Randomized Multi-Channel Interrogation Algorithm for Large-Scale RFID Systems

Transcription

1 andomized Multi-Channel Inteogation Algoithm fo Lage-Scale FID Systems Ami-Hamed Mohsenian-ad, Vahid Shah-Mansoui, Vincent W.S. Wong, and obet Schobe Depatment of Electical and Compute Engineeing, The Univesity of Bitish Columbia, Vancouve, Canada {hamed, vahids, vincentw, Abstact A adio fequency identification (FID) system consists of a set of eades and seveal objects, equipped with small compute chips, called tags. In a dense FID system, whee seveal eades ae placed togethe to impove the ead ate and coectness, eades and tags can fequently expeience packet collision. A common appoach to avoid collision is to use a distinct fequency channel fo inteogation fo each eade. Vaious multi-channel anti-collision potocols have been poposed fo FID eades. Howeve, due to thei heuistic natue, most algoithms may not fully utilize the achievable system pefomance. In this pape, we develop an optimization-based distibuted andomized multi-channel inteogation algoithm, called FDFA, fo lage-scale FID systems. Fo this pupose, we develop elaboate models fo eade-to-tag and eade-toeade collision poblems. FDFA algoithm is guaanteed to find a local optimum of a max-min fai esouce allocation poblem to balance the pocessing load among eades. Simulation esults show that FDFA has a significantly bette pefomance than the existing heuistic algoithms in tems of the numbe of essful inteogations. It also bette utilizes the fequency spectum. I. INTODUCTION adio fequency identification (FID) is an emeging wieless technology which allows objects to be identified automatically. An FID tag is a small and inexpensive electonic device designed fo wieless data tansmission. Each tag has a unique ID. It tansmits data ove the ai in esponse to inteogation signals by an FID eade. Multiple eades can connect to a back-end system to tansfe data fo pocessing o stoage [1]. In an FID system with one eade and seveal tags, since the eade and the tags shae the same wieless channel, tagto-tag collision can occu when multiple tags tansmit signals simultaneously to the same eade. This pevents the eade fom ecognizing any tag. Vaious tag anti-collision potocols ae poposed in [2], [3]. An Aloha-based tag anti-collision scheme has also been standadized by EPCglobal in [4] whee the eade begins each inteogation ound by infoming all the tags about the fame size. Each tag then chooses a time slot at andom and tansmits only within that time slot. In seveal FID applications (e.g., fo inventoy checking in a lage-scale waehouse), it is necessay to deploy seveal eades to achieve complete inteogation coveage. In this case, apat fom tag collision, eade-to-tag and eade-to-eade collisions may also occu. eade-to-tag collision occus when a tag eceives signals of compaable stengths fom moe than one eade simultaneously. This can cause the tag to espond abitaily to the eades, leading to incoect inteogation. eade-to-eade collision occus when a eade, which is in the midst of listening to a tag s eply, eceives stonge signals fom one o moe neighboing eades opeating at the same fequency simultaneously. This intefeence can pevent the eade s eceive fom decoding the tag s eply essfully. In a stationay FID system with synchonized eades and a centalized contolle, eade-to-tag and eade-to-eade collisions can be pevented by using a combination of fequency and time division multiple access (FDMA and TDMA) [5]. Howeve, in many pactical lage-scale FID systems, centalized coodination and synchonization ae difficult to achieve. Theefoe, it is of inteest to design distibuted andomized inteogation schemes such that each eade andomly selects the stat time of its inteogation ounds and its opeating channel to educe the pobabilities of eade-to-tag and eadeto-eade collisions. elated algoithms include naive, andom, and caie sensing potocols [6], the distibuted intefeence avoidance (DIA) algoithm with detect and abot [7], and the slotted listen-befoe-talk (S-LBT) scheme [8]. Howeve, given the heuistic natue of all of these algoithms, some may not be able to fully utilize the potential capacity of FID systems. This motivates us to study the multi-channel andom access poblem in FID systems within an optimization-based theoetical famewok. Ou contibutions ae as follows: We fomulate an optimization-based multi-channel andomized inteogation poblem fo lage-scale FID systems. Ou objective is to achieve max-min fainess among eades by taking into account eade-to-eade and eade-to-tag intefeence. Max-min fainess balances the pefomance and pocessing load among eades. We popose a distibuted algoithm to solve the max-min fai optimization poblem. Ou algoithm, efeed to as FDFA, woks based on the iteative coodinate ascent mechanism and is guaanteed to each a local optimal solution of the optimization poblem. Simulation esults show that the FDFA algoithm has a significantly bette pefomance compaed to the peviously poposed heuistic eade anti-collision algoithms in tems of the numbe of coect inteogations and fainess among eades. It bette utilizes the fequency spectum. It also conveges asynchonously and fast. The max-min fainess objectives at the obtained local optima ae vey close to the globally optimal values. The est of this pape is oganized as follows. The system model is descibed in Section II. Ou poposed FDFA algoithm is pesented in Section III. Simulation esults ae pesented in Section IV. Conclusions ae given in Section V.

2 p,c p,c I Slot Queyep Tag eply d u n d u n Quey 1st Fame (Estimation of N) 2nd Fame Last Fame eade Tag T (a) Type 1 (b) Type 2 Fig. 3. An inteogation inteval, with length T time units, fo eade. Fig. 1. Fig. 2. eade-to-tag collision in an FID system. d u I n eade Tag eade-to-eade collision in an FID system. II. SYSTEM MODEL A. eade Collision Poblem Let, with size =, denote the set of all eades. Fo each,letd denote the ead ange (o inteogation ange) of eade. eade can collect infomation only fom those tags which ae located within its ead ange. Let d I denote the intefeence ange of eade. eade s tansmission can intefee the inteogation pocess of othe eades on any tag within its intefeence ange. Two types of eade-to-tag collisions can be distinguished. The fist type is shown in Fig. 1(a), whee tag u is within the ead ange of eade and the intefeence ange of eade n (but not within the ead ange of eade n). If both eades use the same channel and tansmit simultaneously, tag u cannot coectly decode the message fom its eade, i.e., eade. Let I denote the set of eades with thei intefeence aea (but not thei ead aea) ovelapping the ead aea of eade. Wehave I = {n : d + d n < <d + d I n, n }, (1) whee denotes the Euclidean distance between an. The fist type of eade-to-tag collision can be avoided if eades an opeate at diffeent fequencies o times [5]. The second type of eade-to-tag collision is shown in Fig. 1(b), whee tag u is within the ead ange of both eades an. In FID systems, tags have low functionality and do not have fequency selectivity [4]. Even if eades an opeate at diffeent channels, tag u cannot decode the inteogation message coectly when both eades tansmit simultaneously. Let S denote the set of eades which have shaed (i.e., ovelapped) ead aea with eade. Wehave S = {n : <d + d n, n }. (2) The second type of eade-to-tag collision can be avoided by having neighboing eades opeate at diffeent time slots. Notice that having neighboing eades opeate at diffeent fequencies cannot avoid this type of collision. Also notice that since d I n d n fo all n,wehave I S = {},. (3) Futhemoe, eade-to-eade collision occus when eade n tansmits while eade is eceiving a message fom tag u (see Fig. 2). Let V denote the set of eades which have eade within thei intefeence ange. That is, V = {n : <d I n, n }. (4) eade-to-eade collision can be avoided if eades opeate at diffeent fequencies o time slots [5]. B. FID Multi-Channel Medium Access Contol Let C, with size C = C, denote the set of available othogonal fequency channels. Fo multi-channel andom access, we assume that the FID medium access contol (MAC) laye complies with the EPCglobal Class-1 Geneation-2 (C1G2) standad [9]. Each eade attempts to pefom the inteogation pocess evey T time units. At the beginning of each inteogation inteval (see Fig. 3), eade andomly chooses to stat an inteogation pocess ove fequency channel c C with inteogation pobability p,c [0, 1]. Wehave c C p,c 1,. (5) In ou model, the poposed andomized multi-channel inteogation algoithm (see Section III) contols eade-to-eade and eade-to-tag collisions, while C1G2 MAC is used to avoid tag-to-tag collision. Accoding to C1G2 MAC [9], if eade decides to stat an inteogation ound, it initiates its 1st inteogation fame by boadcasting a Quey message, which includes the numbe of time slots within the fame. The est of the fame is then divided into seveal small time slots, each stating with a Queyep message to coodinate the timing of sending the eply messages by the tags. Each tag andomly chooses to send its eply back to eade at one of the available time slots. By the end of the 1st fame, eade eceives the eplies fom a subset of the existing tags. Based on that, it estimates the numbe of tags inside its ead aea, denoted by N, e.g., using the technique in [2]. Given the estimate of N, eade continues initiating moe inteogation fames (i.e., 2nd fame, etc) until it can assue, with a cetain statistical confidence, that it has obtained the infomation fom all the tags inside its ead aea. We denote the duation of each inteogation pocess by τ (N ) as shown in Fig. 3. In pactice, we can assume that the inteogation inteval is the same fo all eades: T = T fo all whee T>0. Howeve, the inteogation intevals of diffeent eades

3 p,c, n p n,c n(n n) Fame Ovelapping T (a) Δ,n <τ (N ) p,c p,c, n p n,c T n(n n) No Fame Ovelapping (b) Δ,n >τ (N ) p,c Fig. 4. Asynchonism between inteogation intevals fo eades an. In scenaio (a), thee is a time ovelapping between the inteogation pocesses of eades an. In this case, asynchonism facto Δ,n <τ (N ). In scenaio (b), thee is no time ovelapping between the inteogation pocesses of eades an. In this case, asynchonism facto Δ,n >τ (N ). may not be synchonized. Theefoe, the inteogation pocess fo diffeent eades may not stat at the same time. We can assume that fo each pai of eades, n, thee exists a time diffeence Δ,n, called the asynchonism facto, between the inteogation intevals of eades an. The asynchonism facto Δ,n is shown in Fig. 4. Clealy, T <Δ,n <T,, n. (6) Note that Δ,n = Δ n,. In geneal, depending on the FID application, eade may not always be able to estimate Δ,n. In this pape, unless stated othewise, we conside the geneal case, whee the asynchonism factos ae not known. In Fig. 4, eade andomly decides to stat an inteogation on one of the channels evey T time units. If the inteogation intevals of eades an have time ovelapping, as in Fig. 4(a), then a eade-to-tag collision (type 1 o type 2) occus fo n I o n S, espectively. eade-to-eade collision can occu if n V. If inteogation intevals have no ovelapping, as in Fig. 4(b), eade collisions will not occu. Let p =(p,c,, c C) denote the inteogation (p) denote the pobability of completing a essful inteogation inteval by eade. That is, the pobability that eade stats an inteogation inteval without expeiencing eithe eade-to-eade o eadeto-tag collisions. We can show the following key esult. pobability vecto. Let P Theoem 1: Assume that the inteogation inteval T is selected lage enough to be at least twice as lage as the length of any inteogation pocess among eades. That is, let max τ (N ) T/2. (7) In that case, fo each eade,wehave P (p) = ( ( n S 1 γ,n e C p )) n,e ( c C p (,c m I (1 γ,m p m,c ) )) (8), whee γ,n denotes the pobability that the inteogation pocesses of eades an have any time ovelapping: γ,n = τ (N )+τ n (N n ). (9) T If Δ,n is known (by clock synchonization of eades), then γ,n =1 if τ n (N n )<Δ,n <τ (N ); and γ,n =0 othewise. The poof of Theoem 1 is given in Appendix A. Since V I S fo each,ifneithe type 1 no type 2 eade-to-tag collisions occu, then eade-to-eade collision will not happen eithe. Thus, the set V does not appea in (8). It is easy to veify that if τ (N ) >T/2 fo all, then we can simply set γ,n =1 fo each and any n I. Thee ae seveal ways to detemine the duation of an inteogation pocess. In the EPCglobal C1G2 system, we have τ (N )=en τ slot 2.72N τ slot fo. Hee, τ slot denotes the duation of each time slot in a C1G2 fame. On aveage, we have τ slot = 850 μs [4]. Using enhanced dynamic famed slotted Aloha [10] with known N,wehaveτ (N )=3N τ slot. C. Poblem Fomulation Let P denote the feasible set of inteogation pobabilities: P = { p: c C p,c 1, p,c [0, 1],,c C }. (10) In this pape, ou goal is to select p P to incease the pobability of essful inteogation fo all eades to achieve max-min fainess among eades. As a esult, the pocessing load is evenly distibuted among all the eades. A vecto of feasible inteogation pobabilities p P is defined to be max-min fai if any ess pobability P cannot incease without deceasing some Pn which is smalle than o equal to P. To obtain max-min fainess, it suffices to solve the following non-linea optimization poblem [11, Lemma 3]: f α (P (p)) (P1) maximize p P whee f α (P )= α 1 (P ) α, (11) and α>0 is lage (e.g., α 10). Function f α is called a utility function. Next, we popose an algoithm to solve poblem (P1). III. FDFA ALGOITHM Although the utilities in (11) ae concave functions in P fo α>0, poblem (P1) is not a convex optimization poblem with espect to p, due to the poduct foms in (8). Thus, finding the optimal solutions of poblem (P1) is not easy in geneal. Fo each eade, we define p =(p,c c C). Hee, the key idea is to use the iteative coodinate ascent method [12] to locally update the inteogation pobabilities fo each eade. In this method, we fix all of the components of vecto p to some values, except fo those components coesponding to one andomly selected eade (e.g., p fo eade ). Then, in a local optimization pocedue, we maximize the objective function of poblem (P1) with espect to p. This pocedue is epeated, leading to an iteative algoithm. A. Local Poblem Let p =(p n,c, n \{},c C) denote the vecto of inteogation pobabilities of all eades othe than eade. Conside the following local poblem fo eade : maximize p 0 subject to ( p, p )) ( + ( ( )) n: S n n p, p + ( ( )) m: I m m p, p + ( ( )) k:/ S k I k ( k p ) p, p P, (Local-P1)

4 Algoithm 1 - FDFA: Executed by each eade. 1: Allocate memoy fo p and p. 2: andomly choose p and p such that ( ) p, p P. 3: Set clock time t. 4: epeat then 6: Solve poblem (Local-P1) using IPM [13]. 7: Update p accoding to the solution. 8: Boadcast a contol message to announce p. 5: if t T update then 10: Stat an inteogation inteval, using C1G2 MAC, 11: ove fequency channel c C, with pobability p,c. 12: if a contol message is eceived then 13: Update p accodingly. 14: until eade device stops opeation. 9: if t T inte whee denotes coodinate-wise inequality. Note that the objective functions in poblems (Local-P1) and (P1) ae the same. Fo the objective function in (Local-P1), the fist tem is inceasing in p. The second and thid tems ae deceasing in p. The last tem does not depend on p as fo each eade k, such that / S k I k, pobability Pk does not depend on p. By solving poblem (Local-P1), eade can select p such that the objective function in poblem (P1) is maximized assuming that p is fixed (i.e., none of the othe eades change thei inteogation pobabilities). Theoem 2: Fo each eade, poblem (Local-P1) is a convex optimization poblem. The poof of Theoem 2 is given in Appendix B. Fom Theoem 2, we can use vaious convex pogamming techniques to solve poblem (Local-P1) in each eade. In paticula, poblem (Local-P1) can be solved using the inteio-point method (IPM) [13, Ch. 11] via local iteations if each eade can obtain the infomation on the inteogation pobabilities of all eades within d I max =max n d I n distance away. Thus, the iteative coodinate ascent method is applicable. B. FDFA Algoithm Ou poposed fully distibuted fequency allocation (FDFA) algoithm fo FID systems is given in Algoithm 1. It is executed in each eade.lett inte be an unbounded set of time instances at which eade may stat an inteogation inteval based on its inteogation pobability vecto p.fo any two consecutive elements t 1, t 2 T inte,wehave t 2 t 1 = T.LetT update be an unbounded set of time instances at which eade updates its inteogation pobability vecto p by solving poblem (Local-P1) using IPM. We assume that: (a) The updates ae asynchonous acoss the eades. That is, fo any, n such that n, wehave:t update Tn update = {}. (b) Thee is a global constant T update such that fo any, thee exist t 1,t 2 T update such that t 1 t 2 T update.in othe wods, all eades update thei inteogation pobabilities at least once evey T update time units. C. Convegence and Optimality In this section, we analytically investigate the convegence and optimality featues of Algoithm 1. At each time instance t 0, letf α (t) denote the cuent value of the objective function fo poblem (P1). We can show the following: Theoem 3: Fo any choice of system paametes and stating fom any initial point: (a) Function F α (t) is uppe bounded; i.e., F α (t) α 0. (b) Function F α (t) is non-deceasing at time t T update. That is, fo each t 1,t 2 T update such that t 1 t 2, F α (t 1 ) F α (t 2 ). (c) Algoithm 1 conveges. That is, thee exists a Fα such that Fα = lim t F α (t). The poof of Theoem 3 is given in Appendix C. Theoem 3 guaantees the convegence of Algoithm 1 fo any choice of system paametes. We can futhe show that: Theoem 4: Any fixed point of Algoithm 1 is a stationay point of poblem (P1). That is, it is at least a locally optimal solution fo the non-convex optimization poblem (P1). The poof of Theoem 4 is given in Appendix D. Fom Theoems 3 and 4, convegence and local optimality of Algoithm 1 ae guaanteed. Clealy, the obtained inteogation pobabilities may not necessaily be globally optimal. Howeve, simulation esults in Section IV show that Algoithm 1 usually esults in nea globally optimal pefomance, making it a pactical distibuted fequency selection and andomized inteogation algoithm fo lage-scale FID systems. IV. PEFOMANCE EVALUATION In this section, we evaluate the pefomance of the poposed FDFA algoithm and compae it with fou othe distibuted andomized inteogation schemes. In the simulation model, the total coveage aea is a lage squae waehouse with each side equal to 1 km. Thee ae 25 eades andomly deployed in the field such that complete inteogation coveage is achieved. Clealy, some spots can be within the inteogation aeas of multiple eades. Fo each eade, the inteogation ange d and the intefeence ange d I ae 50 m and 100 m, espectively [14]. On aveage, thee ae 1,000 mobile tags in the inteogation ange of each eade. The inteogation inteval T is 10 sec. We also set α =10. The aveage time fo a complete essful inteogation pocess is 3 sec, assuming that the enhanced dynamic famed slotted Aloha [10] is being used. We also set τ slot equal to 1 ms as in [4]. Poblem (Local- P1) is solved by using the MOSEK optimization softwae [15]. We stat by fist compaing the FDFA algoithm with DIA [7], S-LBT [8], and andom anaive algoithms [6] in tems of the aveage pobability of a essful inteogation among all eades (i.e., the atio of the essfully inteogated tags compaed to the total numbe of tags in the system). The simulation time is 1,000 sec. The esults when the numbe of channels C vaies fom 1 to 15 ae shown in Fig. 5. In this figue, each point epesents the aveage esults of simulating 20 diffeent andomly geneated topologies. We can see that, fo all algoithms, the pobabilities incease as moe channels become available. Howeve, the FDFA algoithm always outpefoms the othe heuistic algoithms. Notice that all algoithms each some satuation levels, which ae diffeent fo all algoithms. ecall fom Section II that eade-to-eade and type 1 eade-to-tag collisions can be avoided if the neighboing eades opeate on diffeent channels. Howeve, having the eades opeate on diffeent channels cannot avoid type 2 eade-to-tag collision. Thus, if the numbe of channels

5 Aveage Pobability of Successful Inteogation FDFA with known AFs FDFA with unknown AFs DIA S LBT andom Naive Numbe of Fequency Channels, C Numbe of Successful Inteogations in the eade with Minimum Successful Inteogations FDFA with known AFs FDFA with unknown AFs DIA S LBT andom Naive (sec) Fig. 5. Compaison between the poposed FDFA algoithm and DIA [7], S-LBT [8], and also andom, anaive [6] algoithms when the numbe of channels vaies fom 1 to 15. FDFA outpefoms all othe heuistic algoithms. Fig. 6. Compaison between FDFA algoithm with the DIA [7], S-LBT [8], and also andom, anaive [6] algoithms in tems of max-min fainess. is high, diffeent algoithms diffe depending on thei type 2 eade-to-tag collision avoidance. Fom Fig. 5, FDFA bette avoids type 2 eade-to-tag collisions than the othe algoithms. Next, we investigate fainess (i.e., balanced pefomance) among eades. Following the same simulation model as in [7], [8], we assume that the numbe of channels C is 10. The simulation time is 1,000 sec. We compae the minimum numbe of essful inteogations among eades fo diffeent algoithms. esults ae shown in Fig. 6. Each cuve epesents the aveage esults of simulating 20 diffeent andomly geneated topologies. We can see that the FDFA algoithm pefoms bette than all the heuistic algoithms as max-min fainess is indeed the design objective fo FDFA. Using FDFA, if the asynchonism factos (AFs) (i.e., Δ,n fo all, n )ae unknown, theminimum numbe of essful inteogations among all eades, which is measued at the end of the simulation, i.e., afte 1000 time slots, is 23. We notice that since each inteogation inteval T takes 10 sec, thee ae in total 1000/10 = 100 inteogation attempts duing the simulation time. Since, in the wost case, 23 out of 100 attempts ae essful, the minimum pobability of essful inteogation among all eades is This is 29%, 47%, 63%, and 121% highe than a simila value fo DIA, S-LBT, andom, and naive algoithms, espectively. If the AFs ae known, then the minimum pobability of essful inteogation futhe inceases by 11%. Notice that by achieving max-min fainess, the pocessing load becomes simila fo all eade devices. Finally, we investigate optimality. ecall that the FDFA algoithm aims to maximize the objective value of the max-min (p))). We detemine the pecentage diffeence of the optimal values obtained fom the FDFA algoithm to the global optimal value of poblem (P1). We conside 20 andom topologies, each has 25 eades and 10 channels. The global optimum of poblem (P1) is appoximately obtained by unning the FDFA algoithm 100 times, with each time stating fom a diffeent andomly selected initial point. The global optimal value is then selected to be the maximum obseved local optimum among all 100 simulations. esults ae shown in Fig. 7. On aveage, FDFA achieves 95.1% fainess poblem (i.e., f α(p Nomalized Objective Value (%) fo Max Min Fainess Poblem FDFA with unknown AFs Topology Numbe Fig. 7. Optimality of FDFA algoithm in tems of maximizing the objective value fo max-min fainess poblem (P1). of the global optimal value of poblem (P1). This implies nea optimal pefomance fo the FDFA algoithm. V. CONCLUSION In this pape, we systematically studied the andomized multi-channel inteogation poblem fo lage-scale and dense FID systems. We fist modeled the eade-to-eade collision and eade-to-tag collisions (both types) in FID systems. We then deived the pobability of pefoming a essful inteogation by each eade, whee the eades opeate asynchonously. The joint channel selection and andomized inteogation poblem was fomulated as a max-min fai esouce allocation poblem. We poposed a distibuted algoithm, called FDFA, to solve the optimization poblem. FDFA is guaanteed to convege to at least a local optimal solution of the max-min fainess poblem. Simulation esults show that FDFA has a significantly bette pefomance compaed to the peviously poposed heuistic multi-channel anti-collision algoithms in tems of the numbe of coect inteogations and fainess among eades. It also bette utilizes the fequency spectum and has a fast convegence speed.

6 APPENDIX A. Poof of Theoem 1 We fist pove (9). Given (7), fo each pai, n,an inteogation cycle (which consists of all inteogation fames within an inteogation inteval) fo eade n (with duation τ n (N n )) may ovelap with only one inteogation cycle of eade (with duation τ (N )). Fom Fig. 4, ovelapping of inteogation cycles between eades an occus when τ n (N n ) < Δ,n <τ (N ). (12) Since eades andomly and independently stat up thei opeation, Δ,n has a unifom distibution ove T/2 and T/2. Thus, the pobability of (12) happening is as in (9). Next, we notice that fo each eade, the pobability of completing a essful inteogation is obtained as P (p) = c C p,c P ( ) A No,c A NoT1,c A NoT2,c, (13) whee P(A No,c A NoT1,c A NoT2,c ) denotes the pobability that no eade collision occus while eade is pefoming an inteogation on channel c. A No,c, A NoT1,c, and A NoT2,c coespond to the events whee no eade-to-eade collisions, no type 1 eade-to-tag collisions, ano type 2 eade-to-tag collisions occu, espectively. Since V I S, if eadeto-tag collisions do not happen, then eade-to-eade collision cannot happen eithe. Thus, we have P(A No,c A NoT1,c A NoT2,c )=P(A NoT1,c A NoT2,c ). (14) Since A NoT1,c and A NoT2,c ae independent events due to (3), we have: P(A NoT1,c A NoT2,c )=P(A NoT1,c ) P(A NoT2,c ). Finally, fo all c C,wehaveP(A NoT1,c )= n I (1 γ,n p n,c ) and P(A NoT2,c )= ( n S 1 γ,n e C p n,e). eplacing these pobabilities in (13), we obtain (8). B. Poof of Theoem 2 Fom (8), the objective function in (Local-P1) becomes ( f α c C θ ) (,c p,c + n: S n f α ϑ,n (1 γ,n c C p,c) ) + m: I m f α ( c C ζ,m,c(1 γ,m p,c ) ) + ξ, whee θ,c, ϑ,n, ζ,m,c, and ξ only depend on p and can be teated as constants in poblem (Local-P1). Fo example, θ,c = ( i S (1 γ,i e C p i,e) ) ( ) j I (1 γ,j p j,c ). Since f α is concave fo, the objective function of poblem (Local-P1) is a summation of concave-affine compositions ove p. Thus, it is concave. Since the constaint in (Local-P1) is linea, poblem (Local-P1) is a convex poblem. C. Poof of Theoem 3 Pat (a): Since P 1, f α (P ) f α (1) = 1/α. Pat (b): We pove this pat by contadiction. Fist, we assume that F α (t 1 ) >F α (t 2 ). In that case, thee exists a time instance t [t 1,t 2 ] such that unning Algoithm 1 esults in educing the value of the objective function of poblem (P1) at time t. In othe wods, thee exists a eade such that t T update and F α is educed by executing line 6 of Algoithm 1 in eade. Howeve, this is impossible as the objective function in poblem (P1) is the same as that in poblem (Local-P1). Thus, we indeed have F α (t 1 ) F α (t 2 ). Pat (c): The limit in this pat diectly esults fom Pats (a) and (b). Notice that any uppe boundeon-deceasing sequence of eal numbes conveges to a fixed point. D. Poof of Theoem 4 Let p denote any fixed point of Algoithm 1. Given p = p fo, p = p is the optimal solution of poblem (Local-P1). Since (Local-P1) is convex, p should satisfy the Kaush-Kuhn-Tucke (KKT) conditions [13, p. 244] coesponding to (Local-P1) fo all. By definition, each stationay point [16, p. 194] of non-convex poblem (P1) should also satisfy all the KKT conditions fo poblem (P1). Since the objective functions in poblems (P1) and (Local- P1) ae the same and the set of constaints in (P1) is the union of the set of constaints in (Local-P1) fo all, the KKT conditions fo non-convex poblem (P1) ae equal to the union of the KKT conditions fo poblem (Local-P1) fo all. Thus, since p satisfies the KKT conditions of poblem (Local-P1) fo all eades, it also satisfies the KKT conditions fo poblem (P1). That is, each fixed point p is a local optimal solution fo poblem (P1). 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