r. Ths s h xprsson for V (µ) corrspondng o h scnaro whn hr s balkng bu no ovrm; s maxmzd a ( µ b = λ 0 c r + w ). For h cas n whch cusomrs ar rlavly mpan, h srvr s prof funcon can b xprssd as follows. Lmma 5. If w ( 0 )( 0 )/( 0 ), hn v a (µ), 0 µ µ C V (µ) = v b (µ), µ C < µ < µ ( N rλ0 cµ ), µ µ N. Furhr, V (µ) s connuous and concav n µ. Bcaus of h concavy of V (µ), h opmal capacy lvl can only b on of µ a, µ b, µ C, and µ N. Whn cusomrs ar pan, µ N < µ T and µ C > µ N. Hnc, h scnaro whr hr s balkng bu no ovrm nvr happns n hs cas. In hs cas, h srvr s prof funcon can b xprssd as follows. Lmma 6. If w > ( 0 )( 0 ) 0, hn V (µ) = v a (µ), λ 0 (r c) cµ, λ Furhr, V (µ) s connuous and concav n µ. 0 µ µ N ( rλ 0 cµ), µ µ T. µn < µ < µ T Th frs scnaro for V (µ) s agan on whr boh balkng and ovrm occur and µ a s agan h mnmzr n hs rgon whou consdrng h rang consran. In h scond scnaro hr s no balkng bu h srvr works ovrm and n h hrd hr s no balkng or ovrm. I s sraghforward o chck ha V (µ) s dcrasng for µ µ N and hnc h canddas for h opmal capacy lvl n hs cas ar µ a and µ N only. Now w ar rady o characrz h srvr s opmal capacy lvl, whch w wr as µ. Proposon. () Suppos w ( 0 )( 0 )/( 0 ). ( ). If 0 c ( + 0 ) 0 + 0 r 0 and + c r [ r c ( + w )] w + w ; V (µ ) = λ 0 w ( 0 )( 0 ) 0, hn µ = µ N wh
Wang and Olsn: Srvc Capacy Compon wh Pak Arrvals Arcl submd o Managmn Scnc; manuscrp no. { ( ( r. If w mn c )( 0 ) + 0, + c 3. If ( + 0 ) 0 + 0 0 hn µ = µ C wh V (µ ) = λ 0 w 4. If ( + 0 ) 0 + 0 ( 0 ) V (µ ) = λ 0(r c) ( + w) [(r c) +c 0 ]. ) r }, hn µ = µ b wh V (µ ) = λ 0 ( c + w r ) r ; { c and ( r c)( 0 ) ( r r + 0 w mn c)( 0 ) ( r c) +(+ r c, ( 0 )( 0 ) ) 0 0 }, [ ] r( + w) c 0 r w ( 0 ; ) 0 +3 0 c r and () In h cas ha w > ( 0 )( 0 )/( 0 ),. If c r [ c r ( 0 sasfs ( 0 )( c r ( 0 ) 0 ) 0 < ( r c)( 0 ) w ( 0 )( 0 ) ( r c) +(+ r c) 0 0 [ c r ( 0 c r ( 0 ) ) ( 0 ) c r +]/, hn µ = µ a wh V (µ ) = λ 0(r c). Ohrws, µ = µ N wh V (µ ) = λ 0 ) ( 0 ) c r +]/ and ( 0)( ( + w) [(r c) +c 0 ; ] [(r c) c 0 ( + w w + w ) ]., hn µ = µ a wh 0 ) 0 < w As an llusraon, Fgur 5 shows h opmal capacy dcsons n h cas ha = 0,.., h cusomr arrval ra ncrass (bfor h pak m pon) and dcras (afr h pak m pon) a h sam ra. Accordng o our dfnon, n hs cas, whn w 0 >, cusomrs ar mpan, ohrws, cusomrs ar pan. W can s ha h spac s dvdd no fv rgons. Th dvson of h rgons s ndpndn of h arrval slop λ, alhough h opmal capacy lvl n ach rgon s proporonal o λ. For rgons () and (), h srvr ss a hgh nough capacy o prvn cusomr balkng. No ha µ N s a funcon of w (as shown xplcly n h fgur) and n rgon (), whr cusomrs ar pan, hr wll sll b cusomrs wang a m ; whras, n rgon (), whr cusomrs ar mpan µ N s hgh nough so ha all cusomrs ar clard a. For rgon (v), h quu s clard xacly a m so w rfr o as on m. From Fgur 5, w can s ha wh rlavly mpan cusomrs only whn h rao of cos ra ovr rvnu ra s no hgh and cusomrs ar no oo mpan, s opmal for h srvr o s a hgh nough capacy lvl o srv all cusomrs. Ohrws, s opmal o forgo som cusomrs. To mak hs mor concr, suppos ha h m o rach pak arrvals 0 = hr and h prm m prod = hr, hn h hrshold bwn panc and mpanc occurs around 8 mnus. Thus, for a foodcour h lowr poron (mpanc) sms mor applcabl. In hs cas, so long as cos o rvnu raos ar no vry clos o (abov 0.8 whn panc s s o 0 mns), suffcn capacy should b plannd so ha h quu s clard by h nd of h pak prod. Howvr, only f cos o rvnu raos ar blow around 0.4 for 0 mnu panc (whch sm lk hgh margns for a foodcour) should suffcn capacy b plannd o srv all cusomrs a h pak.
Wang and Olsn: Srvc Capacy Compon wh Pak Arrvals Arcl submd o Managmn Scnc; manuscrp no. 3 0.8 0.6 () µ ( c r, w) = µn ( w) no balkng, ovrm w 00.4 (v) µ ( c r, w) = µa ( c r, w) balkng, ovrm 0. () µ ( c r, w) = µn ( w) no balkng, no ovrm (v) µ ( c r, w) = µc ( w) balkng, on m () µ ( c r, w) = µb ( c r, w) balkng, no ovrm 0 0 0. 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 c r Fgur 5 Opmal Capacy Lvl In h followng, w ar nrsd n how h opmal capacy lvl changs as h valus of h wo man facors vary. I s sraghforward o show ha h opmal capacy lvl µ s connuous n c/r and w. Frs, w fx cusomrs panc lvl and sudy how h opmal capacy lvl changs wh h cos/rvnu rao. No ha, for a fxd panc lvl w, h mnmum capacy lvl ha prvns balkng µ N dos no chang wh h cos rvnu rao c. Furhr, h capacy r lvl a whch som cusomrs balk and h quu s clard xacly a m,.., µ C, dos no chang wh c/r hr. I s sraghforward o show ha µ a and µ b ar dcrasng n c/r for a fxd w. Thrfor, for a fxd valu of w, h opmal capacy lvl µ s non-ncrasng n c/r. Fgur 6 shows how h opmal capacy lvl changs wh h rao c/r for dffrn w valus n h cas ha = 0. For a fxd valu of c/r, h opmal capacy lvl µ s no ncssarly monoonc n w (alhough s connuous n w). Fgur 7 plos h opmal capacy lvl as funcons of w/ 0 for dffrn valus of c/r for h cas ha = 0. No ha µ N s dcrasng n w whl µ a, µ b and µ C ar ncrasng n w. So whn srvng h whol mark s opmal, h opmal capacy lvl s dcrasng n w bcaus as cusomrs accpabl wang m ncrass, h srvr nds a lowr capacy lvl o accommoda all h cusomrs. Bu whn parally srvng h mark s opmal, h opmal capacy lvl s ncrasng n w. Tha s, as cusomrs bcom mor pan, bcoms cos ffcv for h srvr o srv a largr mark. No ha h curvs ar no concav.
Wang and Olsn: Srvc Capacy Compon wh Pak Arrvals 4 Arcl submd o Managmn Scnc; manuscrp no..4. w 0 = 0. w 0 = w 0 = 0.5 Op. Capacy 0.8 0.6 0.4 0. 0 0 0. 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 c r Fgur 6 Opmal Capacy Lvl as a funcon of cos o rvnu rao c/r.6.4 c r = 0.3. Op. Capacy 0.8 c r = 0.5 c r = 0.7 0.6 0.4 c r = 0.9 0. 0 0. 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 w/ 0( 0 = 4 ) Fgur 7 Opmal Capacy Lvl as a funcon of panc w In summary, w hav provdd a back-of-h-nvlop calculaon for capacy sng n a monopoly. For paramr rangs ha w hypohsz ar rasonabl for a foodcour (and could b asly chckd for a ral sng, parcularly h margnal cos and rvnu rao) w rcommnd sng suffcn capacy o clar h quu by h nd of h pak prod bu no suffcn o prvn all balkng. W fnd ha whl h opmal capacy s monooncally dcrasng n h cos o rvnu rao no monoon n h cusomr panc; n fac, s unmodal. Afr a cran
Wang and Olsn: Srvc Capacy Compon wh Pak Arrvals Arcl submd o Managmn Scnc; manuscrp no. 5 pon capacy dcrass n panc bcaus cusomrs wll say no mar wha, bu bfor ha pon capacy s acually ncrasng n cusomr panc, n ffc bcaus hs ncrasng panc cras ncrasd valu o h frm. 4. Duopoly In hs scon, w sudy a duopoly capacy gam whr m-snsv cusomrs may swch from on srvr o h ohr. Th qusons w sk o answr ar h followng. How dos compon affc h cusomrs xprncs? How dos compon affc h srvrs profs? In a symmrc suaon can an asymmrc qulbrum form? In an asymmrc suaon, whr a largr group of cusomrs prfrs on srvr, how dos ha affc qulbrum dlays? Wha f h cos srucurs ar asymmrc? In ordr o answr hs qusons w agan assum a flud modl of h sysm. W also mak h clans (and smpls) s of assumpons ha w can n ordr o sola h spcfc compv ffcs ha w ar ryng o sudy. W assum ha all cusomrs hav h sam wang cos ra u (pr un of m) and cusomrs dffrn as prfrncs ar rflcd by hr valuaons on h srvcs. W assum ha hr ar wo classs of cusomrs, o corrspond o h wo srvrs, whr cusomrs n h sam class hav h sam valuaons on h srvcs. L v k ( =,;k =,) dno class s valuaon on srvc k. W assum v > v and v < v. Tha s, class cusomrs valu srvc hghr han srvc ( ). Cusomrs rcv srvc only whn h n valu of h srvc s nonngav, ohrws, hy balk. Furhr, w assum ha h cusomrs hav ndpndn prfrncs and wh probably p (0 p ), a cusomr s a class cusomr. (Corrspondngly, on un of cusomr blongs o class wh probably p.) Hnc, h cusomr arrval ras o h wo srvrs ar pλ(),( p)λ(), rspcvly. Cusomrs as prfrncs ar xognous and known o boh srvrs. On h srvrs sd, w assum ha srvr ( =,) rcvs rvnu r from ach un of cusomr srvs. To avod rval cass, w assum ha r > v. Srvr ncurs a srvc cos c pr un of capacy pr un m. Dno h srvc ras or capacy lvls of h wo srvr as µ,µ, rspcvly. Thy ar facng h problm of sng h rgh capacy lvls o srv h non-saonary mark wh cusomr prfrncs n ordr o achv h hghs prof for a cran prm m prod. Alhough w mploy a flud modl o undrsand h cusomrs bhavor, w can magn ha whn an ndvdual nw cusomr arrvs, h obsrvs h quu lnghs of boh srvrs and compars h wang m and hn dcds whch srvr o choos or o lav whou rcvng
Wang and Olsn: Srvc Capacy Compon wh Pak Arrvals 6 Arcl submd o Managmn Scnc; manuscrp no. srvc from hr srvr (.., balkng). Cusomrs swch from hr favor srvr o h nfror srvr only whn h wang m a h nfror srvr s shorr by mor han hr dsas for ha srvr (n a mannr o b mad prcs blow). L w = (v r )/u, =, b h longs a yp cusomr s wllng o wa a hs prfrrd srvr and w = (v r )/u, ( ), b h longs a yp cusomr s wllng o wa a hs nfror srvr. Dfn δ = w w. As w hav sn n h monopoly modl, w ndcas cusomrs panc for hr favor srvr. Th facor δ rflcs cusomrs as prfrncs, as ach cusomr s wllng o wa δ mor uns of m for hs favor srvr. Largr valus of δ man cusomrs hav srongr prfrncs for hr favor srvr. 4.. Symmrc Compon W consdr a symmrc scnaro n hs subscon. Tha s, w assum ha h wo srvrs charg h sam prc, dnod by r, and hy ncur h sam capacy cos ra, dnod by c. In addon, w assum ha p = and v = v,v = v,v > v. L w = (r v )/u and δ = (v v )/u. Thn ach cusomr s wllng o wa a mos w uns of m a hs prfrrd srvr and w δ uns of m a hs nfror srvr. In h absnc of compon, h wo srvrs would s hr capacy lvls accordng o Proposon. (Rcall ha h dvson of h rgons s ndpndn of h arrval slop λ.) Bcaus h arrval slop o ach srvr bfor m 0 s half of h arrval slop n h monopoly cas, h opmal capacy lvl, dnod as µ, s n fac half of h opmal capacy lvl ha s s by a monopols. Dfn µ a = µa, µ b = µb, µ C = µc, and µ N = µn. Thus, undr h assumpon of no cusomr swchng, µ a, µ b, µ C, and µ N ar h possbl opmal capacy lvls for ach srvr. Rcall ha µ µ N. In h followng, w procd wh h analyss of cusomrs swchng and balkng bhavor for a cran par of capacy lvls s by h wo srvrs. W hn nvsga h wo srvrs qulbrum bhavor whn makng hr capacy dcsons. In h duopoly cas, by balkng w man ha cusomrs lav h sysm whou rcvng srvc from hr srvr (no us h prfrrd srvr, as would b mor sandard). W wll rfr o h suaon whr cusomrs lav hr prfrrd srvr bu on h ohr srvr as swchng As n Scon for h sngl srvr cas, cusomr balkng occurs whn h xpcd dlay a h prfrrd srvr rachs w, wh h addd rqurmn ha h xpcd dlay a h nfror srvr mus b no lss han w δ. A such a pon, h ffcv arrval ra of h corrspondng cusomr class rducs o h srvc ra of h prfrrd srvr. Cusomr swchng occurs whn h dlay dffrnc a h wo srvrs rachs δ and h dlay a h nfror srvr (of h swchng cusomr class) s no grar han w δ. As w wll s n h followng analyss, cusomr balkng
Wang and Olsn: Srvc Capacy Compon wh Pak Arrvals Arcl submd o Managmn Scnc; manuscrp no. 7 and swchng may occur smulanously. Tha s, for a cran class of cusomrs, som of hm on h prfrrd srvr, som of hm swch o h nfror srvr, and h rs lav whou rcvng srvc. Dno h ffcv arrval ras o h wo srvrs a m as λ () and λ (). Clarly, whn boh srvrs s a rlavly hgh capacy lvl, for xampl, µ = µ = λ 0, hr s no cusomr balkng or swchng and λ () = λ () = Λ() for [0, ]. To analyz how cusomr swchng and cusomr balkng affc h ffcv arrval ras o h wo srvrs, w assum ha µ µ λ 0 n h followng analyss on cusomr swchng and cusomr balkng bhavor. Th analyss for ohr cass of capacy pars can b mpld. No ha bcaus h arrval ras of h wo classs of cusomrs ar h sam a any m pon, s only possbl ha class cusomrs swch o srvr whl class cusomrs nvr swch o srvr. For any par of capacy lvls, w can dvd h cusomr arrval horzon (.., [0, ]) no svn nrvals as follows.. [0,µ /λ]: no cusomr balkng or swchng happns.. [µ /λ,(µ + µ )/λ]: cusomr swchng may or may no occur. No cusomr balkng occurs. 3. [(µ +µ )/λ,µ /λ]: any of cusomr swchng bu no balkng, swchng and balkng smulanously, or balkng bu no swchng may occur. 4. [µ /λ, µ ( 0 )/(λ 0 )]: hr cusomr swchng bu no balkng or balkng bu no swchng may occur. Swchng and balkng canno happn smulanously durng hs nrval. 5. [ µ ( 0 )/(λ 0 ), (µ + µ )( 0 )/(λ 0 )]: any of cusomr swchng bu no balkng, swchng and balkng smulanously, or balkng bu no swchng may occur. 6. [ (µ + µ )( 0 )/(λ 0 ), µ ( 0 )/(λ 0 )]: cusomr swchng may or may no occur. No cusomr balkng occurs. 7. [ µ ( 0 )/(λ 0 ), ]: no balkng happns bu swchng may occur. In addon o h cass oulns abov, s possbl ha hr s no swchng nor balkng n any nrval, whch s omd for as of xposon. A full analyss of hs rgons wh h rasons bhnd ach rsul s gvn n h onln appndx. Ths analyss s llusrad by Fgur 8, n whch arrows ndca h possbl sa ransons. Th sas dffrn horzonal posons ndca hr possbl prcdnc rlaonshp. Basd on h gvn analyss, w hav h followng mporan obsrvaon. Lmma 7. Whn µ µ, λ () µ µ +µ Λ() [0, ]. Lmma 7 sas ha on srvr may arac som cusomrs who orgnally prfr h ohr srvr by sng a hghr capacy lvl. Howvr, h ffcv arrval ra o h fasr srvr s
Wang and Olsn: Srvc Capacy Compon wh Pak Arrvals 8 Arcl submd o Managmn Scnc; manuscrp no. [, ] [, ( 0) ] 0 Swchng, no balkng () () () No Swchng, no balkng () () () () No swchng, no balkng () () () () Swchng, no balkng () () () () Swchng, balkng () () Swchng, no balkng () () () () No swchng, no balkng () () () () [, ] No swchng, balkng () () () No swchng,,balkng () () [, ] ( 0) ( )( 0) 0 0 Swchng, no balkng () () () () No swchng, no balkng () () () () Swchng, balkng () () No swchng, balkng () [, ] Swchng, no balkng () () () () No swchng, balkng () () () () () ( )( 0) 0 No swchng, no balkng () () () () Fgur 8 Swchng Parn
Wang and Olsn: Srvc Capacy Compon wh Pak Arrvals Arcl submd o Managmn Scnc; manuscrp no. 9 boundd by µ Λ()/(µ +µ ). Ths fac s usd n h analyss of h srvrs qulbrum bhavor lar. Havng a horough undrsandng of h cusomr swchng and balkng bhavor, w ar rady o analyz h srvrs capacy dcsons. As w hav sn n h abov analyss and as dpcd n Fgur 8, many dffrn scnaros may occur du o cusomrs swchng and balkng bhavors. In parcular, swchng may happn bfor or afr whn h oal arrval ra s qual o µ +µ ; swchng may happn bfor or afr 0 ; swchng may happn for a whl and hn balkng may occur; balkng may occur arlr han swchng; c. Ths complxs mak ffcvly nracabl o wr h prof funcons xplcly for a gvn par of capacy lvls. Bfor provdng analyc rsuls, w numrcally valua h srvrs prof basd on h analyss of cusomr swchng and balkng bhavor. To do so, w dscrz h cusomr arrval horzon by small ncrmnal sps. Accordng h analyss abov, w dvd h cusomr arrval horzon no svn nrvals and hn w numrcally calcula h dlays a boh srvrs basd on h ffcv arrval ras drvd abov. In gnral, h xsnc of a capacy Nash qulbrum s no guarand. In parcular, whn δ s small (.., h cusomrs do no hav a srong prfrnc), hr may no xs a Nash qulbrum. Fgur 9 shows h bs rspons curvs of wo compng srvrs, n whch r = 0, c =, w = 0.8, and δ = 0.075. Th rd ln rprsns srvr s opmal capacy lvls as srvr s capacy changs; h grn ln rprsns srvr s bs rsponss o srvr s capacy lvls. Whr h wo curvs cross ndcas a Nash qulbrum. In hs xampl, h wo bs rspons curvs do no cross ach ohr. Hnc, hr dos no xs a pur sragy Nash qulbrum. (No ha h vrcal ln n h fgur s acually no par of srvr s bs rspons curv. In fac, corrspondng o srvr s capacy lvl a h vrcal ln, srvr has wo bs capacy rsponss, whch ar h wo nds of h vrcal ln. Smlarly for srvr, h horzonal ln s no par of s rspons curv.) As mnond n h nroducon, h raonal bhnd hs non-xsnc of an qulbrum s smlar o ha n prc compon bwn wo srvrs ovr homognous cusomrs wh obsrvabl quus. Spcfcally, h raonal s as follows. Frs, whn δ s small, a symmrc qulbrum s no susanabl. No ha, h only possbl symmrc capacy qulbrum s (µ,µ ). Bu snc δ s small, on srvr can ncras prof by ncrasng h capacy lvl a ll b o arac cusomrs from h ohr srvr wh no much cos. An asymmrc qulbrum s no susanabl hr. If hr was an asymmrc pur sragy Nash qulbrum, hn swchng would happn n h qulbrum. Snc h srvrs hav dncal rvnu/cos paramrs and hy acually comp
Wang and Olsn: Srvc Capacy Compon wh Pak Arrvals 0 Arcl submd o Managmn Scnc; manuscrp no. ovr homognous cusomrs (sam olranc w and as dffrnc δ), f sng a hghr capacy lvl and grabbng cusomrs from h ohr srvr s opmal for on srvr, h ohr srvr has h ncnv o ncras h capacy lvl o rduc h numbr of swchng cusomrs..8.7.6.5 µ.4.3.....3.4.5.6.7.8 µ Fgur 9 Nonxsnc of Nash qulbrum Whn δ s larg nough, (µ,µ ) s a Nash qulbrum. Tha s, srong cusomr prfrncs can susan a symmrc qulbrum. To s hs, w procd wh h analyss of h bs rspons of ach srvr. Frs of all, whn h opponn ss h capacy lvl no hghr han µ, s subopmal o s a capacy lvl lowr han µ, as sad n h followng lmma. Lmma 8. If µ = µ, hn any µ < µ s domnad by µ for srvr. Ths ylds h followng proposon. Proposon. () If δ ( r c)[( r c) +(+ r c) 0 ( r c) 0 + r c 0 ] ( + w), hn (µ a [( r c) + r c,µ a ) s a Nash 0] qulbrum for hos rgons n h { w,c/r} spac whr µ = µ a. () If δ ( r c + w )( 0+ 0 ), hn (µ b,µ b ) s a Nash qulbrum for hos rgons n h { w,c/r} spac whr µ = µ b. () If δ ( r c )( 0 ) ( c r )( 0 ) 0 w 0 w 0, hn (µ C,µ C ) s a Nash qulbrum for hos rgons n h { w,c/r} spac whr µ = µ C. (v) If δ λ( spac whr µ = µ N. s µn λ µ N 0 4 c r λ ) µ N, hn (µ N,µ N )s a Nash qulbrum for hos rgons n h { w,c/r}
Wang and Olsn: Srvc Capacy Compon wh Pak Arrvals Arcl submd o Managmn Scnc; manuscrp no. To hav a concr da of h condons on δ, w plo h lowr bound on δ/ 0 as a funcon of c/r for h cas ha = 0. In hs spcal cas, h abov condons can b smplfd as follows. Corollary. whn = 0, () If δ 0 (3 c r c r )( r c ) r c { w,c/r} spac whr µ = µ a. (+ w 0 ), hn (µ a,µ a ) s a Nash qulbrum for hos rgons n h () If δ 0 3 ( c r + w 0 ), hn (µ b,µ b ) s a Nash qulbrum for hos rgons n h { w,c/r} spac whr µ = µ b. () If δ 0 ( c r ) ( c r ) w 0 ( w 0 ), hn (µ C,µ C ) s a Nash qulbrum for hos rgons n h { w,c/r} spac whr µ = µ C. δ (v) If 0 3 w ( + 0 ) + w 0 + w 4 0 4c w ( + r 0 w 0 + w 4 0 qulbrum for hos rgons n h { w,c/r} spac whr µ = µ N. ) c r, hn (µn,µ N )s a Nash In h plo w s w/ 0 = /6 (for xampl, h m o rach pak arrvals 0 = hr and cusomrs hav 0 mnus panc o hr favor srvr). For hos rgons whr µ = µ a, a vry small as prfrnc can susan a symmrc Nash qulbrum as w can s from Fgur 0 whr h lowr bound on δ/ 0 s around 0 3. (No ha for w/ 0 = /6, whn c/r 0.8, µ = µ a.) Th lowr bounds on δ/ 0 ar plod for 0 c/r 0.8 n Fgur..4 x 0 3. 0 lowr bound on δ 0.8 0.6 0.4 0. 0 0.8 0.8 0.84 0.86 0.88 0.9 0.9 0.94 0.96 0.98 c r Fgur 0 Lowr Bound on δ 0 o Susan (µ a,µ a ) as a Nash qulbrum
Wang and Olsn: Srvc Capacy Compon wh Pak Arrvals Arcl submd o Managmn Scnc; manuscrp no. 0.45 0.4 0.35 for h rgon whr µ = µ N 0.3 0 lowr bound on δ 0.5 0. 0.5 0. 0.05 for h rgon whr µ = µ b for h rgon whr µ = µ C 0 0 0. 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 c r Fgur Lowr Bounds on δ 0 o Susan a Symmrc Nash qulbrum Whn µ N s h opmal capacy lvl for a monopols, no cusomr balks, whl n h ohr cass, allowng som cusomrs o balk s opmal for ach srvr n a monopoly. In h duopoly, f on srvr ss capacy a µ N and h ohr ss an vn hghr capacy lvl, hn balkng nvr occurs. Thn w only nd o consdr cusomr swchng bhavor whn sudyng such a duopoly. Basd on hs obsrvaon, w dvlop an alrna lowr bound on δ o susan (µ N,µ N ) as a Nash qulbrum. Proposon 3. If δ w cm rµ N [ λ 0 ( + c r m) µn r cr mλ 0 ( r c mλ 0+λ 0 4µ N ) ], whr m = ( 0 ) ( 0 )+ 0, hn (µ N,µ N ) s a Nash qulbrum for hos rgons n h { w,c/r} spac whr µ N s h opmal capacy lvl for a monopols. I s nrsng o fnd ha wh cusomr prfrncs, compon dos no drora h srvrs profs and dos no affc cusomrs xprncs. An asymmrc qulbrum dos no appar o b susanabl. Th ky o hs rsuls s ha boh frms fac h sam prod of pak arrvals, so whn on frm s ovrloadd h ohr s lkly o b ovrloadd as wll. If h mark could b sgmnd so ha on frm s pak was a a dffrn m o h ohr frm s hn h rsuls could look markdly dffrn (and may rsul n a wn-wn suaon for boh h frms and hr cusomrs). Ths s lf as h subc of fuur rsarch.
Wang and Olsn: Srvc Capacy Compon wh Pak Arrvals Arcl submd o Managmn Scnc; manuscrp no. 3 4.. Asymmrc Compon In hs subscon, w sudy compon n asymmrc suaons. In parcular, w consdr wo dmnsons of asymmry, on s asymmry of mark dvson, h ohr s asymmry bwn srvrs. To hav a clar undrsandng of how hs asymmry facors affc h qulbrum rsuls, w analyz hm sparaly. In compon wh asymmrc mark dvson (.., p /), w assum ha h wo srvrs charg h sam prc r and ncur h sam cos ra c. Whl n compon wh asymmrc srvrs, h wo srvrs hav dffrn cos o rvnu raos bu hy hav h sam ponal marks (.., p = /). In h duopoly wh asymmrc mark dvson, w kp h assumpon ha v = v,v = v,v > v. Tha s, ach cusomr s wllng o wa a mos w for hs favor srvr and w δ for hs nfror srvr. Whou loss of gnraly, w assum ha p < /. Dno p by p for noaonal smplcy. As n h analyss for symmrc compon, w frs analyz h possbl cusomr swchng and balkng bhavor and hn analyz h srvrs qulbrum. Dno h capacy lvls of srvr and srvr as µ and µ, rspcvly. If µ > pλ (µ > pλ), hn no quu bulds up a srvr () and h cusomr swchng drcon s clar. Th followng lmma shows h swchng drcon for h cas whr µ < pλ,µ < pλ. Lmma 9. If µ /p < µ / p < λ, hn s possbl ha som class cusomrs swch o srvr bu no class cusomrs swch o srvr. In ohr words, h wang m a srvr s always no lss han h wang a srvr. If µ /p > µ / p, hn s only possbl ha som class cusomrs swch o srvr. In fac, h cusomrs swchng and balkng bhavor n h suaon wh asymmrc mark dvson s smlar as ha n h symmrc cas analyzd n Scon 4.. In parcular, dpndng on whhr h wo srvrs ar n hr pak prod and whhr h oal capacy s grar han h oal arrval ra, h cusomr arrval prod can b dvdd no svral nrvals. Smlar swchng/balkng scnaros occur. Whn swchng happns bu no cusomrs balk, h ffcv arrval ras o h wo srvrs ar µ Λ()/(µ +µ 3 ) ( =,). Swchng and balkng may happn smulanous, n whch cas h ffcv arrval ras o h wo srvrs ar qual o hr capacy lvl. Rcall h analyss on a monopols s capacy dcson, hr sng a hgh capacy lvl o prvn cusomr balkng or opmally forgong som cusomrs dpnds on h cos rvnu rao (c/r) and cusomrs panc ( w) bu no h slop λ. Thrfor, h wo srvrs hav h sam opmal scnaro n h absnc of compon vn hough hy hav dffrn mark shars (..,
Wang and Olsn: Srvc Capacy Compon wh Pak Arrvals 4 Arcl submd o Managmn Scnc; manuscrp no. dffrn slops). For nsanc, f s opmal for srvr o s h capacy lvl a µ N p balkng happns a h opmal oucom, hn s opmal for srvr o s µ = µ N p absnc of compon. No ha µ N p /µ N p = p/ p. and no n h In h symmrc compon modl, w drvd suffcn condons ha suppor a symmrc Nash qulbrum. Howvr, wh asymmrc mark shar, numrcal suds show a Nash qulbrum may no xs wh h sam cusomr prfrnc dffrnc. As s mpraccal o mahmacally xprss h srvrs prof funcons, w mploy numrcal xampls o undrsand h compon bwn srvrs wh asymmrc mark shar. In h frs xampl h paramrs ar as follows: r = 0, c =, 0 = 4, = 8, λ =, w = 0.8, and δ = 0.3. W consdr four dffrn mark dvsons: p = 0.4, p = 0.3, p = 0.5, and p = 0.. I can b vrfd ha wh hs paramrs, n h absnc of compon, µ N p (µ N p ) s h opmal capacy lvl for srvr (). I can also b vrfd ha δ s grar han h lowr bounds drvd n Proposon and Proposon 3, whch mans ha n symmrc compon (p = 0.5), (µ N,µ N ) s a Nash Equlbrum. W fnd ha hr s no pur-sragy Nash qulbrum for p = 0., 0.5, or 0.3 bu (µ N p,µ N p ) s h unqu Nash qulbrum for p = 0.4. Furhr dals on h xac form of h opmal rsponss may b found n h onln appndx, whch graphs h rspons funcons n ach cas and xplans h srvrs compv rsponss. W also sudy h cas whr µ b p(µ b p) s h opmal capacy lvl for srvr () n h absnc of compon. For som mark dvsons, (µ b p,µ b p) s h unqu Nash qulbrum. Th srvrs bs rspons curvs ar mor complcad. Fgur llusras h cas whr r = 0, c = 6, 0 = 4, = 8, λ =, w = 0.8, δ = 0., and p = 0.3. As can b sn from h curvs, hy ar hghly non-concav (rcall ha vn n h monopoly cas h opmal capacy was no concav n h sysm paramrs); hrfor, provng anyhng analycally abou xsnc or non-xsnc of qulbra appars chncally nfasbl. Fnally, w xamn compon bwn wo asymmrc srvrs by numrcal xampls alon. In hs cas, w kp all h assumpons as w hav n h symmrc compon cas xcp ha h wo srvrs hav dffrn cos o rvnu raos. For convnnc, w rfr o h srvr wh lowr cos o rvnu rao as h (rlavly) ffcn srvr whl h srvr wh hghr cos o rvnu rao as h (rlavly) nffcn srvr. In parcular, w ar nrsd n h suaon whr s opmal for h ffcn srvr o s h capacy a µ N, whl s opmal for h nffcn srvr o forgo som cusomrs (.., balkng occurs) as monopolss. W hav qu a fw nrsng obsrvaons. Frs, hr may b mulpl Nash qulbra or a unqu Nash qulbrum (as llusrad n h followng xampls). Scond, a qulbrum h ffcn srvr
Wang and Olsn: Srvc Capacy Compon wh Pak Arrvals Arcl submd o Managmn Scnc; manuscrp no. 5 3.5 3.5 srvr s capacy lvl.5 0.5 0 0 0.5.5.5 3 3.5 srvr s capacy lvl Fgur Bs Rspons Curvs: (µ b p,µ b p) s a Nash qulbrum ss a hghr capacy han µ N, whl h nffcn srvr ss a lowr capacy han wha would s as a monopols. Thrd, a qulbrum, balkng dos no happn, only swchng happns. Fourh, whn hr ar mulpl Nash qulbra, h oal capacy s fxd. Las, a qulbrum, h oal capacy s hghr han ha n h monopoly cas. In h frs xampl, h paramrs ar as follows: r = r = 0, c =, c = 8, 0 = 4, = 8, λ =, w = 0.8, and δ = 0.05. Hnc, srvr s h ffcn on n hs xampl. I can b vrfd ha, as monopolss, s opmal for srvr o s h capacy a µ N =.8 and s opmal for srvr o s h capacy a µ a = 0.73. In h duopoly, from h bs rspons curvs (s Fgur 3), w can s ha mulpl Nash qulbra xs. A qulbrum, h ffcn srvr s capacy lvl s approxmaly n h rang of (.5,.) and h oal capacy s around.60, whch s hghr han µ N +µ a (=.0). Wh hs oal capacy, no cusomrs balk. Thrfor, h compon bnfs cusomrs. In h scond xampl, h paramrs ar as follows: r = r = 0, c =.5, c = 8.5, 0 = 4, = 8, λ =, w = 0.8, and δ = 0.05. Sll, srvr s h ffcn on and s opmal capacy lvl as a monopols s µ N =.83. Srvr s opmal capacy lvl as a monopols s µ a = 0.5739. Fgur 4 shows h bs rspons curvs. Thr s a unqu Nash qulbrum (.86,0.37). In fac, s nuv ha hr s no cusomr balkng a qulbrum n h duopoly wh on srvr s opmal monopols capacy lvl a µ N. Rcall our analyss n h monopols s capacy dcson scon, whr was shown ha whhr s opmal o prvn balkng s drmnd by h cos o rvnu rao (c/r) and cusomrs dlay olranc ( w), bu s ndpndn of h cusomr arrval slop (λ). For h ffcn srvr, swchng only changs h cusomr arrval
Wang and Olsn: Srvc Capacy Compon wh Pak Arrvals 6 Arcl submd o Managmn Scnc; manuscrp no. 4 3.5 3 srvr s capacy lvl.5.5 0.5 0 0 0.5.5.5 3 3.5 4 srvr s capacy lvl Fgur 3 Bs Rspons Curvs: asymmrc srvrs, mulpl Nash qulbra 4 3.5 3 srvr s capacy lvl.5.5 0.5 0 0 0.5.5.5 3 3.5 4 srvr s capacy lvl Fgur 4 Bs Rspons Curvs: asymmrc srvrs, unqu Nash qulbrum slop. Thrfor, a qulbrum, s opmal for h ffcn srvr o prvn balkng. Snc mor cusomrs ar srvd n h qulbrum, h oal capacy s hghr han µ N + µ a. In summary, whn frms ar smlar hr s no vdnc ha compon affcs hr cusomr xprnc or srvr profs. In hs cass, f a Nash qulbrum xss s a h pon whr a monopols would s capacy. An asymmrc qulbrum for a symmrc gam dos no appar o b susanabl. Evn whn mark shar s dssmlar bwn h wo frms, f coss ar smlar, hn h frms wll agan s capacy a h monopols lvl bcaus h arrval paks ar a h sam pon for boh frms. Howvr, f frms ar asymmrc n hr capacy coss h mor ffcn frm may ncras mark shar o h drmn of h lss ffcn frm. In hs cas
Wang and Olsn: Srvc Capacy Compon wh Pak Arrvals Arcl submd o Managmn Scnc; manuscrp no. 7 cusomr xprnc s mprovd bcaus oal capacy n h sysm nds o b hghr han ha s by wo monopolss. 5. Conclusons and Exnsons In hs papr, w sudy capacy compon for obsrvabl quus n a non-saonary nvronmn. To h bs of our knowldg, hs s h frs such sudy. To mak h non-saonary racabl, w modl h sysm usng a flud modl (whch pror rsarch has shown o frqunly approxma ovrloadd sysms wll). Bfor consdrng capacy compon, w consdr a monopols s capacy dcson n hs nvronmn, whch provds nrsng nsghs n s own rgh. W hn analyz h duopoly boh analycally and numrcally. For a monopols, w gv an xplc back-of-h-nvlop calculaon for capacy sng. W fnd ha h opmal capacy lvl dpnds on boh h rao of margnal cos o rvnu and also on cusomr panc. W fnd ha whl capacy s dcrasng n h formr facor s unmodal n h lar, frs ncrasng n panc and hn dcrasng. W clam ha, for a rang of paramrs ha sm a pror rasonabl (and ar asly chckd n pracc), capacy should b s so ha h quu s clard by h nd of h pak prod (bu no so larg ha no cusomrs balk). Ths s a smpl hursc ha could b appld o sysms wh varably, bu on ha should probably b sd horoughly frs o s how prforms. Ths s lf as h subc of fuur rsarch. W fnd ha, whn frms ar smlarly ffcn, compon has no ffc on hr frm s dcsons or cusomrs xprncs. Howvr, whn on frm s sgnfcanly mor ffcn han h ohr, compon can sgnfcanly hur h nffcn frm bu bnf h cusomr (so long as h nffcn frm s no drvn from h mark). Ths lar nsghs ar gand numrcally and fuur work should focus on provdng analyc usfcaon for hm. W hav rsrcd anon o xsnc of pur-sragy Nash qulbra. I s no clar wha nsghs could b gand from a mxd sragy qulbra n hs on-sho capacy gam. On h ohr hand, s lkly ha smlar boundng chnqus o hos usd on cusomr prfrncs δ could b usd o drv xsnc of ɛ-qulbra for h cas of small o zro cusomr prfrncs; nuvly, sms lk h nsghs would b smlar. Ths s lf as h subc of fuur rsarch. Hr w consdrd boh a monopols and a duopols. I appars plausbl ha h nuon from h duopoly compon should carry ovr o an olgopoly. In hs cas, cusomr swchng bhavor bwn frms would nd o b carfully consdrd. Agan, hs s lf as h subc of fuur rsarch.
Wang and Olsn: Srvc Capacy Compon wh Pak Arrvals 8 Arcl submd o Managmn Scnc; manuscrp no. Our modl conans no randomnss. If hr was also no pak hn h flud modl would b lss nrsng bcaus hr arrvals xcd capacy and h quu bulds ndfnly (or says fxd undr cusomr balkng) or hr s no quu. Modls wh randomnss hav complx sochasc dynamcs and a horough modl would nd o ncorpora cusomr ockyng and rngng (whch do no occur hr bcaus h modl s drmnsc). In gnral, capacy compon undr obsrvabl quus s a vry lmd lraur ha could b furhr xplord. Rfrncs []Allon, G., and A. Fdrgrun. 007. Compon n Srvc Indusrs. Opraons Rsarch. 55 (): 37 55. []Allon, G., and A. Fdrgrun. 008. Srvc Compon wh Gnral Quung Facls. Forhcomng n Opraons Rsarch. [3]Allon, G., and I. Gurvch. 007. Compon n Larg-Scal Srvc Sysms: Do Wang Tm Sandards Mar? Workng papr, Kllogg School of Managmn, Chcago, IL. [4]Alason, J., M. A. Eplman, and S. G. Hndrson. 004. Call cnr saffng wh smulaon and cung plan mhods. Annals of Opraons Rsarch 7: 333 358. [4]Cachon, G.P., and F. Zhang. 007. Obanng fas srvc n a quung sysm va prformanc-basd allocaon of dmand. Managmn Scnc. 53 (3): 408 40. [4]Chn, H., and Y.W. Wan. 005. Capacy compon of mak-o-ordr frms. Opraons Rsarch Lrs. 33 (): 87 94. [4]Chang, J., H. Ayhan, J.G. Da and C.H. Xa. 004. Dynamc schdulng of a mul-class flud modl wh ransn ovrload. QUESTA. 48: 63 307. [4]Chay, S., and W.J. Hopp. 007. Squnal Enry wh Capacy, Prc, and Ladm Compon. Workng Papr, Oln Busnss School, S. Lous, MO. [4]Chrs, D., and B. Av-Izhak. 00. Sragc qulbrum for a par of compng srvrs wh convx cos and balkng. Managmn Scnc. 48 (6): 83 80. [4]Corsn, H., and S. Suhlmann. 998. Capacy managmn n srvc organzaons. Tchnovaon. 8 (3): 63 78. [4]Dobson, G., and E. Savrulak. 007. Smulanous prc, locaon, and capacy dcsons on a ln of m-snsv cusomrs. Naval Rsarch Logscs. 54 (): 0. [4]Ed, L.C. 954. Traffc Dlays a Toll Boohs. Journal of h Opraons Rsarch Socy of Amrca. (): 07 38. [4]Eck, S.G., W.A. Massy, and W. Wh. 993a. M /G/ Quu wh Snusodal Arrval Ras. Managmn Scnc. 39 (): 4 5.
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