A Packet Dropping Mechanism for Efficient Operation of M/M/1 Queues with Selfish Users

A P ack et Dropping Mechanism for Ef ficient Operation of M/M/ 1 Qu eues with Sel fish Users Y i Gai, Hua Liu, an d Bhaska r Kr ishnama chari Departmen t of Electrical Engin eering University of So uthern California, L os Angeles, CA 9008 9, USA Email: { ygai, h ual, b krishna } @usc. edu Abstract —W e co nsider a fundamental g ame theoretic problem concerning selfish users contributing p ackets to an M/M/1 q ueue. In this game, each user co ntrols its o wn inpu t rate so as to optimize a desired t radeoff between throughput and delay . W e first show that the original game has an inefficient Nash Equilib rium (NE), with a Price of An arch y (PoA) that scales linearly or worse i n the n umber of users. In order to i mpro ve the outcome efficiency , we propose an easily implementabl e mechanism design wh ereby th e server randomly drops p ackets with a p robability that is a function of the total arriv al rate. W e show that th is r esults in a modi fied M/M/1 queueing game that is an or dinal potential game with at least one NE. In particular , f or a linear p acket dropping function, which is similar to the Random Early Detection (RED) algorithm used in Inter net Congestion Control, we prov e that there is a uniqu e NE. W e also show that the si mple best response dynamic conv erg es to th is uniqu e equilibriu m. F inally , for this scheme , we prove that the social welfare (expressed either as the summation of u tilities of all players, or as the summation of the logarithm of utilities of all players) at the equ ilibrium point can be arbitrarily close to the social welfare at the global optimal p oint, i.e. the PoA can be made arbitrarily close to 1. W e also study the impact of arrival rate estimation error on th e PoA through simulations. I . I N T R O D U C T I O N In the past twenty years, the usage of the I nternet h as transitioned fr om be ing primarily academic/research -oriented to on e that is pr imarily comm ercial in natu re. In th e curr ent Internet en viron ment, each comme rcial entity is inh erently interested only in its own profit. Developing network mecha- nisms that are designed to hand le s elfish behavior has therefore gained increa sing attention in rece nt years. Th e game th eoretic approa ch, which was origin ally d esigned to m odel and guide decisions in econ omic markets, provides a valuable set of tools for dealing with selfish behavior [2]– [7]. In this work, we consider the network co ngestion problem at a single intermediate store-and-for warding spot in the n etwork. Sev eral users send th eir packets to a sing le server with Poisson arriv al rate. The server processes th e packets on a first come first serve (FCFS) basis with an exp onentially distributed service time. T his is an M/M/1 qu eueing model [8]. There exists a trad e-off in this M/M/1 qu eueing mod el between This research was sponsored in part by the U.S. Army Research Labor atory under the Network Science Collaborati ve T echno logy Alliance, Agreement Number W 911NF-09-2-005 3, and by the U.S. National Science Foundatio n under CNS-0831545. This work is an exten ded versi on of the conference paper [1]. throug hput (r epresenting the be nefit fro m serv ice), and delay (represen ting the waiting cost in the queue). In the gatew ay congestion contro l con text [ 9], a measure th at is widely used to describe this trade-off is called “Power”, which is defin ed as the weig hted r atio of the th roughp ut to the delay . When the users are selfish, we can formu late a basic M/M/1 q ueueing game. In this game, we assume th at the users ar e selfish, and each con trol their o wn inpu t arriv al rate to the server . Each user’ s utility is m odeled to b e the power r atio fo r tha t user’ s packets. This classic M/M/1 qu eueing game has been formu lated and studied in [10]– [14]. T he results fro m these prior works and our own results in this work are in ag reement that the basic M/M/1 q ueuing game has a n inefficient Nash Eq uilibrium. W e are ther efore motiv ated to design an incentive mech anism to force the users to o perate at an equilibrium that is globally efficient. In pa rticular, we fo cus on the design of a packet dropp ing scheme implem ented at the server for this pu rpose. Our o bjectiv e is that the dr opping scheme should be as simple as possible, and it should minimize the Price of Anarchy (PoA, the ra tio of th e social op timum welfare to the welfare of the worst Nash equ ilibrium) to b e as close to 1 as possible. A key contribution of this work is the f ormulatio n of a modified M/M/1 q ueuing gam e with a random ized packet dropp ing policy at the server . W e consider a simple and low overhead p olicy in our form ulation, wherein the server need only monitor the sum of the rates of all users in the system. W e show that this mo dified game with a p acket dro pping scheme is an ordinal poten tial g ame [15], which imp lies the existence of at least one pure Nash Equilib rium. W e show first that u tilizing a step-fu nction for p acket dropp ing wh ereby the server d rops all th e packets wh en the sum-rate is greater than a thr eshold ( and none when the sum - rate is be low the threshold ), results in infinite n umber of undesired Nash Equilibria which harms the PoA. This raises the q uestion whether a mo re so phisticated ap- proach can do better . W e show that indeed this is possible. In p articular, we develop an incentive mech anism with a linear packet dropp ing that can imp rove the sy stem efficiency to be arbitrarily close to the g lobal optimal p oint (i.e., a PoA arb itrarily close to 1). This mechan ism is similar to the Random Early Detection (RED) used for congestion a voidance on the Internet [16]. W e prove the uniq ueness of NE of the 2 game with this mechanism. W e also show that best response dynamics will converge to the u nique NE. Our pap er is organized as fo llows . Section II summarizes the related work. W e present the model o f an M /M/1 q ueue game in Section II I. The soc ial welfare and Pr ice of Anarchy are descr ibed in section IV to investigate the efficiency o f the NE. Th en, in section V, we pro pose to design an incentive packet dro pping scheme implemen ted at the server to im prove the efficiency . Section VI proves tha t the game d efined with packet dro pping po licy is an ord inal po tential game by g iving the poten tial function. Section VII shows the best respo nse function . In section VIII, we show the b ehavior when utilizing a simple step-fun ction for p acket d roppin g. In section IX we propo se the RED-like linear packet drop ping ince ntiv e sche me. W e show that with this schem e, it is possible to make the Price of Anarch y arbitrarily c lose to the optimal po int. The uniquen ess of NE of such a game is p roved in sectio n X. In sectio n XI, we show that th e best response dynamics will conv erge to the uniqu e Nash Equilib rium. In section XII, we undertake simulations to see how th e p rocess of statistically estimating the in put arriv al rates in a real system wou ld impact the PoA. W e conclu de th e work in section XIII. I I . R E L AT E D W O R K Throu ghput-d elay trad eoffs in M/M/1 queu es with selfish users hav e been previously stu died in [10]–[14]. A utility func- tion f or each user is defined as the co rrespond ing application’ s power an d each user is tre ated as a player in such a game and adjusts its arriv al rate to handle th e trade- off between throug hput and delay . Every user is a ssumed to be selfish and only wants to maximize its own utility function in a distributed manner . Bharath-Kumar and Jaffe [1 0] wro te one of the earliest papers on the fo rmulation o f throug hput-d elay tradeo ffs in M/M/1 qu eues with selfish users. The paper discusses the proper ties of p ower as a network perfo rmance objective f unc- tion. A class of gree dy alg orithms where e ach user update s its sending rate syn chrono usly to the b est respo nse o f all o ther users’ rates to maximize th e power is propo sed. Conver gence of th e best response to an equilib rium po int is shown in this paper . Douligeris and Mazu mdar [ 11] extended Bhar ath-Kumar and Jaffe’ s work to the case with different weig hting fac- tors d efined in the power fun ction for different user s and provided analy tical results describin g the Nash E quilibrium . They showed that the equilibr ium point that the g reedy best response d ynamic algor ithm conv erged to was a uniq ue Nash Equilibriu m. The work by Zhang and Dou ligeris [ 12] proved the con- vergence o f th e best re sponse dy namics for this basic M/M/1 queuein g game un der th e multiple users case. T hus all these prior works ( [10]–[1 2]) studied on ly variants of the b asic game. T heir work, along with ours, shows that this b asic game results in a n in efficient outcom e. Our work is the first to develop a mechan ism design f or this prob lem tha t add resses this shortcom ing by showing h ow to achieve n ear-optimal perfor mance u sing a p acket-drop ping p olicy . Dutta et al. [13] studied a re lated problem in v olving a server that em ploys an o blivious active q ueue man agement scheme, i.e. drop s p ackets de pending on the total queue occu pancy with th e same probab ility regardless of which flow they come from . They also con sider an M/M/1 setting with u sers offering Po isson traffic to a server with e xpon ential serv ice time. T he u sers’ actions are the inpu t rates and the utilities the go odput/ou tput rates. The existence and the qu ality of symmetric Nash equilibria are studied for different pac ket dropp ing po licies. Althoug h o ur work also explores oblivious packet d roppin g schemes, it is different from a nd somewhat more challengin g to analy ze than [1 3], b ecause ou r utility function reflects th e tradeoff between good put and delay . In another, m ore recent work, [14], Su and van der Schaar have discussed linear ly c oupled com munication g ames in which users’ utilities are linear ly imp acted b y their c ompeti- tors’ action s. An M/M/1 FCFS queuin g game with the power as the utility function is one illustrative example of lin early coupled comm unication g ames. Th ey also quantify the Price of Anarchy in this case, and in vestigate an alternativ e solu- tion concept called Conjectural Equilibriu m, wh ich requires users to maintain a nd operate upon a dditional beliefs about competitor s. There h av e been also several other papers r elated to que ue- ing gam es, albeit with different for mulations. Havi v and Roughga rden [1 7] c onsidered a system with m ultiple servers with h eterogen eous serv ice rates. Arriv als f rom customer s are routed to on e of the servers, an d the rou ting decision s are analyzed based on NE or so cial optimizatio n scheme s. PoA is shown to be upp er bou nded by the n umber of ser vers for the social op timum. W u and Starobin ski [18] analy zed the PoA of N parallel lin ks where the delays of links are char acterized using unbo unded delay functions such as M/M/1 o r M/G/1 queuein g function s. Ec onomide s an d Silvester [19] studied a mu ltiserver two-class q ueueing game an d d ev eloped the routing policy . For mor e general surveys on game theor etic for mulations of networking pro blems, we r efer th e rea der to [4 ], [ 20]. I I I . P RO B L E M F O R M U L AT I O N W e consider a M/M/1 FCFS queue game as shown in Fig. 1. Ther e are m users with independen t Poisson arr iv als an d the arrival rates are λ 1 , λ 2 , . . . , λ m . There is a single server and th e serv ice time is expon entially distributed with mea n 1 µ . W e consider each user as a p layer f or th is g ame and the users are selfish. Each play er wants to maximize its own utility function by adjusting its rate sending to the q ueue. Note that there is a trade off between the thr oughp ut and delay fo r each u ser , i.e., giv en the rates of all o ther users, if the inp ut rate incr eases, the delay in creases too. In this paper, we co nsider the m easurement of th is tradeo ff between the thro ughpu t an d delay of the e ach user, and it is k nown as the “p ower”, which is widely u sed in the gateway cong estion 3 1 2 m ! Fig. 1. An M/M/1 queue control context [9]. W e co nsider the power as the utility function of eac h user to measure its thr oughp ut-delay tradeoff. For a given user i , the p ower is defin ed as: Power = Throu ghput α i Delay , (1) where α i is a p arameter ch osen based on the r elati ve empha sis placed on throu ghput versus d elay . α i > 1 when throug hput is more importan t, while 0 < α i < 1 whe n we want to emp hasis delay more, and α i = 1 when the throug hput and delay are emphasized equally . For M/M/1 queu e, the throu ghput for user i is T i = λ e i where λ e i is the effecti ve rate served b y th e server . The delay for user i is calculated as: D i = 1 µ − m P i =1 λ e i . So the p ower for user i can be expressed as: P i = T α i i D i = ( λ e i ) α i ( µ − m X i =1 λ e i ) . (2) In this M/M/1 game, each play er is selfish and wants to adjust its arr i val rate λ i to maximize its own utility f unction. Throu ghout the pap er , we a ssume th at the queue is stable and thus 0 ≤ m P i =1 λ e i < µ . When there is no drop ping policy implemented at the s erver, λ e i = λ i , and the optimizatio n prob lem for each play er i is: max U i ( λ i , λ − i ) = λ α i i ( µ − m X i =1 λ i ) (3) s.t. X λ i < µ λ i ≥ 0 ∀ i = 1 , 2 , . . . , m I V . S O C I A L W E L FA R E A N D P R I C E O F A N A R C H Y In [11] th e abov e M /M/1 queue game is studied and a u nique pure NE is proved to be: λ N E i = µα i m P k =1 α k + 1 , ∀ i. (4) When α i = α, ∀ i , this uniq ue NE is expressed as λ N E i = µα αm +1 , ∀ i . No w suppose a ll users cooperate to achieve the maximal system utility . W e consider two ways to d efine the social optim al fu nction: the sum of th e utilities of all the users and the sum of the logarith m of the utilities o f all the users. Defin ing the social o ptimal fun ction as th e sum of the utilities of all the users is a co mmon way for evaluating the system efficienc y and we present the analysis results under this definition fir st. H owe ver , the fairness am ong the users shou ld also b e co nsidered and it is not revealed u nder this defin ition; so we also conside r a log- sum-utility social welfare fu nction which provides for utility fairne ss. W e c an measu re the efficiency of the system usin g two well known m easures called the Price of Anarch y (PoA) and Price of Stability ( PoS), th at respectively comp are the perfo rmance of selfish u sers in the worst and best case Nash Equ ilibrium with the g lobal optimum achiev able with non -selfish user s. The definition of PoA and PoS of a gam e G is: P oA ( G ) , max a ∈E ( G ) U ( a OP T ) U ( a ) . (5) P oS ( G ) , min a ∈E ( G ) U ( a OP T ) U ( a ) . (6) where E is the set of all the Nash Equilibr iums in game G . A. Su m-utility The optimization prob lem is defined as: max P λ α i ( µ − P λ i ) (7) s.t. 0 ≤ P λ i < µ λ i ≥ 0 ∀ i = 1 , 2 , . . . , m Here w e consider two cases: 1) α > 1 First calculate U OP T . U OP T = max λ i X λ α i ( µ − X λ i ) ≤ max λ i ( X λ i ) α ( µ − X λ i ) = max λ λ α ( µ − λ ) (8) W e can get λ ∗ = µα α +1 . Equality ho lds in equation 8 wh en λ i = λ ∗ for som e i , and λ i = 0 , ∀ j 6 = i . Hence this is also the solution for P OP T sys . U OP T = α α µ α +1 ( α + 1) α +1 . Then we calculate U N E when players are selfish: U N E = mα α µ α +1 ( αm + 1) α +1 . Note that the re is on ly o ne NE in the ga me, so PoA an d PoS are the same and they a re d eriv ed as below: P oA ( G ) = P oS ( G ) = U OP T U N E = ( αm + 1) α +1 m ( α + 1) α +1 . (9) In this case we find that th e PoA a nd PoS are prop ortional to m α . 2) α < 1 The calcu lation is similar as ab ove, an d details a re omitted . U OP T = max λ i X λ α i ( µ − X λ i ) ≤ m max λ i ( P λ i m ) α ( µ − X λ i ) = max λ m 1 − α λ α ( µ − λ ) . 4 W e also g et λ ∗ = µα α +1 , U OP T = m 1 − α α α µ α +1 ( α + 1) α +1 . PoA and PoS are: P oA ( G ) = P oS ( G ) = U OP T U N E = ( αm + 1) α +1 m α ( α + 1) α +1 . (10) In this case we find that th e PoA and PoS ar e pr oportion al to m . Thus in both cases, we fin d tha t the PoA and PoS degrade linearly or worse with the numb er o f u sers. B. Su m-log-utility Now let’ s consider the sum of the lo garithm of the utilities of all th e users. The r eason we con sider the logar ithm function in the social we lfare is becau se when all user s cooperate to achieve th e optimu m, fairness am ong the users should a lso be considered , and a logarithm ic fun ction would en sure this. The social welfare o ptimization problem is: max m X i =1 log " λ α i ( µ − m X i =1 λ i ) # (11) s.t. 0 ≤ m X i =1 λ i < µ λ i ≥ 0 ∀ i = 1 , 2 , . . . , m Note that f or ea ch player, max imizing the log arithm of its utility f unction is equiv alent to ma ximizing the utility fu nction itself. Theref ore the NE remains the same as b efore. Denote λ = m P i =1 λ i . W e h av e the following theorem for finding o ut th e social optimu m: Theorem 1: T he so lution for the social welf are optimiza- tion problem is: λ ∗ i = µα m ( α +1) . Pr o of: see Append ix A. Note that λ N E i is shown in (4) and by substituting it into (3), we get the power fo r user i as: U N E i = α α µ α +1 ( αm + 1) α +1 In general, the lo g-utility term s c an b e negativ e. T o ensure that both the numera tor an d denominator terms in the PoA and PoS are non-negative in this case, we use a monotonic exponential map ping. Note that ther e is only o ne NE in the game, so PoA an d PoS are the same, and they are d eriv ed as below: P oA ( G ) = P oS ( G ) = e U OP T e U N E =  α α µ α +1 m α ( α +1) α +1  m  α α µ α +1 ( αm +1) α +1  m =  ( αm + 1) α +1 m α ( α + 1) α +1  m > 1 . (12) From (12) we can see that PoA inc reases mono tonically as m incr eases and goes to infinity as m g oes to infinity . So we want to imp lement an in centive mechanism to impr ove the PoA. V . A N I N C E N T I V E P AC K E T D RO P P I N G S C H E M E Note th at λ N E i = µα αm +1 > µα m ( α +1) = λ ∗ i . This inspires u s to find an in centive packet d ropping m echanism implemented at the server an d we wish this packet d roppin g mechanism to be as simple as possible. So we con sider th e case w here th e server need only m onitor th e sum of the rate s of all users in the system and impleme nt the p acket dr opping p olicy based only on this infor mation. Th en the packet dro pping functio n could be expre ssed as P d ( P λ i ) . So th e optimiza tion p roblem for each user i with a dro pping policy in the system is: max U i ( λ i , λ − i ) = ( λ i (1 − P d ( X λ i ))) α i ( µ − X ( λ i (1 − P d ( X λ i )))) s.t. X λ i (1 − P d ( X λ i )) < µ λ i ≥ 0 ∀ i = 1 , 2 , . . . , m (13) T o f acilitate the deriv ation, denote P ( P λ i ) = 1 − P d ( P λ i ) and thus P ( · ) is the prob ability o f keeping packets in the system. T hen the o ptimization problem for each playe r i is: max U i ( λ i , λ − i ) = ( λ i P ( X λ i )) α i ( µ − X ( λ i P ( X λ i )))) (14) s.t. X λ i P ( X λ i ) < µ (15) λ i ≥ 0 ∀ i = 1 , 2 , . . . , m (16) W e den ote th e above gam e as G p = ( N , {A i } , { U i } ) . V I . P O T E N T I A L G A M E In this section, we p rove that when the d roppin g fu nction is a fu nction which o nly depen ds o n the sum of total inco ming rates, the game is a potential game and thus there exists at least one pure NE. Definition 1: a ga me G = ( N , {A i } , { U i } ) is called an or dinal potential game if there e xists a global function φ : A − → R such that for every player i ∈ N , f or every a − i ∈ A − i and fo r every a ′ i , a ′′ i ∈ A i , sg n ( U i ( a ′ i , a − i ) − U i ( a ′′ i , a − i )) = sg n ( φ ( a ′ i , a − i ) − φ ( a ′′ i , a − i )) (17) where sg n ( x ) is the sign functio n that takes on the value -1 when x < 0 , 0 when x = 0 , an d 1 when x > 0 . Also, the fo llowing Theo rem 2 holds f or the existence of NE in a potential game : Theorem 2: [ Monderer-Shapley , 19 96 [ 15]] Every poten tial game with finite-player s, continu ous utilities, an d co mpact strategy sets po ssesses at least o ne pu re-strategy equilibriu m. Now we will prove that the M/M/1 queu eing game with a packet d ropping f unction P d ( P λ i ) is a pote ntial game. Theorem 3: G p is a ordinal poten tial game with poten tial function φ ( λ 1 , λ 2 , . . . , λ m ) =  µ − P ( P λ i ) m P i =1 λ i   m Q i =1 ( λ i P ( P λ i )) α i  5 Pr o of: φ ( λ ′ i , λ − i ) − φ ( λ ′′ i , λ − i ) =   µ − P ( λ ′ i + λ − i )( m X j 6 = i λ j + λ ′ i )     m Y j 6 = i ( λ j P ( X λ j )) α j   λ ′ α i i P ( λ ′ i + λ − i ) α i −   µ − P ( λ ′′ i + λ − i )( m X j 6 = i λ j + λ ′′ i )     m Y j 6 = i ( λ j P ( X λ j )) α j   λ ′′ α i i P ( λ ′′ i + λ − i ) α i =   m Y j 6 = i ( λ i P ( X λ i )) α i   [( µ − P ( λ ′ i + λ − i ) ( m X j 6 = i λ j + λ ′ i ))( λ ′ i P ( λ ′ i + λ − i )) α i − ( µ − P ( λ ′′ i + λ − i )( m X j 6 = i λ j + λ ′′ i ))( λ ′′ i P ( λ ′′ i + λ − i )) α i ] =   m Y j 6 = i ( λ i P ( X λ i )) α i   ( U i ( λ ′ i , λ − i ) − U i ( λ ′′ i , λ − i )) Note that G p has a finite nu mber of players and contin uous utilities. Howe ver its strategy sets are no t com pact in (15) so we c ould not directly ap ply Th eorem 2 to claim there exists at least one NE in G p . But we modify the G p to be the equiv alent game as follows: max U i ( λ i , λ − i ) = ( λ i P ( X λ i ))) α i ( µ − X ( λ i P ( X λ i )))) s.t. X λ i P ( X λ i ) ≤ µ λ i ≥ 0 ∀ i = 1 , 2 , . . . , m (18) since any so lution to the maximization problem in G p will not satisfy P λ i P ( P λ i ) = µ . Now stra tegy sets of G p are compact a nd thus there exists at least one NE in G p . Note th at in the f ollowing when we describe PoA and PoS for the packet dro pping game G p , we respectively compare the worst and best NE obtained fo r this game with r espect to the social welfare (global optim um) that can be obtain ed throug h cooper ation without packet dropp ing. V I I . B E S T R E S P O N S E F U N C T I O N From now , for tra ctability , we consider the case α i = α, ∀ i for our pro posed incentive p acket dro pping scheme . ∀ i , let ∂ U i ( λ i , λ ′ − i ) ∂ λ i = 0 . If P ( P λ i ) is differentiab le with respect to λ i , we will have αP µ − ( α + 1) P P ′ λ i λ − i − ( α + 1) P P ′ λ 2 i − αP 2 λ − i − ( α + 1) P 2 λ i + αP ′ λ i µ = 0 (19) where P ′ is th e deriv ative o f P ( P λ i ) with respect to λ i . The above d efines an implicit best response function F ( λ i , λ − i ) = 0 which shows the relationship between λ i and λ − i . V I I I . S T E P D R O P P I N G F U N C T I O N An intuitiv e dr opping policy that first come s to mind is a step functio n as shown in Fig. 2. ) ( i d P * 1 Fig. 2. Step dropping function P d ( P λ i ) The expression of P d ( P λ i ) is: P d ( X λ i ) =  0 : P λ i ≤ λ ∗ 1 : P λ i > λ ∗ (20) W e h av e th e following result for the correspo nding packet dropp ing game. Theorem 4: λ ′ is a N.E. if an d on ly if P λ ′ i = λ ∗ . Pr o of: see Append ix B . Based on Theorem 4, the NEs o f the gam e with the step dropp ing function are no t uniq ue. PoS = 1 since there exists a NE with λ i = λ ∗ i , ∀ i . Howev er , in the sum -utility case, PoA = m α − 1 when α > 1 , and PoA = m 1 − α when α < 1 . Mor eover , PoA is infin ite in the sum-log -utility case since ther e exists a NE which has on e user i with λ i = 0 . Hence this is not a desira ble r esult fo r improving the efficiency . W e theref ore next consider a slightly more soph isticated drop ping func tion that h as a linear profile. I X . L I N E A R D R O P P I N G F U N C T I O N ) ( i d P 1 r 1 2 r ) ( i P 1 r 1 2 r Fig. 3. Illustrati on of P d ( P λ i ) and P ( P λ i ) W e consider the g ame with the following linear function o f P ( P λ i ) (and thus the packet d roppin g fu nction P d ( P λ i ) = 1 − P ( P λ i ) is also a linear fu nction.) Fig. 3 illustrates P d ( P λ i ) and P ( P λ i ) . P ( X λ i ) =    1 : 0 ≤ P λ i ≤ r 1 A ( P λ i ) + D : r 1 ≤ P λ i ≤ r 2 0 : P λ i ≥ r 2 (21) 6 where    A = 1 r 1 − r 2 D = − Ar 2 P ′ = ∂ P ∂ λ i = A. For linear dro pping schem e, (1 9) beco mes: αP µ − ( α + 1 ) P Aλ i λ − i − ( α + 1) P Aλ 2 i − αP 2 λ − i − ( α + 1) P 2 λ i + αAλ i µ = 0 (22) The above also defines an implicit fun ction F ( λ i , λ − i ) = 0 . Denote λ e i = P ( P λ i ) λ i and λ e − i = P ( P λ i ) λ − i . First we want find out if we cou ld design a dropp ing policy in this linear form such that the system could h a ve a NE th at is the same as the social op timum. If not, we will then explor e h ow much efficiency it c ould ach ie ve. Theorem 5: T here doe s n ot exist a lin ear packet dr opping policy such th at P oA = 1 . Pr o of: Assume the above Th eorem does n ot hold, when P 6 = 0 , substituting λ i and λ − i with λ e i /P and λ e − i /P we have, αP µ − 1 P ( α + 1) Aλ e i λ e − i − 1 P ( α + 1) A ( λ e i ) 2 − αP λ e − i − ( α + 1) P λ e i + 1 P αAλ e i µ = 0 = ⇒ P ( αµ − αλ e − i − ( α + 1) λ e i ) = 1 P [( α + 1) Aλ e i λ e − i + ( α + 1) A ( λ e i ) 2 − αAλ e i µ ] Since λ ∗ = µα α +1 implies αµ = ( α + 1) λ ∗ , we have P [( α + 1) λ ∗ − αλ e − i − ( α + 1) λ e i ] = 1 P Aλ e i [( α + 1)( λ e − i + λ e i ) − ( α + 1) λ ∗ ] = ⇒ P [( α + 1)( λ ∗ − λ e − i − λ e i ) + λ e − i ] = 1 P Aλ e i ( α + 1)( λ e − i + λ e i − λ ∗ ) (23) Note that P λ i + P λ − i = λ ∗ implies that λ e i + λ e − i = λ ∗ . So the righ t-hand side of the eq uality is 0 . While th e left-h and side o f the equality is P λ e − i . Sin ce λ e − i 6 = 0 , so P λ e − i 6 = 0 . Thus the left-h and side o f the equality is n ot 0 and th is lea ds to a contr adiction. Therefo re Theor em 5 holds. Theorem 5 shows that we co uld not design a linear packet dropp ing policy with P oA = 1 . Th e following theor em shows that we cou ld design an incentive packet dro pping policy such that PoA could be arbitrar ily close to 1 . Theorem 6: Given any ǫ , ther e exists a linea r p acket drop - ping policy such that 1 < P oA ≤ 1 + ǫ . Pr o of: Note ( 23) in the proo f of Theo rem 5 implies: P 2 [( α + 1)( λ ∗ − λ e − i − λ e i ) + λ e − i ] = Aλ e i ( α + 1)( λ e − i + λ e i − λ ∗ ) (24) The right-han d side of (24) is gre ater than 0 only whe n λ e − i + λ e i < λ ∗ (note that A < 0 ). Then given A and λ e i , λ e − i such that λ e − i + λ e i < λ ∗ , we will have a solution for P 2 and thus we could get the value of D . This mean s that we can design a packet dropp ing scheme such that it has a NE that satisfies pλ i + p λ − i − → λ ∗ from the left side (left app roximation ). If we can fur ther pr ove that the NE is unique in this g ame (see Theorem 8), then g iv e any ǫ > 0 , we could find a linear p acket dropp ing policy at the server such that 1 < P oA ≤ 1 + ǫ . W e propo se A lgorithm 1 to show how to design the pa- rameters r 1 and r 2 in o ur prop osed incentive p acket dropp ing policy to achieve a desired PoA suc h that 1 < P oA ≤ 1 + ǫ giv en any ǫ . W e denote λ e = P λ e i . L ine 1 ensur es tha t the sum of the rates will be less than µ before the server starts to drop packets. ˜ p is the value of P ( P λ ) at the desired NE which is d eriv ed fro m th e desired PoA. ˜ p = 1 − P r { th e pac ket drop ping pro bability at desired NE } . Line 2 is the calculation of d esired NE. The cho ice of e λ is based on the desired value of PoA, i.e., given ǫ > 0 , we co uld accordin gly d eriv e the value of a desired sum r ate such that 1 < P oA ≤ 1 + ǫ . Since ( f λ e , e p ) is a solution of P ( P λ ) , line 3 shows how to therefo re get the expression of A a nd D . Then at line 4, we cou ld solve the equation (22) given all the values above and g et the value of r 2 . Based on the re sult o f r 2 , the value o f r 1 is c alculated. Algorithm 1 P arameter Calculation for Incentive Packet Drop- ping Sch eme Input: PoA bou nd parameter ǫ Output: r 1 and r 2 of ou r pro posed incen ti ve p acket drop ping policy in (2 1) such that 1 < P oA ≤ 1 + ǫ . 1: Pick any ˜ p suc h tha t a a +1 < ˜ p < 1 . 2: Calculate a desired sum rate e λ , of which e λ = { f λ e m e p , ( m − 1) f λ e m e p } . (25) is th e desired NE such that 1 < P oA ≤ 1 + ǫ . Note that λ e = ˜ pλ . 3: Supp ose P ( P λ i ) = A P λ i + D pass th rough the po int ( e λ, e p ) . Then we have A = e p e λ − r 2 , D = − e pr 2 e λ − r 2 . (26) 4: Insert the a bove values of the variables into (2 2) and g et the value of r 2 . 5: Insert the value of r 2 into (2 6) a nd get the value o f A . Then r 1 = 1 A + r 2 Note that (22) is a qua dratic equ ation fo r the param eters A and D gi ven the v alues of all the other variables. But with Algorithm 1, we co uld always fin d a uniq ue solution as stated in Th eorem 7 . Theorem 7: Alg orithm 1 yields a u nique linear packet dropp ing sch eme, i.e., unique values for r 1 and r 2 for any PoA b ound. 7 Pr o of: After in serting the v alue e p , and e λ = { f λ e m e p , ( m − 1) f λ e m e p } in to (22) a t lin e 4, we have the equa lity e p 2 [( α + 1)( λ ∗ − f λ e ) + ( m − 1) f λ e m e p ] = e p f λ e / e p − r 2 f λ e m e p ( α + 1)( f λ e − λ ∗ ) (27) It is obvious that the above is a linear equ ation of th e variable r 2 . An d thus we could get a unique solution of r 2 . Therefo re, ther e is a alw ays a uniqu e solu tion of r 1 and r 2 provided by Algo rithm 1. Our propo sed packet d roppin g sch eme is similar to the Random Early Detection ( RED) algorithm. I t is simp le and easy to be implem ented with low overhead at the server . Fig. 4 shows an examp le of ou r linear pa cket dr opping policy with µ = 6 , m = 2 an d α = 2 . (1 1) and (12) are used to calculate the utility and PoA. Point A represents λ ∗ , w hich is th en calculated to be 4 ( λ 1 = λ 2 = 2 ). W e assume ˜ p = 0 . 9 , PoA b ound p arameter ǫ = 0 . 05 . Implementing Algorith m 1 with Matlab, we pick f λ e = 3 . 9 , we the n ge t r 1 = 4 . 3012 , r 2 = 4 . 6 22 , A = − 3 . 11 54 , D = 1 4 . 400 0 . Point B r epresents the Nash Equilib rium with ou r pro posed p ackets dropp ing policy . Point C repr esents the Na sh Equ ilibrium of th e orig inal game without packet dro pping po licy . The shad ed area shows the cases where p acket dro pping happen s. F or the co mparison, the utilities are sh own in the fig ure and we can see that P oA is im proved f rom 1 . 339 6 to 1 . 045 5 . 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 4 3.5 4.5 5 A ( U = 4.16) A B ( U = 4.11) B C ( U = 3 .87) C 1 l 2 l Fig. 4. An example of our incenti ve packe t dropping scheme. X . U N I Q U E N E S S O F N E If we use the packet d roppin g sch eme in algorithm 1 we are guar anteed th at the gam e G p always has a NE with the desired PoA boun d. N ow our question is whether the scheme yields a unique NE. T his is impo rtant not on ly for finding out whether our pr oposed sch eme is efficient but also for the conv ergence issues. As surveyed in [2 1], there are n ot many general r esults o n eq uilibrium un iqueness. W e were unab le to find any existing th eorem that we cou ld use dir ectly to prove the uniq ueness of NE in our M/M/1 queueing game. This makes the analysis of th is ince ntiv e design p roblem m ore challengin g. Theorem 8: T here is a uniqu e NE for the M/M/1 Game with the linear packet d ropping scheme d escribed in Algo- rithm 1. T o prove 8, we first p rove th e following th ree lem mas. Lemma 1 : | A | increases mon otonically as ( λ ∗ − f λ e ) de- creases wh ere f λ e is the total rate of all u sers at d esired NE (as in Algorithm 1). Pr o of: Note th at (27) is equivalent to: e p 2 [( α + 1)( λ ∗ − f λ e ) + ( m − 1) f λ e m e p ] = A f λ e m e p ( α + 1)( f λ e − λ ∗ ) This m eans ( m − 1) f λ e m e p ( λ ∗ − f λ e ) = ( α + 1) | A | f λ e m e p − e p 2 ! . (28) Note th at a s ( λ ∗ − f λ e ) decreases, f λ e increases an d 1 λ ∗ − f λ e increases, so the left-hand side of (2 8) increase. This implies | A | increases, and thus Lemm a 1. Lemma 2 : ∀ λ − i < r 1 , ∂ U i ( λ i ,λ − i ) ∂ λ i > 0 at λ i = ( r 1 − λ − i ) + . Pr o of: Note th at P = 1 at r 1 , the n ∂ U i ( λ i , λ − i ) ∂ λ i = αµ − ( α + 1) Aλ i λ − i − ( α + 1) Aλ 2 i − αλ − i − ( α + 1) λ i + αAλ i µ = αµ − αλ − i − ( α + 1) λ i − Aλ i [( α + 1)( λ − i + λ i ) − αµ ] = αµ − ( α + 1)( λ − i + λ i ) + λ − i − Aλ i [( α + 1)( λ − i + λ i ) − αµ ] = [ αµ − ( α + 1)( λ − i + λ i )](1 + Aλ i ) + λ − i = [ αµ − ( α + 1) r 1 ](1 + Aλ i ) + λ − i = [( α + 1 ) λ ∗ − ( α + 1) r 1 ](1 + Aλ i ) + λ − i = ( α + 1 )( λ ∗ − r 1 )(1 + A ( r 1 − λ − i )) + λ − i Denote (29) as g ( λ − i ) . Then, ∂ g ∂ λ − i = − ( α + 1 )( λ ∗ − r 1 ) A + 1 . (29) When | A | is large enou gh such that r 1 > λ ∗ and | A | > 1 ( α +1)( r 1 − λ ∗ ) , ∂ g ∂ λ − i < 0 . From Lemm a 1 w e know that as f λ e gets closer to λ ∗ , | A | increases. This mean s that when we d esign a drop ping p olicy with P oA appro aching to 1 , | A | cou ld be large enou gh such that r 1 > λ ∗ and | A | > 1 ( α +1)( r 1 − λ ∗ ) . So g ( λ i ) achie ves the minimum v alue when λ − i = r 1 . Then ∂ U i ( λ i , λ − i ) ∂ λ i > ( α + 1)( λ ∗ − r 1 ) + r 1 = ( α + 1) λ ∗ − αr 1 = αµ − αr 1 > 0 . (30) 8 Lemma 2 implies that giv en λ − i , U i ( λ i , λ − i ) which is a function of λ i , h as a maxima at r 1 − λ − i < λ i < r 2 − λ − i as shown in Fig. 5. i U 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 i Local Maximum Point Fig. 5. Illustrati on of local maximum point of U i ( λ i , λ − i ) 0 1 2 3 4 5 6 0 5 10 15 20 25 30 i U i 1 max , i U 2 max , i U Fig. 6. Illustrati on of local maximum point of U 1 i,max and U 2 i,max Giv en λ − i , if λ − i + ( µ − λ − i ) α α +1 < r 1 , then U i ( λ i , λ − i ) will reach a lo cal m aximal p oint when λ i = ( µ − λ − i ) α α +1 as shown in Fig. 6. No te th at P ( P λ i ) = 1 a t this point. W e denote this local maximal value as U 1 i,max . T hen U i ( λ i , λ − i ) will rea ch another local maximal point with λ e i = ( µ − λ − i ) α α +1 . No te that P ( P λ i ) < 1 . W e den ote this lo cal maxim al value as U 2 i,max . Lemma 3 : Given λ − i , if λ − i + ( µ − λ − i ) α α +1 < r 1 , U 1 i,max < U 2 i,max . Pr o of: Note th at U 1 i,max = ( λ 1 i ) α ( µ − ( λ 1 i ) − λ − i ) , where λ 1 i = ( µ − λ − i ) α α +1 . U 2 i,max = ( P λ 2 i ) α ( µ − ( P λ 1 i ) − P λ − i ) . Denote λ e i = P λ 2 i . Note that λ e i ranges fr om r 1 to 0 and w e have r 1 > λ 1 i . Also note that max λ e i ( λ e i ) α ( µ − λ e i − P λ − i ) > ma x λ e i ( λ e i ) α ( µ − λ e i − λ − i ) since P λ − i < λ − i . Thus, U 2 i,max > U 1 i,max . W e show in Lem ma 3 th at g i ven λ − i , U i ( λ i , λ − i ) achieves the m aximal poin t when P < 1 under th e con dition that λ − i + ( µ − λ − i ) α α +1 ≤ r 1 . Wh en λ − i + ( µ − λ − i ) α α +1 > r 1 , th ere is o nly on e maximal point for U i ( λ i , λ − i ) . When λ − i + ( µ − λ − i ) α α +1 = r 1 , U 1 i,max and U 2 i,max will overlap, an d since r 1 > λ ∗ , this case could n ot resu lt in a N E and thus we d o n ot co nsider this case. Lemma 1, Lem ma 2 and Lemma 3 sh ows that giv en λ − i , U i ( λ i , λ − i ) a chieves th e max imal point when P < 1 . Then we will prove the uniqu eness of NE ba sed on the expression P ( P λ i ) = A ( λ i + λ − i ) + D . Pr o of o f Theor em 8 : Suppose the G p has mo re th an one NE. No te that the game is symmetric, so there must exist o ne NE λ = { λ 1 , λ 2 , . . . , λ m } , such that ∃ i , j, λ i 6 = λ j . This mean s there exists a NE λ = { λ i , λ − i } and a co nstant c such that λ = { λ i + c, λ − i − c } is also a NE. W e insert λ = { λ i , λ − i } in to (2 4) and we get: [ A ( λ i + λ − i ) + D ] 2 [( α + 1)( λ ∗ − λ e − i − λ e i ) + λ e − i ] = Aλ e i ( α + 1)( λ e − i + λ e i − λ ∗ ) = ⇒ ( α + 1)( λ ∗ − λ e − i − λ e i ) + λ e − i λ e i = 1 [ A ( λ i + λ − i ) + D ] 2 A ( α + 1)( λ e − i + λ e i − λ ∗ ) (31) W e also insert λ = { λ i + c, λ − i − c } in to (24) and den ote ˜ c = c A ( λ i +˜ c + λ − i − ˜ c )+ D = c A ( λ i + λ − i )+ D . W e have: [ A ( λ i + ˜ c + λ − i − ˜ c ) + D ] 2 [( α + 1)( λ ∗ − λ e − i − ˜ c − λ e i + ˜ c ) + λ e − i − ˜ c ] = A ( λ e i + ˜ c )( α + 1 )( λ e − i − ˜ c + λ e i + ˜ c − λ ∗ ) = ⇒ ( α + 1)( λ ∗ − λ e − i − λ e i ) + λ e − i − ˜ c λ e i + ˜ c = 1 [ A ( λ i + λ − i ) + D ] 2 A ( α + 1)( λ e − i + λ e i − λ ∗ ) (32) Note that th e right-h and side of (31) and (32) are the same and thu s the left-h and side o f (31) an d (32) ar e equ al to each other . So, ( α + 1)( λ ∗ − λ e − i − λ e i ) + λ e − i λ e i = ( α + 1)( λ ∗ − λ e − i − λ e i ) + λ e − i − ˜ c λ e i + ˜ c = ⇒ ( λ e i + ˜ c )( α + 1 )( λ ∗ − λ e − i − λ e i ) + ( λ e i + ˜ c ) λ e − i = λ e i ( α + 1)( λ ∗ − λ e − i − λ e i ) + λ e i ( λ e − i − ˜ c ) = ⇒ ˜ c ( α + 1)( λ ∗ − λ e − i − λ e i ) + ˜ c ( λ e − i + λ e i ) = 0 = ⇒ ˜ c [( α + 1) λ ∗ − α ( λ e − i + λ e i )] = 0 = ⇒ ˜ c [ αµ − α ( λ e − i + λ e i )] = 0 = ⇒ ˜ cα [ µ − ( λ e − i + λ e i )] = 0 = ⇒ cα [ µ − ( λ e − i + λ e i )] A ( λ i + λ − i ) + D = 0 . 9 Note that µ − ( λ e − i + λ e i ) > 0 . This im plies tha t c = 0 and therefor e Theore m 8 holds. X I . B E S T R E S P O N S E D Y N A M I C S A N D C O N V E R G E N C E In this section , we show tha t the best respon se d ynamic [7], a simple learning mechanism, will lead the queu ing game to conv erge to th e pu re Nash equilibrium . Best response dynamic is a straightforward up dating rule which pro ceeds as follows: wh enever p layer i h as an opp or- tunity to revise her strategy , she will ch oose the best response to the actions of all the other p layers in the p revious r ound. Mathematically , for a game G = ( N , {A i } , { U i } ) , let a t i denotes the action of player i in iteratio n t , a t i = arg max a ′ i ∈A i U i ( a ′ i , a t − 1 − i ) . (33) In g eneral, th e best respon se dyn amic is not guarante ed to co n verge. Howe ver , if the proce ss d oes converge, it is guaran teed to conver ge to a NE . Now , we want to investigate the conver gence of our propo sed M/M/1 Game with the p acket dropp ing scheme, deno ted as G p in the pr e vious section. Theorem 9: Best r esponse dynamic will conver ge to the unique NE for the M/M/1 Game with th e pr oposed packet dropp ing scheme. Pr o of: There is an impor tant result abo ut the c on vergence for ordinal potential ga me as shown in Th eorem 21 in [7]: if G is an ord inal po tential game with a compa ct action space an d a continuo us p otential fu nction, then the best response dynam ic will (almost surely) either conv erge to a NE or e very limit point of the sequence will be a NE. W e h av e showed in Theo rem 3 that G p is an o rdinal potential game, and althou gh the o riginal d efinition o f the game does no t ha ve a compact action s pace, the equi valent modification as shown in (1 8) has a compact action sp ace. W e can also see that the potential f unction is co ntinuou s. W e have also proved in Theorem 8 that th ere is a un ique NE. Thus Theorem 9 holds. 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 λ 1 λ 2 Fig. 7. Quiv er plot for a tw o user ex ample with µ = 10 , α = 2 , r 1 = 7 . 0321 and r 2 = 7 . 8222 . The vector length are scaled to 1 14 of the origina l length. Figure 7 is an illustration of Th eorem 9 by the quiver plot. In Fig ure 7, on the lower trian gle o f a grid (i.e., the f easible operating domain) , we plot vector summation for a two-u ser rate co ntrol qu euing game with µ = 10 , α = 2 , r 1 = 7 . 0 321 and r 2 = 7 . 8 222 . At each po int, the vectors’ pro jections on λ 1 and λ 2 represent the best respon se for the cor respondin g users in next iteration . T o make th e plot neat, th e length o f each vecto r is scaled to 1 14 of th e original len gth. The figure shows that at each po int, the p layers in the best response dynamic move towards the equilib rium po int. The leng th of the best respo nse vectors are pr oportion al to the distance fr om the equilibriu m point. At the equilib rium poin t, the step size of the movement in the next iteration tends to zero, which implies th e conv ergence of the be st re sponse dy namic. X I I . I M PAC T O F A R R I V A L R AT E E S T I M A T I O N For a real system implem entation, the server nee ds to estimate the total ar riv al ra te f rom users in or der to apply the incentive packet drop ping sche me. The pa ckets arrive random ly over time, so there will be a difference betwee n the estimated total rate and the average to tal r ate. This in accuracy will cause a loss in the PoA. While ap plying the pac kets dropp ing scheme, we note th at the closer to 1 th e desired PoA is, the steeper (on the linear par t) the packet droppin g scheme is, and th erefore, the greater the impact of estimation inaccuracy will be. So as th e desired Po A appro aches 1, on the one hand the PoA o f the real system should incre ase due to the impleme ntation of the incentive scheme, but on th e other hand, the sensiti vity to estimation error will r educe th e g ain in PoA. Theref ore, the achieved PoA in practice may no t be arbitrarily close to 1. W e show this fact in Fig ure 8 to 1 1 whe re we show simulation results f rom a 3-user qu eue with α = 2 . In the se simulations, we discretize time into slots. Th e server estimates the m ean arriv al rate from th e pr e vious time slot and app lies the packet dropp ing functio n corresp onding to a desire d PoA to all p ackets in the c urrent slot ( the u sers contr ibute arr i vals at a constant rate that co rrespond s to the equ ilibrium in put fo r this desired PoA). The running ti me for all the simulations is 1 0 5 time slots. Figur e 8 an d 9 show the simulation r esults of PoA un der different service rates for both the sum- utility definition an d sum-lo g-utility definition with the in stantaneous arriv al rate to the server a s th e estimated arrival rate. W e can see that as the service rate varies from 500 to 50 00 packets per time slo t, the op timal poin t of PoA (i.e. the lowest achiev able PoA) is g etting closer to 1 because the estimation inaccur acy decreases. Also, the em pirically achieved PoA is ge tting closer to th e desired PoA. Note th at when th e service rate is low , the ach ie vable PoA we g et co uld be very bad as shown in figure 8(a) a nd 9(a). In these c ases, using m ore histor y data/lon ger estimation lengths will help to in crease the estimation accuracy and improve PoA. The compar ison r esults under different estimation length s for µ = 600 p ackets per time slot are sho wn in figure 10 and 11. These simulatio n results illustrate a tr adeoff between the optimal PoA and the overhead in computin g and storage: while estimating with mo re history data w ill increase the estimation 10 accuracy and therefor e increase PoA, it increases the overhead in terms of compu ting and storag e. X I I I . C O N C L U S I O N In this paper, we ha ve design ed a novel incentive mechanism for M/M/1 queueing games with th rough put-delay trade offs. Because the origin al gam e yield s an inefficient Nash equi- librium, we pr opose to implement a linear packet drop ping mechanism at the rou ter . W e show h ow the par ameters of this mechanism can be optim ized to ensure sy stem efficiency that is ar bitrarily clo se to th e social welfare solution. Fu rther, we prove th at the proposed modification has a unique NE, and that the simple best respo nse dyn amics co n verges to this solution. Future work could consider extensions of this work to consider non-h omogen eous u sers, other queuin g mo dels beyond th e M/M/1 mod el, more complex arrangements of multiple rou ters in a network , as well as oth er system issues that may arise in practical implemen tations. A C K N O W L E D G M E N T W e would like to th ank Professor Rahu l Jain at Un i versity of Southern California for his valuable co mments. A P P E N D I X A P RO O F O F T H E O R E M 1 Pr o of: m X i =1 log " λ α i ( µ − m X i =1 λ i ) # = α lo g( Y λ i ) + m log( µ − λ ) ≤ α lo g( P λ i m ) m + m lo g( µ − λ ) (34) = m lo g( λ m ) α ( µ − λ )) Denote f ( λ ) = λ α ( µ − λ ) . So m aximize (1 1) is equ i valent to maximize f ( λ ) . T ake the d eriv ati ve of f ( λ ) and let it e quals 0. W e get: ∂ f ∂ λ = 0 ⇒ λ ∗ = µα α + 1 . Note th at equality h olds in (34) only w hen λ 1 = λ 2 = · · · = λ m . This implies λ ∗ i = λ ∗ m = µα m ( α +1) A P P E N D I X B P RO O F O F T H E O R E M 4 Pr o of: ( ⇐ =) Suppose P λ ′ i = λ ∗ . ∀ i , let ∂ U ( λ i ,λ ′ − i ) ∂ λ i = 0 . W e get the optimal point λ ∗∗ i = ( µ − P j 6 = i λ ′ j ) α i α i +1 . Note th at λ ′ i = λ ∗ − P j 6 = i λ ′ j = αµ α +1 − P j 6 = i λ ′ j = ( µ − P j 6 = i λ ′ j ) α i − P j 6 = i λ ′ j α i +1 < ( µ − P j 6 = i λ ′ j ) α i α i +1 = λ ∗∗ i . Also, ∀ λ i < λ ∗∗ i , ∂ U ( λ i ,λ ′ − i ) ∂ λ i > 0 , which mean s U ( λ i , λ ′ − i ) increases mon otonically with respect to 0 ≤ λ i < λ ∗∗ i , so U ( λ ′ i , λ ′ − i ) > U ( λ i , λ ′ − i ) , ∀ λ i ∈ [0 , λ ′ i ) . Also n ote th at U ( λ i , λ ′ − i ) = 0 , ∀ λ i ∈ ( λ ∗ i , µ − P j 6 = i λ ′ j ). Hence λ ′ i ∈ B i ( λ ′ − i ) . Therefo re, λ ′ is a N.E. (= ⇒ ) Suppose λ ′ is a N.E., ∀ i, λ ′ i ∈ B i ( λ ′ − i ) . ∀ λ ′ − i , consider the following two cases: 1) P j 6 = i λ ′ j ≤ λ ∗ : Denote λ ′′ i = λ ∗ − P j 6 = i λ ′ j . Then < λ ′′ i , λ ′ − i > is a N.E., λ ′′ i ∈ B i ( λ ′ − i ) . U ( λ ′′ i , λ ′ − i ) = U ( λ ′ i , λ ′ − i ) > 0 . So λ ′ i ≤ λ ∗ − P j 6 = i λ ′ j = λ ′′ (if n ot so, U ( λ ′ i , λ ′ − i ) = 0 ). Note th at U ( λ i , λ ′ − i ) increases monoto nically with respec t to 0 ≤ λ i < λ ′′ i . Theref ore, λ ′ = λ ′′ . P λ ′ i = λ ∗ . 2) P j 6 = i λ ′ j > λ ∗ : Under this case, since P λ ′ > λ ∗ , we ha ve B i ( λ ′ − i ) = 0 , ∀ i . Then P j 6 = i λ ′ j ≥ λ ∗ holds for all i . ∀ i, λ ′ i < µ − P j 6 = i λ ′ j < µ − λ ∗ = µ − µα α +1 = µ α +1 . So m X i =1 λ ′ i < µm α + 1 < µ ⇒ m < α + 1 . (35) Howe ver , no te that λ ∗ = µα α + 1 < X j 6 = i λ ′ j < µ ( m − 1) α + 1 ⇒ m > α + 1 (36) Since (35) an d (36) contradict each o ther, th ere is no N.E. λ ′ such that P j 6 = i λ ′ j > λ ∗ . R E F E R E N C E S [1] Y . Gai, H. Liu and B. Krishnamachar i, “ A packet dropping-base d incent iv e mechan ism for M/M/1 queue s with selfish users”, th e 30 th IEEE Interna tional Confer ence on Computer Communicati ons (IEEE INFOCOM 2011) , April, 2011. [2] V . Sriv asta va , J. Neel, A. MacK enzie, R. Menon, L. A. DaSilva, J. Hicks, J. H. Reed , and R. Gilles, “Using game th eory to analyze wire less ad hoc netw orks”, IEEE Communicat ions Survey s & T utorial s , vol. 7, no. 4, pp. 46-56, 2005. [3] A. Ozdagla r and R. 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Kapadia, “Game Theoretic T ools Applied to Wirel ess Netwo rks”, chapter in Encyclope dia of Ad Hoc and Ubiquito us Computing , W orld Scienti fic Publishers, 2008. [21] S. Lasaulce, M. Debbah and E. Altman, “Methodologi es for analyzing equili bria in wireless games”, IEEE Signal Proc essing Magazine , v ol. 26, no. 5, pp. 41-52, 2009. 12 1 1.02 1.04 1.06 1.08 1.1 1.12 1 1.1 1.2 1.3 1.4 1.5 Desired PoA PoA Got from Simulation (a) µ = 500 . 1 1.02 1.04 1.06 1.08 1.1 1.12 1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.18 Desired PoA PoA Got from Simulation (b) µ = 5000 . 1 1.02 1.04 1.06 1.08 1.1 1.12 1 1.02 1.04 1.06 1.08 1.1 1.12 Desired PoA PoA Got from Simulation (c) µ = 500 00 . Fig. 8. Simulation results of PoA under differe nt service rates of a 3-user system with α = 2 (sum-utilit y definition). 1 1.001 1.002 1.003 1.004 1.005 1.006 1.007 1.008 1 1.005 1.01 1.015 1.02 1.025 1.03 1.035 Desired PoA PoA got by Simulation (a) µ = 500 . 1 1.001 1.002 1.003 1.004 1.005 1.006 1 1.001 1.002 1.003 1.004 1.005 1.006 1.007 1.008 Desired PoA PoA got by Simulation (b) µ = 5000 . 1 1.0005 1.001 1.0015 1.002 1.0025 1.003 1.0035 1.004 1 1.0005 1.001 1.0015 1.002 1.0025 1.003 1.0035 1.004 Desired PoA PoA got by Simulation (c) µ = 500 00 . Fig. 9. Simulation results of PoA under differe nt service rates of a 3-user system with α = 2 (sum-log-util ity definition). 1 1.02 1.04 1.06 1.08 1.1 1.12 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 Desired PoA PoA Got from Simulation (a) The estimated total rate got based on the instan- taneous arri va l rate. 1 1.02 1.04 1.06 1.08 1.1 1.12 1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 Desired PoA PoA Got from Simulation (b) The estimated total rate got by avera ging the continu ous arriv al rates in 10 slots. 1 1.02 1.04 1.06 1.08 1.1 1.12 1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 Desired PoA PoA Got from Simulation (c) The estimated total rate got by av eraging the continu ous arriv al rates in 100 slots. Fig. 10. Impact of estimati on length on PoA of a 3-user system with µ = 600 , α = 2 (sum-utility definition ). 1 1.001 1.002 1.003 1.004 1.005 1.006 1.007 1.008 1 1.005 1.01 1.015 1.02 1.025 1.03 Desired PoA PoA Got from Simulation (a) The estimated total rate got based on the instan- taneous arri va l rate. 1 1.001 1.002 1.003 1.004 1.005 1.006 1.007 1.008 1 1.001 1.002 1.003 1.004 1.005 1.006 1.007 1.008 1.009 1.01 Desired PoA PoA Got from Simulation (b) The estimated total rate got by avera ging the continu ous arriv al rates in 10 slots. 1 1.001 1.002 1.003 1.004 1.005 1.006 1.007 1.008 1 1.001 1.002 1.003 1.004 1.005 1.006 1.007 1.008 Desired PoA PoA Got from Simulation (c) The estimated total rate got by av eraging the continu ous arriv al rates in 100 slots. Fig. 11. Impact of estimati on length on PoA of a 3-user system with µ = 600 , α = 2 (sum-log-utili ty definiti on).