The Optimality of Two Prices: Maximizing Revenue in a Stochastic Network

Transcription

1 PROC. OF 45TH ANNUAL ALLERTON CONFERENCE ON COMMUNICATION, CONTROL, AND COMPUTING (INVITED PAPER), SEPT The Opimaliy of Two Prices: Maximizing Revenue in a Sochasic Newor Longbo Huang, Michael J. Neely Absrac This paper considers he problem of pricing and ransmission scheduling for an Access Poin (AP) in a wireless newor, where he AP provides service o a se of mobile users. The goal of he AP is o maximize is own ime-erage profi. We firs obain he opimum ime-erage profi of he AP and prove he Opimaliy of Two Prices heorem. We hen develop an online scheme ha joinly solves he pricing and ransmission scheduling problem in a dynamic environmen. The scheme uses an admission price and a business decision as ools o regulae he incoming raffic and o maximize revenue. We show he scheme can achieve any erage profi ha is arbirarily close o he opimum, wih a radeoff in erage delay. This holds for general Marovian dynamics for channel and user sae variaion, and does no require a-priori nowledge of he Marov model. The model and mehodology developed in his paper are general and apply o oher sochasic seings where a single pary ries o maximize is ime-erage profi. Index Terms Wireless Mesh Newor, Pricing, Queueing, Dynamic Conrol, Lyapunov analysis, Opimizaion I. INTRODUCTION In his paper, we consider he profi maximizaion problem of an access poin (AP) in a wireless mesh newor. Mobile users connec o he mesh newor via he AP. The AP receives he user daa and ransmis i o he larger newor via a wireless lin. Time is sloed wih inegral slo boundaries 0, 1, 2,...}, and every imeslo he AP chooses an admission price p() (cos per uni pace) and announces his price o all presen mobile users. The users reac o he curren price by sending daa, which is queued a he AP. While he AP gains revenue by acceping his daa, i in urn has o deliver all he admied paces by ransmiing hem over is wireless lin. Therefore, i incurs a ransmission cos for providing his service (for example, he cos migh be proporional o he power consumed due o ransmission). The mission of he AP is o find sraegies for boh pace admission and pace ransmission so as o maximize is ime erage profi while ensuring queue sabiliy. We assume ha he expeced number of new paces sen o he AP is deermined every imeslo by a demand sae variable M() and a user demand funcion F (M(), p()). Specifically, he sae variable M() represens he curren condiion of he user populaion ha affecs is aggregae spending abiliy. For example, M() can represen he ineger Longbo Huang ( longbohu@usc.edu) and Michael J. Neely (web: hp://www-rcf.usc.edu/ mjneely) are wih he Deparmen of Elecrical Engineering, Universiy of Souhern California, Los Angeles, CA 90089, USA. This wor is suppored in par by one or boh of he following: The DARPA IT-MANET program gran W911NF , he Naional Science Foundaion gran OCE number of users presen a ime, or can be a rough esimae of he aggregae willingness-o-pay (such as Low, Medium, and High ). The demand funcion F (M(), p()) is equal o he expeced number of paces ha arrive on slo under a given user condiion M() and a given price p(). We assume he AP nows he curren demand sae M() and he demand funcion F (M(), p()) for each slo. However, M() is assumed o vary according o a general finie sae ergodic Marov chain, and he ransiion and seady sae probabiliies of M() may be unnown. Similarly, he condiion of he wireless channel from AP o he mesh newor is poenially ime varying and is deermined by a Marov modulaed channel sae process S(). The AP is assumed o now he curren channel sae S() on each imeslo, alhough he ransiion and seady sae probabiliies of S() are poenially unnown. We develop a join pricing and ransmission scheduling algorihm (PTSA) for he AP. The PTSA algorihm has low complexiy and can be viewed as maing greedy decisions every imeslo. Despie is simpliciy, we show ha PTSA is able o dynamically reac o he ime varying newor condiions. I yields an erage ne profi ha can be pushed arbirarily close o he opimum, wih a corresponding radeoff in erage queueing delay. Many exising wors on revenue maximizaion can be found. Wor in [1] [2] models he problem of maximizing revenue as a dynamic program. Wor in [3] and [4] model revenue maximizaion as saic convex opimizaion problems. A game heoreic perspecive is considered in [5], where equilibrium resuls are obained. Wors [6], [7] and [8] also use game heoreic approaches wih he goal of obaining efficien sraegies for boh he AP and he users. The paper [9] loos a he problem from a mechanism design perspecive, and [10], [11] consider profi maximizaion wih Qos guaranees. Early wor on newor pricing in [12], [13], and [14] consider hroughpuuiliy maximizaion raher han revenue maximizaion. There, prices play he role of Lagrange mulipliers, and are used mainly o faciliae beer uilizaion of he shared newor resource. This is very differen from he revenue maximizaion problem, where he service provider is only ineresed in is own profi. Indeed, he revenue maximizaion problem can be much more complex due o non-convexiy issues. The above prior wor does no direcly solve he profi maximizaion problem for APs in a wireless newor for one or more of he following reasons: (1) Mos wors consider ime-invarian sysems, i.e. he newor condiion does no change wih ime. (2) Wors ha model he problem as an opimizaion problem rely heily on he assumpion ha he

2 PROC. OF 45TH ANNUAL ALLERTON CONFERENCE ON COMMUNICATION, CONTROL, AND COMPUTING (INVITED PAPER), SEPT user uiliy funcion or he demand funcion is conce. (3) Many of he prior wors adop he flow rae allocaion model, where a single fixed operaing poin is obained and used for all ime. However, in a wireless newor, he newor condiion can easily change due o channel fading and/or node mobiliy, so ha a fixed resource allocaion decision may no be efficien. Also, as has been poined ou in [15], he user uiliy funcion does no always he he conciy propery. Indeed, profi maximizaion problems are ofen non-convex in naure. Hence, hey are generally hard o solve, even in he saic case where he channel condiion, user condiion, and demand funcion is fixed for all ime. I is also common o loo for single-price soluions in hese saic newor problems. Our resuls show ha single-price soluions are no always opimal, and ha even for saic problems he AP can only maximize ime erage profi by providing a regular price some fracion of he ime, and a reduced price a oher imes. Moreover, mos newor pricing wor considers flow allocaion ha neglecs he pace-based naure of he raffic, and neglecs issues of queueing delay. An excepion is he recen wor in [16] ha considers a pace-based model for a free mare wireless newor. However, [16] focuses on neworwide efficiency and on guaranees of non-negaive profi o all paricipans, and does no consider he very differen problem of maximizing revenue for a single AP. In order o enable he AP o beer reac o he varying newor condiion, o overcome he difficuly of solving nonconvex/non-conce opimizaion problems, and o beer operae in a pace-based seing, we propose a novel join pricing and ransmission scheduling algorihm (PTSA). PTSA has he same naure as he schemes proposed in [16], which are saedependen [12], alhough i solves a very differen problem. As we will see laer, PTSA bypasses he non-conciy/nonconvexiy difficuly by urning he saic opimizaion problem ino a sochasic opimizaion problem. Our analysis of he performance of PTSA uses he Lyapunov echniques and general uiliy-opimizaion framewor developed in [17] [18] [19]. We show ha PTSA can achieve a ime-erage profi ha is arbirarily close o opimum, and obain an explici radeoff beween profi and queuing delay. In he nex secion we describe he newor model. In Secion III we characerize he opimal ime erage profi and prove he Opimaliy of Two Prices heorem. The PTSA algorihm is presened in Secion IV, where performance opimaliy is proven. Preliminary simulaion resuls are provided in Secion V. II. NETWORK MODEL We consider he newor as shown in Fig 1. The newor is assumed o operae in sloed ime, i.e. 0, 1, 2,...}. A. Arrival Model: The Demand Funcion We firs describe he pace arrival model. Le M() be he demand sae a ime. M() migh be he number of presen mobile users, or could represen he curren demand siuaion, such as he demand being High, Medium or Low. We assume ha M() evolves according o a finie sae Fig. 1. AP F(M(), p()) Newor Φ(cos(), S()) An Access Poin (AP) ha connecs mobile users o a larger newor. ergodic Marov chain wih sae space M. Le π m represen he seady sae probabiliy ha M() = m. The value of M() is assumed nown a he beginning of each slo, alhough he ransiion and seady sae probabiliies are poenially unnown. Every imeslo, he AP firs maes a business decision by deciding wheher or no o allow new daa (his decision can be based on nowledge of he curren M() sae). Le Z() be a 0/1 variable for his decision, defined as: 1 1 if he AP allows new daa on slo Z() = 0 else If he AP chooses Z() = 1, i hen chooses a per-uni price p() for incoming daa and adverises his price o he mobile users. We assume ha price is resriced o a compac se of price opions P, so ha p() P for all. We assume he se P includes he consrain ha prices are non-negaive and bounded by some finie maximum price p max. Le R() be he oal number of paces ha are sen by he mobile users in reacion o his price. The income earned by he AP on slo is hus Z()R()p(). The arrival R() is a random variable ha depends on he demand sae M() and he curren price p() via a demand funcion F (M(), p()): (1) F : (M(), p()) E R()} (2) Specifically, he demand funcion maps M() and p() ino he expeced value of arrivals E R()}. We furher assume ha here is a maximum value R max, so ha R() R max for all, regardless of M() and p(). The higher order saisics for R() (beyond is expecaion and is maximum value) are arbirary. The random variable R() is assumed o be condiionally independen of pas hisory given he curren M() and p(). The demand funcion F (m, p) is only assumed o be coninuous and o saisfy 0 F (m, p) R max for all m M and all p P. The se P is assumed only o be compac (i.e., closed and bounded), and may consis of a finie discree se of prices. Example: In he case when M() represens he number of mobile users in range of he AP a ime, a useful example model for F (M(), p()) is: F (M(), p()) = M() ˆF (p()) where ˆF (p) is he expeced number of paces sen by a single user in reacion o price p, a curve ha is possibly obained 1 The Z() decisions are inroduced o allow sabiliy even in he possible siuaion where user demand is so high ha incoming raffic would exceed ransmission capabiliies, even if price were se o is maximum value p max.

3 PROC. OF 45TH ANNUAL ALLERTON CONFERENCE ON COMMUNICATION, CONTROL, AND COMPUTING (INVITED PAPER), SEPT via empirical daa. In his case, we assume ha he number of users is bounded by some value M max and he maximum number of paces sen by any single user is bounded by some value Rmax single, so ha R max = M max Rmax single. In Secion IV, we show ha his ype of demand funcion (i.e, F (m, p) = m ˆF (p)) leads o an ineresing siuaion where he AP can mae demand sae blind pricing decisions, where prices are chosen wihou nowledge of M(). B. Transmission Model: The Rae-Cos Funcion Le S() represen he channel condiion of he wireless lin from AP o he mesh newor on slo. We assume ha he channel sae process S() is a finie sae ergodic Marov chain wih sae space S. Le π s represen he seady sae probabiliy ha S() = s. The ransiion and seady sae probabiliies of S() are poenially unnown o he AP, alhough we assume he AP nows he curren S() value a he beginning of each slo. We assume ha he ransmission rae of he AP s ougoing lin is deermined every imeslo by a resource allocaion decision (such as power) and by he curren channel sae S(). We model his decision compleely by is cos o he AP, and define cos() as he cos of he ransmission decision on slo. We assume ha cos() is chosen wihin some compac se of coss C, and ha C includes he consrain 0 cos C max for some finie maximum cos C max. The ransmission rae is hen given by rae-cos 2 funcion µ() = Φ (cos(), S()). In our problem, we assume ha Φ (cos, S()) is coninuous in he variable cos for every given S(), and ha Φ(0, S()) = 0 for all S(). Furher, we assume here is a finie maximum ransmission rae, so ha: Φ (cos(), S()) µ max for all cos(), S(), (3) We assume ha paces can be coninuously spli, so ha µ() = Φ (cos(), S()) deermines he porion of paces ha can be sen over he lin from AP o he newor on slo (for his reason, he rae funcion can also be viewed as aing unis of bis). Of course, he se C can be resriced o a finie se of coss ha correspond o inegral unis for Φ(cos(), S()) in sysems where paces canno be spli. C. Queueing Dynamics and oher Noaions Le U() be he queue baclog of he AP a ime, in unis of paces. 3 Noe ha his is a single commodiy problem as we do no disinguish paces from differen users. 4 We assume he following queueing dynamics for U(): U ( + 1) = max [U() µ(), 0] + Z()R() (4) where µ() = Φ(cos(), S()). Throughou he paper, we adop he following noion of queue sabiliy: E U} lim sup 1 1 E U(τ)} < (5) 2 This is essenially he same as he rae-power curve in [18]. 3 The pace unis can be fracional. Alernaively, he baclog could be expressed in unis of bis. 4 Our analysis can be exended o rea muli-commodiy models, alhough ha is omied for breviy. III. CHARACTERIZING THE MAXIMUM PROFIT In his secion, we characerize he opimal erage profi ha is achievable over he class of all possible conrol polices ha sabilize he queue a he AP. We show ha i suffices for he AP o use only wo prices for every demand sae M() o maximize is profi. A. The Maximum Profi To describe he maximum erage profi, we use an analysis ha is similar o he analysis of he minimum erage power for sabiliy problem in [18]. Noe ha he heorem in [18] considers he problem of using minimum erage power o serve he given incoming raffic, while in our case, he AP needs o balance beween he profi from daa admission and he cos for pace ransmission. The following heorem shows ha opimaliy can be achieved over he class of saionary randomized pricing and ransmission scheduling sraegies wih he following srucure: Every slo he AP observes M() = m, and maes a business decision Z() by independenly and randomly choosing Z() = 1 wih probabiliy φ (m) (for some φ (m) values defined for each m M). If Z() = 1, hen he AP allocaes a price randomly from a counable collecion of prices p (m) 1, p (m) 2, p (m) 3,...}, wih probabiliies } =1. Similarly, he AP observes S() = s and maes a ransmission decision by choosing cos() randomly from a se of coss cos (s) } =1 wih probabiliies β(s) } =1. Theorem 1: (Maximum Profi wih Sabiliy) The opimal erage profi for he AP, wih is queue being sable, is given by P rofi op, where P rofi op is he soluion o he following opimizaion problem: max P rofi = Income Cos (6) } s.. Income = E m φ (m) F (m, p (m) )p (m) (7) Cos = E s λ = E m φ (m) µ = E s =1 =1 =1 } β (s) cos(s) =1 } F (m, p (m) ) (8) (9) β (s) Φ ( cos (s), s ) } (10) 0 φ (m) 1 m M (11) µ λ (12) p (m) P, m M (13) cos (s) C,, s S (14) =1 =1 = 1 m M (15) β (s) = 1 s S (16) where E s and E m denoe he expecaion over he seady sae disribuion for S() and M(), respecively, and φ (m),, p (m), β (s), and cos(s) are auxiliary variables wih he inerpreaion given in he ex preceding Theorem 1.

4 PROC. OF 45TH ANNUAL ALLERTON CONFERENCE ON COMMUNICATION, CONTROL, AND COMPUTING (INVITED PAPER), SEPT The proof of Theorem 1 conains wo pars. Par I shows ha no algorihm ha sabilizes he AP can achieve an erage profi ha is larger han he opimal soluion of he problem (6)-(16). Par II shows ha we can achieve a profi of a leas ρp rofi op (for any ρ such ha 0 < ρ < 1) wih a paricular saionary randomized algorihm ha also yields erage arrival and ransmission raes λ and µ ha saisfy λ < µ. The formal proof is given in Appendix A. Because he ses P, C are compac and he funcions F (m, p) and Φ(cos, s) are coninuous for all m M and s S (where M and S he finie sae space), i can be shown ha P rofi op can be achieved by a paricular saionary randomized algorihm. 5 The following imporan corollary o Theorem 1 is somewha simpler and is useful for analysis of he online algorihm described in Secion IV. Corollary 1: There exiss a conrol algorihm ST AT ha maes saionary and randomized business and pricing decisions Z () and p () depending only on he curren demand sae M() (and independen of queue baclog), and maes saionary randomized ransmission decisions cos () depending only on he curren channel sae S() (and independen of queue baclog) such ha: EZ ()R ()} Eµ ()} (17) EZ ()p ()F (M(), p ())} Ecos ()} = P rofi op (18) where P rofi op is he opimal ime erage profi, and where µ () = Φ(cos (), S()). The above expecaions are aen wih respec o he seady sae disribuions for M() and S(). Specifically: E Z ()R ()} = E m Z ()F (m, p ())} E µ ()} = E s Φ(cos (), s)} B. The Opimaliy of Two Prices The following wo heorems show ha insead of considering a counably infinie collecion of prices p (m) 1, p (m) 2,...} for he saionary algorihm of Corollary 1, i suffices o consider only wo price opions for each disinc demand sae M() M. Theorem 2: Le (λ (m), Income (m) ) represen any raeincome uple formed by a saionary randomized algorihm ha chooses Z() 0, 1} and p() P, so ha: E Z()F (M(), p()) M() = m} = λ (m) E Z()p()F (M(), p()) M() = m} = Income (m) Then: a) (λ (m), Income (m) ) can be expressed as a convex combinaion of a mos hree poins in he se Ω (m), defined: Ω (m) = (ZF (m, p), ZpF (m, p)) Z 0, 1}, p P} b) If (λ (m), Income (m) ) is on he boundary of he convex hull of Ω (m), hen i can be expressed as a convex combinaion 5 The same heorem can be shown for disconinuous funcions and noncompac ses, bu he opimum is hen a supremum over all saionary algorihms ha saisfy he consrains of Theorem 1. This generalizaion would no affec he resuls or analysis of our PTSA algorihm. of a mos wo elemens of Ω (m), corresponding o a mos wo business-price uples (Z 1, p 1 ), (Z 2, p 2 ). c) If he demand funcion F (m, p) is coninuous in p for each m M, and if he se of price opions P is conneced, hen any (λ (m), Income (m) ) poin (possibly no on he boundary of he convex hull of Ω (m) ) can be expressed as a convex combinaion of a mos wo elemens of Ω (m). Proof: Par (a): I is nown ha for any vecor random variable X ha aes values wihin a se Ω, he expeced value E X} is in he convex hull of Ω (see, for example, Appendix 4.B in [17]). Therefore, he 2-dimensional poin (λ (m) ; Income (m) ) is in he convex hull of he se Ω (m). By Caraheodory s heorem (see, for example, [20]), any poin in he convex hull of he 2-dimensional se Ω (m) can be achieved by a convex combinaion of a mos hree elemens of Ω (m). Par (b): We now from par (a) ha (λ (m), Income (m) ) can be expressed as a convex combinaion of a mos hree elemens of Ω (m) (say, ω 1, ω 2, and ω 3 ). Suppose hese elemens are disinc. Because (λ (m), Income (m) ) is on he boundary of he convex hull of Ω (m), i canno be in he inerior of he riangle formed by ω 1, ω 2, and ω 3. Hence, i mus be on an edge of he riangle, so ha i can be reduced o a convex combinaion of wo or fewer of he ω i poins. Par (c): We now from par (a) ha (λ (m), Income (m) ) is in he convex hull of he 2-dimensional se Ω (m). An exension o Caraheodory s heorem in [21] shows ha any such poin can be expressed as a convex combinaion of a mos wo poins in Ω (m) if Ω (m) is he union of a mos wo conneced componens. The se Ω (m) can clearly be wrien: Ω (m) = (0; 0)} (F (m, p); pf (m, p)) p P} ˆΩ (m) which corresponds o he cases Z = 0 and Z = 1. Le represen he se on he righ hand side of he above union, so ha Ω (m) = (0; 0)} ˆΩ (m). Because he F (m, p) funcion is coninuous in p for each m M, he se ˆΩ(m) is he image of he conneced se P hrough he coninuous funcion (F (m, p), pf (m, p)), and hence is iself conneced [22]. Thus, Ω (m) is he union of a mos wo conneced componens. I follows ha (λ (m) ; Income (m) ) can be achieved via a convex combinaion of a mos wo elemens in Ω (m). Theorem 3: (Opimaliy of Two Prices) Le (λ, Income ) represen he rae-income uple corresponding o any saionary randomized policy Z (), p (), cos (), possibly being he policy of Corollary 1 ha achieves an opimal profi P rofi op. Specifically, assume he algorihm yields an erage profi P rofi (defined by he lef hand side of (18)), and ha: λ = E m Z ()F (m, p ())} Income = E m Z ()p ()F (m, p ())} Then for each m M, here exiss wo business-price uples (Z (m) 1, p (m) 1 ) and (Z (m) 2, p (m) 2 ) and wo probabiliies

5 PROC. OF 45TH ANNUAL ALLERTON CONFERENCE ON COMMUNICATION, CONTROL, AND COMPUTING (INVITED PAPER), SEPT q (m) 1, q (m) 2 (where q (m) 1 + q (m) 2 = 1) such ha: λ = 2 [ Income π m m M i=1 2 [ π m m M i=1 q (m) i q (m) i Z (m) i F (m, p (m) i ) Z (m) i ] ] p (m) i F (m, p (m) i ) Tha is, a new saionary randomized pricing policy can be consruced ha yields he same erage arrival rae λ and an erage income ha is greaer han or equal o Income, bu which uses a mos wo prices for each user demand sae m M. 6 Proof: For he saionary randomized policy Z () and p (), define: λ (m) = E Z ()F (m, p ()) M() = m} Income (m) = E Z ()p ()F (m, p ()) M() = m} Noe ha he poin (λ (m), Income (m) ) can be expressed as a convex combinaion of a mos hree poins ω (m) 1, ω (m) 2, ω (m) 3 in Ω (m) (from Theorem 2 par (a)). Then (λ (m), Income (m) ) is inside (or on an edge of) he riangle formed by ω (m) 1, ω (m) 2, ω (m) 3. Thus, for some value δ 0 he poin (λ (m), Income (m) + δ) is on an edge of he riangle. Hence, he poin (λ (m), Income (m) + δ) can be achieved by a convex combinaion of a mos wo of he ω (m) i values. Hence, for each m M, we can find a convex combinaion of wo elemens of Ω (m), defining a saionary randomized pricing policy wih wo business-price choices (Z (m) 1, p (m) 1 ), (Z (m) 2, p (m) 2 ) and wo probabiliies q (m) 1, q (m) 2. This new policy yields exacly he same erage arrival rae λ, and has an erage income ha is greaer han or equal o Income. Mos wor in newor pricing has focused on achieving opimaliy over he class of single-price soluions, and indeed in some cases i can be shown ha opimaliy can be achieved over his class (so ha wo prices are no needed). However, such opimaliy requires special properies of he demand funcion. Theorem 3 shows ha for any demand funcion F (m, p), he AP can opimize is erage profi by using only wo prices for every demand sae m M. In fac, he following example shows ha he number wo is igh, in ha a single fixed price does no always suffice o achieve opimaliy. C. Example Demonsraing Necessiy of Two Prices For simpliciy, we consider a saic siuaion where he ransmission rae is equal o µ = 2.28 wih zero cos for all (so ha Φ(cos(), S()) = 2.28 for all S() and all cos(), including cos() = 0). The demand sae M() is also assumed o be fixed for all ime, so ha F (m, p) can be simply wrien as F (p). Le P represen he inerval 0 p p max, wih p max = 10. We consider he following F (p) funcion: F (p) = 4 0 p 1 6p < p p (19) 3 2 < p 10 6 Because he new erage income is greaer han or equal o Income, he new erage profi is greaer han or equal o P rofi when his new pricing policy is used ogeher wih he old cos () scheduling policy F(p) C 0 p F(p)p B Fig. 2. F (p) and F (p)p, A=(5, 50 ),B=(1.2867, ),C=(1.2867, 2.28). 17 The F (p) and pf (p) funcions corresponding o (19) are ploed in Fig. 2. Now consider he siuaion when he AP only uses one price. Firs we consider he case when Z() = 1 for all ime. Since µ = 2.28, in order o sabilize he queue, he AP has o choose a price p such ha λ = F (p) < Thus we obain ha p has o be greaer han (poins B and C in Fig. 2 show F (p) and F (p)p for p = ). 7 I is easy o show ha in his case he bes single-price is p = 5 (poin A in Fig. 2), which yields an erage profi P rofi single of P rofi single = 50/ However, we see ha in his case he erage arrival rae F (p) is only 10/ , which is far smaller han µ. Now consider an alernaive scheme ha uses wo prices p 1 = 31/30 and p 2 = 5, each wih probabiliy 0.5. Then he resuling profi is: P rofi T wo = 0.5F (p 1 )p F (p 2 )p 2 = 0.5( ) > P rofi single (20) Furher, he resuling arrival rae is only: λ T wo = 0.5F (p 1 ) + 0.5F (p 2 ) = 0.5( ) which is sricly less han µ = Therefore he queue is sable under his scheme [19]. Now consider he case when he AP uses a varying Z() and a single fixed price. From Theorem 1 we see ha his is equivalen o using a probabiliy 0 < φ < 1 o decide wheher or no o allow new daa for all ime. 8 In order o sabilize he queue, he AP has o choose a price p such ha F (p)φ < µ. Thus he erage profi in his case would be F (p)pφ < pµ. If p 1.5, hen F (p)pφ < = 3.42 (noe ha his is indeed jus an upper bound); else if 1.5 < p 10, F (p)pφ < F (5) 5 = 50/17. Boh are less han P rofi T wo obained above. One may hin ha he wo opimum prices chosen by he AP are he wo prices ha generae he wo local maximums in he funcion F (p)p, i.e. p = 1 and p = 5. However, i is easy o show ha if one only uses prices p = 1 and p = 5, he 7 Throughou he paper, numbers of his ype are numerical resuls and are accurae enough for our argumens. 8 The case when φ=0 is rivial and hus is excluded. A p

6 PROC. OF 45TH ANNUAL ALLERTON CONFERENCE ON COMMUNICATION, CONTROL, AND COMPUTING (INVITED PAPER), SEPT maximum erage profi is On he oher hand, one can achieve profi wih prices p = 1 and p = Thus we see ha even in his simple example, nowing he exac F (p) funcion does no guaranee a simple way of finding he opimal pricing sraegy. IV. ACHIEVING THE MAXIMUM PROFIT Even hough Theorem 2 and 3 show he possibiliy of achieving he opimum erage profi by using only wo prices for each demand sae, in pracice, we sill need o solve he problem in Theorem 1. This ofen involves a very large number of variables and would require he exac demand sae and channel sae disribuions, which are usually hard o obain. To overcome hese difficulies, here we develop he dynamic Pricing and Transmission Scheduling Algorihm (PTSA). The algorihm offers a conrol parameer V > 0 ha deermines he radeoff beween he queue baclog and he proximiy o he opimal erage profi. Admission Conrol: Every slo, he AP observes he curren baclog U() and he user demand M() and chooses he price p() o be he soluion of he following problem: Max : V F (M(), p)p 2U()F (M(), p) s.. p P (21) If for all p P he resuling maximum is less han or equal o zero, he AP sends he CLOSED signal (Z() = 0) and does no accep new daa. Else, i ses Z() = 1 and announces he chosen p(). Cos/Transmission: Every slo, he AP observes he curren channel sae S() and baclog U() and chooses cos() o be he soluion of he following problem: Max : 2U()Φ(cos, S()) V cos s.. cos C (22) The AP hen sends ou paces according o µ() = Φ (cos(), S()). The conrol policy is hus decoupled ino separae algorihms for pricing and ransmission scheduling. Noe from (21) ha a larger U() increases he negaive erm 2U()F (M(), p) in he opimizaion meric, and hence ends o lead o a higher price p(). Inuiively, such a slow down of he pace arrival helps alleviae he congesion in he AP. Noe ha he meric in (21) can be wrien as F (M(), p) ( V p 2U() ). This is posiive only if p is larger han 2U()/V. Thus, we he he following simple fac: Lemma 1: Under he PTSA algorihm, if 2U()/V > p max, hen Z() = 0. A. Performance Analysis In his secion we evaluae he performance of PTSA. The following heorem summarizes he performance resuls: Theorem 4: PTSA sabilizes he AP and achieves he following bounds (assuming U(0) = 0): U() U max = V p max /2 + R max (23) P rofi P rofi op B V (24) where: P rofi = lim inf 1 1 E Z(τ)P (τ)r(τ) cos(τ)} and where P rofi op is he opimal profi characerized by (6) in Theorem 1, and B is defined in equaion (41) of he proof, and B = O(log(V )). The V parameer can be increased o push he profi arbirarily close o he opimum value, while he wors case baclog bound grows linearly wih V. In fac, we can see from (21) and (22) ha hese resuls are quie inuiive: when using a larger V, he AP is more inclined o admi paces (seing p() o a smaller value and only requiring p() 2U()/V ). Also, a larger V implies ha he AP is more careful in choosing he ransmission opporuniies (indeed, Φ (cos(), S()) mus be more cos effecive, i.e. larger han V cos()/2u()). Therefore a larger V would yield a beer profi, a he cos of larger baclog. We firs prove (23) in Theorem 4: Proof: ((23) in Theorem 4) We prove his by inducion. I is easy o see ha (23) is saisfied a ime 0. Now assume U() V p max /2 + R max for some ineger slo 0. We will prove ha U( + 1) V p max /2 + R max. We he he following wo cases: (a) U() V p max /2: In his case, U( + 1) V p max /2 + R max by he definiion of R max. (b) U() > V p max /2: In his case, 2U()/V > p max. Hence, by Lemma 1 he AP will decide no o admi any new daa. Therefore U( + 1) U() V p max /2 + R max. In he following we prove (24) in Theorem 4 via a Lyapunov analysis, using he framewor of [19]. Firs define he Lyaponov funcion L(U()) o be: L(U()) U 2 () (25) Define he one-sep uncondiional Lyapunov drif as () =EL(U( + 1)) L(U())}. Squaring boh sides of (4) and rearranging he erms, we see ha he drif saisfies: () B E2U() [ Φ(cos(), S()) Z()R() ] } (26) where B = R 2 max + µ 2 max. For a given number V > 0, we subrac from boh sides he insananeous profi (scaled by V ) and rearrange erms o ge: () V E Z()p()R() cos() } B E 2U()Φ(cos(), S()) V cos() } E Z() [ V p()r() 2U()R() ]} (27) Now we see ha he PTSA algorihm is designed o minimize he righ hand side of he drif expression (27) over all alernaive conrol decisions ha could be chosen on slo. Thus, we he ha he drif of PTSA saisfies: P () V E Z P ()p P ()R P () cos P () } B E 2U P ()Φ(cos (), S()) V cos () } E Z () [ V p ()R () 2U P ()R () ]} (28)

7 PROC. OF 45TH ANNUAL ALLERTON CONFERENCE ON COMMUNICATION, CONTROL, AND COMPUTING (INVITED PAPER), SEPT where he decisions Z (), p (), and cos () (and he resuling random arrival R ()) correspond o any oher feasible conrol acion ha can be implemened on slo (subjec o he same consrains p () P and cos () C). Noe ha we he used noaion P (), Z P (), p P (), R P (), and cos P () on he lef hand side of he above inequaliy o emphasize ha his lef hand side corresponds o he variables associaed wih he PTSA policy. Noe also ha, because he PTSA policy has been implemened up o slo, he queue baclog on he righ hand side a ime is he baclog associaed wih he PTSA algorihm and hence is also denoed U P (). We emphasize ha he righ hand side of he drif inequaliy (28) has been modified only in hose conrol variables ha can be chosen on slo. Noe furher ha R () is a random variable ha is condiionally independen of he pas given he p () price and he curren value of M(). Now consider he alernaive conrol policy ST AT described in Corollary 1, which chooses decisions Z (), p () and cos () on slo as a pure funcion of he observed M() and S() saes and yields: P rofi op = E m Z ()R ()p ()} E s cos () } (29) λ E m Z ()R () } µ E s µ () } (30) where P rofi op is he opimal erage profi defined in Theorem 1, µ () = Φ(cos (), S()), and R () is he random arrival for a given p () and M(). Recall ha E m } and E s } denoe expecaions over he seady sae disribuions for M() and S(), respecively. Of course, he expecaions in (29) and (30) canno be direcly used in he righ hand side of (28) because he M() and S() disribuions a ime may no be he same as heir seady sae disribuions. However, regardless of he iniial condiion of M(0) and S(0) we he: 1 1 lim E Z (τ)p (τ)r (τ) cos (τ)} = P rofi op (31) Le f P () represen a shor-hand noaion for he lef-hand side of (28), and define g () as he righ hand side of (28), so ha: g () =B E 2U P ()[µ () Z ()R ()] } V E Z ()p ()R () cos () } (32) where we he rearranged erms and he used µ () o represen Φ(cos (), S()). Thus, he inequaliy (28) is equivalen o f P () g (). To compue a simple upper bound on g (), noe ha for any ineger d 0, we he: U P () U P ( d) + dr max U P () U P ( d) dµ max These inequaliies hold since he baclog a ime is no smaller han he baclog a ime d minus he maximum deparures during he inerval from d o, and is no larger han he baclog a ime d plus he larges possible arrivals during his inerval. Plugging hese wo inequaliies direcly ino he definiion of g () in (32) yields: g () B + 2d(µ 2 max + R 2 max) E 2U P ( d)[µ () Z ()R ()] } V E Z ()p ()R () cos ()} (33) Also noe ha (by he law of ieraed expecaions): E U P ( d) [ µ () Z ()R () ]} = E U P ( d)e [ µ () Z ()R () ] χ( d) }} (34) where χ() =[M(), S(), U()] is he join demand sae, channel sae, and queue sae of he sysem. Since M() and S() are Marovian and boh he well defined seady sae disribuions, and he ST AT policy maes p () and cos () decisions as a saionary and random funcion of he observed M() and S() saes (and independen of queue baclog), we see ha he resuling processes µ () and Z ()R () are Marovian and he well defined seady sae erages. Furher, hey converge exponenially fas o heir seady sae values [23] (one such example is provided in Appendix B). Of course, we now he seady sae erages are given by µ and λ, respecively. Therefore here exis posiive consans θ 1, θ 2, γ 1, and γ 2 wih 0 < γ 1, γ 2 < 1, such ha: E µ () χ( d) } µ θ 1 γ d 1 (35) E Z ()R () χ( d) } λ + θ 2 γ d 2 (36) Plugging (35) and (36) ino (34) yields: E U P ( d) [ µ () Z ()R () ]} E U P ( d) [ θ 1 γ1 d + θ 2 γ2] } d (37) where we he used he fac ha λ µ (from (30)). Plugging (37) direcly ino (33) yields: g () B 1 + 2E U P ( d)(θ 1 γ d 1 + θ 2 γ d 2) } V E Z ()p ()R () cos ()} (38) where B 1= B + 2d(µ 2 max + Rmax). 2 However, he queue baclog under PTSA is always bounded by U max (by (23) in Theorem 4). We now choose d large enough so ha θ i γi d 1/(2U max ) for i 1, 2}. Specifically, by choosing: ( )} log 2θi U max d = max i=1,2 log ( 1/γ i ) (39) we he 2U max [θ 1 γ d 1 + θ 2 γ d 2] 2. Inequaliy (38) becomes: g () B V E Z ()p ()R () cos ()} (40) Now define B as follows: B =B = (2d + 1)(R 2 max + µ 2 max) + 2 (41) where d is defined in (39). Because U max = V p max /2+R max (by (23) in Theorem 4), he value of d is O(log(V )), and hence B = O(log(V )). Recalling ha f P () g (), where f P () is he lef hand side of (28), we he: P () V E Z P ()p P ()R P () cos P () } B V E Z ()p ()R () cos ()}

8 PROC. OF 45TH ANNUAL ALLERTON CONFERENCE ON COMMUNICATION, CONTROL, AND COMPUTING (INVITED PAPER), SEPT The above inequaliy holds for all. Summing boh sides over τ 0, 1,..., 1} and using P () = E L(U P ( + 1)) L(U P ()) }, we ge: E L(U P ()) } E L(U P (0)) } 1 V E Z P (τ)p P (τ)r P (τ) cos P (τ) } 1 B V E Z (τ)p (τ)r (τ) cos (τ)} Dividing by V, using he fac ha L(U P ()) 0, L(U(0)) = 0, and aing limis yields: lim inf 1 1 E Z P (τ)p P (τ)r P (τ) cos P (τ) } P rofi op B/V (42) where we he used (31). The lef hand side of he above inequaliy is he liminf ime erage profi of he PTSA algorihm. This complees he proof of Theorem 4. 9 B. Demand Blind Pricing In he special case when he demand funcion F (m, p) aes he form of F (m, p) = m ˆF (p), PTSA can in fac choose he curren price wihou looing a he curren demand sae M(). To see his, noe in his case ha (21) can be wrien: Max : M() [ V ˆF (p)p 2U() ˆF (p) ] s.. p P (43) Thus we see ha he price se by he AP under PTSA is independen of M(). So in his case, PTSA can mae decisions jus by looing a he queue baclog value U(). V. SIMULATION In his secion, we provide simulaion resuls for he PTSA algorihm. We simulae he same example ha we use in Secion III. Tha is, he sysem has ransmission rae µ = 2.28 for all ime (wih zero cos), and has a saic demand funcion F (p) given by (19). We compare wo cases of arrival processes. In he firs case, he arrival R() is deerminisic and is exacly equal o F (p()). In he oher case, we assume ha R() is a Bernoulli random variable, and saisfies: 10 2F (p()) w.p. 0.5 R() = (44) 0 w.p. 0.5 A each ime slo, he AP chooses a price according o (21) and admis all incoming paces. The simulaion is conduced wih conrol parameers V 1, 2, 5, 10, 100, 200} and we run each simulaion over 5, 000, 000 imeslos. Figure 3 and 4 show he baclog and he erage profi performances. 9 In he case when F (m, p) is no necessarily coninuous, he same proof holds wih P rofi replacing P rofiop, where P rofi represens he profi of any paricular saionary randomized algorihm. The bound (42) can hen be opimized by aing a supremum over all such P rofi. 10 For simpliciy here, we assume R() can ae fracional values. Alernaively, we could resric pace sizes o inegral unis and mae he probabiliies be such ha E R() p()} = F (p()). U Fig. 3. Profi Fig Random Arrival Deerminisic Arrival V Average Baclog v.s. V Average Profi v.s. V V Random Arrival Deerminisic Arrival We see from Fig. 3 ha he erage baclog grows linearly in V and he erage baclog is smaller han he wors case bound V p max /2+R max. We also see ha he resuls for boh arrival processes are very close o each oher. Fig. 4 shows he achieved erage profi versus he parameer V. We see ha he erage profi converges quicly as V grows. The erage profis are almos indisinguishable from he opimal value when V 100. We also observe an ineresing fac ha in boh cases, he prices chosen by he PTSA algorihm exhibi a wo-value propery, i.e. he prices swich beween values ha are eiher 1 or close o This fac is shown in Fig. 5. In he deerminisic arrival case, we observe ha he price jumps from one o he oher every ime slo wih some occasional phase inversions, i.e., he price occasionally says a he same level for one more slo and hen sars jumping again. While in he random arrival case, he price someimes says a a value for a few slos before jumping o he oher value, and when he price near 5.15 does no jump o he oher side, i gradually decreases unil i maes he jump. Inuiively, his happens in he random arrival case because of he following: There is a probabiliy of 1/2 ha he AP ges no new daa even if p < 10. If a one slo he AP ses a price p = 1 bu does no ge any new daa, hen he AP only serves some daa a ha slo. Since p = 1, i follows ha he AP plans o admi new daa. The plan will be preserved a he nex slo, since he AP jus sends ou some paces and furher reduces is queue. Thus he price would sill be 1 in

9 PROC. OF 45TH ANNUAL ALLERTON CONFERENCE ON COMMUNICATION, CONTROL, AND COMPUTING (INVITED PAPER), SEPT he nex slo. If insead he AP ses p near 5.15, i in fac needs o reduce he number of new paces. Thus if he AP does no ge any incoming pace a his slo, i can use ha slo o serve some paces. Therefore in he nex slo, i can lower he price and allow more new daa Price Random Price Deerminisic Fig. 5. The chosen prices in he firs 50 slos in boh cases wih V=100 VI. CONCLUSION In his paper, we firs characerized he opimum erage profi for he AP, and proved he Opimaliy of Two Prices heorem. We hen developed a join pricing and ransmission scheduling algorihm, PTSA, which can achieve any erage profi ha is arbirarily close o he opimum while ensuring queue baclog is bounded. PTSA uses he admission price and he business decision as ools o regulae he incoming raffic. I also provides a parameer V o radeoff he wors case baclog wih he profi loss. The analysis uses a Lyapunov drif echnique which joinly aes ino accoun he sabiliy issue and he performance opimizaion issue. APPENDIX A PROOF OF THEOREM 1 Proof: (Par I) We prove he firs par by using a similar analysis as in [18]: Consider any rule for choosing he business decision Z() 0, 1} and price p() P, and any rule for choosing cos() C. If he policy sabilizes he AP, hen: lim sup 1 1 E Z(τ)p(τ)R(τ) cos(τ)} P rofi op (45) where P rofi op is defined by he opimizaion in Theorem 1. To show his, le Alg1 be a pricing and scheduling algorihm ha sabilizes he queue. Le (Z(0), p(0), cos(0)), (Z(1), p(1), cos(1)),...} be he sequence of conrol decisions used by Alg1 over ime. Then here is a sub-sequence of imes H i } such ha H i and such ha he limiing ime erage expeced profi over imes H i is equal o he lim sup erage profi under Alg1 (defined by he lef hand side of (45)). Now define he condiional erage of income and pace rae over H slos: ( Income m (H); λ m (H) ) = 1 H 1 H E ( Z(τ)R(τ)p(τ); Z(τ)R(τ) ) M(τ) = m } The above can be rewrien as: ( Income m (H); λ m (H) ) H 1 1 = E ( Z(τ)F (m, p(τ))p(τ) H ; Z(τ)F (m, p(τ) ) M(τ) = m } Lemma 2: For every H, here exis probabiliies φ (m) (H), (H) and price values p (m) (H) P such ha: Income m (H) = 3 φ (m) (H) =1 λ m (H) = φ (m) (H) (H)F (m, p (m) (H))p (m) (H) (46) 3 (H)F (m, p (m) (H)) (47) =1 Proof: Define Ψ m (Z, p) ( ZF (m, p)p; ZF (m, p) ) as a funcion mapping from R 2 ino R 2. Then (Income m (H), λ m (H)) is equal o: H 1 1 E Ψ m (Z(τ), p(τ)) M(τ) = m} H The above is an expecaion over poins in he image of Ψ m (Z, p), and hence is in he convex hull of he image. Hence, i can be expressed as a convex combinaion of a mos hree elemens of he image (by Caraheodory s heorem). Indeed, noe ha he image of Ψ m (Z, p) consiss of wo ses: (0, 0)} and (pf (m, p), F (m, p)) p P}, corresponding o Z=0 and Z=1, respecively. Thus he convex combinaion can be expressed as ρ 0 (H)(0, 0) + 3 =1 ρ (H)(p (m) (H)F (m, p (m) ha 3 =0 ρ (H) = 1 and p (m) (H)), F (m, p (m) (H))), so (H) P. Define φ (m) (H) = for all 1 ρ 0 (H). If ρ 0 (H) 1, define (H) = ρ (H) 1 ρ 0(H) 1; else if ρ 0 (H) = 1 define (H) = 0 for all 1, we see ha he lemma follows. Now define: (Income (H), λ (H)) = m M π m (Income m (H), λ m (H)) Using coninuiy of F (p, m) and compacness of P and following he same line of analysis as in [18], we see ha we can find a sub-subsequence H i } of he subsequence of imes H i } such ha Hi as i, and here exis probabiliies φ (m), p (m) ( H i ) p (m), and price values p (m) P such ha: ( H i ), φ (m) ( H i ) φ (m) λ m ( H i ) λ m, F (m, p (m) ( H i )) F (m, p (m) ) (48) I is easy o see ha he p (m) values saisfy (13), he φ (m) probabiliies saisfy (11), and he values saisfy (15). Furher, because π m } are he seady sae values for M(), he corresponding λ and Income values saisfy (9) and (7). Similarly, one can ae limiing ime erage expeced values over he same sequence of imes H i (possibly passing o a convergen subsequence if necessary), o define limiing probabiliies β (s) and cos (s) and a limiing ime erage service rae µ ha saisfy consrains (16), (14), (10), (8). Finally, as in [18], because Alg1 is sable we can infer ha λ µ for hese paricular limiing values. Thus, we now he a saionary randomized policy ha saisfies all consrains (7)-(16) of he opimizaion problem of Theorem 1, and yields an erage profi ha is idenical o he lim sup erage profi of Alg1. However, P rofi op is defined as he maximum ime erage profi over any such saionary

10 PROC. OF 45TH ANNUAL ALLERTON CONFERENCE ON COMMUNICATION, CONTROL, AND COMPUTING (INVITED PAPER), SEPT randomized policy ha saisfies he consrains (7)-(16), and hence (45) holds. The proof of Par I is hus compleed. Proof: (Par II) We wan o show ha: if P rofi op > 0, hen for any arbirary small ɛ > 0, he profi P rofi op ɛ can be achieved wih an algorihm, which yields an erage arrival rae λ and an erage ransmission rae µ ha saisfy λ < µ. Firs le Alg be he saionary randomized algorihm ha solves (6) wih P rofi op > 0. Now le λ and µ denoe he erage arrival rae and he erage ransmission rae ha Alg yields, le φ m } denoe he probabiliy values used by Alg, and le Income and Cos denoe he erage income and erage of Alg. We see from (12) ha µ λ. Since P rofi op > 0, we see ha λ > 0. Consider he following algorihm Alg2: Alg2 is a saionary randomized algorihm ha is exacly he same as Alg, excep ha i uses probabiliy values φ m } = ρφ m }, wih some consan 0 < ρ < 1. I is easy o see ha Alg2 will yield he same erage ransmission rae µ and erage cos Cos since he business facors do no affec he ransmission under boh Alg and Alg2. Bu we can see ha under Alg2, he erage income and erage inpu rae become: Income Alg2 = ρincome, λ Alg2 = ρλ Now i is easy o see ha λ Alg2 < µ = µ Alg2 P rofiloss = Income Income Alg2 = (1 ρ)income and ha: To achieve an erage profi no smaller han P rofi op ɛ for some ɛ 0, we only need: (1 ρ)income ɛ (49) Since Income R max p max, we see ha (49) can easily ɛ be saisfied by choosing ρ 1 R maxp max. Thus we see ha Alg2 sabilizes he AP (λ Alg2 < µ Alg2 ) and achieves an erage profi P rofi Alg2 Also, by aing ɛ, we see ha P rofi Alg2 This proves Par II. > P rofi op ɛ. P rofi op APPENDIX B AN EXAMPLE OF EXPONENTIAL CONVERGENCE FOR MARKOV PROCESSES. Here we provide a simple example abou he exponenial convergence of a Marov process. We consider he wo sae Marov chain in Fig. 6. Le he iniial disribuion be given by [P ON (0), P OF F (0)]. 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