Provably Near-Optmal LP-Based Polces for Revenue Management n Systems wth Reusable Resources Retsef Lev Ana Radovanovc 21 February 2007 Abstract Motvated by emergng applcatons n workforce management, we consder a class of revenue management problems n systems wth reusable resources. The correspondng applcatons are modeled usng the well-known loss network systems. We use an extremely smple lnear program (LP) that provdes an upper bound on the best achevable expected long-run revenue rate. The optmal soluton of the LP s used to devse a conceptually smple control polcy that we call the class selecton polcy (CSP). Moreover, the LP s used to analyze the performance of the CSP polcy. We obtan the rst control polcy wth unform performance guarantees. In partcular, for the model wth sngle resource and unform resource requrements, the CSP polcy s guaranteed to have expected long-run revenue rate that s at least half of the best achevable. More generally, as the rato between the capacty of the system and the maxmum resource requrement grows to nnty, the CSP polcy s asymptotcally optmal, regardless of any other parameter of the problem. The asymptotc performance analyss that we obtan s more general than exstng results n several mportant dmensons. It s based on several novel deas that we beleve wll be useful n other settngs. retsef@mt.edu. Sloan School of Management, MIT, Cambrdge, MA, 02139. Part of ths research was conducted whle the author was a postdoctoral fellow at the IBM, T. J. Watson Research Center. aradovan@us.bm.com. IBM T. J. Watson Research Center, P.O. Box 218, Yorktown Heghts, NY 10598.
1 Introducton In ths paper, we consder a class of revenue management problems that arse n systems wth reusable resources. The paper s motvated by several applcaton domans, and, n partcular, by several emergng applcatons n workforce management. In many ndustres, a sgncant part of the workforce s hred adhoc to perform a specc project. Thus, professonal manpower servces s a growng market that brngs up new challenges n workforce revenue management. Smlar problems arse n large corporatons, such as IBM that need to manage ther nternal workforce n the face of dynamc and evolvng tasks. The major ssue n all of these scenaros s how to manage capactated resources over tme n dynamc envronments wth many uncertantes, speccally, how to choose the most protable customers/projects to maxmze the resultng revenue. Other notable applcatons are hotel room bookng and car rentals. Typcally, these systems consst of several capactated resources that are used to serve multple classes of customers, each of whch has dfferent characterstcs, such as arrval rate, prce, resource and servce tme requrements. The goal s to devse a polcy that selects protable customers and maxmzes the resultng revenue. There are three key characterstcs of these systems. The rst characterstc s the reusablty of resources. That s, resources that are allocated to serve a certan customer/project wll become avalable to serve other customers after the servce/project s over. The second characterstc s that the decson whether to serve customers should be made upon ther arrval. In partcular, f a customer s not served upon arrval, ether because the system decdes she/he s not protable enough, or because the avalable capacty n the system s not sufcent to satsfy her/hm, she/he s assumed to be lost and leaves the system. (In many of the correspondng applcatons customers are not wllng to wat or only wllng to wat a very short tme relatve to the servce tmes.) The thrd characterstc s that the arrval process of customers, as well as ther servce tme requrements, are stochastc. Ths generates stochastc optmzaton models that are usually computatonally challengng. In ths paper, we model the correspondng revenue management problems as loss network models. These are well-known models that have been ntroduced over four decades ago, and have been studed extensvely n the context of communcaton networks (see, for example, the survey paper by Kelly ([13])). The classcal loss network model conssts of a system wth several capactated resources that faces multple classes of customers. Customers of dfferent classes arrve accordng to mutually ndependent homogenous Posson processes, each of whch requres a certan combnaton of resources for a tme that s a-pror random (wth nte mean), and s wllng to pay a certan prce per unt of servce tme. Customers must be served upon ther arrval, or otherwse, they leave the system. If a customer s served, the requred combnaton 1
of resources must be engaged for the (random) duraton of the servce tme, and can not be used by other customers untl the servce s over. The system may deny servce from customers n order to keep the capacty free for more protable future customers. A customer can be served only f at the moment of arrval the avalable capacty n the system s sufcent to satsfy her/hs specc requrements. The goal s to nd an admsson polcy that maxmzes the long-run revenue rate. Lke many stochastc optmzaton models, one can formulate the problem usng a dynamc programmng approach. However, even n specal cases (e.g., wth exponentally dstrbuted servce tmes), the resultng dynamc program seems computatonally ntractable as the correspondng state-space grows very fast. (Ths s known as the `curse of dmensonalty'.) Thus, ndng provably good polces s a very challengng task. We rst focus on revenue management model wth sngle reusable resource, where there s only a sngle resource n the system that s used to serve multple classes of customers as descrbed above. (In the lterature on loss network models ths s sometmes called the stochastc knapsack problem.) We use a smple knapsack-type lnear program (LP) that provdes an upper bound on the expected long-run revenue rate. The LP can be easly solved, and the optmal soluton s used to construct a conceptually smple admsson control polcy for the orgnal model; the polcy s called the class selecton polcy (CSP). The LP optmal soluton gudes the polcy to select the more protable classes. The CSP polcy admts all the customers of the selected (protable) classes as long as capacty permts, and always rejects customers from other classes. Moreover, the LP s used to analyze the performance of the CSP polcy. We use the fact that the CSP polcy nduces a stochastc process that can be reduced to a classcal loss network model. Facltatng the results from [21, 11, 10, 4, 23], whch characterze the statonary dstrbutons of the correspondng loss network models, we are able to develop explct expressons for the resultng blockng probabltes nduced by the CSP polcy. That s, for each one of the protable classes, we derve an exact expresson for the statonary probablty that a customer arrves at some random tme, and the avalable capacty n the system s not sufcent to satsfy her/hs requrement. We then bound the customer blockng probabltes and analyze ther asymptotc behavor as the capacty of the system grows large. In partcular, the bounds on the blockng probabltes are used to obtan unform and asymptotc performance guarantees. For the case, where all the classes requrements are dentcal, we obtan an explct lower bound on the rato between the expected long-run revenue rate of the CSP polcy and the best achevable rate. The bound s a functon only of the capacty of the system, regardless of the other parameters, such as arrval rates, number of classes, prces and servce tme dstrbutons. It s shown that ths bound s at least 0.5, unformly for all capacty values, and that t approaches 1 as the capacty of the system grows to nnty. That s, the CSP polcy s guaranteed to have expected long-run revenue rate that s at least half of the best achevable, 2
and t s asymptotcally optmal as the capacty of the system grows large. To the best of our knowledge, ths s the rst proof of unform performance guarantees that hold for all capacty values. These results are then extended to the more general case wth arbtrary, possbly non-dentcal, resource requrements. In ths case, the underlyng combnatorcs of the blockng states s more complex, and hence, t s much harder to bound the correspondng blockng probabltes. In partcular, the lower bounds on the rato between the long run average revenue rate of the CSP polcy and the best achevable rate depend only on the rato between the capacty of the system and the maxmum resource requrement of a class rased to the power of seven. Moreover, f the rato between the capacty and the maxmum resource requrement rased to the power of seven grows to nnty, the CSP polcy s asymptotcally optmal, regardless of the other parameters of the problem. For each xed value of maxmum resource requrement, t s possble to derve unform performance guarantees that depend only on the system capacty value. The CSP polcy and the asymptotc performance analyss can be extended to the revenue management model wth multple reusable resources as long as the number of resources s bounded. In ths case, the resultng lnear program s a packng-type LP that s agan very easy to solve. Fnally, we ncorporate statc prcng to the sngle resource model. In ths model we rst determne the respectve prces that each class s charged. The respectve arrval rate of each class depends on the prce t s charged. After the prces are set, we wsh to nd the best admsson control polcy that maxmzes the expected long-run revenue rate. The CSP polcy and ts performance analyss can be extended to ths more general model. However, the polcy s derved based on a non-lnear program (NLP). We show how to smplfy the resultng NLP, and dscuss several scenaros n whch t can be solved efcently. As we already mentoned loss networks have been studed extensvely n the context of communcaton networks, and there s a huge body of lterature. The study of loss networks has been focused on two major ssues, the study of heurstcs and senstvty analyss. Snce t s apparent that computng optmal polces s lkely to be ntractable, researchers have proposed dfferent heurstcs, studed ther propertes and analyzed ther performance (see, for example, [17, 19, 14, 13, 8, 18, 7]). The knapsack-type LP used n ths paper has been dscussed by several researchers (see for example, [14, 8]). In fact varants of the CSP polcy have been dscussed by Key [14] and Kelly [13], who proposed the randomzed thnnng polcy. Moreover, Key [14] has shown that the varant of the CSP polcy for the sngle resource case s asymptotcally optmal, but n a very specc heavy trafc regme. (We dscuss ths further n the next paragraph.) Iyengar and Sgman [9] have also used an LP dentcal to the one used n ths paper to devse a heurstc for the same model. However, the polcy they have proposed s very dfferent than ours. Speccally, they have used the LP to generate a `desrable' target performance mode, and, then, 3
explot exponental penalty functons to mantan the system as close as possble to the target mode. Another polcy that has been studed extensvely s the trunk reservaton polcy. Accordng to ths polcy, each class of customers s assocated wth a trunk reservaton level, and a customer of that class s admtted to the system only f upon arrval the avalable capacty n the system exceeds the correspondng trunk reservaton level. Key [14] has shown that for the sngle resource model ths polcy s asymptotcally optmal n the correspondng heavy trafc regme. (It s nterestng to note that the CSP polcy can be vewed as a specal trunk reservaton polcy, where the trunk reservaton of a class s ether 0 or the capacty of the system.) Other mathematcal-programmng-based approxmaton have been used to study models smlar to the one dscussed n ths paper (see, for example, Adelman [2]). However, the performance analyss of the polces descrbed above and ther asymptotc optmalty are obtaned only n very specc regmes, usually called heavy trafc regmes. Speccally, the capacty and the arrval rates are scaled smultaneously at the same (lnear) rate, whle all the other parameters of the problem, such as servce tme dstrbutons, resource requrements, number of classes and prce rates are kept xed. In some scenaros one also needs to assume that the servce tmes are exponentally dstrbuted. Moreover, n models wth multple resources t was usually requred to assume a very specc structure of the dfferent class resource requrements. The performance of the correspondng polces s then analyzed n the resultng lmtng regmes. Whle these mght be reasonable assumptons n the context of communcaton networks, they are less lkely to hold n the applcaton domans that motvate ths paper. In contrast, our analyss provdes unform performance guarantees that hold for any capacty value. Our asymptotc performance analyss holds under very general assumptons. Speccally, the asymptotc analyss holds for general servce tme dstrbutons, and only requres that the rato between the capacty of the system and the maxmum resource requrement (rased to the power of seven) grows to nnty, allowng all the other parameters of the problem to change arbtrarly. In addton, we can easly characterze the correspondng rate of convergence. Thus, as a by-product of our work, we obtan generalzatons for some of the results n [14, 13]. The second major ssue that has been studed s the senstvty of the correspondng loss system to changes n varous parameters, especally the capacty and the arrval rates (see, for example, [20]). The man effort has been to study changes n the resultng blockng probabltes. (By blockng we refer to the event that a customer arrves at some random tme, and can not be served upon arrval because the avalable capacty n the system s not sufcent.) Snce computng blockng probabltes s known to be #P -Hard [16], there have been efforts to propose methods to approxmately compute blockng probabltes and bound them (see, for example, [6, 21, 10, 4, 23, 12, 13, 25]). We note that there have been several approaches that 4
use lnear and non-lnear programs to bound blockng probabltes (see, for example, [15, 3]). One of the key features n our performance analyss s the bounds that we develop on the correspondng blockng probabltes nduced by the CSP polcy. The technques that we use are sgncantly dfferent than the ones used n the exstng lterature, and we beleve that they wll have applcatons n other settngs. It s nterestng to note that our asymptotc analyss n the multple resources case captures the specc regme analyzed by Kelly n [13], where he analyzed the blockng probabltes n loss network models wth multple resources. The rest of the paper s organzed as follows. In Secton 2, we provde the mathematcal formulaton of the revenue management model wth a sngle reusable resource. In Secton 3, we develop the LP and descrbe the CSP polcy. In Secton 4, we dscuss the performance analyss of the CSP polcy. Fnally, n Secton 5, we dscuss the extensons to multple resources and statc prcng. 2 Model Formulaton In ths secton, we provde a mathematcal formulaton of the revenue management model wth a sngle reusable resource dscussed n ths paper. Consder a system wth a sngle resource pool of nteger capacty C < that s facng demands from M dfferent classes of customers. The customers of each class = 1,..., M, arrve accordng to an ndependent Posson process wth rate λ. Each class- customer requests A Z + unts of the resource for a certan perod of tme that s a-pror random and has nte mean µ. Durng the tme a class- customer s served, the requested A unts can not be used by other customers; after the servce s over, the unts become avalable agan to serve other customers. (Whle a customer s served we only know the condtonal dstrbuton of the resdual servce tme of ths customers.) In partcular, we allow generally dstrbuted servce tmes, and assume that servce tmes are ndependent of the customer arrval process and among dfferent customers. If served, a class- customer s wllng to pay r dollars per tme unt of servce. A customer can be served only f the avalable capacty n the system at the moment of her/hs arrval s sufcent to satsfy ts requrement. That s, a class- customer can be served only f there are currently at least A unts avalable n the system. However, customers can be rejected even f the avalable capacty s sufcent to serve them. (Rejectng a customer now possbly enables servng more protable customers n the future.) The assumpton s that customers that are not satsed upon arrval, ether due lack of sufcent capacty or because they are rejected, are lost and leave the system mmedately. The goal s to nd an admsson polcy that maxmzes the expected long-run revenue rate. For each polcy π, let R π (T ) be the revenue acheved by polcy π over the nterval [0, T ]. Next dene 5
R(π), the expected long-run revenue rate of a polcy π as R(π) =: lm T E π [R π (T )], (1) T where the expectaton E π s taken wth respect to probablty measure nduced by polcy π. At any pont of tme t, the state of the system s speced by the class of each customer currently beng served, as well as the tme that elapsed from the moment of her/hs arrval. Wthout loss of generalty, we restrct attenton to state-dependent polces. That s, polces that are represented as measurable functons from the state-space dened above to actons. (The actons are whether to accept a class- customer n the current state, for each = 1,..., N.) It s straghtforward to verfy that for each feasble polcy, there exsts a state-dependent polcy that acheves at least the same expected long-run average revenue. Note that each state-dependent polcy nduces a Markov process over the state-space. Moreover, usng analogous arguments to those used n Theorems 4, 5 and 6 of [21], one can show that condtons n the statement of Theorem 1 n [21] hold and, therefore, the Markov process homogeneous n tme that s nduced by the state dependent polcy has a unque statonary dstrbuton whch s ergodc. Thus, for each state-dependent polcy π, the lmt n (1) above s well-dened and the expectaton n the numerator of (1) can be omtted. As n most stochastc control optmzaton models, dynamc programmng framework s the most common way to formulate the problem. However, t s straghtforward to see that the correspondng state-space of the underlyng dynamcs program s very large even for smple specal cases. In partcular, t seems computatonally ntractable to solve the dynamc program and compute an optmal polcy. For example, consder the specal case wth servce tmes that follow exponental dstrbutons. The state n ths case s speced merely by the number of customers of each class currently beng served. However, the correspondng statespace grows exponentally fast n the capacty C, and becomes computatonally ntractable. 3 LP-Based Approach In ths secton, we construct a smple lnear program (LP) that provdes an upper bound on the achevable long-run average revenue rate. Our LP s dentcal to the one used by Key [14] and Iyengar and Sgman [9], and t s also smlar to the one used by Adelman [2] n the queueng networks framework wth unt resource requrements. We shall show how to use the optmal soluton of the LP to construct a smple admsson control polcy that s called class selecton polcy (CSP). Moreover, the LP wll be used to analyze the performance of the proposed CSP polcy. In partcular, we shall show that the expected long-run revenue rate of the CSP polcy s guaranteed to be near-optmal for any capacty value C (above a certan threshold) 6
and that the polcy s asymptotcally optmal. More speccally, we shall show that R(CSP ) β(c/a 7 )R(OP T ), (2) where OPT denotes the optmal control polcy, A denotes the maxmum resource requrement (.e., A = max,...,m A ) and β(c/a 7 ) s a postve scalar for each value of C/A 7. (If no optmal polcy exsts, we thnk about R(OP T ) as the correspondng supreme of the achevable expected revenue rate.) Furthermore, f the resource requrements of all classes are dentcal,.e., A = 1 for each = 1,..., M, then β(c) 1/2 for all C 1. Moreover, as C grows larger the CSP polcy becomes asymptotcally optmal. That s, β(c) approaches 1 as C grows to nnty, and ths occurs rrespectve of other model parameters, such as the number of classes, arrval rates, servce duratons and prce rates. For the model wth non-unform resource requrements, we establsh a smlar results. The CSP polcy s asymptotcally optmal as C/A 7 grows large. That s, we show that β(c/a 7 ) approaches 1 as C/A 7 grows to nnty. In addton, for each xed value A, t s possble to compute a threshold C, such that for each capacty values C C, the CSP polcy s guaranteed to have expected long-run revenue rate at least half of the best achevable. 3.1 An LP We have already dscussed n Secton 2 that any state-dependent polcy nduces a Markov process on the state-space of the system wth a unque statonary dstrbuton that s ergodc. In partcular, for each class and a gven state-dependent polcy π, there exsts a statonary probablty α (π) for acceptng a class- customer, whch s equal to the long-run proporton of accepted customers of class whle runnng the polcy π. Thus, any state-dependent polcy π s assocated wth the statonary probabltes α (π) 1, α(π) 2,..., α(π) M. Furthermore, by applyng Lttle's law, we can use the statonary probablty α (π) to express the expected number of class- customers beng served n the system under state-dependent polcy π as λ µ α (π) = ρ α (π). (Note that ρ = λ µ s the expected number of class- customers beng served n the system wth nnte capacty and no rejectons; n the context of communcaton networks t s usually called the trafc ntensty.) It follows that the expected long-run revenue rate of polcy π can be expressed as M r α (π) λ µ = M r α (π) ρ. Smlarly, the overall expected long-run number of resource unts beng engaged to serve customers can be expressed as M α (π) ρ A. 7
The physcal constrans of the system dscussed n ths paper mply that, for any feasble polcy, t s not possble to nd more than C unts beng used to serve customers. We conclude that M for any feasble state-dependent polcy π. Ths suggests the followng LP: max α 1,...,α M s.t. α(π) ρ A C M r α ρ (3) M α ρ A C (4) 0 α 1, 1 M. (5) Note that, for each feasble state-dependent polcy π, the correspondng vector α (Π) = (α (π) 1, α(π) 2,..., α(π) ) s a feasble soluton for the LP dened by (3)-(5) above, and has objectve M value that s equal to the expected long-run revenue rate of polcy π. In fact, the LP enforces the capacty constrant of the system only n expectaton, whle n the orgnal problem ths constrant has to hold for every sample path. It follows that the LP dened by (3)-(5) relaxes the orgnal problem and provdes an upper bound on the optmal expected long-run revenue rate. Moreover, the LP dened above s a knapsack LP. Thus, t can be solved optmally by applyng the followng greedy rule. Wthout loss of generalty, assume that classes are renumbered such that r 1 /A 1 r 2 /A 2 r M /A M. Then, for each = 1,..., M, we set α = 1 as long as constrant (4) s satsed. In partcular, the optmal soluton has the followng structure: α 1 = α 2 = = α M 1 = 1, for some 1 M M; for M the correspondng value of α M s possbly a fracton,.e., 0 < α M 1; and, for = M + 1,..., M, we have α = 0. Next we shall use the optmal soluton of the knapsack LP dened above to construct an extremely smple admsson polcy, and show that ts expected long-run revenue rate s guaranteed to be near-optmal n the sense dened n (2) above. 3.2 The Class Selecton Polcy Let α = (α1,..., α M ) be the optmal soluton of the knapsack LP dened by (3)-(5) above. We propose the followng polcy that we call class selecton polcy: Consder an arrval of a class- customer ( = 1,..., M): For each = 1,..., M 1, accept the customer as long as the avalable capacty n the system upon arrval s sufcent (.e., greater than A unts); If = M, accept wth probablty 0 < α M 1 and as long as there avalable capacty n the system upon arrval s sufcent (.e., greater than A M unts); 8
For each = M + 1,..., M, reject. The CSP polcy has a very smple structure. It always admts customers from the classes for whch the correspondng value α n the optmal LP soluton equals to 1, as long as capacty permts; t never admts customers from classes for whch the correspondng value α equals to 0; t ps a con for the possbly one class wth fractonal value αm. The CSP polcy s conceptually very ntutve n that t splts the classes nto protable and non-protable that should be gnored. 4 Performance Analyss In ths secton, we analyze the performance of the CSP polcy. The specal propertes of the CSP polcy nduce a well-structured stochastc process. Each class = 1,..., M generates an ndependent Posson arrval stream wth respectve rate α λ. Thus, each class wth α = 1 generates the orgnal process, each class wth α = 0 can be gnored, and possbly one class wth fractonal 0 < αm < 1 generates a thnned Posson process. Moreover, the nduced stochastc process can be descrbed as a classcal loss network model wth a sngle resource. There are C servers that are used to serve M ndependent Posson streams of requests. The requests of stream (class) arrve at rate λ, and each requres A servers for some random servce tme wth mean µ ; whenever a request arrves and the number of dle servers s not sufcent to serve t, the request s lost and leaves the system. It can be easly vered that the loss network model descrbed above s dentcal to the stochastc process nduced by the CSP polcy. One of the natural questons studed n the context of loss networks s what s the statonary blockng probablty of a gven request. That s, what s the statonary probablty that a certan request arrves at some random tme and the number of dle servers n the system s not sufcent to serve t. The latter queston s drectly related to the performance analyss of the CSP polcy. Focus on the classes wth postve α, say there are M of them. Wthout loss of generalty, assume that there s no fractonal varable n the optmal soluton α,.e., for each = 1,..., M, α = 1. (If αm s fractonal, we thnk of class M as havng an arrval rate λ M = α M λ and then elmnate the fractonal varable from α.) For each = 1,..., M, let P be the statonary probablty of rejectng a class- customer under the CSP polcy. It s straghtforward to verfy that probablty P s equal to the statonary blockng probablty of a stream- request n the correspondng loss network model descrbed above. Speccally, let X be the the statonary (random) number of class- (stream-) customers beng served n the system at some random tme under the 9
CSP polcy. Then by the PASTA property (see [24]), t follows that, for each = 1,..., M : P = P ( M ) X k A k > C A. (6) k=1 Moreover, snce the correspondng stochastc process s ergodc, t follows that the long-run proporton of class- customers beng served equals to 1 P. Thus, the expected long-run revenue rate of the CSP polcy can be expressed as r α ρ (1 P ) = r ρ (1 P ). (7) M However, M r α ρ s the optmal value of the LP, whch s an upper bound on the achevable expected long-run revenue rate for any feasble polcy. Thus, a key aspect of the performance analyss of the CSP polcy s lower boundng probabltes 1 P, or equvalently, upper boundng the probabltes P. In partcular, (7) above mples that any unform constant bound on these probabltes can be drectly translated to a performance guarantee of the CSP polcy. Speccally, f 1 P β for each = 1,..., M, t follows that R(CSP ) = r α ρ (1 P ) r α ρ β βr(op T ). M In the rest of the analyss, we shall establsh upper bounds on the correspondng blockng probabltes P 1,..., P M, and analyze ther asymptotc behavor. M Recall that the CSP polcy nduces a stochastc process that can be reduced to a classcal loss network wth a sngle resource. A central tool n our analyss s the result of Burman et. al. [4], who characterzed the statonary probabltes for general loss network models (see Theorem 2 n [4]). In fact, the result of Burman et. al. [4] mples that the statonary probabltes of the correspondng loss network model can be expressed through the counterpart system wth no capacty constrans. That s, consder an nnte capacty system that faces Posson streams of requests/customers of class 1,..., M, wth respectve rates λ 1,..., λ M and servce tme dstrbutons wth respectve means µ 1,..., µ M, and accept all the requests/customers. In partcular, for each = 1,..., M, let Y be the statonary number of class- customers beng served n the nnte capacty system descrbed above. Then, for each n = 0,..., C, we have ( M ) ( M ) P k=1 A ky k = n P A k X k = n = ( M ). (8) k=1 P k=1 A ky k C M Equaton (8) s very useful, snce the varables Y 1,..., Y M are ndependent of each other, and for each = 1,..., M, the random varable Y follows a Posson dstrbuton wth parameter ρ. Thus, n conjuncton 10
wth (6) above, we get that, for each = 1,..., M P = C n=c A y Y(n) C n=0 y Y(n) ρ y 1 1 ρ y 1 1 y 1!... ρ y 1!... ρ y M M y M! y M M y M!, (9) where we dene Y(n) = {y Z M + : M j=1 A jy j = n}. Next we use the explct expresson n (9) and the LP Constrant (4) to unformly bound customer rejecton probabltes, and obtan unform and asymptotc performance guarantees of the CSP polcy. 4.1 Identcal Resource Requrements In ths secton, we dscuss the specal case, n whch the resource requrements of all classes are dentcal; wthout loss of generalty, A = 1, for each = 1,..., M. In ths specal case, n vew of (6), the rejecton probabltes are dentcal for all classes 1,..., M. Speccally, P = P( M k=1 X k = C), where X k, as before, denotes the statonary (random) number of class-k customers beng served n the system under the CSP polcy. Lemma 4.1 Consder the revenue management model wth a sngle reusable resource and dentcal resource requrements,.e., A 1 = A 2 = = A M = 1. Wthout loss of generalty, assume that there s no fractonal varable n the soluton α, and that for each class = 1,..., M (M M), we have α = 1. Then, for each = 1,..., M, the blockng probabltes P has the followng propertes: () P 0.5 for all capacty values C 1. () The probablty P dmnshes to 0 as C grows to nnty, regardless of other parameters of the problem Proof: such as the prce rates, servce tme dstrbutons and number of classes. Recall that, n vew of dscusson above, probabltes P s are the same and equal to blockng probabltes n the correspondng loss network model descrbed above. That s, P = P( M k=1 X k = C). We use the followng dentty y Y(n) ρ y 1 1 y 1!... ρym M y M! = (ρ 1 + + ρ M ) n, n! that holds for each n = 0,..., C. In conjuncton wth (9) above we obtan P = (ρ 1 + +ρ M ) C C! C (ρ 1 + +ρ M ) n n=0 n!. 11
Consder the functon f(z) = zc /C! P C. By lookng on the dervatve t s straghtforward to check that k=0 zk /k! f(z) s ncreasng n z on (0, C]. Ths and Constrant (4) mply that P C C C! C. (10) n=0 Cn n! However, from the propertes of the Posson dstrbuton, t follows that functon g(c) e C CC P C! C Cn n=0 e C n! s decreasng n C. One way to show ths s by replacng x! usng dentty 6.1.38 from [1] rst, and, then, showng that functon log(g(c)) s decreasng snce ts dervatve wth respect to C s negatve for all C 1. More speccally, for any x > 0, for some θ (0, 1). Then, t s not hard to show that, for all C 1, x! = 2πx x+1/2 e x+ θ 12x, (11) d dc log(g(c)) 1 2C + 1 12C 2 < 0. Thus, n vew of the prevous observaton, the maxmum value of g(c) s g(1) = 1/2. Moreover, expresson C n=0 Cn n! e C approaches 0 as C grows to nnty (see 6.5.34 of [1]). Usng Strlng approxmaton, we can characterze the rate of convergence. Speccally, we know that C C C! C k=0 Ck k! 2 1 π C for large values of C. Ths concludes the proof of the lemma. We have obtaned the followng theorem. Theorem 4.2 Consder the revenue management model wth a sngle reusable resource and dentcal resource requrements,.e., A 1 = A 2 = = A M expected revenue rate of the CSP polcy s guaranteed to be at least g(c) = = 1. Then, for each capacty value C the long-run e C C C P C! C Cn n=0 e C n! of the best achevable long-run expected revenue rate. In partcular, the long-run expected revenue rate of the CSP polcy s guaranteed to be at least half of the best achevable, for all capacty values C 1. Moreover, the CSP polcy s asymptotcally optmal as capacty C grows to nnty. In lght of Theorem 4.2 above, we note that f C = 1, t s usually straghtforward to compute the optmal polcy. In fact, one can mprove the performance guarantee of the CSP polcy by solvng the problem exactly for small values of C, and apply the LP-based polcy only for values of C, where t becomes computatonally ntractable to compute the optmal polcy. Takng ths strategy wll mprove the overall performance guarantee. 12
We also note that the bound n Theorem 4.2 above s tght wth respect to the LP dened by (3)-(5) above. Speccally, consder the case where there s a sngle class wth ρ 1 = 1 = C. Clearly, the optmal polcy s to accept every customer as long as capacty permts. It can be vered that the expected long-run revenue rate of the optmal polcy s half the optmal value of the LP. Thus, there s no hope of provng unform performance guarantees stronger than half usng the LP as the only upper bound. Fnally, note that the performance analyss presented above stll holds f the prce rate of each class s a random varable wth mean r. 4.2 Non-dentcal Resource Requrements In ths secton, we consder the case, where each class may have dfferent resource requrement A Z +. Agan, we assume wthout loss of generalty that there s no fractonal varable n α, the optmal soluton of the LP, and that there are M M classes wth postve α varable. That s, α = 1, for each = 1,..., M. It s now clear that the respectve rejecton (blockng) probabltes P 1,..., P M are not dentcal. In partcular, by Theorem 2 of [4], for each = 1,..., M, the rejecton probablty P can be expressed as ( M ) P (C A < ) M k=1 Y ka k C P = P X k A k > C A = ( M ). (12) k=1 P k=1 Y ka k C It follows that f A A j then P P j. However, the analyss n ths case s more nvolved compared to the analyss n the case wth dentcal resource requrements. The man dfculty comes from the fact that f there are several classes wth dfferent resource requrements, then M k=1 Y ka k follows a compound Posson dstrbuton, whereas n the case of dentcal requrements t follows a Posson dstrbuton. (In the latter case we can assume that all the resource requrements are equal 1.) Ths makes the analyss sgncantly harder. Nevertheless, we wll show that the CSP polcy s asymptotcally optmal, even f the resource requrements are not dentcal. Let A = max,...,m A be the maxmum resource requrement by a class. We shall show that all blockng probabltes P 1,..., P M dmnsh to zero as rato C/A 7 grows to nnty. In lght of (12) and the monotoncty of the blockng probabltes n the respectve class resource requrement, t s sufcent to show that P(C A<P M k=1 Y ka k C) P( P M dmnshes to 0 as the rato C/A 7 grows to the nnty. Speccally, k=1 Y ka k C) we wll show that P(C A < M k=1 Y ka k C) dmnshes to 0, and that P( M k=1 Y ka k C) s asymptotcally at least 0.25 as the rato C/A 7 grows to nnty. Indeed, ths mples that the blockng probabltes P 1,..., P M dmnsh to 0 as the rato C/A 7 grows to nnty. We rst focus on the case where λ = M ρ C 2A. That s, the total trafc ntensty s relatvely hgh. 13
Lemma 4.3 Assume that λ C 2A. Then probablty P(C A < M k=1 Y ka k C) dmnshes to 0 as the rato C/A 4 grows to nnty. Proof: Wthout loss of generalty, we assume that M A. Otherwse, there are two classes wth the same resource requrement, and we can consder them n the analyss as one class wth ntensty equal to the sum of the ntenstes of the orgnal classes. (Ths wll not change the probablty P(C A < M k=1 Y ka k C).) Thus, A max ρ λ C 2A and, therefore, there exsts at least one class wth ρ C 2A 2 ; wthout loss of generalty, assume that M s that class. Let S be the set of all feasble states. That s, S = {y Z M + : 0 M y A C}. Let B S be the set of all blockng states. That s, { } M B = y Z M + : C A < y A C. Let S M be the projecton of S onto the space of the varables (y 1,..., y M 1). Speccally, { } S M = y Z M 1 + : 0 M 1 y A C For each y S M, let P (y) = P r(y 1 = y 1,..., Y M 1 = y M 1). Focus now on y S M, and consder the set of values of y M, for whch state (y, y M ) s a blockng state,.e., (y, y M ) B. Denote ths set by B M (y ). By denton B M (y ) = {y M. Z + : C A < M 1 y A + y M A M C}. We conclude that the set B M (y ) conssts of at most A A A consecutve ntegers. Moreover, snce the M random varables Y 1,..., Y M are ndependent, t follows that ( ) M P C A < Y A C = P (y) y S M y M B M (y) P(Y M = y M ) (13) max P(z A < Y M M z) AρρM z 0 ρ M! e ρ M. The rst nequalty follows from the fact that for each y S M, the set B M (y) conssts of at most A consecutve ntegers and y SM P (y) 1. (Ths s the sum of probabltes of dsjont events.) The second nequalty follows from the propertes of the Posson dstrbuton. In partcular, the maxmum probablty assgned to a specc value s at most ρρ M M ρ M! e ρ M, and as we have already seen, the set B M (y) contans at most A values. Fnally, t s straghtforward to check that functon h(z) zz z! e z s monotone decreasng n z. Ths would follow from dentcal arguments used to justfy monotoncty of functon g(z) n the proof of Lemma 4.1. Then, snce ρ M C, we obtan 2A 2 ( ) M P C A < Y A C A 14 C 2A 2 C 2A 2 C 2A 2! e C 2A 2, (14)
where, for any postve x, x! satses dentty (11). Moreover, usng Strlng approxmaton, we know that the functon C 2A 2 C 2A 2 C 2A 2! e C 2A 2 behaves lke 1 2π 1 q C 2A 2 as C/A 2 grows to nnty. Therefore, as C/A 4 grows to nnty, probablty P(C A < M Y A C) dmnshes to 0. Ths concludes the proof of the lemma. Next, we show that as C/A 7 grows to nnty, probablty P(0 M Y A C) s asymptotcally at least 0.25. Lemma 4.4 Assume that λ C 2A. Then probablty P(0 M k=1 Y ka k C) s asymptotcally at least 0.25 as C/A 7 grows to nnty. Proof: Recall that random varable M k=1 Y ka k follows a compound Posson dstrbuton. Let W be a Posson random varable wth parameter λ = M ρ, and let Q be a dscrete random varable that takes value A k wth probablty ρ k /λ, for each k = 1,..., M. Next, we express random varable M k=1 Y ka k as W Q, where {Q } s a sequence of ndependent and dentcally dstrbuted random varables equal n dstrbuton to Q, and are ndependent of W. (It s easy to verfy that random varable W Q has the same dstrbuton as M k=1 Y ka k.) In order to prove the lemma, we develop two lower bounds for probablty P(0 W Q C); one of the bounds s based on the Central Lmt Theorem and the other on Chebyshev nequalty. Then, we show that, as the rato C/A 7 grows to nnty, at least one of the two lower bounds s asymptotcally at least 0.25. Clearly, ( W ) P Q C ( W ) P(W λ)p Q C W λ λ P(W λ)p Q C W λ λ = P(W λ)p Q C. The rst nequalty follows from the fact that we restrct attenton only to event [W λ]. The second nequalty follows from the fact that Q 's are nonnegatve, W s an nteger-valued random varable and we restrct attenton to event [W λ]. The equalty follows from the fact that W s ndependent of λ Q. Frst, consder the last expresson n (15) above. Focus on term P(W λ). By the propertes of the medan of the Posson dstrbuton (see [22]), we know that P(W λ) 0.5 0.5P(W = λ ). However, we have already seen that probablty P(W = λ ) dmnshes to 0 as λ C 2A grows to nnty. Thus, t follows that as C 2A grows to nnty, probablty P(W λ) s asymptotcally at least 0.5. 15 (15)
Next, focus on term P( λ Q C). From Constrant (4) t follows that C λe[q 1 ] λ E[Q 1 ], and we obtan λ λ P Q C P (Q E[Q 1 ]) 0. However, the random varables Q s are ndependent and dentcally dstrbuted and λ C A grows to nnty. Thus, a Central Lmt argument can be appled. In partcular, we use the well-known Berry- Essen bound to express devaton from the Normal dstrbuton (see, for example, [5]). Speccally, let σ 2 = V ar[q] be the varance of Q and γ = E[Q 3 ] ts thrd moment. (Both exst and are nte.) Then, λ P (Q E[Q 1 ]) 0 γ Φ(0) τ 0 σ 3 (16) λ = 0.5 τ 0 γ σ 3 λ, where τ 0 > 0 s a unversal constant ndependent of Q, and Φ(x) s the cumulatve dstrbuton functon of the standard normal dstrbuton. Snce γ A 3, we can extend (16) above to get P( λ (Q E[Q 1 ]) 0) 0.5 τ 0A 3 σ 3 λ. Snce λ C 2A t follows that, as C/A7 grows to nnty, and unless σ 3 s gong to 0 faster than (C/A 7 ) 1/2, the lower bound developed n (15) above approaches 0.25. Note that we only assume that the rato C/A 7 grows to nnty, rrespectve of other parameters of the model (whch can be arbtrary). In partcular, we can not assume that σ s a bounded from below. Indeed the bound n (16) above becomes meanngless when σ 3 approaches zero faster than (C/A 7 ) 1/2. Thus, we develop the second lower bound for P( W Q C) that s based on Chebyshev nequalty, and s strong when σ s close to 0. Usng smlar arguments to those n (15) above, we can condton on event [W λ λ], and obtan ( W ) P Q C ( P W λ ) λ P Focus on the rst term P(W λ λ). We clam that, as λ λ λ C 2A Q C. (17) grows to nnty, the probablty P(W λ λ) approaches = 0.5 1 2π. We have already seen that P(W λ) approaches 0.5. Thus, t s sufcent to show that P(λ λ < W λ) s asymptotcally at most 1 2π. However, by arguments dentcal to the one used n the proof of Lemma 4.3 above, we obtan that P(λ λ < W λ) λλ λ! e λ λ. Usng Strlng approxmaton, we conclude that, ndeed, P(λ λ < W λ) approaches 1 2π as λ C 2A grows to nnty. ( λ ) λ Now, we focus on the second term P Q C. Observe that C λe[q], the expectaton [ λ ] λ E Q = λ [ λ ] λ λ E[Q] and the varance V ar Q = λ λ σ 2. Thus, by 16
applyng Chebyshev nequalty, we derve λ λ P Q > C P λ λ (λ λ)σ 2 ( λ 1) 2 (E[Q]) 2 λσ 2 /( λ 1), where the last nequalty follows from the fact that E[Q] 1. (Q E[Q 1 ]) > λ E[Q 1 ] (18) Now, f λσ 2 /( λ 1) 1 0.25/, t follows that the lower bound developed n (17) above approaches 0.25 as C/A grows to nnty. Otherwse, we get λ 1 σ 2 > (1 0.25/ ), λ and the lower bound developed n (15) above s asymptotcally at least 0.25 as C/A 7 grows to nnty. Ths concludes the proof of the lemma. Lemmas 4.3 and 4.4 mply the followng theorem. Theorem 4.5 Assume that λ C 2A. Then the blockng probabltes P 1,..., P M dmnsh to 0 as the rato C/A 7 grows to nnty. Fnally, we dscuss the case where λ = M ρ < C 2A. Usng the notaton ntroduced above, we wsh to show that probablty P( W Q > C A) dmnshes to 0 as rato C/A grows to nnty. Indeed, P( W Q > C A) P(W > C A C A ). (If W, t s clear that W Q C A snce Q A A.) However, by the propertes of Posson random varable (see 6.5.34 of [1]), t follows that probablty P(W > C A A ) dmnshes to 0 as C A grows to nnty. (The mean of W s at most C 2A.) We have obtaned the followng theorem. A Theorem 4.6 The CSP polcy s asymptotcally optmal for the revenue management model wth sngle reusable resource as the rato C/A 7 grows to nnty, regardless of other parameters of the problem. We note that, for each xed value of A one can use Lemmas 4.3 and 4.4 to derve unform performance guarantees smlar to the one developed n Theorem 4.2. 4.3 A Comparson of the Performance Analyss wth Exstng Approaches Next we would lke to contrast our performance analyss wth exstng lterature. As already mentoned, there are several results that establsh the asymptotc optmalty of dfferent polces. However, most f not 17
all of the exstng results assume at least one of the followng: () smultaneous scalng of the arrval rates and the capacty (That s, we set C n = nc and λ n = nλ, and let n grow to nnty.); () other parameters of the problem, such as the number of classes and resource requrements and servce tme dstrbutons, are kept xed; () (n some cases) exponentally dstrbuted servce tme. In contrast, our analyss holds for general servce tme dstrbutons, and t only requres that the rato C/A 7 grows to nnty, lettng other parameters of the problem, such as arrval rates, number of classes and resource requrements, be arbtrary. The analyss hghlghts the fact that the most mportant characterstc of the problem s the rato between the capacty and the resource requrements of dfferent classes. Moreover, the fact that our performance analyss reles on the LP-based upper bound enables us to compute explct and unform performance bounds. As we shall show n the next secton ths analyss extends to models wth multple resources. 5 Extensons 5.1 Revenue Management Model wth Multple Reusable Resources In ths secton, we show how to use deas analogous to those used n the sngle resource case to extend the CSP polcy and the performance analyss to models wth multple resources. We present a packng-type LP that provdes an upper bound on the best achevable revenue rate and use ts optmal soluton to construct the CSP polcy and analyze ts performance. As before we can use general Erlang formulas (see, for example, Burman et. al. [4]) to express the statonary class rejecton (blockng) probabltes. Ths s agan used to show the asymptotc optmalty of the CSP polcy n a way smlar to the analyss of the sngle resource case dscussed n Secton 4 above. Let C j be the capacty of resource j = 1,..., N, and let A j Z + be the number of unts of resource j requested by class. One can wrte a packng-type LP that provdes a smlar upper bound on the best achevable revenue rate. Speccally, the objectve s the same as (3), but we have a constrant M α ρ A j C j, for each j = 1,..., N. Ths LP can be solved and one can derve the CSP polcy that accepts class- customer wth probablty α as long as there s sufcent amount of avalable resources to satsfy her/hs requrement. In vew of the dscusson at the begnnng of Secton 4, one can extend the same arguments to ths case as well, and express the rejecton probablty of a class k M customer as ( P N j=1 {C j A kj < ) M k=1 Y ka kj C j } P k = ( ), (19) P N j=1 { M k=1 Y ka kj C j } where Y k, 1 k M, are ndependent Posson random varables wth mean values α k ρ k, 1 k 18
M. Next, note that for every 1 j N, random varable M k=1 Y ka kj follows a compound Posson dstrbuton. Let W be a Posson random varable wth parameter λ = M ρ α, and let Q (j), 1 j N, be a dscrete random varable that takes value A kj wth probablty ρ k /λ. Next, we express random varable M k=1 Y ka kj, 1 j N, as W Q(j), where for each 1 j N, {Q (j) } s a sequence of ndependent and dentcally dstrbuted random varables equal n dstrbuton to Q (j) and ndependent of W. Frst, we show that the numerator n (19) dmnshes to zero as mn j C j /A 4 j approaches nnty, where A j max 1 k M A kj. By unon bound, ( { }) M P C j A j < Y k A kj C j N j=1 k=1 ( N W P C j A j < Usng arguments analogous to one used n the proof of Lemma 4.3, each summand on the rght hand sde of the prevous expresson approaches zero as mn 1 j N (C j /A 4 j ) approaches nnty. Snce the number of resources N s a nte constant, we obtan that P( N j=1 {C j A j < M k=1 Y ka kj C j }) dmnshes to zero as mn 1 j N (C j /A 4 j ) grows to nnty. Next, we show that the denomnator n (19) can be bounded by some postve constant as mn 1 j N C j /A 7 j approaches nnty. In vew of the prevous dscussons, ( { M }) ( P Y k A kj C j = P N j=1 k=1 j=1 N j=1 { W and, smlarly as n (15), we lower bound ths expresson by condtonng on event [W λ f(n) λ], ( ) wth f(n) = N a, where a s the smallest postve real number such that N max j Φ f(n) A j < 1. (We use Φ(x) to denote the tal of the standard normal dstrbuton at x.) Therefore, we obtan ( { W }) P N j=1 Q (j) l C j P(W λ f(n) λ f(n) λ λ)p N j=1 Q (j) C j. (20) l=0 Furthermore, by applyng the unon bound, we further lower bound the left hand sde of (20) and get ( { W }) P C j P(W λ f(n) λ f(n) N λ λ) 1 P > C j. (21) N j=1 l=0 Q (j) l Next, we show that both terms on the rght hand sde of (21) can be lower bounded by postve constants as mn j C j /A 7 j j=1 l=0 Q (j) l l=0 C j Q (j) l }), Q (j) C j approaches nnty. Before proceedng, we wll dfferentate between two cases: () λ mn j C j /2A j, and () λ < mn j C j /2A j. We rst focus on the rst case. Snce W s a Posson random varable, the probablty P(W λ f(n) λ) P( λ f(n) λ ɛ λ), where {ɛ } 1 s a sequence of ndependent exponentally 19 ).
dstrbuted random varables wth mean 1. Thus, P(W λ f(n) λ) P P λ f(n) λ λ f(n) λ (ɛ 1) f(n) λ λ f(n) λ = P (ɛ 1) λ f(n) λ Φ f(n) λ λ f(n) λ ɛ λ λ f(n) λ f(n) λ 1 τ 0 λ f(n), λ 1 where n the last nequalty we appled Berry-Essen nequalty analogously as n (16). Now, f mn j C j /A 7 j grows to nnty, then λ mn j C j /2A j grows to nnty as well, mplyng that P(W λ f(n) λ) approaches Φ(f(N)). Fnally, snce the number of resources N s nte, Φ(f(N)) > 0. Next, we shall show that the sum n the second term of (21) can be upper bounded by a constant less than 1 as mn j C j /A 7 j approaches nnty. In partcular, we upper bound each summand P( λ f(n) λ Q (j) > C j ), 1 j N. Dene σ 2 j V ar[q (j) ]. Then, smlarly as n the proof of Lemma 4.4, dependng on whether σ 3 j approaches zero faster than (C j/a 7 j ) 1/2, or not, the upper bound s based on Chebyshev nequalty, or Berry-Essen bound. In partcular, by Berry-Essen nequalty and λe[q (j) ] C j, we obtan λ f(n) λ λ f(n) λ P Q (j) > C j P (Q (j) E[Q (j) ]) > f(n) λe[q (j) ] (22) λ f(n)λ (Q (j) E[Q (j) ]) P σ j λ f(n) λ Φ f(n) λe[q (j) ] σ j λ f(n) + τ 0 λ σ 3 j > f(n) λe[q (j) ] σ j λ f(n) λ A 3 j λ f(n). λ Now, when λ C j /2A j and σj 3 does not go to zero faster than (C j/a 7 j ) 1/2, we apply the bound n (22) ( λ f(n) ) λ and obtan that P Q (j) > C j does not exceed Φ(f(N)/A j ) as C j /A 7 j approaches nnty. On the other hand, f σ 3 j approaches zero faster than (C j/a 7 j ) 1/2, we apply Chebyshev nequalty (usng the facts that λe[q (j) ] C j and E[Q (j) ] 1) to obtan λ f(n) λ P > C j P Q (j) λ f(n) λ (Q (j) λ f(n) (f(n)) 2 λ σ2 j. E[Q (j) ]) > f(n) λe[q (j) ] (23) 20
It follows that f σ 2 j satses ( λ f(n))σ 2 j ( Φ f(n) (f(n)) 2 λ ( λ f(n) ) λ then bound (23) mples that the probablty P Q (j) > C j does not exceed Φ(f(N)/A j ) as C j /A 7 j approaches nnty. Otherwse, the same asymptotc upper bound holds by (22). Fnally, we show that when λ mn j C j /2A j and mn j C j /A 7 j approaches nnty, then for each 1 ( λ f(n) ) λ j N, the probablty P Q (j) > C j does not exceed asymptotcally Φ(f(N)/A j ). Gven the assumpton that N max j Φ(f(N)/Aj ) < 1, the asymptotc lower bound for P(W λ f(n) λ) ( { W }) derved above, and (21) above, t follows that P N j=1 l=0 Q(j) l C j can be lower bounded by a postve constant as mn j C j /A 7 j approaches nnty. In a way smlar to one used n the case wth a sngle resource wth non-unform requrements, we show that when λ < mn j C j /2A j, then for each 1 j N, the probablty P( W l=1 Q(j) A j ), l > C j ) P(W > ( N j=1 C j /A j ) dmnshes to zero as C j /A j approaches nnty. Ths mples that the probablty P 1 ( N j=1 P W ) l=1 Q(j) l C j approaches 1 as mn j C j /A 7 j approaches nnty. Ths concludes the analyss n the multple resource case. { W }) l=1 Q(j) l C j 5.2 Prce-Drven Customer Arrvals In ths secton, we consder an extenson of the model dscussed n Secton 2, n whch the arrval rates of the dfferent classes of customers are affected by prces. Speccally, consder a two stage decson. At the rst stage we set the respectve prces r 1,..., r M for each class. Ths determnes the respectve arrval rates λ 1 (r 1 ),..., λ M (r M ). (The rate of class s affected only by prce r.) Then, gven the arrval rates, we wsh to nd the optmal admsson polcy that maxmzes the expected long-run revenue rate. In partcular, we assume that λ (r ) s nonnegatve, dfferentable and decreasng n r for every 1 M. In addton, there exsts a prce, say r, such that for each = 1,..., M, we have λ (r) = 0 for r r. Usng arguments analogous to the dscusson n Secton 3, we construct an upper bound on the achevable expected long-run revenue rate through the followng nonlnear program (NLP1): max α 1,...,α M,r 1,...,r M s.t. M r α ρ (r ) (24) M α ρ (r )A C (25) 0 α k 1, 1 M. (26) 21