Any buyer that depends on suppliers for the delivery of a service or the production of a make-to-order

Transcription

1 MANAGEMENT SCIENCE Vol. 53, No. 3, March 2007, pp ssn essn nforms do /mnsc INFORMS Obtanng Fast Servce n a Queueng System va Performance-Based Allocaton of Demand Gérard P. Cachon Operatons and Informaton Management, The Wharton School, Unversty of Pennsylvana, Phladelpha, Pennsylvana 19104, cachon@wharton.upenn.edu Fuqang Zhang Paul Merage School of Busness, Unversty of Calforna, Irvne, Calforna , fzhang@uc.edu Any buyer that depends on supplers for the delvery of a servce or the producton of a make-to-order component should pay close attenton to the supplers servce or delvery lead tmes. Ths paper studes a queueng model n whch two strategc servers choose ther capactes/processng rates and faster servce s costly. The buyer allocates demand to the servers based on ther performance; the faster a server works, the more demand the server s allocated. The buyer s obectve s to mnmze the average lead tme receved from the servers. There are two mportant attrbutes to consder n the desgn of an allocaton polcy: the degree to whch the allocaton polcy effectvely utlzes the servers capactes and the strength of the ncentves the allocaton polcy provdes for the servers to work quckly. Prevous research suggests that there exsts a trade-off between effcency and ncentves,.e., n the choce between two allocaton polces a buyer may prefer the less effcent one because t provdes stronger ncentves. We fnd consderable varaton n the performance of allocaton polces: Some ntutvely reasonable polces generate essentally no competton among servers to work quckly, whereas others generate too much competton, thereby causng some servers to refuse to work wth the buyer. Nevertheless, the trade-off between effcency and ncentves need not exst: It s possble to desgn an allocaton polcy that s effcent and also nduces the servers to work quckly. We conclude that performance-based allocaton can be an effectve procurement strategy for a buyer as long as the buyer explctly accounts for the servers strategc behavor. Key words: game theory; onng behavor; Nash equlbrum; procurement; sourcng; suppler management Hstory: Accepted by Wallace J. Hopp, stochastc models and smulaton; receved February 1, Ths paper was wth the authors 8 months for 2 revsons. Fast servce s clearly mportant. Less obvous s how to go about obtanng fast servce from supplers or servce provders. One technque s to make servers compete by allocatng busness to them based on ther performance,.e., the faster server s rewarded wth a greater share of demand. For example, Sun Mcrosystems mantans multple memory chp supplers and allocates demand wth a scorecard system. A score that depends on a number of factors, delvery lead tme among them s perodcally assgned to each suppler, and a suppler s allocaton of Sun s busness ncreases as they mprove ther score relatve to the other supplers (Farlow et al. 1996). GE Lghtng and Ar Products and Chemcals also allocate demand towards better-performng supplers (Pyke and Johnson 2003). Ths paper studes, n the context of a stylzed queueng model, the ssue of how performance-based demand allocaton can nduce competton among supplers to obtan faster servce or delvery lead tmes. A precursor to ths lne of research s the extensve body of work on queue-onng behavor, poneered by Naor (1969). That lterature focuses on the behavor of strategc customers/obs: e.g., whether or not to on a queue (e.g., Naor 1969), or whch of several queues to on (Bell and Stdham 1983). It s generally found that the behavor of ndvdual obs creates externaltes on other obs (e.g., overcongeston of the faster server). (See Hassn and Havv 2003 for a revew of the queue-onng lterature wth strategc customers/obs.) Those externaltes do not occur n our settng because a sngle buyer controls all of the obs. Instead, we have strategc servers servers that can regulate how fast they work, and workng faster s costly. In our model the buyer pays a fxed amount for each ob, so the buyer s task s to choose an allocaton polcy to mnmze the average lead tme to complete obs. We study allocaton polces that can be classfed nto two groups, state-dependent polces (the 408

2 Management Scence 53(3), pp , 2007 INFORMS 409 allocaton of a ob to a server depends on the servers current workload) and state-ndependent polces (the allocaton of a ob does not depend on the number of obs currently n the servers queues). Wth nonstrategc servers t s clear that a statedependent polcy can delver faster lead tmes than a state-ndependent polcy because, n part, a state-ndependent polcy rsks allocatng obs to busy servers whle other servers reman dle,.e., a state-dependent polcy can do a better ob of poolng the servers capactes. 1 However, are state-dependent polces better when the servers are strategc? Suppose a statendependent polcy nduces servers to work more quckly than a state-dependent polcy. Then the buyer may be better off wth a state-ndependent polcy even though the system s capacty s not as effectvely utlzed. In other words, ncentves may trump effcency. In fact, Glbert and Weng (1998) arrve at that concluson. Nevertheless, there are several reasons why ths mght not be the best concluson. Frst, we show that there s an error n ther equlbrum exstence proof, so t s not always meanngful to compare ther two allocaton polces. Second, and more mportantly, they do not compare optmal polces. We compare the buyer s best state-dependent polcy wth the buyer s best state-ndependent polcy and fnd that the buyer s better off wth the state-dependent polcy,.e., the buyer can have both ncentves and effcency. In general, we fnd that there s consderable varaton n the performance of ntutvely reasonable polces. For example, the buyer s optmal state-ndependent polcy wth nonstrategc servers s found to perform poorly n the presence of strategc servers, and proportonal allocaton, whch s probably the most ntutve allocaton polcy, can be the worst performer of the polces we consder. The next secton descrbes our model n detal. Secton 2 expands upon the related lterature. Secton 3 studes the buyer s allocaton polcy choce and the competton between servers under several dfferent allocaton polces. Secton 4 dscusses several extensons to the model. The fnal secton concludes wth a summary of our results. 1. The Model A buyer procures a good (e.g., a make-to-order component, as n the Sun Mcrosystems example) or a servce. For ease of exposton, we assume a servce s procured. There are two servers. (Most of our results extend to more than two servers; see Zhang Poolng s not necessarly a good dea f servers have sgnfcantly dfferent capactes. Rubnovtch (1985a) characterzes the condtons under whch a ob should never be allocated to the slow server n a two-server queueng system. for detals.) Demand for the servce arrves accordng to a Posson process wth rate. Each demand s referred to as a ob and all obs are eventually completed. Server s average servce rate s and servce tmes are exponentally dstrbuted. We refer to as server s capacty and = 1 2 denotes the capacty vector. A server wth capacty ncurs a capacty cost at rate c, no matter whether the capacty s utlzed or dle, where c0 = 0, c >0, and c 0 are assumed. The servers varable cost per ob s normalzed to zero. We say that a ob s allocated to a server when t s certan that server wll complete the ob. The buyer pays R per allocated ob. We assume R>r 1, where r 1 = c/2//2 because t s the mnmal requrement for the supplers to earn a nonnegatve proft and delver fnte lead tmes (see Zhang 2004). We assume R s exogenous: There could be an ndustry standard prce that the buyer s unable to negotate away from, or the prce could be set va negotatons that nvolve ssues beyond the scope of ths model. The buyer controls her allocaton polcy (.e., how obs are allocated to servers) and the servers choose ther capactes. The buyer seeks to mnmze the average delvery lead tme over an nfnte-horzon subect to the constrant that each server earns a nonnegatve proft, and the servers seek to maxmze ther average proft: = R c (1) where s the rate at whch server s allocated obs. 2 Hence, we assume that the buyer and the servers do not dscount future cash flows and that they expect a long-term relatonshp. We focus on equlbra n whch the servers adopt open-loop strateges,.e., strateges that are ndependent of the hstory of play. As a result, ths nfnte-horzon capacty game among servers can be analyzed as a sngledecson capacty game. Prevous research on strategc servers also restrcts attenton to open-loop strateges. In 33 we dscuss lead tme-based allocaton rather than capacty-based allocaton. 2. Lterature Revew Kala et al. (1992) were the frst to study strategc servers, but they only consder a smple state-dependent polcy n whch obs are allocated to dle 2 Note that servers are pad for allocated obs rather than completed obs. If they were pad for completed obs then ther proft functon would be = R mn c. The equlbrum analyss of ths proft functon s sgnfcantly more complex due to the knk created by the mn functon. Nevertheless, our qualtatve results are not dfferent. See Zhang (2004) for detals.

3 410 Management Scence 53(3), pp , 2007 INFORMS servers wth equal probablty. Glbert and Weng (1998) expand upon ther model to nclude a statendependent allocaton polcy that allocates obs to servers mmedately upon arrval. They conclude that a state-ndependent polcy can be better for the buyer than a state-dependent polcy. Our results are dfferent, as we explan n detal n the subsequent sectons. Chrst and Av-Itzhak (2002) extend those models to nclude customer balkng, but we do not have balkng. Ha et al. (2003) study the competton between two supplers servng one buyer, n whch delvery frequency s an element of the buyer s allocaton decson. However, they study determnstc demand, so although they consder ssues smlar to ours, a drect comparson between ther work and ours s not meanngful. There are papers that compare sole sourcng versus dual sourcng, whereas we assume that a dualsourcng strategy has been adopted: e.g., Anton and Yao (1989, 1992), Anupnd and Akella (1993), Benaafar et al. (2007), Seshadr (1995), and Seshadr et al. (1991). See Mnner (2003) and Elmaghraby (2000) for revews of the lterature on sourcng strateges. There are papers that study a buyer s procurement polcy when there are multple supplers wth exogenously determned characterstcs: e.g., Bonser and Wu (2001), Chen et al. (2001), L and Kouvels (1999), Martnez de Albenz and Smch-Lev (2003), Sedarage et al. (1999), and Tallur (2002). In our model the servers lead tmes depend on ther choces and the buyer s allocaton polcy. Several papers study coordnaton and competton n supply chans wth multple supplers: Bernsten and DeCrox (2004); Wang and Gerchak (2003); and Nagaraan and Bassok (2003). In these papers, lmted capacty leads to demand truncaton rather than slower delvery tmes. Bernsten and de Vercourt (2005) consder a market wth multple supplers and multple buyers. Ther supplers have fxed processng rates and compete by offerng dfferent lead tmes to buyers, whch they obtan va holdng nventory. There are a number of papers that study server competton n whch frms choose operatonal strateges to adust ther delvery tmes: e.g., Allon and Federgruen (2003), Cachon and Harker (2002), Chayet and Hopp (2002), Lederer and L (1997), and So (2000). In those papers the structure of how frms compete s exogenous, whereas n our model t s determned by the buyer va her allocaton polcy. There s lterature on capacty allocaton (e.g., Cachon and Larvere 1999a, b, c; Deshpande and Schwarz 2002), n whch a sngle manufacturer allocates scarce capacty among multple buyers. Although allocaton polces smlar to ours are mplemented, those models are analytcally qute dfferent. L (1992) and Armony and Plambeck (2005) study models n whch a buyer submts duplcate orders to multple supplers. In our model, each ob s allocated to a sngle server, but we brefly dscuss order duplcaton n Allocaton and the Servers CapactyGame Our model can be analyzed n two nterdependent parts. The frst part s the buyer s allocaton polcy choce.e., how wll the buyer allocate obs among the two servers. The second part s the capacty choce game played between the servers, whch clearly depends on the partcular allocaton polcy the buyer has selected. Furthermore, the attractveness of an allocaton polcy to the buyer depends on the capactes chosen by the servers, as well as how obs are routed through the system. We treat these two parts sequentally. The set of allocaton polces can be dvded nto two broad classes: state-ndependent polces and state-dependent polces. Wth a state-ndependent polcy, the buyer allocates obs to servers based only on ther capactes (whch are nferred from past allocatons and resultng delvery tmes) and not on the current state of the system (e.g., how many obs are allocated to each server, whch server s dle, etc.). Because no current nformaton s utlzed wth a state-ndependent polcy, the buyer mmedately allocates a ob to a server upon ts arrval,.e., there s no beneft n watng to allocate a ob f watng does not change the allocaton decson process. In contrast, wth a state-dependent allocaton polcy the buyer allocates obs based on the current state of the system. For example, the buyer may choose to allocate obs only to dle servers. Gven a fxed-capacty vector, the buyer s optmal state-dependent polcy s clearly never worse (and can be strctly better) than the buyer s optmal state-ndependent polcy because state-ndependent polces are a subset of the set of state-dependent polces. To be more specfc, assume both servers choose capacty so that t s optmal for the buyer to allocate half of the obs to each server. The optmal statedependent polcy allocates obs only to dle servers, and so the average lead tme, W sd, s equvalent to an M/M/2 queueng system, W sd = 2 /22 The optmal state-ndependent polcy allocates obs upon arrval to servers wth equal probablty, whch yelds an average lead tme, W s, that s equvalent to two M/M/1 systems, W s = 1 /2

4 Management Scence 53(3), pp , 2007 INFORMS 411 Assumng stable systems, >/2, t s ntutve that the state-dependent lead tme s faster than the statendependent lead tme, W sd <W s, because the state-dependent polcy does a better ob of poolng the servers capactes. Wth the state-dependent polcy a ob s never watng whle there s an dle server, but that neffcent outcome can occur wth the statendependent polcy. In addton to how obs are routed through the system, the buyer s lead tme depends on the capactes chosen by the servers. Agan assumng that the servers choose dentcal capactes, t s easy to see that both W sd and W s are decreasng n,.e., the buyer s lead tme wth ether type of allocaton s reduced as the servers work faster. Because workng faster s costly to the servers, there exsts a maxmum rate,, at whch the servers earn zero proft gven that they are allocated half of the obs,.e., s the soluton to c = R/2. From the buyer s perspectve, the deal state-dependent allocaton polcy nduces the servers to choose capacty and routes obs so that the resultng lead tme s W sd. Smlarly, the deal state-ndependent polcy nduces the servers to choose capacty and routes obs so that the lead tme s W s. 3 It remans to be determned whether those deals can be acheved,.e., does there exst an allocaton polcy that acheves as an equlbrum outcome of the servers capacty game? If so, then clearly the optmal state-dependent polcy would be strctly better for the buyer than the optmal statendependent polcy State-Independent Allocaton Polces Bell and Stdham (1983) dentfy the state-ndependent allocaton polcy, whch we call Bell-Stdham allocaton, that mnmzes the buyer s lead tme for any fxed-capacty vector, : ( / )( n n ) 1/2 1/2 for n = =1 =1 0 for >n (2) where the servers capactes are sorted n decreasng order and n 2 s the largest nteger, such that 3 We assume the buyer desres to have two symmetrc servers. Gven that the servers have the same capacty cost functon, t s ether optmal for the system to have one server that s allocated all obs or two servers that are allocated half of the obs, where the latter s more lkely as the capacty cost functon becomes more convex. There could be other reasons for mantanng multple servers even f the capacty cost functon suggests one server would be optmal. We do not attempt to model those alternatve reasons, so we assume throughout that the buyer desres to dual source and equally dvde obs between the servers. n 0. 4 Ths allocaton rule equates the margnal change n the average number of obs at each queue wth respect to the arrval rate. Naturally, Bell- Stdham allocaton assgns half of the obs to each server when the servers have the same capacty, bs, thereby achevng the lead tme W s bs. Bell-Stdham allocaton was desgned for nonstrategc servers. Wth strategc servers, accordng to Theorem 1, a symmetrc equlbrum exsts n ths capacty game only under certan condtons. The capacty cost functon restrcton s relatvely mld, but the restrctons on R are sgnfcant. (All proofs are n the appendx.) Theorem 1. Wth Bell-Stdham allocaton, (2), f R> r 2 = 2c /2, c 0, and bs 0, where bs s the unque soluton to ( )( R 1 + /2 ) c 4 bs = 0 (3) bs then = bs >/2 s the unque symmetrc Nash equlbrum. An equlbrum (wth fnte lead tmes) may fal to exst wth Bell-Stdham allocaton because the buyer s prce may be too low, R r 2 : The servers do not feel the need to buld enough capacty to provde a stable system (.e., they prefer to work at 100% utlzaton than to compete for addtonal demand by workng more quckly and operatng at less than 100% utlzaton). (Note that because c s convex, t s straghtforward to show that r 1 <r 2.) Alternatvely, an equlbrum may fal to exst because the buyer pays too much, thereby causng so much competton between the servers that they both cannot earn a postve proft. 5 Furthermore, t s apparent from (3) that the servers may not choose n equlbrum the buyer s deal capacty,.e., bs s possble. Although Bell-Stdham allocaton s optmal for the buyer for any gven capacty vector, t does not take nto consderaton the behavor of strategc servers, and, as a result, t does not necessarly provde the correct ncentves for servers to choose a desrable capacty vector. Wth strategc servers t s mportant to recognze that the buyer s allocaton polcy need not be optmal for all capacty vectors (as s Bell-Stdham). The role of the allocaton polcy s to establsh ncentves for the servers to converge 4 They also provde results for M/G/1 queues and allow watng tme costs to vary across queues. In ths applcaton the watng tme cost s naturally the same across all queues. We dscuss n 4 our results wth nonexponental servce tmes. 5 For example, wth a quadratc capacty cost functon t can be shown that there exsts an upper bound, r 3, such that there does not exst an equlbrum wth R>r 3.

5 412 Management Scence 53(3), pp , 2007 INFORMS to a partcular capacty equlbrum that s desrable for the buyer, deally. As a result, t s worthwhle to consder other allocaton polces that acheve an equal dvson of obs n equlbrum, as wth Bell-Stdham, but allocate obs dfferently than Bell-Stdham for nonequlbrum/nonsymmetrc capactes. Glbert and Weng (1998) propose balanced allocaton: Wth balanced allocaton the buyer attempts to equalze (.e., balance) the servers lead tmes for all capacty vectors (and only fals to do so f all obs are allocated to one server because of a large dsparty n ther processng rates): = { + ( ) + otherwse Theorem 2. Wth balanced allocaton, f R r 2 = 2c /2, c >0, and c b c b /, where b s the unque soluton to c b = R/2, then b b s the unque Nash equlbrum and the servers average lead tmes are fnte. Otherwse, there does not exst an equlbrum wth fnte lead tmes. As wth Bell-Stdham allocaton, balanced allocaton leads to a symmetrc equlbrum, but the two allocaton polces need not result n the same capacty, bs b, and balanced allocaton also generally results n less than the buyer s desred capacty, b. Furthermore, three condtons are needed for an equlbrum to exst wth balanced allocaton. Frst, balanced allocaton requres that the buyer s prce s suffcently hgh, R r 2, otherwse the reward for workng fast s nsuffcent to provde an ncentve to work. Second, the capacty cost functon must be strctly convex, c >0, whch rules out the mportant case of lnear capacty costs. Glbert and Weng (1998) correctly recognzed those frst two condtons, but dd not recognze the necessary thrd condton, c b c b /, whch requres the servers to earn a nonnegatve proft n equlbrum (e.g., wth a quadratc cost functon c = a 2 + b, a>0, ths condton translates nto R 22a + b 2 + 4a 2 2 ). They erred by belevng that each server s proft functon s globally concave. In fact, t s concave and decreasng for 0 and concave for >. Hence, each server s global optmum s ether the maxmum of the frst concave range, = 0, or the maxmum of the second concave range, >. As a result, each server s reacton functon (the optmal capacty gven the capacty of the other server) harbors a dscontnuty, whch creates the possblty of no equlbrum. However, f an equlbrum exsts, then Glbert and Weng (1998) correctly dentfy t. An alternatve allocaton polcy s needed that can be parameterzed so as to adust up or down, as (4) needed, the level of competton between the servers. We offer two such polces: lnear allocaton and proportonal allocaton. Wth lnear allocaton, 1 ( ) n = n for n =1 (5) 0 for >n where the servers capactes are sorted n decreasng order, >0, 0 < 1, and n 2 s the largest nteger such that n 0 and n > 0. A server does not necessarly receve a postve allocaton even f the server bulds some capacty, but a server surely receves no allocaton f the server bulds no capacty. If = 1 and = 1, then lnear allocaton s almost dentcal to balanced allocaton: The only excepton s the addtonal n > 0 requrement to receve a postve allocaton. (That reasonable requrement facltates the unqueness equlbrum proof.) Hence, lnear allocaton can be consdered a generalzaton of balanced allocaton. The parameters and could potentally enable lnear allocaton to acheve many dfferent capacty vectors as an equlbrum to the servers capacty game. However, as already dscussed, the buyer s desred outcome from the servers capacty game s wth an even dvson of obs between the servers. Accordng to the next theorem, lnear allocaton can acheve that obectve. Hence, lnear allocaton s an optmal state-ndependent allocaton polcy. Theorem 3. Gven lnear allocaton: () If c >0, = 2c l /R, and = 1, then = l = for all s a unque Nash equlbrum and the average lead tmes are fnte. () If c = b b > 0, = 4 1/2 l c l /R, and = 1/2, then = l = for all s the unque Nash equlbrum and the average lead tmes are fnte. The parameters provded n Theorem 3 are not the only ones that acheve our obectve (that s the unque Nash equlbrum), so we choose ntutve values for : Wth strctly convex capacty cost the parameter s not necessary (hence, set to = 1), but wth a lnear capacty cost <1 s necessary to create an nteror optmum for each server. Proportonal allocaton s another polcy that can be parameterzed to adust the level of competton between the servers. Wth proportonal allocaton, server s share of the buyer s obs s ( ) = 1 + (6) 2 where 1 s a parameter. In partcular, ncreasng rases the ntensty of competton, thereby allowng the buyer to acheve the desred capacty vector,

6 Management Scence 53(3), pp , 2007 INFORMS 413. Hence, proportonal allocaton can also be an optmal state-ndependent allocaton polcy. However, because the servers proft functons are not necessarly well behaved as s ncreased, Theorem 4 provdes results only for a quadratc capacty cost functon. Theorem 4. Gven proportonal allocaton and a quadratc capacty cost functon c = a 2 +b, a 0, b 0, a + b>0, f = 2 c c where c = R/2 (.e., s the server s break-even capacty), and R>r 1 = c/2//2, then = for all s the unque Nash equlbrum and average lead tmes are fnte. Although >1 s desrable for the buyer, t s worthwhle to menton that = 1 yelds an ntutvely appealng allocaton mechansm: Wth = 1 a server s demand share equals the server s share of total capacty and the servers utlzatons are equated (.e., each server has the same number of obs on average). Recall that Bell-Stdham allocaton equates the margnal change n the number of obs at each server wth respect to that server s arrval rate. However, exstence of an equlbrum wth = 1 requres the buyer to pay a suffcently large prce and the servers capactes are less than deal for the buyer, p <. Theorem 5. Wth proportonal allocaton and = 1, f R>r 2 = 2c /2, then = p for all s a unque Nash equlbrum wth fnte lead tmes, where p s the unque soluton to ( )( ) R c p = 4 p Otherwse, a Nash equlbrum does not exst wth fnte lead tmes State-Dependent Allocaton The smplest state-dependent polcy s common-queue allocaton, frst studed by Kala et al. (1992): Jobs are only allocated to dle servers, where each dle server s equally lkely to be allocated a ob, and obs are mantaned on a queue f both servers are occuped. For convenence, the followng lemma repeats ther results. Lemma 6. Gven that c >0 and the buyer mplements common-queue allocaton, let c be the unque soluton to c R 2 c = 2 c 2 c + If R>r 2 = 2c /2, then c c s the unque Nash equlbrum n the capacty game and the servers average lead tmes are fnte. If R r 2, then there does not exst an equlbrum wth fnte lead tmes. Common-queue allocaton has the desrable feature that t pools the capactes of the servers (there are never watng obs and dle servers at the same tme). Hence, wth nonstrategc and dentcal servers, common queue s n fact optmal for the buyer. However, an equlbrum wth fnte lead tmes does not exst wth common-queue allocaton f the prce s too low, R r 2. Furthermore, Glbert and Weng (1998) demonstrate that wth strategc servers common queue can be worse for the buyer than balanced allocaton because t does not provde suffcent ncentves for the servers to work quckly. Hence, a state-dependent allocaton polcy may actually perform worse than a statendependent polcy. Although common queue s optmal for the buyer gven symmetrc capactes, t s not optmal for the buyer wth asymmetrc capactes. Intutvely, f one server s much slower than the other server, then the buyer may be better off allocatng a ob to the busy fast server than to the dle slow server; e.g., a fast server may be able to complete two obs faster than the slow server can complete one ob. Ths ntuton suggests a threshold allocaton polcy that s mplemented as follows. One server s labeled the prmary server and the other the secondary server. A sngle parameter, m 0 1 2, regulates how obs are allocated to the prmary and secondary servers: allocate a ob to the prmary server f the prmary server s dle or f the prmary server has fewer than m obs n queue; allocate a ob to the secondary server only f the secondary server s dle, the prmary server s busy, and has m obs n queue. It s natural to thnk of the faster server as the prmary server, but the polcy can also be mplemented wth the slower server desgnated as the prmary. Gven nonstrategc servers, Rubnovtch (1985b) provdes a numercal method to evaluate the system s performance under threshold allocaton, and Ln and Kumar (1984) prove that a threshold polcy s the buyer s optmal allocaton wth two servers,.e., the average tme n the system for each unt s mnmzed. Addtonal proofs are avalable from Koole (1995) and Walgrand (1984). It s ntutve that as the threshold parameter, m, ncreases, the prmary server s share of the buyer s demand ncreases and the secondary server s share decreases. Wth m =, the prmary server earns the buyer s entre demand, whle the secondary server s never allocated a ob. Hence, by varyng whch server s desgnated the prmary and by randomzng between dfferent m values, the buyer s able to allocate to the faster server any porton of the buyer s demand. 6 As a result, t s possble to desgn a threshold polcy n whch server s allocaton exactly equals 6 Even wth m = 0, the faster server, when desgnated the prmary, can earn more than 50% of the buyer s demand. Threshold

7 414 Management Scence 53(3), pp , 2007 INFORMS hs allocaton wth lnear allocaton for any chosen capactes. Servers only care about ther share of the buyer s obs, not how that allocaton s acheved or the resultng lead tme for the buyer. Therefore, f the descrbed threshold polcy s used, the equlbrum n the capacty game s equvalent to the equlbrum wth lnear allocaton. Furthermore, n equlbrum the servers have equal capacty, so the threshold s m = 0,.e., n equlbrum the servers buld capacty as f lnear allocaton were mplemented, but the system actually acheves the same lead tme as common-queue allocaton. Although the technques n Rubnovtch (1985b) allow for the evaluaton of the proper thresholds, a threshold polcy s clearly not as smple to evaluate as the other allocaton polces we dscuss. However, n theory, t provdes n equlbrum the maxmum capacty lke lnear allocaton, whle also provdng the operatonal effcency of common-queue allocaton. Hence, t s an optmal state-dependent polcy for the buyer. We conclude that there need not exst a trade-off between ncentves and effcency: The optmal state-dependent polcy, threshold allocaton, performs better than the optmal state-ndependent polcy, lnear allocaton. Addtonal comparsons among the polces can be made va some graphcal examples. For each allocaton polcy, Fgures 1 and 2 show the relatonshp between R and the equlbrum lead tmes wth two examples: c = 4 and = 1; and c = 4 2 and = 1. We see from these fgures that for a gven prce the buyer s lead tme can vary consderably. In all cases, common-queue allocaton and proportonal allocaton wth = 1 perform poorly. Bell-Stdham allocaton gves ntermedate performance. Balanced allocaton performs reasonably well when an equlbrum exsts, but an equlbrum exsts for a relatvely lmted range of prces (t never exsts wth lnear capacty cost). Overall, threshold allocaton s clearly the best, but lnear allocaton, especally gven ts smplcty, s a good second choce. The next lemma further explores the dfference between lnear and threshold allocaton. Lemma 7. Defne zr = W sd t R/W s l R where t R and l R are the equlbrum capactes under threshold and lnear allocatons, respectvely, when the prce s R. Recall that t R = l R,.e., for a fxed wholesale prce, threshold and lnear allocatons generate the same capacty. The rato zr s concave and ncreasngfrom 1/2 to 1. allocaton can assgn less demand to the faster server only f the faster server s desgnated the secondary server. Fgure 1 Lead tme The Lead Tme Receved by the Buyer as a Functon of the Prce Pad, R, and the Allocaton Polcy wth Capacty Cost c = 4 and = 1 t l bs p c R Note. t = threshold polcy, l = lnear allocaton, bs = Bell-Stdham, c = common queue, p = proportonal allocaton wth = 1. Balanced allocaton s not ncluded because an equlbrum does not exst n ths settng. The comparson between threshold and lnear allocaton s ntutve: If system utlzaton s qute hgh because R s low, then threshold allocaton has a sngle queue wth a large number of obs, whereas lnear allocaton has two queues wth a large number of obs (.e., threshold s lead tme s half of lnear s lead tme). However, f system utlzaton s qute low because R s hgh, then obs never wat wth ether allocaton polcy. Although Lemma 7 ndcates that lnear allocaton s sgnfcantly worse than threshold allocaton when the buyer s prce s low, ths result s somewhat ms- Fgure 2The Lead Tme Receved by the Buyer as a Functon of the Prce Pad, R, and the Allocaton Polcy wth Capacty Cost c = 4 2 and = 1 Lead tme t l b bs c p R Note. t = threshold polcy, l = lnear allocaton, b = balanced allocaton, bs = Bell-Stdham, c = common queue, p = proportonal allocaton wth = 1.

8 Management Scence 53(3), pp , 2007 INFORMS 415 leadng. Now suppose that the buyer s able to modfy her prce somewhat. Let R t be the prce wth threshold allocaton and let R l be the prce wth lnear allocaton and choose these prces such that they lead to the same delvery lead tme, W sd t R t = W s l R l. Accordng to the next lemma, f R t s ether low or hgh, then there s a small prce premum needed wth lnear allocaton to acheve the same lead tme. Lemma 8. Let be the system s utlzaton n equlbrum. R l /R t 1 as ether 1 or Lead Tme-Based Allocaton Ths secton consders whether the buyer could do better (or at least as well) wth an allocaton polcy based on the servers lead tmes rather than based on ther capactes. In a lead tme-based allocaton, the buyer announces the demand share functon n terms of servers lead tme vector W = W 1 W 2 >0, the servers submt ther bds on lead tmes, demand shares are determned, and each server bulds capacty W W to fulfll ts lead-tme bd, where s a decreasng functon of W. Assume 1 W 1 W < 1 W 1 1 W 1 W W (7) where W = W 1 + W 2 and >0: f server 1 promses a longer lead tme, then server 1 s requred capacty to acheve that lead tme decreases faster than server 1 s demand allocaton. The analogous assumpton s taken for the other server as well. Ths assumpton holds, for example, when each server operates an M/M/1 queue, n whch case W W = 1/W + W (8) Lead tme-based allocaton s analytcally cleaner than capacty-based allocaton because there s no ssue wth the stablty of the queues: By defnton, the buyer s lead tme s postve and fnte for any strategc choce vector of the servers, whereas wth capactybased allocaton the servers may fal to choose a suffcent capacty to yeld a fnte lead tme for the buyer. However, accordng to the next lemma, analytcal tractablty can come wth a prce. Lemma 9. Consder any contnuous lead-tme allocaton wth decreasngn W. If W 1 W 2 s a Nash equlbrum wth correspondngdemand shares 1 2 and W W satsfes (7), then W W ˆ for all, where ˆ s the soluton to c ˆ = R. (If c s lnear, let ˆ =.) Recall that s the servers maxmum capacty (.e., c = R/2) and the capacty acheved wth lnear or threshold allocaton based on capactes. It s possble that the maxmum achevable capacty wth a lead tme-based allocaton polcy, ˆ, s less than the maxmum wth a capacty-based allocaton polcy,. We demonstrate ths wth two examples n whch the relatonshp between a server s lead tme and ts capacty s gven by (8),.e., a state-ndependent allocaton polcy s mplemented. Frst, suppose the capacty cost functon s quadratc, c = a 2 + b and a>0. Then > ˆ for all R r 1 a+ b 2 + a 2,.e., for suffcently small R n the feasble range (R>r 1 the buyer cannot desgn a contnuous allocaton polcy based on the servers lead tmes that acheves the maxmum capacty,. Next suppose c = a for a>0 and >1. In ths case, ( ( ) 1/ ) 1/ 1 ˆ = R/2a1/ R/a = r1 1/ 1 R where recall that r 1 = c/2//2 = a/2 1. It follows that > ˆ when R r 1 r 1,.e., lead tmebased allocaton s lkely to be nferor to capactybased allocaton because the buyer s prce s low and as the capacty cost functon becomes more convex ( ncreases). Despte the one-to-one relatonshp between a server s lead tme and the server s capacty for a fxed allocaton, lead tme-based allocaton may not be as effectve as capacty-based allocaton because lead tme-based allocaton has a self-restranng property that dampens competton among the servers: Commttng to a hgher servce level requres more nvestment than commttng to a hgher capacty. A smlar result s obtaned n Cachon and Zpkn (1999) n the context of nventory management n a seral supply chan wth two ndependent frms: Wth nonstrategc frms the optmal polcy can be mplemented as ether a set of nstallaton base-stock polces or as a set of echelon base-stock polces (brefly, these polces dffer n what nformaton they use), but wth strategc frms these two approaches yeld dfferent equlbrum results. 4. Dscusson Ths secton dscusses several modelng ssues. Although we assume exponental processng tmes, some of our results extend to more general processng tme dstrbutons. As n Bell and Stdham (1983), suppose s the servce rate and the servce tme S has frst moment ES = 1/ and second moment ES 2 = b 2. The varance s then b 1 2 and the coeffcent of varaton s constant, b 1 1/2. For an M/G/1 queue wth the above servce tme dstrbuton, the average lead tme (duraton n the system) s W= 1 + b 2

9 416 Management Scence 53(3), pp , 2007 INFORMS Balanced, lnear, proportonal, Bell-Stdham, and threshold allocatons readly extend to ths general dstrbuton because demand s allocated based only on the servers capactes and not on ther lead tmes. However, the extenson s not straghtforward for common queue because then the servers shares of demand are endogenously determned. Throughout our analyss we have assumed that each ob s processed by only one server. In practce, there are examples n whch frms duplcate ther orders across multple supplers or servers (see Armony and Plambeck 2005, L 1992, Yoffe 1990). If order duplcaton s feasble, then t s deal from the pont of vew of system effcency: Even f there s only one ob n the system all servers are workng at ther full rate. However, as we have demonstrated, t s also mportant for an allocaton polcy to provde suffcent ncentves for strategc servers to work hard. Zhang (2004) demonstrates that order duplcaton performs poorly on ncentves, so poorly that ts overall performance tends to be worse than lnear allocaton. Hence, even f operatng condtons are deal for order duplcaton, a buyer should avod order duplcaton. We use demand allocaton as the motvator to provde fast servce, but other motvators may exst. For example, f the buyer has some control over the prce, R, then rasng the prce, as we see n Fgures 1 and 2, generates faster servce (but wth some allocaton polces t also elmnates the exstence of an equlbrum). The buyer could make a trade-off between the hgher prce pad and the faster servce receved. Nevertheless, unless the prce pad s extremely hgh, there remans consderable varaton n the performance of the allocaton polces. Instead of allocatng demand, the buyer could try to motvate faster servce by postng a payment schedule that s contngent on the servers capactes or lead tmes. For example, suppose the buyer wants each server to buld >/2 capacty. Ths s achevable wth the followng prce schedule, R, R <0, R /2 = c, and R /2 = c : The frst condton ensures a unque maxmzes the server s proft, the second condton ensures that s optmal for the server, and the thrd condton makes the server s proft condton bndng. It s also possble that the R schedule could be mplemented wth a fxed prce and late fees, because then the late fees pad are contngent on the chosen capacty. (See Cachon and Zhang 2006 for a smlar model wth sole sourcng and late fees.) Our model does not address whether demand allocaton s preferable to these or other contractng methods. However, we pont out that these contractng methods requre the buyer to possess sgnfcant barganng power over the servers the buyer must be able to control the prcng schedule used and ts parameters, whereas demand allocaton can be mplemented by the buyer even f the buyer has lttle barganng power. Therefore, because allocaton polces are smple to mplement and observed n practce, we suspect they are desrable vs-à-vs other technques along at least some dmensons. Overall, addtonal research s needed to dentfy the stuatons n whch demand allocaton s the best opton for the buyer. In our analyss we assume the supplers have dentcal cost functons, whch s reasonable n markets that have homogeneous technologes. Ths naturally leads to symmetrc equlbra. Wth heterogeneous cost functons, equlbrum analyss s more challengng. Zhang (2004) provdes some ntal results and fnds that exstence s less lkely as costs become more heterogeneous: As one suppler gans a cost advantage t becomes necessary to dampen the competton among the supplers to prevent one suppler from drvng the other suppler out of the market, ust lke the problem we see when R s too hgh n the symmetrc cost case. Hence, performance-based allocaton of demand appears to be most effectve when supplers have comparable costs. Our model assumes that each suppler only serves the buyer, as n the case when a suppler bulds or reserves dedcated capacty for the buyer. In some cases each suppler may cater to multple buyers, thereby creatng two strategc decsons for each suppler: how much capacty to buld and how to prortze that capacty across buyers. Furthermore, there may be dfferent prces for dfferent prortes. The analyss of these systems s clearly beyond the scope of ths research, but we agan suspect that the buyer could use a smartly desgned allocaton polcy to obtan hgher prorty from supplers. We conduct our analyss n the context of a queueng system, but there are also stuatons that may be better modeled as an nventory system: e.g., each suppler could choose a base-stock polcy and the buyer s concerned wth some dmenson of the suppler s delvery lead-tme dstrbuton. Whle the specfcs of the analyss would dffer, we suspect that demand allocaton would agan be a useful tool for the buyer to motvate for better relablty among her supplers. Our analyss s conducted exclusvely n steady state. For example, we assume that the buyer s able to nfer each server s capacty from the servers delvery tmes so that the correct demand share can be mplemented. In practce the buyer would only obtan an estmate of each server s capacty. The sgnfcance of samplng error on our results s an open queston. Fnally, we have mplctly assumed that the buyer s able to credbly commt to mplement the chosen allocaton polcy. Wthout that ablty, the buyer s set of allocaton polces to choose from s qute lmted. For example, f the buyer must mplement a

10 Management Scence 53(3), pp , 2007 INFORMS 417 state-ndependent polcy, then only Bell-Stdham allocaton s credble because t mnmzes the buyer s watng tme for any set of capactes chosen. If the buyer mplements a state-dependent polcy, then only threshold allocaton s credble, but not necessarly the same threshold polcy dscussed n 3.2. Agan, the threshold polcy must be chosen so as to mnmze the buyer s watng tme for any capacty vector. Hence, the ablty to credbly commt to an allocaton polcy s mportant to the buyer. We note that ths same ssue occurs n many other settngs. For example, n the supply chan contractng lterature, many coordnatng contracts are studed and observed that requre commtments: a buy-back contract s an a pror commtment by a suppler to pay a retaler for unts returned by the retaler after stochastc demand occurs even though the suppler has no ex post ncentve to do so. 5. Concluson In ths paper, two queueng servers strategcally choose ther capactes/processng rates n response to a buyer s demand allocaton polcy. The buyer s obectve s to desgn the allocaton polcy to acheve the shortest possble average delvery tme from the servers, ether by motvatng the servers to buld more capacty or by ensurng that the avalable capacty s effectvely utlzed. We focus on allocaton polces based on the servers capactes because we show that lead tme-based allocaton polces may not perform as well. Prevous research suggests that there may exst a trade-off between ncentves and effcency: An allocaton polcy that effcently utlzes the servers capacty may provde weak ncentves for them to work quckly, and an allocaton polcy wth strong ncentves to work quckly may not effectvely utlze the servers capacty. We ndeed demonstrate that there s consderable varaton n the performance across allocaton polces. Many allocaton polces ether provde absolutely no ncentve for the servers to delver quckly or provde too much competton among servers, thereby leadng to unpredctable behavor. Even polces that are optmal for the buyer wth nonstrategc and symmetrc servers can perform poorly wth strategc servers. However, we show that there need not be a trade-off between ncentves and effcency,.e., there exsts an allocaton polcy, threshold allocaton, that nduces the servers to work at ther maxmum rate and mnmzes the buyer s lead tme, gven the resultng capactes. Unfortunately, threshold allocaton s complex. For example, ts optmal parameters cannot be determned n closed form. We offer lnear allocaton as an alternatve. Lnear allocaton also nduces the servers to work at the maxmum possble rate, but lnear allocaton does not utlze the servers capacty as effectvely as threshold allocaton. In partcular, because lnear allocaton allocates obs mmedately upon arrval and the assgnment of obs does not depend on the current state of the system (t s a state-ndependent allocaton polcy), lnear allocaton may allocate a ob to a busy server whle the other server s dle. Nevertheless, we show that lnear and threshold allocatons converge n performance at hgh utlzatons, whch suggests that lnear allocaton s attractve along many dmensons. To conclude, a buyer should not gnore demand allocaton as a strategy to obtan faster servce, especally gven ts smplcty: There s no need to negotate new contract terms or prcng wth the servers because demand allocaton can be mplemented by a buyer wthout the explct consent of the servers. However, creatng competton among servers va ther past performance requres some sophstcaton; a haphazard applcaton of ths strategy could have lttle mpact. Acknowledgments The authors thank Saf Benaafar, Noah Gans, Martn Larvere, Wllam Loveoy, Sergue Netessne, Erca Plambeck, and Yong-Pn Zhou for ther helpful comments, as well as the semnar partcpants at Columba, Cornell, New York Unversty, Northwestern Unversty, Unversty of Calforna at Irvne, Unversty of Mnnesota, Unversty of Washngton, the Second MIT Symposum n Operatons Research: Procurement and Prcng Strateges to Improve Supply Chan Performance, and the Competton wth Delays meetng at Washngton Unversty. An electronc verson of ths paper s avalable from the authors webpages. The prevous verson of ths paper was ttled Procurng Fast Delvery, Part I: Mult-sourcng and Scorecard Allocaton of Demand va Past Performance. Appendx Proof of Theorem 1. Server s proft functon s = R c. Let 0 be defned such that 0 > 0 and 0 = 0 or 0 = 0. s then concave and decreasng for 0 0. Now dfferentate, ( = R 1 1/2 1/2 ( ) 1/2 ) ( 1/2) 2 c 2 1/2 and, for notatonal convenence, let B =, 2 2 = R1/2 4 ( 1/2 ) 3 ( 1/2 B 3/2 + 3B 1 3 1/2 1/2 1 ) c Defne f = B 1/2 3/2 + 3B 1 3 1/2 1/2 1. If B 0, then 0 and s concave. Otherwse, t can be shown that df /d = 0 has only one postve soluton. Moreover, f and df /d < 0as 0 and f<0as. Thus, f decreases from the postve doman to the negatve

11 418 Management Scence 53(3), pp , 2007 INFORMS doman. Because c 0, there exsts a 1 0 such that s concave and decreasng for 0 0, convex for 0 1, and concave for 1 <. Because 0 = 0, t follows that any nteror soluton to server s frst-order condton s a global optmum f at that soluton proft s nonnegatve. The followng equaton provdes the unque soluton to the frst-order condtons gven the constrant = : ( )( R 1 + /2 ) c 4 = 0 (Because c s convex, the left-hand sde s decreasng, so there s a unque soluton.) The lower bound on R ensures that bs >/2. The condton bs 0 ensures that bs s ndeed a global optmum for all servers. Proof of Theorem 2. There are two sgnfcant complcatons to ths analyss that prevent the use of standard exstence and unqueness results. (1) s not unmodal (f 2 >, then 1 s concave and decreasng for and concave for 1 > 2, but not globally concave), whch may create a dscontnuty n the servers best reply functons. (2) s not dfferentable at =, whch prevents the uncondtonal use of frst-order condtons to determne the global maxmzer of. Let s frst establsh when b b s a Nash equlbrum under the gven condtons. The servers frst-order condtons are satsfed when c b = R/2, whch yelds a fnte lead tme only f b >/2, whch smplfes to the frst condton. However, because = 0 can be optmal for a server, b s an optmal response only f b b 0,.e., f R/2 c b, whch can be wrtten as c b c b / (the second condton). Now let s rule out other equlbra. Suppose s an equlbrum,. Several cases need to be consdered. () +.If > 0, then server earns a negatve proft, so ths s not an equlbrum. If = 0, then t must be that =. For server we have b = c b b c b >0, breakng the equlbrum. () + <, whch mples </2. From server s frst-order condton we get c = R>2c /2, whch mples >/2 because c s ncreasng. Hence, we have a contradcton, so no equlbrum. () + =. For ths to be optmal for both servers t must be that R 2c and R 2c, whch cannot both be satsfed because R>2c /2. (v) + > and <+. Now the only soluton to the frst-order condtons s b b. To obtan an equlbrum wth fnte lead tmes, we need + >. Because + cannot be an equlbrum, there must be <+, whch mples that b b s the only soluton to the frst-order condtons. However, b b s not an equlbrum f c b <c b /, and b </2f R<r 2. Proof of Theorem 3. () For Nash equlbrum we need to show that l maxmzes server s proft f the other server chooses = l. The prmary complcaton s due to the revenue term, R, n the proft functon. Server s allocaton s = mn /2 l /2 + /2 + {( = mn /2 ( c l + } l 1)) 2 c l The second term s negatve, so there exsts some 0 such that = 0 for all 0.If = l, then = /2. The condton R>r 1 ensures that /2 < l, so t follows that 0 < l. The server s proft functon s concave and decreasng for 0 and concave and contnuous for > 0, although possbly not dfferentable when =. For > 0,by constructon of the parameters, s maxmzed wth = l and = 0 wth = l. Therefore, l s optmal for server. Lead tmes are fnte because l >/2. Next we concentrate on unqueness. Suppose s a Nash equlbrum of the capacty game. The proof frst rules out asymmetrc equlbrum wth postve capacty for all servers and then equlbrum wth = 0 for some are ruled out. Suppose n some equlbrum > 0 for all. It must be, then, that > 0 for all (otherwse server would make negatve proft). Thus, the frst-order condton for each server must be satsfed gven n = 2, but that yelds = l for all because the soluton to each server s frstorder condton depends only on. Now suppose there exsts a = 0 n equlbrum. All servers choosng = 0 cannot be an equlbrum because then one server could buld a small amount of capacty and earn postve proft. If server 1 has the only postve capacty, then server 1 receves 1 = as long as 1 > 0; ths cannot be an equlbrum because then server 1 s optmal capacty s some arbtrarly small capacty, whch then allows the other server to buld postve capacty and earn postve proft. (The condton > 0 for all n n the allocaton functon s crtcal to ths result.) () The proof s smlar to (), so t s omtted. Proof of Theorem 4. For server = R 1 ( ) 2 2a + b = 1 2 Gven R and, smple algebra reveals that = p s a symmetrc soluton to the frst-order condtons, and t s the only soluton. If each server chooses p, then each server earns a zero proft, so t s an equlbrum f max = 0. Dfferentate: 2 2 = R a Note that = 2 c /c > 2. It can be shown (see Zhang 2004 for detals) that s concave-convex-concave f = p. Because = 0 and < 0 when = 0, t must be that max = 0. Therefore, 1 = 2 = p s the unque Nash equlbrum of the capacty game. Proof of Theorem 5. Frst demonstrate that = p for all s a unque Nash equlbrum when R>r 2. The frstorder condtons must be satsfed: = R ) 2 c ( = 0 = 1 2 (9) The frst-order condtons mply =, so the only soluton must be = p for all. The condton R>r 2 ensures that the soluton to the frst-order condtons has >/2, whch provdes fnte lead tmes. Now suppose R r 2 and there s an equlbrum wth fnte lead tmes. If the lead tmes are fnte, then the frst-order condtons (9) hold and only = satsfy them. Agan, because lead tmes are fnte, t must