Cross-Layer Scheduling for Cooperative Multi-Hop Cognitive Radio Networks

Cr oss-La yer Sc heduling f or Cooperative Multi-Hop Cognitive Radio Netw orks Dongyue Xue, Eylem Ekici Depar tment of Electrical and Computer Engineering, Ohio State University , Columb us, OH 43202, USA xued@ece .osu.edu, ekici@ece.osu.edu ABSTRA CT The pap er aims to design cross-la yer optimal sc heduling al- gorithms for coop erativ e multi-hop Co gnitive Radio Net- w orks (CRNs), where secondary users (SUs) assist primary user (PU )’s multi-hop transmissions and in return gain au- thorization to access a share of the spectrum. W e build tw o mo dels fo r tw o differen t t yp es of PUs, corresp onding to elastic and inelastic service classes. F or CRNs with elastic service, the PU maximizes its throughput while assigning a time-share of the channel to S Us prop ortional to SUs’ assis- tance. F or the inelastic case, the PU is guaran teed a mini- mum utility . The prop osed algorithm for elastic PU model can ac hieve arbitrarily close t o the optimal PU th roughput, while the prop osed algorithm for inelastic PU mo del can ac hieve arbitrarily close to the optimal SU u tilit y . Both al- gorithms provide deterministic u pp er-boun ds for PU qu eue backlo gs. In addition, we sho w a tradeoff b etw een th rough- put/utility and PU’s a verage end- to-end dela y upp er-b ounds for b oth algorithms. F urthermore, the algorithms work in b oth bac klogged as well as arbitrary arriv al rate systems. Keyw ords Congestion control, netw ork scheduling, multi-hop wireless netw orks, cognitive radio netw orks, end-to-end d elay guar- antees 1. INTR ODUCTION In traditional n et w orks, sp ectrum bands or channels are allocated to licensed users. How ever, suc h fixed sp ectrum assignmen t giv es rise to the sp ectrum un der-utilization prob- lem as was reported by F ederal Communication Commissi on (FC C) [1]. Cognitive R ad io Netw orks (CRNs) [2 ] hav e re- cently emerged as a technology for unlicensed users, referred to as secondary users (SUs), to opportun isticall y utilize the sp ectru m assig ned to licensed users, referred to as primary users (PUs). R esearc hers ha ve b een w orking on optimizing data rate and t h roughput of CRNs in single-hop settings Permission to m ak e digital or hard copi es of all or part of this work for personal or classroom use is grante d without fee provide d that copies are not made or distribut ed for profit or commercia l adv antag e and that copies bear this notice and the full cita tion on the first page. T o cop y otherwise, to republi sh, to post on serv ers or t o redistrib ute to li sts, requires prior spe cific permission and/or a fee. Copyri ght 20XX A CM X-XXXXX-XX-X/XX/XX ...$10.00. [3]-[6]. How ever, these w orks are not readily extendable to multi-hop CRNs, since m ulti-hop transmission requ ires that the CR p olicies take into acco unt scheduling and routing issues. Bac k-pressure sc heduling algorithms with Lyapuno v opti- mization to ols ha ve b een extensively investig ated for generic wireless netw orks [11][12]. In addition to the seminal wo rk [11], distributed and lo w-complexit y a lgorithms ha ve b een prop osed in the literature such as [14][15]. This technique has b een applied to CRNs in [7]-[10]. Specifically , in [7], an optimal cross-la y er scheduling algorithm have b een prop osed in a single-hop setting to maximize SU throughp u t sub ject to PU collision constraints. This single-hop setting is ex- tended in [8] where aggregated utilit y is maximized sub ject to PU pow er constraints. In [9], a coop erative CRN is con- sidered t o optimize PU and SU utilit y , where SUs assist PU transmission in a tw o-hop relay scenario, which is not readily extendable to generic multi-hop CRNs. A multi-hop CRN sc heduling algorithm is prop osed in [10], without consider- ing coop eration b etw een PUs and SUs. T o the b est of our knowl edge, no throughput/utilit y-optimal scheduling algo- rithms hav e b een prop osed in the literature for co op erativ e multi-hop CR N s. In th is p ap er, we prop ose tw o optimal cross-la yer sc hedul- ing algorithms for a multi-hop coop erative CRN, where SUs rela y data for a PU pair to gain access to th e licensed sp ec- trum. These tw o algorithms aim to solve the throughpu t/utilit y maximization problem und er the so-called inelastic and el as- tic PU mo dels. In the inelastic PU mo del, the PU pair is guaran teed a minimum utility and the SU utility is maxi- mized. In this mo del, we consider an adaptive-routing sce- nario where the routes of th e PU flow are not determined a priori , whic h is more general than a fixed- routing scenario. In the elastic PU m o del, the PU throughput is maximized using fi x ed routes while the SUs are guaranteed a through- put prop ortional to the PU data th at they relay . Salien t contri butions of our work with resp ect to the liter- ature can be listed as follo ws: (1) Both inelastic and elastic algorithms can achiev e a throughput/ut ilit y arbitrarily close to the optimal v alues. (2) The algo rithms guarantee deter- ministically upp er-b ounded finite buffer sizes for PU queues in t he CRN. (3) W e identify a tradeoff b etw een the through- put/utility and the a verage end-to-end d ela y u pp er-boun ds for PU data: th e inelastic algorithm ac hieve s a PU d elay upp er-b ound of order O ( N 2 ǫ ), i.e., p olynomial delay [27] is ac hieved, where N denotes the num b er of nodes inv olved in PU relay and ǫ characterizes th e difference b etw een the ac hieved utilit y and the optimal utility; The elastic algo- rithm ac h iev es or der optimal delay [21][22], i.e., th e delay is u pp er-bou n ded by the first order of the num b er of hops in a route. ( 4) Both algorithms are extended from a back- logged source mo del to a mo del with arbitrary arriv al rates at transp ort la yer. The rest of the pap er is organized as follo ws: Section 2 introduces the netw ork and PU mo dels for the coop erativ e multi-hop CRN. In Section 3, we p rop ose and analyze the inelastic algorithm. The elastic algorithm and its p erfor- mances are provided in Section 4. In Section 5, we extend b oth algorithms to the mo del with arbitrary arriv al rates at transp ort la yer. W e conclude our w ork in Section 6. 2. NETWORK MODEL In this section, we first present th e o verall multi-hop co- operative CRN m o del, fo llo wed by analysis of the tw o PU mod els. 2.1 Overall Network Elements and Constraints In this p aper, w e consider a multi-hop coop erative CRN where SUs relay PU d ata in return for th e right to use the wireless sp ectrum. T he multi-hop coop erative CRN in qu es- tion can be div ided into t wo subnetw orks: a “PU relay sub- netw ork” and an “SU subnetw ork” . The PU rela y subnet- w ork is comp osed of one primary source node ( s P ), a corre- sp on d ing primary destination nod e ( d P ), and a set of SU s S that relay the PU traffic b etw een s P and d P o ver p ossibly multiple h ops, where | S | = N . W e assume that the chan- nel cond ition cannot supp ort direct transmission b et we en the PU p air, and thus PU data will b e solely rela yed by secondary no des. Denote th e no de set of th e PU relay sub- netw ork b y N = { s P , d P } ∪ S . Denoting the set of links in PU relay subnetw ork as L , we can represen t the PU re- la y subnetw ork as ( N , L ). N ote that our mo del is readily extendable to the scenario of m ultiple PU pairs. The SU subnetw ork is composed of th e set of SUs S that participate in PU data rela ying, and the set of their one-h op secondary n eigh b ors S ′ with whic h they communicate. F or notational simplicity , we assume th at S ∩ S ′ = ∅ and that there is a distinct SU l ′ ∈ S ′ that corresponds to eve ry SU l ∈ S . T hen, th e SU subnetw ork is represented by ( S ∪ S ′ , L ′ ), where L ′ = { ( l, l ′ ): l ∈ S , l ′ ∈ S ′ } is t h e set of links in the SU su b netw ork. Note that ou r analysis can readily b e extended to cases where S ∩ S ′ 6 = ∅ . Let V = L ∪ L ′ . Then t he CRN top ology is represented by an interference graph G = ( V , E ) (or sometimes referred to as conflict graph). There is an edge in E b etw een tw o links in V if the links in terfere with each other when sched- uled simulta neously . F u rthermore, let µ mn b e the sc heduled link rate for PU data o ver link ( m, n ) ∈ L , and den ote th e sc heduled SU link rate as s l o ver link ( l, l ′ ) ∈ L ′ . F or an- alytical simplicit y , we assume a scheduled link rate takes a v alue from { 0 , 1 } . A link schedule represented by a vector (( µ mn ) ( m,n ) ∈L , ( s l ) l ∈S ) ∈ { 0 , 1 } |L| + N is said to b e f e asible iff a ny p air of t w o sc heduled li nks does not b elong to the interf erence edge set E . Let I b e the set of feasible link sc hedules. Then , a feasible link sc heduler chooses a feasi- ble link sc hedule (( µ mn ( t )) ( m,n ) ∈L , ( s l ( t )) l ∈S ) ∈ I for each time slot t . In addition, we assume th at each no de only p ossesse s one transceiver t h at can only send or receive d ata from one neighbor node. Th us, ∀ n ∈ N \{ s P } , the follo wing inequality h olds: X j :( j,n ) ∈L µ j n ( t ) + X i :( n,i ) ∈L µ ni ( t ) + 1 { n ∈S } s n ( t ) ≤ 1 , ∀ t, (1) where 1 { x } is the indicator function for even t x . Note that since s P is the sender of the PU pair, w e must ha ve X n ∈S µ ns P ( t ) = 0 , ∀ t. In the follo wing tw o subsections, we build tw o PU mo d- els correspondin g t o different PU service classes, namely , inelastic PU mo del and elastic PU mo del. In the inelastic PU mo del, adaptive-routing scenarios are considered and w e maximize SU utilit y while PU is guaranteed a minimum utilit y . In the elastic mo del, w e ass ume fixed-routing sce- narios and maximize the PU throughpu t while SUs ob t ain a throughput prop ortional to the PU data th at they rela y . 2.2 Queueing Structu re and Constraints for Inelastic PU Model In the inelastic PU mod el, we denote by U n ( t ) the queue backlo g for PU pac kets at nod e n ∈ N , where U d P ( t ) = 0 ∀ t . Let Q l ( t ) b e the queue backlog for SU packe ts corresponding to the SU pair asso ciated w ith l ∈ S . Now we defi n e t h e stabilit y of a generic qu eue with queue backlog X ( t ): X ( t ) is said to b e stable if lim sup T →∞ 1 T T − 1 X t =0 E { X ( t ) } < ∞ . Therefore, the netw ork is stable if queues U n ( t ) and Q l ( t ) are stable ∀ n ∈ N and ∀ l ∈ S . F or the time being, w e assume that PU and SU traffics are backlogg ed a t the transp ort lay er. Thus, a congestion contro ller is n eed ed to admit packe ts into th e netw ork la yer. Let µ ps P ( t ) be the admitted PU arriv al rate in time slot t . Note th at w e can consider p in the subscript of µ ps P ( t ) as the virtual no de representing the PU t ransport lay er and consider ( p, s P ) as the virtual link from transp ort la yer to source PU, so w e construct a new link set as L c , L ∪ { ( p, s P ) } . Let A l ( t ), l ∈ S , b e the admitted SU arriv al rates to th e SU pair associated with secondary no de l in time slot t . W e assume µ ps P ( t ) ≤ µ M and A l ( t ) ≤ A M ∀ l ∈ S , where µ M and A M are the upp er-b ounds for admitted PU and SU arriv al rates, resp ectively . F or analytical simplicit y , w e assume that admitted pack ets are added to the q ueues at the end of time slot t . F rom th e ab ov e analysis, we can d ev elop the q u eueing dynamics for U n ( t ), n ∈ N \{ d P } , as follo ws: U n ( t + 1) ≤ [ U n ( t ) − X i :( n,i ) ∈L µ ni ( t )] + + X j :( j,n ) ∈L c µ j n ( t ) , (2) where [ x ] + = max { x, 0 } and P i :( n,i ) ∈L µ ni ( t ) stand s for the sc heduled service rate. Note t h at (2 ) is an in eq ualit y since a feasible scheduler can b e designed indep endent of the queue backlo g informatio n. Sp ecifically , the inequ alit y holds when the actual arriv al rate at node n is less than the scheduled ar- riv al rate P j :( j,n ) ∈L c µ j n ( t ), i.e., some neighbor n o de j does not hav e pac kets for the sc heduled transmission µ j n ( t ) = 1. Similarly , Q l ( t ), l ∈ S , evolv es as follo ws: Q l ( t + 1) = [ Q l ( t ) − s l ( t )] + + A l ( t ) . (3) W e denote by f ( x ) an d g l ( x ) with l ∈ S , respectively , t he PU and SU utility functions of th e time-av erage transmission rate. As con ven tion, w e assume that t h e utility functions are positive-v alued, conca ve, strictly increasing and contin- uously d ifferen tiable, with f ( 0) = 0 and g l (0) = 0 ∀ l ∈ S . Examples of utility functions include θ ′ log (1 + x ) and θ ′ x , where θ ′ > 0 i s a w eigh t for the utilit y functions. W e as- sume the inelastic PU imp oses a minimum utilit y constrain t a P , i.e., the utility of the time-av erage PU transmission rate must b e greater than or equal to a P . According to [11 ][12], we define the capacit y region Λ I of the inelastic CRN as the closure of all feasible arriv al rate vectors consisting of an admitted PU arriv al rate and N admitted SU arriv al rates, where eac h feasible arriv al rate vector is stabilizable by some scheduler. With out loss of generality , w e assume th at t h ere exists an SU rate vec- tor ( r l ) l ∈S such that ( f − 1 ( a P ) , ( r l ) l ∈S ) is strictly inside Λ I , where f − 1 ( x ) is the inv erse function of th e u tilit y function f ( x ). T o assist th e analysis, we let ( r ∗ l,ǫ ) l ∈S b e a solution to the follo wing optimization problem: max ( r l ) l ∈S :( f − 1 ( a P )+ ǫ, ( r l + ǫ ) l ∈S ) ∈ Λ I X l ∈S g l ( r l ) , where ǫ > 0 can b e chosen arbitrarily small. Then according to [13], we ha ve: lim ǫ → 0 + X l ∈S g l ( r ∗ l,ǫ ) = X l ∈S g l ( r ∗ l ) , where ( r ∗ l ) l ∈S is a solution to th e follow ing optimization: max ( r l ) l ∈S :( f − 1 ( a P ) , ( r l ) l ∈S ) ∈ Λ I X l ∈S g l ( r l ) . In Section 3, w e will p ropose an algorithm that satisfies the PU minimum utilit y constrain t and can achiev e SU util- it y arbitrarily close to the optimal v alue P l ∈S g l ( r ∗ l ), with a tradeoff b et ween the SU utility and the av erage PU dela y upp er-b ound. 2.3 Routing and Queu eing Structur e f or Elas- tic PU Model F or the elastic PU model, we consider a fixed multi-path routing scenario, where the PU data t ransmissio n ha ve K loopless pre-determined routes. W e denote the path for the k - th route as P k = ( v 0 k , v 1 k , ..., v H k k , v H k +1 k ), where ( H k + 1) is the total number of hops in the PU relay subnetw ork for route k , where v m k ∈ N , ∀ m ∈ { 0 , 1 , ..., H k + 1 } , ∀ k ∈ { 1 , 2 , ..., K } . Without loss of generalit y , we assume that eac h nod e l ∈ S is in at least one of the K routes, that is: ∀ l ∈ S , ∃ k , m s.t. v m k = l . Note that w e alwa y s ha ve v 0 k = s P and v H k +1 k = d P , ∀ k ∈ { 1 , 2 , ..., K } . According to this routing structure, w e constru ct PU q ueues U k m ( t ) along the no des in the K routes, where 0 ≤ m ≤ H k + 1 and 1 ≤ k ≤ K . Note that, si nce v H k +1 k = d P , w e ha ve U k H k +1 ( t ) = 0 , ∀ t , ∀ k ∈ { 1 , 2 , ..., K } . Similar to the inelastic mo d el, we assume that PU and SU traffics are bac klogged at the transp ort lay er. Let µ k − 1 , 0 ( t ) b e the admitted arr iv al rate from the P U transp ort la yer to the source PU that is scheduled to pass through the k - th route. Note that, consistent with the elastic mo del, w e assume that the sum of ad mitt ed PU arriv al rates ov er K routes is upp er-b ounded by µ M , i.e, K X k =1 µ k − 1 , 0 ( t ) ≤ µ M , ∀ t . In addition, w e let λ k , k = { 1 , 2 , ... K } , b e the time-av erage of µ k − 1 , 0 ( t ). Let µ k m,m +1 ( t ), 0 ≤ m ≤ H k , b e the sched- uled rate for the hop ( v m k , v m +1 k ) along the k - t h path. T hus, U k m ( t ) evolv es as follo ws: U k m ( t +1) ≤ [ U k m ( t ) − µ k m,m +1 ( t )] + + µ k m − 1 ,m ( t ) , 0 ≤ m ≤ H k , (4) where the inequalit y holds if µ k m − 1 ,m ( t ) = 1 and U k m − 1 ( t ) = 0, 1 ≤ m ≤ H k . Note that a link ( m, n ) ∈ L can b e a hop in multiple routes, and hence w e can only sc hedule the hop with rate 1 on one su c h route in any time slot. Let ρ k b e the reward for S Us when a packe t is admitted to route k , i.e., ρ k µ k − 1 , 0 ( t ) pac kets will b e admitted sim ul- taneously to the S U queues corresp onding to the no des v m k , 1 ≤ m ≤ H k . Here, we assume that ρ k µ k − 1 , 0 ( t ) takes in te- ger v alues. Note that our analysis is readily ext endable to fractional-v alued ρ k µ k − 1 , 0 ( t ) by constructing a counter that only admits ⌊ ρ k µ k − 1 , 0 ( t ) ⌋ packe ts, where ⌊ x ⌋ is th e flo or func- tion. A lso note that the analysis can b e extended to delay ed rew ards, i.e., a rew ard rate ρ k µ k − 1 , 0 ( t ) is admitted to SU queues at t + τ ′ , where τ ′ is the delay in unit of time slots. F rom the ab o ve analysis, the SU queueing dynamics for Q l ( t ) can b e expressed as follo ws: Q l ( t + 1) =[ Q l ( t ) − s l ( t )] + + K X k =1 H k X m =1 ρ k µ k − 1 , 0 ( t ) 1 { v m k = l } =[ Q l ( t ) − s l ( t )] + + K X k =1 ρ k µ k − 1 , 0 ( t ) 1 {∃ m : v m k = l } , (5) where the second equalit y holds since each route is a lo op- free. The n etw ork is stable if qu eues U k m ( t ) and Q l ( t ) are stable ∀ m, k ∀ l . Then, we define the capacit y region Λ E of the elastic CRN as th e closure of all feasible arriv al rate vectors eac h stabilizable by some sc heduler. Note that a feasible arriv al rate v ector is in the form of (( λ k ) k ∈{ 1 , 2 ,...,K } , ( K X k =1 ρ k λ k 1 {∃ m : v m k = l } ) l ∈S ) where ( λ k ) k ∈{ 1 , 2 ,...,K } represents the PU arriv al rates per route and ( P K k =1 ρ k λ k 1 {∃ m : v m k = l } ) l ∈S represents the SU ar- riv al rates according to the rew ard mechanism. T o assist the analysis, we let ( λ ∗ k,ǫ ) k ∈{ 1 , 2 ,...,K } b e a solution to t he follo w- ing optimization problem: max ( λ k ) k ∈{ 1 , 2 ,...,K } K X k =1 λ k s.t. ( λ k ) : (( λ k + ǫ ) , ( K X k =1 ρ k ( λ k + ǫ ) 1 {∃ m : v m k = l } )) ∈ Λ E where ǫ > 0 can b e chosen arbitraril y small. S imilarly , ac- cording to [13], we ha ve: lim ǫ → 0 + K X k =1 λ ∗ k,ǫ = K X k =1 λ ∗ k , where ( λ ∗ k ) k ∈{ 1 , 2 ,...K } is a solution to the follo wing optimiza- tion: max ( λ k ) k ∈{ 1 , 2 ,...,K } K X k =1 λ k s.t. ( λ k ) : (( λ k ) , ( K X k =1 ρ k λ k 1 {∃ m : v m k = l } )) ∈ Λ E In S ection 4, w e will p ropose an algorithm that can ac hieve a PU th roughput arbitrarily close to the optimal value P K k =1 λ ∗ k , with a tradeoff b etw een the PU th roughput and av erage PU/SU delay u pper- boun d. 3. INELASTIC ALGORITHM FOR THE CRN In th is section, we fi rst introduce tw o typ es of v irtual queues and their structu res to assist the developmen t of th e inelastic algo rithm. T he inelastic algorithm is then intro- duced in Subsection 3.2. 3.1 V irtual Queues and A ppr oaches W e construct a virtu al queue U p ( t ) at the PU transp ort la yer with the follo wing q ueue d ynamics: U p ( t + 1) = [ U p ( t ) − µ ps P ( t )] + + R ( t ) , (6) where R ( t ) denotes the v irtual arriv al rate to U p ( t ) in time slot t whic h will b e determined by the inela stic algori thm in the next sub section. F urthermore, let R ( t ) b e upp er- b ounded by µ M . Wh en U p ( t ) is stable, w e know from q ueue- ing theory that th e time-av erage admitted PU arriv al rate µ satisfies: µ , lim T →∞ T − 1 X t =0 µ ps P ( t ) ≥ r , lim T →∞ T − 1 X t =0 R ( t ) . (7) The v irtu al queue U p ( t ), along with R ( t ), regulates the ad- mitted PU arriv al rate in the inelastic algorithm, in an at- tempt to guaran tee an avera ge end-to-end dela y upp er-b ound , as will b e stated in detail in th e next subsection. W e construct another virtual service queue Z ( t ) at t he PU source no de s P with the follo wing queueing d ynamics: Z ( t + 1) = [ Z ( t ) − R ( t )] + + f − 1 ( a P ) . (8) When Z ( t ) and U p ( t ) are stable, we hav e f ( µ ) ≥ f ( r ) ≥ a P . Sp ecifically , the minimum u tilit y constrain t imposed by PU is satisfied when th e t w o virtual queu es are stable. 3.2 Inelastic Algorithm W e design a control parameter q M indicating the buffer size for each PU queue in the CRN, with q M ≥ µ M . The op- timal in elastic algorithm consists of four parts, namely , SU congestion controller, R ( t ) con troller, PU congestion con- troller and a link scheduler, described as follo ws. 1) SU Congesti on Con troller : min 0 ≤ A l ( t ) ≤ A M A l ( t ) Q l ( t ) − V 1 g l ( A l ( t )) , ∀ l ∈ S , (9) where V 1 > 0 is a con trol parameter in the algorithm. Note that w e alwa ys hav e A l ( t ) Q l ( t ) − V 1 g l ( A l ( t )) ≤ 0 u nder the SU congestion controller, since A l ( t ) = 0 is a v alid candidate for the admitted arriv al rate. 2) R ( t ) Regulator : min 0 ≤ R ( t ) ≤ µ M R ( t )( U p ( t ) q M − µ M q M − Z ( t )) . (10) Sp ecifically , when U p ( t ) q M − µ M q M − Z ( t ) > 0, the virtual rate R ( t ) is set to zero; otherwise, R ( t ) = µ M . 3) PU Congestion Contro lle r : max 0 ≤ µ ps P ( t ) ≤ µ M µ ps P ( t )( q M − µ M − U s p ( t )) . (11) Sp ecifically , when q M − µ M − U s p ( t ) ≤ 0, the admitted PU arriv al rate µ ps P ( t ) is set to zero; O therwise, µ ps P ( t ) = µ M . 4) Link Rate Scheduler : max X ( m,n ) ∈L µ mn ( t ) U p ( t ) q M ( U m ( t ) − U n ( t )) + X l ∈S Q l ( t ) s l ( t ) , (12) with the constraint { ( µ mn ( t )) ( m,n ) ∈L , ( s l ( t )) l ∈S } ∈ I . Note that when U m ( t ) − U n ( t ) ≤ 0, ( m, n ) ∈ L , we set µ mn ( t ) = 0 according to (12). The inelastic algorithm has the follow ing property: Pr oposition 1. U n ( t ) ≤ q M , ∀ n ∈ N . (13) Pr oof. W e can prov e Prop osition 1 by induction. Ini- tially when t = 0, U n (0) = 0 ∀ n ∈ N . Now assume in time slot t we h a ve U n ( t ) ≤ q M ∀ n ∈ N . In th e in d uction step, w e consider tw o cases: Case 1: n = s P . If U s P ( t ) ≤ q M − µ M , then since the ad- mitted arriv al rate to U s P ( t ) is bounded b y µ M , we hav e U s P ( t + 1) ≤ U s P ( t ) + µ M ≤ q M . Otherwise, we hav e U s P ( t ) > q M − µ M , and according to the PU congestion contro ller (11) we ha ve µ ps P ( t ) = 0, from which w e obtain U s P ( t + 1) ≤ U s P ( t ) ≤ q M . Case 2: n 6 = s P . I f U n ( t ) ≤ q M − 1, then we hav e U n ( t + 1) ≤ U n ( t ) + 1 ≤ q M according to (1) an d the queueing dy namics (2). Otherwise, w e ha ve U n ( t ) = q M and U n ( t ) ≥ U m ( t ) ∀ m ∈ N , and according to the link scheduler (12) we h a ve µ j n ( t ) = 0 ∀ j such that ( j, n ) ∈ L , from which we obt ain U n ( t + 1) ≤ U n ( t ) = q M by the qu eu eing dynamics (2). Therefore, U n ( t + 1) ≤ q M ∀ n ∈ N , i.e., th e indu ction step holds, and the p roposition is prove d. Now we present the main results of the inelastic algorithm in Theorem 1. Theorem 1. L et ǫ > 0 b e chosen arbitr aril y smal l. Gi ven that q M > µ 2 M + N + 1 ǫ + µ M , (14) the inelastic algorithm ensur es the fol lowing ine quality on queue b acklo gs: lim sup T →∞ 1 T T − 1 X t =0 E { X l ∈S Q l ( t ) + U p ( t ) + Z ( t ) } ≤ B 1 + V 1 g M δ 1 , (15) wher e B 1 , 1 2 µ 2 M + 1 2 ( f − 1 ( µ M )) 2 + µ 2 M ( q M − µ M ) q M + 1 2 N + 1 2 N A 2 M + 1 2 µ M q M ( N + 1) , δ 1 is chosen such that 0 < δ 1 < ǫ ( q M − µ M ) − µ 2 M − N − 1 2 q M , and g M is define d as: g M , lim sup T →∞ 1 T T − 1 X t =0 E { X l ∈S g l ( A l ( t )) } − X l ∈S g l ( r ∗ l,ǫ ) ≤ X l ∈S ( g l ( A M ) − g l ( r ∗ l,ǫ )) . F urthermor e, the inelastic algorithm achieves: X l ∈S g l ( a l ) ≥ X l ∈S g l ( r ∗ l,ǫ ) − B 1 V 1 , (16) wher e a l is define d as the time-aver age ensemble value of A l ( t ) : a l , lim inf T →∞ 1 T T − 1 X t =0 E { A l ( t ) } , l ∈ S . R emark 1 (Network Stability) : T he inequalities (13) from Proposition 1 and (15) from Theorem 1 indicate that the in- elastic algorithm stabilizes the actual an d virtual queues. As an immediate result, t he netw ork is stable and the minim um utilit y constrain t is met. In addition, Prop osition 1 ensures that the actual PU q ueues are dete rministic al l y b ounded by the finite buffer size q M . R emark 2 (Optimal Utility and T r ade off with Delay): The inequality (16) gives the lo wer-bound of the SU utility the inelastic algorithm can achiev e. Since the constant B 1 is indep endent of the control parameter V 1 , the algorithm can ac hieve a utility arbitrarily close to the optimal v alue P l ∈S g l ( r ∗ l ) as ǫ is c hosen arbitrarily small, with a tradeoff in the PU bu ffer size q M whic h is of order O ( N ǫ ) as shown in (14). By Little’s Theorem, t he av erage end-to-end delay upp er-b ound is of order O ( N 2 ǫ ). Note that it is easy to v erify that when applied to a fixed -routing scenario, the inelastic algorithm achieve s an a verag e end-to-end delay upp er-b ound of order O ( H 2 ǫ ), where H denotes the number of h op s in t he route. R emark 3 (Complexity of Algorithm): In the inelastic al- gorithm, the SU congestion controller, the R ( t ) regulator and the PU congestion control ler can op erate locally at SU transp ort la yer and source PU. The l ink rate sc h eduler is essen tially a centra lized maximal w eigh t matc hing problem [11][16]. T o reduce complexity of the link rate sc heduler, sub optimal algorithms can be develo p ed to at least ac hieve a fraction γ of the optimal utility . These su boptimal al- gorithms includ e the well-studied Greedy Maximal Match- ing ( GMM) [1 5] a lgorithm w ith γ = 1 2 and th e maximum w eighted ind epend en t set (MWIS) p roblem such as GW- MAX and GWMIN prop osed in [20] with γ = 1 ∆ , where ∆ is the maximum degree of the CRN top ology . R emark 4 (Distribute d Implementation of the Link Sche d- uler) : D istribu ted implementation can b e devel op ed in muc h the same w ay as in [14] to achiev e a f r action of the optimal utilit y . In order to ac hieve a u tilit y arbitrarily close to the optimal v alue with distributed implementatio n, we can em- plo y random access techniques [24][25] in th e link scheduler with fugacities [26] chose n as exp { α ¯ U p ( t )[ U m ( t ) − U n ( t )] + q M } for link ( m, n ) ∈ L and exp { αQ l ( t ) } for an SU link asso ciated with l ∈ S , where ¯ U p ( t ) is a lo cal estimate of U p ( t ) and α is a p ositive weig ht. It can be shown th at th e d istributed algorithm can still ac hieve an av erage PU end-to-end dela y of order O ( N 2 ǫ ) with th e time-scale separation assumption [22]-[24]. D ue to limited space, a detailed discussion is omit- ted. W e pro ve Theorem 1 in the next subsection. 3.3 Pr oof of Theor em 1 Before we pro ceed, we present Lemma 1 as follo ws to assist us in proving Theorem 1. Lemma 1. F or any fe asible r ate ve ctor ( θ, ( r l ) l ∈S ) ∈ Λ I , ther e exists a stationary r andomize d algorithm SI that stabi- lizes the network with SU admitte d arrival r ate A S I l ( t ) = r l , ∀ t ∀ l ∈ S , and PU admitte d arrival r ate µ S I ps P ( t ) = θ , ∀ t , and sche dule { ( µ S I mn ( t )) ( m,n ) ∈L , ( s S I l ( t )) l ∈S } indep endent of queue b acklo gs satisfying: E { X i :( n,i ) ∈L µ S I ni ( t ) − X j :( j,n ) ∈L c µ S I j n ( t ) } = 0 , ∀ t, ∀ n ∈ N ; E { s S I l ( t ) } = r l , ∀ t, ∀ l ∈ S . Note that it is not necessary for the randomized algorithm SI to pro vide finite buffer size or dela y guarantees. S imilar form ulations of stationary randomized algorithms and exis- tence p roofs hav e b een presented in [8][11]-[13], so we omit the pro of of Lemma 1 for brevity . R emark 5: According to the S I algorithm in Lemma 1, w e assign the virtual input rate as R S I ( t ) = µ S I ps P ( t ) = θ , ∀ t . Hence, the t ime av erage of R S I ( t ) satisfies r S I = θ . Note that ( θ , ( r l ) l ∈S ) can tak e v alues as ( f − 1 ( a ) + 1 2 ǫ, ( r ∗ l,ǫ ) l ∈S ) or ( f − 1 ( a ) + ǫ, ( r ∗ l,ǫ + ǫ ) l ∈S ). W e define the queue vector Q I ( t ) as: Q I ( t ) = ( ( U n ) n ∈N , ( Q l ) l ∈S , U p ( t ) , Z ( t )) and define the Lyapunov function L I ( Q I ( t )) as follo ws: L I ( Q I ( t )) , 1 2 { X l ∈S Q l ( t ) 2 + q M − µ M q M U p ( t ) 2 + Z ( t ) 2 + X n ∈N U n ( t ) 2 U p ( t ) q M } , where the last term of t h e ab ov e Lyapunov function takes a simila r form as in [17][18]. Then, the correspond ing Lya- punov drift is defined b y ∆ I ( t ) , E { L I ( Q I ( t + 1)) − L I ( Q I ( t )) | Q I ( t ) } . By squaring both sides of the qu eueing dynamics (2)(3)(6)(8) and through algebra, we can obtain: ∆ I ( t ) − V 1 X l ∈S E { g l ( A l ( t )) | Q I ( t ) } ≤ B 1 + µ 2 M + N + 1 2 q M U p ( t ) − X n ∈N E { U n ( t ) U p ( t ) q M ( X i :( n,i ) ∈L µ ni ( t ) − X j :( j,n ) ∈L c µ j n ( t )) | Q I ( t ) } − E { Z ( t )( R ( t ) − f − 1 ( a P )) | Q I ( t ) } − E { ( q M − µ M ) U p ( t ) q M ( µ ps P ( t ) − R ( t )) | Q I ( t ) } − E { X l ∈S Q l ( t )( s l ( t ) − A l ( t )) | Q I ( t ) } − V 1 X l ∈S E { g l ( A l ( t )) | Q I ( t ) } , (17) where we also emplo y the follo wing ineq ualities: X n ∈N U n ( t + 1) 2 U p ( t + 1) q M ≤ ( R ( t ) q M + U p ( t ) q M ) X n ∈N U n ( t + 1) 2 ≤ µ M q M ( N + 1) + U p ( t ) q M ( µ 2 M + N + 1) + U p ( t ) q M X n ∈N U n ( t ) 2 − 2 U p ( t ) q M X n ∈N U n ( t )( X i :( n,i ) ∈L µ ni ( t ) − X j :( j,n ) ∈L c µ j n ( t )) } . Through algebra, w e fin d the equiv alence of (17): ∆ I ( t ) − V 1 X l ∈S E { g l ( A l ( t )) | Q I ( t ) } ≤ B 1 + µ 2 M + N + 1 2 q M U p ( t ) + f − 1 ( a P ) Z ( t ) + X l ∈S E { A l ( t ) Q l ( t ) − V 1 g l ( A l ( t )) | Q I ( t ) } + E { R ( t )( ( q M − µ M ) U p ( t ) q M − Z ( t )) | Q I ( t ) } − E { µ ps P ( t ) U p ( t ) q M ( q M − µ M − U ps P ( t )) | Q I ( t ) } − E { X l ∈S Q l ( t ) s l ( t ) + X ( m,n ) ∈L µ mn ( t ) U p ( t ) q M ( U m ( t ) − U n ( t )) | Q I ( t ) } . (18) Note that the last four terms of the R HS of (18 ) are mini- mized by the S U congestion contro ller (9), th e R ( t ) regulator (10), the PU congestion controller (11), and the link sched- uler (12), resp ective ly , over a set of feasible algorithms in - cluding the stationary randomized algorithm S I introduced in Lemma 1 and Remark 5 . Then, w e substitut e into the fourth and fifth terms of the RHS of (18) (i.e., th e t h ird and fourth lines of (18)) a stationary rand omized S I with admitted arriv al rate vector ( f − 1 ( a ) + 1 2 ǫ, ( r ∗ l,ǫ ) l ∈S ) , and w e substitute into th e last t w o terms the S I with admitted ar- riv al rate vector ( f − 1 ( a ) + ǫ, ( r ∗ l,ǫ + ǫ ) l ∈S ). A fter the ab o ve substitutions, we obtain: ∆ I ( t ) − V 1 X l ∈S E { g l ( A l ( t )) | Q I ( t ) } ≤ B 1 − ǫ ( q M − µ M ) − µ 2 M − N − 1 2 q M U p ( t ) − ǫ X l ∈S Q l ( t ) − ǫ 2 Z ( t ) − V 1 X l ∈S g l ( r ∗ l,ǫ ) ≤ B 1 − δ 1 ( X l ∈S Q l ( t ) + U p ( t ) + Z ( t ) ) − V 1 X l ∈S g l ( r ∗ l,ǫ ) , (19) where the second inequality h olds when the condition (14) in Theorem 1 is satisfied. W e take th e exp ectation of b oth sides of (19) o ver Q I ( t ) and take th e time av erage on t = 0 , 1 , ..., T − 1, whic h leads to δ 1 T T − 1 X t =0 E { X l ∈S Q l ( t ) + U p ( t ) + Z ( t ) } ≤ B 1 + V 1 T T − 1 X t =0 E { X l ∈S g l ( A l ( t )) } − V 1 X l ∈S g l ( r ∗ l,ǫ ( t )) . (20) By taking limsup of T on b oth sides of (20 ) , w e can pro ve (15). W e can prov e (16) by taking the liminf of T on b oth sides of (20) and by emplo ying the follo wing fact from the conca vity of the SU utility funct ions: X l ∈S g l ( E { A l ( t ) } ) ≥ X l ∈S E { g l ( A l ( t )) } . Therefore, Theorem 1 is prov ed. 4. ELAS TIC ALGORITHM FOR THE CRN In this section, we desig n the optimal elastic algorithm composed of tw o parts, namely , PU congestion contro ller and a hop/link scheduler, describ ed in Subsection 4.1. Note that according to the fixed- routing structure in PU rela y subnetw ork introduced in S ubsection 2.1, when developing the sc heduler, we focus on the hop/link schedule (( µ k m,m +1 ( t )) m,k , ( s l ( t )) l ∈S )) whic h is composed of a PU hop sc hedule and an S U link sc hedule. Note th at each hop sc hedule ( µ k m,m +1 ( t )) m,k corresponds to a PU link sc hedule ( µ mn ( t )) ( m,n ) ∈L . 4.1 Elastic Algorithm 1) PU Congestion Contro lle r : min K X k =1 µ k − 1 , 0 ( t )( ρ k X l ∈S Q l ( t ) 1 {∃ m : v m k = l } + U k 0 ( t ) − V 2 ) s.t. K X k =1 µ k − 1 , 0 ( t ) ≤ µ M , (21) where V 2 is a con trol p arameter in the algo rithm. F or time slot t , define k ∗ , arg min k ( ρ k P l ∈S Q l ( t ) 1 {∃ m : v m k = l } + U k 0 ( t )). Specifically , from (21), we set µ k ∗ − 1 , 0 ( t ) =      µ M , if ρ k ∗ X l ∈S Q l ( t ) 1 {∃ m : v m k ∗ = l } + U k ∗ 0 ( t ) ≤ V 2 , 0 , otherwise. F or k 6 = k ∗ , we set µ k − 1 , 0 ( t ) = 0. 2) Hop/Link Scheduler : max { K X k =1 H k X m =0 µ k m,m +1 ( t )( U k m ( t ) − U k m +1 ( t )) + X l ∈S Q l ( t ) s l ( t ) } , s.t. { ( µ mn ( t )) ( m,n ) ∈L , ( s l ( t )) l ∈S } ∈ I , (22) where the optimization is taken o ver all feas ible (( µ k m,m +1 ( t )) m,k ,( s l ( t )) l ∈S )) and we note that eac h hop sched- ule ( µ k m,m +1 ( t )) m,k corresponds to a PU link schedule ( µ mn ( t )) ( m,n ) ∈L . F rom (22), when U k m ( t ) − U k m +1 ( t ) ≤ 0, m ∈ { 0 , 1 , ... , H k } , w e set µ k m,m +1 ( t ) = 0. The elastic algorithm has th e follow ing prop ert y: Pr oposition 2. ∀ m ∈ { 0 , 1 , ..., H k } , ∀ k ∈ { 1 , 2 , ..., K } , the fol lowi ng ine quali ty holds: U k m ( t ) ≤ U M , µ M + V 2 . (23) Pr oof. Similar t o the pro of of Prop osition 1, we prove Proposition 2 by induction. Initial ly when t = 0, U k m (0) = 0 ∀ m , ∀ k . No w assume in time slot t we hav e U k m ( t ) ≤ U M , ∀ m , ∀ k . In the induction step, we consider t wo cases: Case 1: m = 0. Given any route k , if U k 0 ( t ) ≤ V 2 , then w e hav e U k 0 ( t + 1) ≤ U k 0 ( t ) + µ M ≤ U M according to queueing dynamics (4), where w e recall that µ k − 1 , 0 ( t ) ≤ µ M from the constrain t in PU congestion controll er (21). O therwise, we hav e V 2 < U k 0 ( t ) ≤ U M , and hence we ha ve ρ k X l ∈S Q l ( t ) 1 {∃ m : v m k = l } + U k 0 ( t ) > V 2 , whic h in duces µ k − 1 , 0 ( t ) = 0 from the PU congestion con- troller (21), and it follow s that U k 0 ( t + 1) ≤ U k 0 ( t ) ≤ U M by the queueing dyn amics (4). Case 2: m ∈ { 1 , 2 , ..., H k } , for an y given route k . If U k m ( t ) ≤ U M − 1, then w e ha ve U k m ( t + 1) ≤ U k m ( t ) + 1 ≤ U M accord- ing to qu eu eing d ynamics (4). Otherwise, w e ha ve U k m ( t ) = U M ≥ U k m − 1 ( t ), and according to th e hop/link sc heduler w e hav e µ k m − 1 ,m ( t ) = 0, from which we hav e U k m ( t + 1) ≤ U k m ( t ) = U M by t he queueing dynamics (4). Therefore, U k m ( t + 1) ≤ U M ∀ m ∈ { 0 , 1 , ..., H k } , ∀ k ∈ { 1 , 2 , ..., K } , i.e., the induction step holds, and the prop osi- tion is prov ed. As a complement to Prop osition 2, recall that given route k , w e alw ays hav e U k H k +1 ( t ) = 0, ∀ t . Now we present the main results of the elastic algorithm in Theorem 2. Theorem 2. L et ǫ > 0 b e chosen arbitr arily smal l . The elastic algorithm ensur es the fol lowing ine quality on queue b acklo gs: lim sup T →∞ 1 T T − 1 X t =0 E { X l ∈S Q l ( t ) } ≤ B 2 + V 2 B R δ 2 , (24) wher e B 2 , 1 2 K ( N + 2) + 1 2 N + 1 2 N µ 2 M max k ρ 2 k , δ 2 , ǫ min k ρ k , and B R is define d as: B R , lim sup T →∞ 1 T T − 1 X t =0 E { K X k =1 µ k − 1 , 0 ( t ) } − K X k =1 λ ∗ k,ǫ ≤ µ M − K X k =1 λ ∗ k,ǫ . F urthermor e, the inelastic algorithm achieves: lim inf T →∞ 1 T T − 1 X t =0 K X k =1 E { µ k − 1 , 0 ( t ) } ≥ K X k =1 λ ∗ k,ǫ − B 2 V 2 . ( 25) R emark 6 (Stability): The inequalities (23) from Prop o- sition 2 and (24) from Theorem 2 ind icate that PU and SU queues are all stable, and hence is the CRN. In a ddition, Proposition 2 ensures that PU q u eues maintained in eac h route are deter ministic al l y b ounded by the finite bu ffer size U M . R emark 7 (Optimal Thr oughput and T r ade off with Delay): The ineq ualit y (25) give s the lo w er-b ound of the throughput the elastic algori thm can achiev e. Since the constant B 2 is indep endent of the control parameter V 2 , the algorithm can ac hieve a PU th roughput arbitrarily close to the optimal v alue P K k =1 λ ∗ k as ǫ can b e chosen arbitrarily small and V 2 can b e chosen arbitrarily large, with the follo wing tradeoffs in PU and SU d ela y: • The PU b uffer size U M is of order O ( V 2 ) as shown in (23). By L itt le’s Theorem, the PU’s ave rage end- to- end dela y o ver an y giv en route k is of order O (( H k + 1) V 2 ) which is b ounded by the first order of H k , i.e., the algorithm has or der-o ptimal delay per route. • F rom (24), the ave rage S U buffer o ccupancy is of order O ( N + V 2 ǫ ). An d so is the SU a verage delay by Little’s Theorem. The a verage S U delay up per- boun d has an extra t erm 1 ǫ in order compared with the av erage PU dela y . R emark 8 (Employing Delaye d Queue Information): The PU congestion control ler (21) is p erformed at the source PU. Thus, in order to accoun t for the propagation d ela y of queue information ( Q l ( t )) l ∈S , we can rep lace ( Q l ( t )) in ( 21) by ( Q l ( t − τ )), where τ is an integer num b er th at is larger than the maximum propagation delay from any nod e to a source. It is not difficu lt to show that T heorem 2 still holds with a different v alue of B 2 , with similar pro of techniques as in [18][19]. W e pro ve Theorem 2 in the next subsection. 4.2 Pr oof of Theor em 2 Before we pro ceed, w e present Lemma 2 as follo ws to assist us in proving Theorem 2. Lemma 2. F or any f e asible r ate ve ctor (( λ k ) k ∈{ 1 , 2 ,...,K } , ( K X k =1 ρ k λ k 1 {∃ m : v m k = l } ) l ∈S ) ∈ Λ E , ther e exists a stationary r andomize d algorithm SE that stabi- lizes the network wi th PU admitte d arrival r ates µ k,S E − 1 , 0 ( t ) = λ k , ∀ t ∀ k ∈ { 1 , 2 , ..., K } and a hop/link sche dule (( µ k,S E m,m +1 ( t )) m,k , ( s S E l ( t )) l ∈S ) indep endent of queue b acklo gs satisfying: E { µ k,S E m − 1 ,m ( t ) − µ k,S E m,m +1 ( t ) } = 0 , ∀ t, ∀ m, k ; E { s S E l ( t ) } = K X k =1 ρ k λ k 1 {∃ m : v m k = l } , ∀ t, ∀ l ∈ S . Similar to Lemma 1, it is n ot necessary for the randomized algorithm SE to provide finite bu ffer size or d ela y guaran- tees. F or brevity , we omit the proof of Lemma 2, and inter- ested readers are referred t o [8][11]-[13] for details. Note that ( λ k ) k ∈{ 1 , 2 ,...,K } can take values as ( λ ∗ k,ǫ ) k and ( λ ∗ k,ǫ + ǫ ) k . W e denote th e q ueue vector Q E ( t ) = (( U k m ) m,k , ( Q l ) l ∈S ) and define the Lyapunov function L E ( Q E ( t )) as follo ws: L E ( Q E ( t )) , 1 2 { K X k =1 H k X m =0 ( U k m ( t )) 2 + X l ∈S Q l ( t ) 2 } Then, the corresp onding Lyapuno v drift is defined as ∆ E ( t ) , E { L E ( Q E ( t + 1)) − L E ( Q E ( t )) | Q E ( t ) } . By squaring b oth sides of the queueing dynamics (4)(5), w e can obt ain: ∆ E − V 2 E { K X k =1 µ k − 1 , 0 ( t ) | Q E ( t ) } ≤ 1 2 K X k =1 H k X m =0 E { ( µ k m,m +1 ( t )) 2 + ( µ k m − 1 ,m ( t )) 2 − 2 U k m ( t )( µ k m,m +1 ( t ) − µ k m − 1 ,m ( t )) | Q E ( t ) } + 1 2 X l ∈S E { s l ( t ) 2 + ( K X k =1 ρ k µ k − 1 , 0 ( t ) 1 {∃ m : v m k = l } ) 2 − 2 Q l ( t )( s l ( t ) − K X k =1 ρ k µ k − 1 , 0 ( t ) 1 {∃ m : v m k = l } ) | Q E ( t ) } − V 2 E { K X k =1 µ k − 1 , 0 ( t ) | Q E ( t ) } , from which we obtain: ∆ E − V 2 E { K X k =1 µ k − 1 , 0 ( t ) | Q E ( t ) } ≤ B 2 − V 2 E { K X k =1 µ k − 1 , 0 ( t ) | Q E ( t ) } − K X k =1 H k X m =0 E { U k m ( t )( µ k m,m +1 ( t ) − µ k m − 1 ,m ( t )) | Q E ( t ) } − X l ∈L E { Q l ( t )( s l ( t ) − K X k =1 ρ k µ k − 1 , 0 ( t ) 1 {∃ m : v m k = l } ) | Q E ( t ) } . (26) Through algebra, we find the equiv alence of (26): ∆ E − V 2 E { K X k =1 µ k − 1 , 0 ( t ) | Q E ( t ) } ≤ B 2 + E { K X k =1 µ k − 1 , 0 ( t ) × ( ρ k X l ∈S Q l ( t ) 1 {∃ m : v m k = l } + U k 0 ( t ) − V 2 ) | Q E ( t ) } − E { K X k =1 H k X m =0 µ k m,m +1 ( t )( U k m ( t ) − U k m +1 ( t )) + X l ∈S Q l ( t ) s l ( t ) | Q E ( t ) } , (27) where we emp lo y the follo wing fact that ∀ k ∈ { 1 , 2 , ..., K } : H k X m =0 U k m ( t )( µ k m,m +1 ( t ) − µ k m − 1 ,m ( t )) = H k X m =0 µ k m,m +1 ( t )( U k m ( t ) − U k m +1 ( t )) − µ k − 1 , 0 ( t ) U k 0 ( t ) . Note that the second and third terms of the RHS of (27) are minimized by the PU congestion controller (21) and the hop/link scheduler (22), respectively , o ver a set of feasible algorithms including the stationary randomized algori thm S E introd u ced in Lemma 2. Then, w e sub stitute into th e second t erm of the RHS of (27) a stationary randomized S E with admitted PU arriv al rate v ector ( λ ∗ k,ǫ ) k ∈{ 1 , 2 ,...,K } and into the t h ird terms the S E with admitted PU arriv al rate vector ( λ ∗ k,ǫ + ǫ ) k ∈{ 1 , 2 ,...,K } . After the above substitutions, w e obtain: ∆ E − V 2 E { K X k =1 µ k − 1 , 0 ( t ) | Q E ( t ) } ≤ B 2 − V 2 K X k =1 λ ∗ k,ǫ − ǫ X l ∈S Q l ( t ) K X k =1 ρ k 1 {∃ m : v m k = l } ≤ B 2 − V 2 K X k =1 λ ∗ k,ǫ − δ 2 X l ∈S Q l ( t ) . (28) W e take the expectation of both sides of (28) ov er Q E ( t ) and take th e time av erage on t = 0 , 1 , ..., T − 1, whic h leads to δ 2 T T − 1 X t =0 E { X l ∈S Q l ( t ) } ≤ B 2 + V 2 T T − 1 X t =0 E { K X k =1 µ k − 1 , 0 ( t ) } − V 2 K X k =1 λ ∗ k,ǫ . (29) By taking limsup of T on b oth sides of (29 ) , w e can pro ve (24). By tak ing t he liminf of T o n b oth sides of (29), we can prov e (25). Theref ore, Theorem 2 is pro ved. 5. ARBITRAR Y ARRI V AL RA TES A T TRANS- POR T LA YE R In th e previous model description and algorithm develop- ment, w e assumed th at PU and S U pack ets are bac klogged at the transp ort la yer. In th is section, we present optimal algorithms for t he inelastic and elastic PU models, re sp ec- tively , for arbitrary arriv al rates at the transp ort lay er. At the transp ort lay er, let E p ( t ) and E l ( t ), l ∈ S , b e the PU and S U arriv al rates at th e b eginning of time slot t , resp ectiv ely . W e assume that E p ( t ) and E l ( t ) ∀ l ∈ S are i.i.d. with resp ect to time. F or si mplicity of analysis, w e assume that the time a verage arriv al rate v ector, formed by the PU and SU a rriv al rates, is in the exterior of the capacit y region, so that a congestion con troller is n eeded. Let W p ( t ) and W l ( t ), l ∈ S , b e th e bac klog of PU and SU data at the transport lay er. PU and SU buffer sizes at the transp ort la yer are denoted by W P and W S , resp ective ly . I n the follo wing subsections, w e present modified algorithms that can han d le arbitrary arriv al rates at transport lay er for inelastic and elastic PU mo dels. 5.1 Inelastic Alg orithm f or Arbitrary Arrival Rates at T ransport Layer In the inelastic PU mo del, recalling that A l ( t ) is the ad- mitted SU rate, we up date the SU backlog W l ( t ) at the transp ort la yer as follow s: W l ( t +1) = min { [ W l ( t ) + E l ( t ) − A l ( t )] + , W S } , ∀ l ∈ S . (30) Note th at W l ( t ) = 0 ∀ t ∀ l ∈ S and W S = 0 if there is no buffer at S U transp ort lay er. Similarly , we up date the PU backlo g W p ( t ) at the transp ort la yer as follo ws: W p ( t + 1) = min { [ W p ( t ) + E p ( t ) − µ ps P ( t )] + , W P } . (31) Note that W p ( t ) = 0 ∀ t and W P = 0 when there is no buffer at PU transp ort lay er. F ollo wing t h e idea introdu ced in [12], w e construct a vir- tual queue Y p ( t ) at the PU transport la y er wi th queueing dynamics: Y p ( t + 1) = [ Y p ( t ) − R ( t )] + + u p ( t ) , where u p ( t ) is an aux iliary v ariable asso ciated with Y p ( t ). Similarly , we construct a v irtual queue Y l ( t ), l ∈ S , at the SU transp ort la yer with queueing dynamics: Y l ( t + 1) = [ Y l ( t ) − A l ( t )] + + u l ( t ) , where u l ( t ) is an auxiliary v ariable associated with Y l ( t ), with u l b eing its time-a verag e. Note that when Y l ( t ) is sta- ble, the time-av erage of SU a dmitted rate A l ( t ) is gre ater than or equal to u l . Thus, when Y l ( t ) is stable, ∀ l ∈ S , if we can ensu re that P l ∈S g l ( u l ) is arbitrarily close to the optimal v alue P l ∈S g l ( r ∗ l ), so is the SU u tilit y . Instead of (8), the queu e state Z ( t ) is n o w up dated as: Z ( t + 1) = [ Z ( t ) − u p ( t )] + + f − 1 ( a P ) . Denote the time-a verage of u p ( t ) as u p . Thus, when Y p ( t ), U p ( t ) and Z ( t ) are stable, w e hav e f ( µ ) ≥ f ( r ) ≥ f ( u p ) ≥ a P , where we recall t hat µ and r are the time-a verage v al- ues of µ ps P ( t ) and R ( t ), resp ectively . Sp ecifically , when Y p ( t ), U p ( t ) and Z ( t ) a re stable, the PU minim um utility constrain t is met. Now w e provide the inelastic algorithm for arbitrary ar- riv al rates at the transp ort la yer: 1) SU Congesti on Con troller : u l ( t ) and A l ( t ), l ∈ S , are u p dated as follo ws: min 0 ≤ u l ( t ) ≤ A M u l ( t ) Y l ( t ) − V g l ( u l ( t )) (32) min A l ( t )( Q l ( t ) − Y l ( t )) s.t. 0 ≤ A l ( t ) ≤ min { W l ( t ) + E l ( t ) , A M } (33) Note that (32) and (33) can b e so lved independently and locally at every S U. 2) R ( t ) Regulator : min 0 ≤ u p ( t ) ≤ µ M u p ( t )( Y p ( t ) − Z ( t )) (34) min R ( t )( q M − µ M q M U p ( t ) − Y p ( t )) s.t. 0 ≤ R ( t ) ≤ min { W p ( t ) + E p ( t ) , µ M } (35) Note that (34) and (35) can b e so lved independently and locally at the source PU. 3) PU Congestion Contro lle r : max µ ps P ( t )( q M − µ M − U s p ( t )) s.t. 0 ≤ µ ps P ( t ) ≤ min { W p ( t ) + E ( p ) , µ M } 4) Link Rate Scheduler : The link scheduler is th e same as (12) in Section 3.2. It is n ot d ifficult to chec k that Prop osition 1 still holds, and we present the follo wing th eorem for th e p erformance of the algorithm: Theorem 3. L et ǫ > 0 b e chosen arbitr aril y smal l. Gi ven that q M > µ 2 M + N +1 ǫ + µ M , the i nelastic algorithm ensur es the f ol lowing i ne quality on queue b acklo gs: lim sup T →∞ 1 T T − 1 X t =0 E { X l ∈S ( Q l ( t ) + Y l ( t )) + U p ( t ) + Y p ( t ) + Z ( t ) } ≤ B 3 + V 1 g M δ 3 , wher e B 3 , B 1 + N A 2 M + µ 2 M and δ 3 is a c onstant. F urther- mor e, the algorithm achieves: X l ∈S g l ( a l ) ≥ X l ∈S g l ( r ∗ l,ǫ ) − B 3 V 1 , wher e we r e c al l that ( a l ) l ∈S is define d in The or em 1. The proof f ollo ws simil ar steps as the pro of of Theorem 1. Due to space limitations, we omit the pro of for Theorem 3 and the choice of δ 3 . Similar statements as presented in Remarks 1-4 also hold for the inelastic algorithm introduced in this subsection. 5.2 Elastic Algorithm f or Arbitrary Arriv al Rates at T ransport Layer In this subsection, the elastic algorithm for arbitrary ar- riv al rates at tra nsp ort la yer and its performance are dis- cussed. S imilar to inelas tic algorithm, w e denote by A l ( t ) the admitted SU rate, whic h is u pp er-b ou n ded by A M . W e up date t h e SU backlog s W l ( t ) at the t ransport la yer as (30). Similarly , w e can up date th e PU backlog W p ( t ) at the tran s- p ort la yer as: W p ( t + 1) = min { [ W p ( t ) + E p ( t ) − K X k =1 µ k − 1 , 0 ( t )] + , W P } . Similar to the previous subsection, we construct virtual queues Y p ( t ) and Y l ( t ), ∀ l ∈ S , with an auxiliary va riable u p ( t ) ass o ciated with Y p ( t ). The virtual queues evo lve as follo ws: Y p ( t + 1) = [ Y p ( t ) − K X k =1 µ k − 1 , 0 ( t )] + + u p ( t ); Y l ( t + 1) = [ Y l ( t ) − A l ( t )] + + K X k =1 ρ k µ k − 1 , 0 ( t ) 1 {∃ m : v m k = l } , ∀ l ∈ S . Note that when Y p ( t ) is stable, if we can ensure that u p is arbitrarily close to t h e optimal v alue P K k =1 λ ∗ k , th en so is the PU throughput, where w e recall t hat u p is the time av erage of u p ( t ). In add ition, when Y l ( t ) is stable, ∀ l ∈ S , the SUs’ throughput is prop ortional to the PU data th at th ey rela y . Now we provide the elastic algorithm for arbitrary arriv al rates at the transp ort la yer: 1) SU Conges tion Con troller : min 0 ≤ A l ( t ) ≤ min { W l ( t )+ E l ( t ) ,A M } A l ( t )( Q l ( t ) − Y l ( t )), ∀ l ∈ S . Note that the SU congestio n con troller can b e solved lo cally at each SU . 2) PU Congestion Contro lle r : min 0 ≤ u p ( t ) ≤ µ M u p ( t )( Y p ( t ) − V 2 ) (36) min K X k =1 µ k − 1 , 0 ( t )( ρ k X l ∈S Y l ( t ) 1 {∃ m : v m k = l } + U k 0 ( t ) − Y p ( t )) s.t. 0 ≤ K X k =1 µ k − 1 , 0 ( t ) ≤ min { W p ( t ) + E l ( t ) , µ M } . (37) Note that (36) and ( 37) can be solved indep endently . 3) Link/Hop Rate Sc heduler : The link/hop sc heduler is the same as (22) in Section 4.1. W e p resen t Prop osition 3 and Theorem 4 to characterize the p erformance of the prop osed algorithm: Pr oposition 3. ∀ m ∈ { 0 , 1 , ..., H k } , ∀ k ∈ { 1 , 2 , ..., K } , the fol lowi ng ine quali ty holds: U k m ( t ) ≤ 2 µ M + V 2 . Theorem 4. L et ǫ > 0 b e chosen arbitr arily smal l . The algorithm ensur es the fol lowing ine quali ty on queue b acklo gs: lim sup T →∞ 1 T T − 1 X t =0 E { X l ∈S ( Q l ( t ) + Y l ( t )) + Y p ( t ) } ≤ B 4 + V 2 B R δ 4 , wher e B 4 , B 2 + µ 2 M + N A 2 M and δ 4 > 0 i s a c onsta nt. F urthermor e, the inelastic algorithm achieves: lim inf T →∞ 1 T T − 1 X t =0 K X k =1 E { µ k − 1 , 0 ( t ) } ≥ K X k =1 λ ∗ k,ǫ − B 4 V 2 . The p roofs of Proposition 3 and Theorem 4 follo w similar steps as the pro ofs of Prop osition 2 an d Theorem 2, respec- tively . Detailed proofs and the c hoice of δ 4 are omitted due to limited space. R emarks 6- 8 hold for the elastic algorithm introduced in this subsection with minor modifications. 6. CONCLUSIONS AND FUTURE WORKS In this pap er, tw o cross-la yer sc heduling algorithms for multi-hop co operative cognitive radio n et wo rks are intro- duced. 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