Space-time Queuing Theoretic Modeling of Opportunistic Multi-hop Coexisting Wireless Networks With and Without Cooperation

Transcription

1 Space-tme Queung Theoetc Modelng of Oppotunstc Mult-hop Coexstng Weless Netwoks Wth and Wthout Coopeaton 1 Dbaka Das, Electcal, Compute and Systems Engneeng Rensselae Polytechnc Insttute Toy, NY Emal: dasd2@p.edu Alhussen A. Abouzed, Electcal, Compute and Systems Engneeng Rensselae Polytechnc Insttute Toy, NY Emal: abouzed@ecse.p.edu Abstact In ths pape, we chaacteze the aveage end-to-end delay n an oppotunstc mult-hop seconday cogntve ado netwok ovelad wth a pmay mult-hop netwok. Nodes n both netwoks use andom medum access contol (MAC) scheme wth exponentally dstbuted back-off. We fst model the netwok as a two-class poty queung netwok and use queung-theoetc appoxmaton technques to obtan a set of elatons nvolvng the mean and second moments of the nte-aval tme and sevcetme of packets at a seconday node. Then, applyng those paametes to an equvalent open sngle-class G/G/1-queung netwok, we obtan expessons fo the aveage end-to-end delay of a packet n the seconday netwok usng dffuson appoxmaton. Next we extend the analyss to a case whee seconday nodes coopeatvely elay pmay packets so as to mpove the own tansmsson oppotuntes. The mathematcal esults ae valdated aganst extensve smulatons. July 31, 2014 DRAFT

2 2 I. INTRODUCTION The exponental gowth n the numbe and demand of weless devces has motvated two paadgm shfts n the desgn of weless netwoks. Fst, futue ( 5G ) netwok desgn s now devatng fom the tadtonal (maco) cellula netwok achtectue to consde small and hybd cells, called mco, pco o femto-cells that may be vewed as base statons wth smalle coveage aea and coveage capacty. Due to the small coveage aea of these cells, mult-hop opeaton s expected to be commonly adopted. Ths new achtectual vson s typcally called heteogeneous netwoks o Hetnets (e.g. [1]) whch ncludes devce-to-devce (D2D) communcatons. Second, the exponental gowth has led to new spectum shang egulatoy ules allowng eal-tme shang of lcensed but unused spectum, and new poposals fo dynamc spectum shang that leveages the advances n softwae/ cogntve ado technologes, pomsng to povde bette spectum utlzaton and aglty [2], [3]. In ths wok, n the context of these futue 5G netwoks, we consde the coexstence of two mult-hop weless netwoks, one composed of lcensed pmay uses (PUs) and the othe composed of lowe poty oppotunstc seconday uses (SUs) that possess spectum-agle functonaltes. We nvestgate the aveage end-to-end delay n a mult-hop seconday netwok coexstng wth anothe mult-hop pmay netwok. Both netwoks use andom access based MAC potocol. The aveage end-to-end delay s the end-to-end delay aveaged ove all eceved packets and netwok topologes and t depends on the taffc patten, numbe of nodes, MAC scheme and outng potocol. To the best of ou knowledge, no pevous wok (except ou pelmnay wok n [4]) addessed the aveage end-to-end delay n such a settng. Both netwoks shae a sngle channel and use backoff and collson avodance schemes smla to IEEE andom access MAC. A modfed potocol model of ntefeence s used to account fo packet loss dung tansmsson due to fadng. We consde two cases. In the non-coopeatve case, thee s no coopeaton between the two netwoks. In the coopeatve case, we consde a potocol wheen seconday nodes elay pmay packets that wee not coectly eceved at a pmay eceve but wee successfully eceved at a neaby seconday node. We assume nfntesmally small sensng ntevals and an deal sensng pocess. The man esult of ths pape s the applcaton of queung theoetc technques fo devng appoxmate mathematcal expessons fo: 1) The aveage end-to-end delay of seconday packets fo the non-coopeatve as well as

3 3 the coopeatve case. 2) The maxmum achevable thoughput of seconday uses unde both coopeatve and non-coopeatve potocols, and condtons unde whch seconday nodes beneft fom coopeaton (n tems of thoughput). The solutons fo the aveage end-to-end delay fo both coopeatve and non-coopeatve potocols have closed fom expessons except fo tems coespondng to the aveage pobablty of successful tansmsson fom pmay and seconday uses. Those tems ae obtaned usng numecal ntegaton. Fom smulaton esults we obseve that the appoxmaton matches closely wth expemental esults fo low to modeate channel utlzaton, whee channel utlzaton s pecsely defned late n the pape. The methodology n ths pape can be summazed as follows. Fo both coopeatve and non-coopeatve scenaos we fst model the netwok as an open netwok of two-class G/G/1 pe-emptve esume sevce Fst Come Fst Seve (FCFS) poty queues. We then use cetan queung appoxmaton technques fom [5] to fnd elatons nvolvng the effectve sevce-tme and nte-aval tme of packets at evey node. Tansmssons fom ntefeng neghbos (to be defned late) consttute hgh-poty class-1 taffc at evey staton. Self-geneated o fowaded packets fom othe nodes and unsuccessful packet tansmsson attempts by a node consttute aval of lowe poty class-2 taffc at the coespondng staton. Next we model the netwok as a collecton of sngle-class G/G/1 (non-poty) queues fo whch the effectve sevce tme and the nte-aval tme of a job at any staton satsfes the coespondng elatons fo class-2 jobs n the poty queung netwok epesentaton (obtaned n the fst step). Ths enables the devaton of expessons fo the aveage end-to-end delay usng dffuson appoxmaton fo an open queung netwok consstng of G/G/1 statons as gven n [6]. Thee exsts seveal elated woks on queung delay n cogntve netwoks. In [7], the authos analyze queung delay n a sngle-hop netwok of multple SUs, that uses andom access, n pesence of multple pmay channels by usng contnuous flud-queue appoxmaton. In [8], the authos study delay pefomance of one SU n the pesence of othe PUs shang the same channel. They popose a tme-theshold scheme fo SU packet tansmsson by developng a Makovan model wheen each state s the numbe of SU packets at the begnnng of evey dle tme-slot. In [9] the authos model the pmay use actvty as an ON-OFF altenatng enewal pocess and used M/G/1 queung analyss to obtan the aveage delay fo a sngle seconday

4 4 use n pesence of pmay actvty. In [10] the authos use a poty queue model to analyze delay pefomance of a seconday use wth multple classes of taffc. In [11] the authos use M/D/1 poty queueng scheme to obtan the aveage delay fo pmay and seconday uses. In [12], the authos use pe-emptve poty queung system to evaluate the aveage watng tme of delay-senstve and delay-nsenstve packets fo two cases- (a) multple PUs and a sngle SU whee the SU senses only at the begnnng of a tme-slot and (b) a sngle PU and a sngle SU whee the SU senses the channel contnuously. Othe woks such as [13], [14] also analyze delay fo sngle-hop SUs by usng a poty queue model fo channel access. Howeve, none of these woks [7] [14] study delay n a mult-hop netwok. Whle we use a smla poty queue-model as [11] [14], n ou case the sevce-tme pocess of an SU s nteupted due to tansmsson pocess of neaby PUs and SUs and the latte n tun depends on the own espectve sevcetme pocesses. As a esult, ou scenao s dffeent fom that n pevous woks. In [15], the authos chaacteze the mnmum mult-hop delay and connectvty of the seconday netwok as a functon of SU and PU denstes. Howeve, ths wok also does not addess the scenao whee dffeent seconday nodes ae contendng fo the channel. Ou coopeaton model s smla to othe woks such as [16] [19] whee seconday uses coopeatvely elay pmay packets. In [16] the authos consde two lnks: a pmay and a seconday; a seconday elay node e-tansmts packets that wee coectly eceved at the elay node but not successfully eceved by the ntended pmay destnaton node. A smla model s used n [17] wheen the authos consde a sngle pmay s-d pa n the pesence of multple seconday nodes that can act as elay. In addton, the queued pmay packets at the elay nodes ae tansmtted wth a hghe poty n evey dle slot. In [18] the authos obtan stable thoughput egon fo the pmay and seconday uses n a 5 node netwok wth 2 pmay tansmttes and one common seconday elay. In [19] the authos consde uplnk of a TDMAbased pmay netwok whee some cogntve nodes assst the pmay netwok by e-tansmttng some pmay packets that wee not eceved successfully at the base-staton. Howeve none of the above woks looked at the aveage delay of seconday packets n a mult-hop scenao. Ou netwok model s based on that n [20] whee the authos obtan closed fom expesson fo the aveage end-to-end delay n a mult-hop netwok of nodes that use a andom-access MAC and pobablstc outng potocol. The authos fst obtaned exact expessons fo the mean and second moments of the effectve sevce-tme of packets explotng the smplstc

5 5 natue of the outng potocol and MAC scheme. Then they used a dffuson appoxmaton fo sngle-class G/G/1 systems to obtan closed fom expessons fo the aveage end-to-end delay. In contast to [20] whch consdeed a sngle netwok, ths wok consdes two co-exstng and nteactng netwoks (pmay and seconday) whee nodes fom one netwok (.e. pmay) have hghe poty of channel access than the nodes fom the second netwok (.e. seconday). Ths couplng of the behavo of the queues n the two netwoks ntoduced new modelng challenges, whch ae analyzed by applyng new appoxmaton technques that have not been used befoe n ths context. In the next secton, we descbe the mult-hop cogntve ado netwok model. Secton III befly summazes the theoetcal queung netwok esults on dffuson appoxmaton fo poty queues [5] and the dffuson appoxmaton fo non-poty sngle-class G/G/1 systems [6], whch ae used n late sectons. In Secton IV we deve expessons fo the aveage delay fo seconday packets wthout coopeaton. In Sectons V and VI we dscuss the coopeatve potocol model and the aveage end-to-end delay fo seconday packets wth coopeaton espectvely. In Secton VII we dscuss when coopeaton benefts seconday uses. In Secton VIII, we compae the analytcal and smulaton esults and fnd that they closely match fo wde ange of channel utlzaton scenaos. Secton IX concludes the pape. II. NETWORK MODEL FOR NON-COOPERATIVE CASE In ths secton we pesent the model of multhop pmay and seconday ad-hoc netwok whee the pmay and seconday nodes do not coopeate wth each othe. Both pmay and a seconday netwoks co-exst togethe and shae a sngle channel. We assume the pmay and seconday netwoks consst of n + 1 and n + 1 nodes espectvely. Nodes ae dstbuted unfomly and ndependently ove a tous of unt aea. In the followng subsectons we dscuss the channel model, ntefeence model, outng model and the MAC behavo. We also pesent a queung netwok epesentaton of the netwok. A. Channel model The weless channel between a tansmttng node and ts destnaton s subjected to Raylegh fadng wth addtve whte nose at the eceve. Let d j and h t j denote the dstance between node and j and the channel fadng coeffcent between and j at tme t, espectvely. h t j s

6 6 a zeo mean ndependent and dentcally dstbuted (..d.) complex Gaussan andom pocess wth vaance σ 2 0. The sgnal eceved at a eceve j fom a node at tme t, denoted as y t j, can be modeled as: y t j = ψ d α j ht jx t + n t j (1) whee ψ denotes the tansmsson powe of, α denotes the path-loss exponent, x t denotes the sgnal of unt powe tansmtted by at tme t and n t j denotes an..d. zeo mean whte Gaussan nose at j wth vaance N 0. A packet tansmsson s successful f the sgnal-to-nose-ato (SNR) at the eceve s above a specfed theshold β. The pobablty of ths event s theefoe, B. Intefeence model P (SNR at j β) = exp( N 0β ψ σ0d 2 α ) (2) j In ode to account fo packet-loss due to fadng and smultaneously ensue tactable analyss 1 we use a modfed veson of the Potocol Model of ntefeence descbed n [21]. Detals about the model s pesented next. Evey pmay and seconday node tansmts at fxed powe of ψ and ψ espectvely. A pmay (o seconday) node can only tansmt to othe pmay (o seconday) nodes wthn ts tansmsson adus R (o R ) and the tansmsson s successful f the eceved SNR s hghe than β. All pmay (o seconday) nodes located wthn tansmsson adus of a gven pmay (o seconday) node ae called tansmsson neghbos to that node. We assume R and R ae such that both pmay and seconday netwoks ae connected. A pmay (o seconday) node can tansmt to ts tansmsson neghbo j only f no othe node k, such that j s wthn tansmsson adus of k, s tansmttng smultaneously. Accodngly we defne ntefeng neghbos of k as the set of all nodes whose tansmssons may collde wth that of k. Pmay (o seconday) ntefeng neghbos to a pmay (o seconday) node k ae all pmay (o seconday) nodes located wthn dstance 2R (o 2R ) fom k. If k s a seconday node, pmay ntefeng neghbos of k ae all pmay nodes located wthn dstance R +R 1 A physcal model of ntefeence s typcally used to study netwoks wth fadng. Howeve pefomng delay analyss fo such model s cumbesome because t eques the analyss to account fo tansmsson powe of adjacent nodes along wth N 0 n (2). On the othe hand by usng a modfed potocol model of ntefeence we captue the effect of channel fadng between a tansmtte-eceve pa on the eceved SNR whle gnong ntefeence fom adjacent nodes. Ths leads to smple analyss.

7 7 fom k. A node feezes ts backoff tmes and pauses any ongong tansmsson actvty f any of ts ntefeng neghbos stats tansmttng. Note that pmay nodes do not have any seconday ntefeng neghbo snce they have hghe poty of channel access than seconday nodes. Defnton of ntefeng neghbos s same as that n [21] obtaned by settng a guad-zone paamete n [21] to zeo. Note that the expesson (1) fo sgnal eceved at any node j gnoes tansmssons fom othe nodes (ncludng nodes located fathe away fom j than ts ntefeng neghbos). Hence, a geneal value of s moe appopate to justfy ths expesson. Howeve, fo convenence of analyss we set as zeo. Delay analyss fo non-zeo wll be smla as that n ths pape except that the expessons fo the aveage numbe of pmay and seconday ntefeng neghbos of any node wll be dffeent than that obtaned usng Lemma 3 (descbed late). The same appoach as n Lemma 3 can be used to deve those values. C. Packet geneaton and MAC model Evey pmay and seconday node geneates packets at a ate λ and λ packets/second, espectvely. The sze of a packet s constant and s equal to L bts. Nodes n both netwoks use andom access MAC wth exponental back-off tmes. The mean duatons of the back-off tme s denoted as 1. If a packet tansmsson fom any node s ξ unsuccessful, the packet emans at the head of ts tansmsson queue. We assume an deal sensng pocess,.e. the seconday nodes can sense tansmsson actvty of pmay ntefeng neghbos almost nstantaneously and pause any of the ongong tansmsson pocesses. We assume a gven pmay node lstenng fo the channel actvty can dffeentate between channel usage by pmay and seconday nodes. If a pmay node senses that the channel s beng used by a seconday node, t stll teats the channel as f t s dle.e. ts backoff pocess s not affected. Ths s because seconday nodes have lowe poty than pmay nodes. The tansmsson ate of both pmay and seconday nodes s W bts/second. The tansmsson tme of evey packet s L W seconds, gnong the tme equed to exchange RTS, CTS and ACK packets. D. Routng model We assume a pobablstc outng model fo both pmay and seconday nodes. On ecevng a packet fom one of ts tansmsson neghbos, a pmay (o seconday) node absobs t wth

8 8 R B A R (S) E C D Fg. 1. A netwok wth two pmay nodes: A,B and 3 seconday nodes: C,D and E. A and B ae both tansmsson and ntefeng neghbos to each othe. C,D and C,E ae tansmsson neghbos to each othe. Intefeng neghbos of C ae D,E and A. Intefeng neghbos of D ae C,E and A. Intefeng neghbos of E ae D and C. pobablty q (o q ) o fowads t to a tansmsson neghbo, pcked wth equal pobablty among all tansmsson neghbos 2, wth pobablty 1 q (o 1 q ). The vaables q and q model localty of taffc between the souce and destnaton of tansmtted packets and ae functons of node denstes. Hghe q (o q ) means, on aveage, souce-destnaton pas n the pmay (o seconday) netwok ae connected va lowe numbe of hops and vce vesa. E. Queung netwok epesentaton We epesent the netwok as an open 2-class queung netwok of G/G/1 FCFS pe-emptve esume sevce poty queues. Each staton of the netwok coesponds to a node. Thoughout the pape we use the tems node and staton ntechangeably wheneve thee s no confuson. Ths queung netwok model accounts fo nteuptons n the on-gong sevce of a packet (ncludes the duatons of both the back-off tme and the tansmsson tme) due to tansmsson fom ntefeng neghbos. All such tansmsson actvtes ae modeled as hgh poty class-1 jobs. Low poty class-2 jobs coespond to physcal packets pesent at any node and the 2 Ths selecton s ndependent of past hstoy of packet tansmsson attempts.e. whethe o not a pevous tansmsson to a gven node was successful.

9 9 (2,1) AD (2,1) EC EC AA Staton A DD (2,1) DC DC Staton D (2,1) BA (2,1) AB (2,1) AB AC CC Staton C (2,1) CD CD BB Staton B (2,1) CE CE EE Staton E BA Class-1 queue Class-2 queue EC (2,1) DE ED (a) Repesentaton of the netwok n Fg 1 as a poty queung netwok. Each staton conssts of 2 queues: hgh poty class- 1 queues and low poty class-2 queues. Black and blue lnes denote outng of class-1 and class-2 jobs espectvely. Statons coespondng to ntefeng neghbos of (1) Class-1 jobs seved at ate W/L and fowaded to extenal snk Statons coespondng to tansmsson neghbos of Flte q (2) ntefeng neghbo Class-2 jobs coespondng to unsuccessful packet tansmssons fom Class-2 jobs seved at ate 1/(L/W + 1/ ) and fowaded to tansmsson neghbos as well as nodes fo whch s an (b) Repesentaton of a node n the seconday netwok as a staton n the poty queung netwok. Seved class-1 jobs ae outed to an extenal snk. λ (m) denotes the aval ate of a class-m (whee m=1,2) job at. Fg. 2. Poty-queung netwok model epesentaton of the netwok n Fg 1. sevce-tme s equal to the sum of tansmsson tme and duaton of backoff tme. Snce, the duaton of evey nteupton event s the packet tansmsson tme L, the sevce-tme of any W class-1 job s L. The model can be summazed as follows: W 1) The tansmsson pocess fom all ntefeng neghbos consttute a vtual aval pocess of hghe poty class-1 jobs at any staton. Ths pocess s appoxmated as the sum of the tansmsson pocess fom those nodes 3. Those pocessed jobs ae then fowaded to 3 Ths s an appoxmaton because n pactce multple ntefeng neghbos may tansmt at the same tme. As a esult the oveall duaton of nteupton events at any node s lowe than sum of tansmsson tmes of ts ntefeng neghbos.

10 10 AA BB DD Staton A DC AB Staton C Staton B BA CC CE Staton E Staton D CD EE EC (a) Repesentaton of the netwok n Fg 1 as a sngleclass G/G/1 queung netwok. Each staton fowads jobs only to statons coespondng to ts tansmsson neghbos. Statons coespondng to tansmsson neghbos of Flte q (2) Jobs coespondng to unsuccessful packet tansmssons fom Jobs seved at ate E[1/ ] and fowaded to tansmsson neghbos of (b) Repesentaton of a node n the seconday netwok as a staton n the G/G/1 queung netwok. Effectve sevce-tme of a job at staton s same as that of a class-2 job n the poty queung netwok epesentaton, denoted as c (2). Fg. 3. Sngle-class G/G/1 queung netwok model epesentaton of the netwok n Fg 1. C (2) a snk wth pobablty 1. 2) Followng events consttute aval pocess of class-2 jobs at any staton : a packet s geneated at (modeled as extenal aval pocess of class-2 jobs at staton ), a tansmsson neghbo fowads a packet to and t s not absobed, and a packet tansmsson attempt fom s unsuccessful. Evey staton fowads pocessed class-2 jobs as both class-1 and class-2 jobs to othe statons. We denote as (k,l) j the pobablty that a class-k job at staton s fowaded as a class-l job to staton j. All pocessed class-2 jobs fom staton ae fowaded wth pobablty 1 to evey staton j fo whch s an ntefeng neghbo.e. (2,1) j = 1. Recall, evey packet s tansmtted to one of the tansmsson neghbos pcked andomly and the tansmsson s successful only f the SNR at the

11 11 eceve s geate than o equal to β. Hence, evey class-2 job fom s fowaded wth pobablty k to a tansmsson neghbo k whee k = P (SNR at k β) P (packet s absobed at k). (3) Numbe of neghbos of A pocessed class-2 job fom staton s fowaded back to wth pobablty = (1 k k ). Snce the pmay nodes do not elay seconday taffc and vce vesa, we have j = 0 f ethe ethe o j s a pmay node and the othe one s a seconday node. The poty queung netwok epesentaton of the netwok n Fg 1 s shown n Fg. 2. The novelty of the model s n the ntoducton of vtual jobs to captue nteupton of sevce pocess of any node due to tansmsson fom ts ntefeng neghbos. The vtual jobs do not coespond to eal packets that ought to be fowaded to anothe node; they ae mmedately tansfeed to an extenal snk afte pocessng. Smla to [20], once the oveall sevce-tme pocess of class-2 jobs s known, the netwok can also be modeled as a sngle class G/G/1 queung netwok. Each job n ths netwok coespond only to class-2 jobs n the poty queung netwok epesentaton. Fo the example n Fg 1 the G/G/1 queung netwok epesentaton s shown n Fg. 3. In Fg 3a the outng pobabltes of jobs ae the same as that of class-2 jobs n the poty queueng netwok epesentaton and fo consstency ae denoted usng same notatons. We deve the aveage delay at any seconday node by calculatng expected numbe of jobs pesent at that staton n the coespondng sngle-class queung netwok epesentaton. The latte paamete coesponds to the aveage numbe of physcal packets pesent at node. The aveage queung delay at node s obtaned fom Lttle s theoem by dvdng the aveage numbe of such jobs by the eal packet aval ate to the tansmsson queue at that node. III. DIFFUSION APPROXIMATION FOR QUEUING NETWORKS In ths secton, we gve a bef ovevew of the dffuson appoxmaton technques used n ou wok, namely, the dffuson appoxmaton fo a geneal sngle-class G/G/1-FCFS netwok (nonpoty) [6] and the dffuson appoxmaton fo poty queues [5]. The dffuson appoxmaton s a useful technque to deve closed fom expessons fo the aveage end-to-end delay n a queung netwok. In [6] a dffuson appoxmaton method s used to obtan the aveage

12 12 numbe of packets stoed at any staton n an open G/G/1 queung netwok. Recently n [5] the authos poposed a dffuson appoxmaton soluton fo mult-class poty queung netwoks wth peemptve-esume sevce. We ae patculaly nteested n one ntemedate step n [5] wheen the authos appoxmated the mean and second-moments of completon-tme 4 and ntedepatue tme of jobs at any staton as functon of mean and second moments of the aval and sevce-tme of hghe poty jobs. We fst use these appoxmatons fom [5] to obtan statstcs of the completon-tme of class-2 jobs and then use the dffuson appoxmaton fom [6], to deve the aveage delay of seconday packets at each node. A. Dffuson appoxmaton fo G/G/1- queung netwok Consde a netwok of n statons wth G/G/1- queues, whee K denote the numbe of jobs at staton. The coeffcent of vaaton of nte-aval tmes of jobs at s denoted by C 2 A. The extenal aval pocess of jobs to the netwok has mean nte-aval tme 1 λ and coeffcent of vaaton C A. The sevce-tme at has mean 1 µ and coeffcent of vaaton C B. The vst-ato (e ) of s defned as the mean numbe of vsts of a job to and s gven by, n e = p 0 + p j (n)e j, (4) j=1 whee p 0 denotes the pobablty that a job enteng the netwok fom outsde fst entes, p j denotes the pobablty that a job completed by staton j s tansfeed to. The aval-ate of jobs to s λe. Usng dffuson appoxmaton, the magnal pobablty of queue length at s gven as, whee 1 ρ k = 0, ˆπ (k ) = k ρ (1 ˆρ ) ˆρ 1 k 1. ρ = λe µ (6) ˆρ = exp( 2(1 ρ ) CA 2.ρ ). (7) + CB 2 (5) 4 Completon tme of a job s defned as the tme-peod between the begnnng and end of sevce of a job. It s same as sevce-tme fo the hghest poty-class, s geate than sevce-tme fo othe classes because the on-gong sevce of a job of a lowe class s nteupted wth the aval of a job of hghe poty class.

13 13 The coeffcent of vaaton of nte-aval tmes at s gven as, n CA 2 = 1 + (CBj 2 1)p 2 je j e 1 (8) whee CB0 2 = C2 A. The mean numbe of jobs at s gven as, j=0 K = B. Dffuson appoxmaton fo poty queues ρ 1 ˆρ. (9) In [5], the authos consde a pe-emptve esume G/G/1-FCFS poty queung netwok wth K classes of jobs ndexed as 1, 2,..., K whee a job of the k-th class (1 k < K) has hghe poty than a job belongng to (k + 1)-th class. Fo a netwok wth sngle class of jobs, the sum of jobs enteng staton j n a tme-nteval t s appoxmated to be nomally dstbuted wth mean and vaance M λ j t = { λ j + λ 0j }t (10) =1 M σajt 2 = { CDλ 2 j + C0jλ 2 0j }t (11) =1 whee j denotes the pobablty that a job seved by staton s tansfeed to j, λ 0j denotes the aval ate of jobs fom a staton extenal to the netwok, M denotes the numbe of statons n the netwok; C D and C 0j denote the coeffcent of vaaton of nte-depatue tme of jobs fom and a staton extenal to the netwok espectvely. Let a (k), b (k) and c (k) denote the nteaval tme, sevce-tme and completon-tme espectvely of a k-th class job n a staton n a mult-class netwok. Let λ (k), C (k) A, µ(k) and C (k) B denote the aval ate, co-effcent of vaaton of nte-aval tme, sevce-ate and co-effcent of vaaton of sevce-tme of a k-th class job n a staton n the netwok espectvely. The change n total numbe of jobs of class 1, 2,..., K at any staton dung the tme-peod [0, t] s appoxmated to be nomally dstbuted wth mean β (K) t and vaance α (K) t wth β (K) and α (K) beng, β (K) = α (K) = K λ (k) k=1 K k=1 K k=1 λ (k) C (k)2 A + ρ (k) R (k) µ(k) (12) K k=1 ρ (k) R (k) µ(k) C (k)2 B, (13)

14 14 whee R (k) = k l=1 ρ(l) and ρ (l) = λ(l) µ (l). When the seve s busy pocessng a k-th class job, the sevce maybe nteupted wth the aval of a hghe poty job. If thee ae n such beaks wthn the sevce tme T of a classk job, then n s appoxmated to be nomally dstbuted wth mean k 1 l=1 λ(l) T and vaance k 1 l=1 λ(l) C (l)2 A T. Each such beak has the dstbuton γ(k 1) (t). The total duaton of such beaks n T has the pobablty densty functon (pdf) ψ (k) (t T ) = P n T γ (k 1) n (t), (14) n=0 whee P n T denotes the pobablty of n beaks n T and γ (k 1) n (t) denotes the n-fold convoluton of γ (k 1) wth tself. The moment and the second moment of the vaable wth dstbuton γ (k) (t) ae appoxmately gven as below, The pdf of the completon tme s appoxmated as, whee f c (k)(t) = E[γ (k) ] = 1 β (k) (15) E[γ (k)2 ] = α(k) β (k)3 + 1 β (k)2 (16) 0 f b (k)(t)ψ (k) (t T T )1(t T )dt (17) 0, f t < 0 1(t) = 1 othewse Let d (k) denote the nte-depatue tme of class-k jobs. Then pdf of d (k) s appoxmated as, f d (k)(t) = ρ (k) 1 R (k 1) f c (k)(t) + (1 ρ (k) 1 R (k 1) )[(1 R(k 1) )f a (k)(t) f c (k)(t) (18) +R (k 1) f a (k)(t) γ (k 1) (t) f c (k)(t)] (19) Usng (19), the squaed coeffcent of vaaton of nte-depatue tme of a k-th class job fom staton j,c (k)2 Dj, can be expessed as a functon of coeffcent of vaaton of nte-aval tme of a l-th (whee (l = 1,..., k)) class job at j, C (l)2 Aj. C(l)2 Aj of nte-depatue tmes of jobs that ae beng outed to staton j as, M K =1 k=1 (k,l) j λ (k) [(C (k)2 1) (kl) j + 1] C (l)2 Aj = λ (l) j D tself can be expessed as a functon + C(l)2 0j λ (l) j λ l 0j, (20)

15 15 whee λ (l) j denotes the aval ate of l-th class jobs at j; λ (l) 0j and C(l)2 0j denotes the aveage aval ate and coeffcent of vaaton of nte-aval tme espectvely of a job of class l at j fom a staton extenal to the netwok. IV. DELAY ANALYSIS FOR NON-COOPERATIVE PROTOCOL In ths secton, we fnd the aveage end-to-end delay when thee s no coopeaton between pmay and seconday nodes. In Lemmas 1, 2 and 3 we fnd the aveage outng pobablty, aval-ate of class-2 jobs at any staton n the poty queung netwok epesentaton and the aveage numbe of pmay and seconday ntefeng neghbos to any node espectvely. Then n Theoem 1 we use these values along wth appoxmaton esults fom [5] and [6] to deve the aveage end-to-end delay. Recall j denotes the outng pobablty of a class-2 job fom staton to j. Lemma 1: If and j ae two pmay nodes, the expected pobablty that a class-2 job s fowaded fom staton to j s gven as, whee and j = (1 q ){1 (1 A R ) n } n d A R 1 {1 (1 A R ) n } R = 0 exp( N 0β ψ σ0z 2 d, f j d, othewse (21) 2πz ) dz (22) α A R = π(r ) 2. (23) If and j ae two seconday nodes the expected pobablty that a class-2 job s fowaded fom staton to j s gven as, whee and j = (1 q ){1 (1 A R ) n } n d A R 1 {1 (1 A R ) n } R = 0 exp( N 0β ψ σ0z 2 d, f j d, othewse (24) 2πz ) dz (25) α A R = π(r ) 2. (26)

16 16 R z+dz z j Fg. 4. Two pmay tansmsson neghbos and j at dstance z fom each othe. Poof: We only pove (21). The poof fo (24) follows smlaly. Let N denote the numbe of tansmsson neghbos of. Let P ( j) denotes the pobablty that a class-2 job fowaded by entes queue of j. Let ( j) denotes the mean of P ( j). Then fo evey j, j = ( j) (27) = ( j d j R )P (d j R ) + ( j d j > R )P (d j > R ) (28) = ( j d j R )P (d j R ) (29) = n n=1 ( j d j R, N = n)p (N = n d j R )P (d j R ) (30) In ode to calculate ( j d j R, N = n) we consde a ccula stp of nfntesmal wdth dz at dstance z fom as shown n Fg 4. Fom (3) the pobablty of successful tansmsson fom to a tansmsson neghbo on ths stp, gven N = n, s exp( N 0β ) (1 q ) ψ σ0 2z α n neghbo of.e. d j R, s 2πzdz A R. Condtonal pobablty of j beng on ths stp, gven j s a tansmsson snce P (d j < R ) = A R. Theefoe, ( j d j R, N = n) = R 0 exp( N 0β ψ σ0z )(1 q ) 2 α n 2πzdz A R (31)

17 17 Substtutng the value of ( j d j R, N = n) fom (31) n (30) we obtan, j = = n { n=1 R 0 ( n 1. n 1 n ( n 1 { n 1 n=1. (1 q ) n exp( N 0β ψ σ0z )(1 q ) 2πzdz 2 α n A R ) (A R ) n 1 (1 A R ) n n A R } (32) ) (A R ) n (1 A R ) n n R 0 = (1 q ){1 (1 A R ) n } n Note that fo any j, j exp( N 0β )2πzdz } (33) ψ σ0z 2 α A R R 0 exp( N 0β 2πz ) dz (34) ψ σ0z 2 α A R s the aveage pobablty of the event: packet tansmted by entes the tansmsson queue at j. The aveage pobablty of a packet tansmsson fom beng unsuccessful s theefoe, = ( ) (35) = 1 j:j ( successfully tansmts a packet to j) (36) = 1 j:j j 1 q (37) = 1 n j (38) 1 q = 1 {1 (1 A R ) n } z=r 0 exp( N 0β 2πz ) dz (39) ψ σ0z 2 α A R Lemma 2: The aval ate of class-2 jobs at a pmay staton s gven as, λ (2) = λ q d and the aval ate of class-2 jobs at a seconday staton k s gven as, λ (2) k = λ q d Poof: We only pove (40). The poof fo (41) follows smlaly., (40). (41)

18 18 We obseve that any pmay staton can eceve class-2 jobs only fom low-poty queues of othe pmay statons o fom the extenal souce. Theefoe, n ode to calculate the vst-ato of class-2 jobs we can teat the netwok of all low poty queues at pmay statons as an open netwok of G/G/1 queues. Vst-ato of class-2 jobs at s obtaned by applyng (4) as, e = 1 n j j e j + e (42) Snce all pmay nodes ae smla, we have ē = ē j fo any two pmay nodes and j. Takng expectaton on both sdes of (42) we have ē = Now the tem 1 (1 A R ) n 1 + n n j ē + ē (43) + 1 n (21) epesents the pobablty that a pmay node has no tansmsson neghbo. Fo a connected netwok ths tem s appoxmately equal to 1. Theefoe 1 q j n d, f j (44) 1 d, othewse Substtutng the value of j fom (44) n (43) and afte some algeba we obtan, ē = 1 (n + 1)q d Total extenal aval ate of class-2 jobs coespondng to pmay taffc s (n + 1)λ. Theefoe aval ate of class-2 jobs at s (n + 1)λ ē.e. λ (2) Let ˆN and ˆN = λ q d denote the numbe of pmay and seconday ntefeng neghbos of node. ˆN, denoted as ˆN s, Lemma 3: The aveage value of The aveage value of ˆN = ˆN 4n π(r ) 2, (n + 1)π(R + R ) 2, othewse, denoted as ˆN ˆN = s, 4n π(r ) 2, f s a pmay node 0, f s a pmay node othewse Poof: We only pove (47). The poof fo (46) follows smlaly. ˆN s zeo f s a pmay node snce a pmay node has no seconday ntefeng neghbo. Next, assume s a seconday node. Snce dstbuton of all nodes s unfom ove a unt aea, (46) (47)

19 19 any seconday node s a seconday ntefeng neghbo of wth pobablty π(2r ) 2. Snce the dstbuton of each node s ndependent of each othe, the numbe of seconday ntefeng neghbos of has a bnomal dstbuton B(n, π(2r ) 2 ). The aveage numbe of seconday ntefeng neghbos of s theefoe, 4n π(r ) 2. We assume, pmay and seconday packet geneaton s a Posson pocess even though ou analyss can be easly extended to non-posson pocesses as well. Theoem 1: The aveage end-to-end delay of a seconday packet, D(n, n ) s gven as D(n, n ) = D k 1 q (n ), (48) whee D k s a functon of known paametes. The functon s closed fom except fo tems coespondng to pobablty of successful tansmsson, d and (25) usng numecal ntegaton. and d whch ae obtaned fom (22) Poof: An outlne of the poof s povded hee. The complete poof s n Appendx A. We fst obtan a set of polynomal equatons nvolvng mean and second moment of nte-aval tme, nte-depatue tme and completon-tme of class-1 and class-2 jobs at each staton n the poty queung netwok usng Lemmas 1, 2, 3 and appoxmaton technques fom [5]. Those set of equatons along wth elaton (8) fom [6] s used to fnd second moments of nte-aval tme and completon-tme of class-2 jobs. Then we apply (9) to fnd the aveage numbe of jobs at each seconday staton 5 n the equvalent sngle-class G/G/1 queue epesentaton. Ths paamete s also equal to the aveage numbe of packets at each seconday node. Usng value of ths paamete and applyng Lttle s Theoem we obtan the aveage system delay at any seconday node. We fnd the aveage end-to-end delay by multplyng the aveage delay at each node wth the aveage numbe of hops tavesed by a pmay packet po to absopton. V. NETWORK MODEL FOR COOPERATIVE CASE In ths secton we consde a coopeatve potocol. Consde the event whee tansmsson fom a pmay node to anothe fals but a neaby seconday node successfully eceves the pmay packet. Assume ths seconday node s close to the pmay eceve and consequently, the pobablty of successful tansmsson of a pmay packet fom the seconday node to the 5 Statons coespondng to pmay and seconday nodes ae efeed as pmay and seconday statons espectvely.

20 20 pmay eceve s vey hgh. Theefoe f the seconday node elays the pmay packet t may educe the facton of tme the channel s occuped by pmay uses. Ths n tun can povde moe tansmsson oppotunty fo seconday uses. The channel model, ntefeence model, and packet geneaton eman the same as n the non-coopeatve case (Secton II). Each pmay (o seconday) node tansmts a pmay (o seconday) packet to one of ts tansmsson neghbos, pcked andomly, whee t s absobed wth pobablty q (o q ). The detaled coopeatve potocol s pesented next. A. Coopeatve potocol Followng steps consttute the coopeatve potocol. 1) Fo any two pmay tansmsson neghbos and j, only seconday nodes located at dstance less than o equal to d j fom and wthn a dstance d el (whee d el R ) fom j ae allowed to elay faled pmay packet tansmssons fom to j. We efe to such seconday nodes as elays of the pa (, j). The pobablty of successful tansmsson fom to any such elay k s at least as hgh as that fom to j snce d k d j. Moeove, selectng small d el ensues vey hgh pobablty of successful tansmsson fom elay node to j (but educes the numbe of possble elays). 2) Consde the event: packet tansmsson fom to j s unsuccessful but one o moe elays of (, j) eceves ths packet successfully. In ths case, exactly one of those elays, andomly pcked, sends an acknowledgement (ACK) packet back to the pmay tansmtte. The ACK ecepton s assumed to be collson-fee and nstantaneous. 3) We assume an ACK s tansmtted nstantaneously and eceved coectly by and othe elays. On heang the ACK all those elays and dop the packet. The sende of the ACK buffes the packet nto ts tansmsson queue. Ths packet wll eventually be elayed to j. 4) Both pmay and seconday packets ae tansmtted n FCFS manne by elay nodes. B. Queung model The queung netwok epesentaton s same as n Secton II except fo the followng dffeence: snce some seconday nodes can elay pmay packets, k and k evey pmay node and seconday node k. ae no longe zeo fo

21 21 Z A(z) j d el z A(z) j d el A(z) z j d el (a) d el 2z. (b) d el 2z (c) 2z > d el > 2z Fg. 5. Aea A(z) epesents possble locatons of a seconday elay fo the pa (, j), whee and j denotes two pmay tansmsson neghbos, as a functon of paamete d el and dstance z between and j. VI. DELAY ANALYSIS FOR COOPERATIVE PROTOCOL We deve the expected pobablty of successful tansmsson of a pmay packet to any elay node, and fom a elay node to pmay eceve n Lemmas 4 and 5 espectvely. In Lemma 6 we fnd the aval-ate of class-2 jobs coespondng to pmay packets at any seconday staton. Theoem 2 follows fom Lemmas 4, 5 and 6 along wth appoxmaton esults fom [5] and [6] and chaactezes the aveage end-to-end delay fo seconday packets. Let P el,(,j) denote the pobablty that a packet tansmtted by pmay node to ts tansmsson neghbo j entes tansmsson queue at some elay of (, j). Fo any seconday node k let P [ k (, j)] denote jont pobablty of the events: k s a elay of (, j) and packet tansmtted by to j s successfully eceved at k. Lemma 4: The mean of P el,(,j) s gven as, el,(,j) = (1 d ){1 (1 [ k (, j)]) n +1 } (49) whee the expesson fo [ k (, j)] s povded n (57). Poof: Gven d j = z, the pobablty that k s a elay of (, j) s the pobablty that k s located wthn dstance z fom and less than d el fom j. Ths s equal to the pobablty that a seconday node s n the shaded egon shown n Fg. 5. The aea of ths egon, denoted as A el (z), s obtaned usng the fomula fo fndng aea of ovelapped egon of two ccles [22]

22 22 as, A el (z) = πz 2, V 1 + V 2, f d el 2z f d el 2z V 3 + V 2, othewse whee V 1 = z 2 cos 1 ( x z ) x z 2 x 2 (50) V 2 = d 2 el cos 1 ( z x ) (z x) d 2 el d (z x)2 (51) el V 3 = πz 2 (z 2 cos 1 ( x z ) + x z 2 x 2 ) (52) x = z d2 el 2z Let Φ (,j),k denote the pobablty of k beng a elay of (, j). The value of Φ (,j),k s obtaned by ntegatng, ove all z fom 0 to R, the poduct of A el (z) and condtonal pobablty of j beng on an nfntesmal ccula stp of wdth dz.e. Φ (,j),k = R 0 (53) A el (z) 2πz dz (54) A R Fom symmety Φ (,j),k s same fo all pmay tansmtte-eceve pas and seconday nodes. Theefoe the ndex s dopped and we efe to Φ (,j),k smply as Φ. Gven d j = z we denote v x and v y as the x and y components, espectvely, of any pont v n shaded aea A el (z) wth ogn of the co-odnate system beng at. Due to (3) the condtonal pobablty of successful tansmsson fom to a elay k of (, j), gven j s located on an nfntesmal ccula stp of wdth dz at dstance z fom (as shown n Fg 6), s N exp( 0 β v aea A el (z) ψ σ 0 2(v x 2 +vy 2 ) α )dxdy 2 A el (z) as, [ k (, j), k s a elay of (, j)] =. Theefoe [ k (, j), k s a elay of (, j)] s obtaned R v aea A el (z) z=0 N 0 β exp( )dxdy ψ σ0 2(vx2 +v 2 y ) α 2 2πz dz(55) A el (z) A R

23 23 R z+dz z v A(z) j d el Fg. 6. A seconday elay of pa (, j), whee and j ae two pmay tansmsson neghbos. Node j s located on an nfntesmal ccula stp of wdth dz at dstance z fom. The ogn of the coodnate system s at ; the seconday elay s located at pont v. The expected value of P [ k (, j)] s, [ k (, j)] = [ k (, j), k s a elay of (, j)]p [k s a elay of(, j)] (56) N exp( 0 β )dxdy R ψ v aea A el (z) σ0 2(v x 2 +v 2 y ) α 2 2πz = Φ dz (57) A el (z) A R z=0 Note that [ k (, j)] s same fo all (, j) and k due to symmety of pmay and seconday nodes. Now P el,(,j) s the jont pobablty of the events: tansmsson fom to j s unsuccessful and tansmsson fom to at least one of the elays of (, j) s successful. Snce those two events ae ndependent, the aveage value of P el,(,j) el,(,j) = (1 s theefoe, d ){1 (1 [ k (, j)]) n +1 } (58) Due to symmety of pmay nodes, el,(,j) s equal fo all (, j) and hence-foth we efe to el,(,j) smply as el. Let P el,(,j),k elay k of (, j) to j. denote the pobablty of successful tansmsson of a pmay packet fom a

24 24 Lemma 5: The mean of P el,(,j),k s gven as, N exp( 0 β )dxdy R ψ v aea A el (z) σ0 2{(v x z) 2 +v 2 y } α 2 2πz el,(,j),k = dz (59) 0 A el (z) A R Poof: Gven j s located on a ccula stp of nfntesmal wdth dz at dstance z fom, the condtonal pobablty of successful tansmsson fom a elay k to j s N exp( 0 β v aea A el (z) ψ σ 0 2{(v x z) 2 +vy 2 } α )dxdy 2 A el (z) s a tansmsson neghbo of s ove z fom 0 to R we obtan (59).. The condtonal pobablty that j s on ths stp gven j 2πz dz. Theefoe, ntegatng the poduct of those two tems A R Due to the symmety of pmay and seconday nodes, el,(,j),k Hence-foth we efe to el,(,j),k smply as el. s equal fo all (, j) and k. Lemma 6: The aval ate of class-2 jobs coespondng to pmay packets at any seconday staton, denoted as λ ext, s gven as, whee Poof: The vaable λ ext = λ el (n + 1) coopq (n + 1) coop = el + el (60) d (61) coop s the aveage pobablty that a packet tansmtted by pmay node to tansmsson neghbo j s successfully eceved by ethe j o any elay of (, j). Fo a stable system, the ate at whch attempts packet tansmsson s equal to the ate of aval of class-2 jobs at staton. Fom (40) we obtan the ate at whch attempts packet tansmsson as λ (2) = λ q coop pmay packet wth the aveage pobablty nodes s theefoe, node s Snce λ q coop el λ q coop n +1. n +1. Gven a packet tansmsson attempt, a elay node eceves the el. Pmay packet aval ate at all seconday el (n +1). Hence, the pmay packet aval ate at evey seconday el s the aveage pobablty of successful tansmsson fom a elay of (, j) to j, then the aveage ate at whch class-2 jobs coespondng to pmay packets ente a seconday staton s λ q coop el n +1 (n +1) el. Theoem 2: Unde the coopeatve potocol, the aveage end-to-end delay of a seconday packet, D(n, n ) s gven as D(n, n ) = D k 1 q (n ), (62)

25 25 whee D k s a functon of of known paametes. The functon s closed fom except fo the tems coespondng to pobablty of successful tansmsson: coop, el and fom (61), (59) and (25) espectvely usng numecal ntegaton.. d whch ae obtaned Poof: The poof s smla to that of Theoem 1 n Secton II except fo the followng dffeence: the aveage outng pobablty between pmay and seconday statons s non-zeo due to coopeaton. Detaled poof s povded n Appendx B. VII. MAXIMUM ACHIEVABLE THROUGHPUT In ths secton we nvestgate unde what condtons coopeaton mpoves the thoughput of seconday uses. We compae the maxmum achevable thoughput of seconday uses both wth and wthout coopeaton usng the poty queung netwok epesentaton. The maxmum achevable thoughput of seconday uses, fo a gven pmay packet geneaton ate λ, s defned as the maxmum packet geneaton ate λ fo whch the queue-length at evey seconday node emans bounded. In the est of ths secton we assume n and n ae suffcently lage such that n + 1 n and n + 1 n. Coolay 1: Unde the coopeatve potocol the maxmum achevable thoughput fo seconday uses, denoted as λ max(λ ), coespondng to pmay packet geneaton ate λ s, whee λ max(λ ) = max{t A T B T C, 0} (63) T A = T B = q d L W + 1 ξ + L W ˆN ˆN Lλ q d W q coop( L L W ξ W T C = q d λ el n n q el ˆN ) Poof: Let j denote a pmay node. Due to symmety of pmay and seconday nodes, the aval ate of class-1 taffc at seconday staton s coop (64) (65) (66) λ (1) = ˆN λ (2) j + ˆN λ (2) (67) = ˆN λ q coop + ˆN ( λ q d + λ ext ) (68)

26 26 = ˆN λ q coop + ˆN { λ q d + λ el n q coopn el )} (69) = λ (H 3 + ˆN H 2 ) + λ H 1 ˆN (70) n whee H 1 = 1 el q, H 2 = d q and H 3 = coopn el q. In (69) we have used the expesson coop of λ ext fom Lemma 6 and then appoxmated n + 1 and n + 1 as n and n espectvely. The completon-tme of a class-2 job at s obtaned usng (17) as E[c (2) ] = 1 to (88) n Appendx A). Fo stablty of the class-2 queue at we must have, By substtutng the values of E[c (2) ] and λ (2) λ (2) 1 µ (2) ( 1 1 λ(1) µ (1).e. λ (2) < µ (2).e. λ (2) < µ (2).e. λ H 2 + λ H 1 < µ (2) ˆN 1 ( µ (2) 1 λ(1) µ (1) ) (efe E[c (2) ]λ (2) < 1. (71) n (71) we get ) < 1 (72) µ(2) µ (1) µ(2) µ (1) µ(2) µ (1) Afte some eaangng, and substtutng µ (1) = W L λ < µ (2) H 1 + ˆN µ H (2) 1 µ (1) H 3 λ H 1 + ˆN µ(2) µ (1) λ (1) (73) {λ (H 3 + ˆN H 2 ) + λ H 1 ˆN } (74) {λ (H 3 + ˆN H 2 ) + λ H 1 ˆN } (75) µ H (2) 1 µ (1) and µ(2) = 1 1 ξ + L W λ H 2 ( ˆN µ (2) µ (1) H 1 + ˆN, we obtan µ H (2) 1 µ (1) + 1). (76) Substtutng the expessons of H 1, H 2 and H 3 n (76) and notng that thoughput cannot be negatve, we obtan (63). The fst tem T A n the ght hand sde (RHS) of (63) s the maxmum thoughput fo seconday uses n absence of any pmay taffc. As expected, λ max nceases wth hghe pobablty of successful tansmsson by seconday nodes q. The tem L W ˆN d and hghe pobablty of taffc beng localzed n denomnato eflects ntefeence fom othe seconday nodes. Consequently thoughput of seconday nodes deceases wth hghe numbe of seconday ntefeng neghbos.

27 coop d el =R no coop coop d el =0.1R λ max,s (0.8725) Numbe of seconday nodes Fg. 7. Compason of maxmum achevable thoughput of seconday nodes wth coopeaton vs non-coopeaton fo two values of d el : R and 0.1R. Value of n s 100 and λ s , the maxmum achevable thoughput fo pmay uses wthout coopeaton. The second tem T B n the RHS of (76) coesponds to educton n thoughput of a seconday node due to packet tansmsson fom pmay ntefeng neghbos. Snce coop > eflects that tansmssons fom pmay ntefeng neghbos decease wth coopeaton. The thd tem T C coesponds to educton n thoughput of a seconday node due to pmay packet tansmsson fom seconday ntefeng neghbos. Note that ths tem s 0 when thee s no coopeaton. Fo the non-coopeatve potocol, the maxmum achevable thoughput fo seconday uses, fo a gven λ s obtaned fom (63) by eplacng coop and el wth d d ths and 0 espectvely. Let λ max denote the maxmum achevable thoughput fo pmay nodes wthout coopeaton 6. Snce λ max(λ ) deceases lnealy wth λ untl t becomes zeo we conclude that seconday uses beneft fom coopeaton f ˆN Lq d W q coop( L L W ξ W ˆN d ) + q n λ max(λ el n el q coop max) > 0 and (77) ˆN Lq d W q d ( L L W ξ W ˆN ) Example: We use a numecal example to llustate how λ max vaes wth numbe of seconday uses. In Fg 7 we plot λ max when n s 100 and n vaes fom 50 to 500. α equals 2; the (78) 6 Clealy, λ max can be obtaned by calculatng λ max(0) fo a seconday netwok wth ts paametes same as that of the pmay netwok. Hence λ max s λ max(0) whee d, q and ˆN equals d, q and ˆN espectvely.

28 28 TABLE I PARAMETERS USED IN FIG 7 Paamete R R q q W Value log(n ) n log(n ) n 0.4R 0.4R L bts/sec paametes N 0, β, ψ and ψ ae selected such that d = d = 0.7. Othe paametes fo the example ae lsted n Table 1 7. λ s selected as , the maxmum achevable thoughput fo pmay uses wthout coopeaton. Two dffeent values of d el : R and 0.1R ae used. In Fg 7 when d el s R, coopeaton educes thoughput fo seconday uses when n s lowe than about 120. At ths ange the dstance between elay nodes and pmay eceves s somewhat lage whch leads to a small pobablty of successful tansmsson fom elay node to coespondng pmay eceve. Any gan n tansmsson oppotuntes fo seconday nodes due to lowe tansmsson fom pmay ntefeng neghbos s offset by ntefeence due to hgh tansmsson attempts of pmay packets by seconday ntefeng neghbos. As n nceases d el becomes pogessvely lowe and el s hgh because elay nodes ae close to coespondng pmay eceves. Theefoe the maxmum achevable thoughput of seconday uses nceases wth coopeaton at hghe n. In Fg 7 when d el s 0.1R coopeaton does not hut at low n because the dstance fom elay nodes to pmay eceves s small. Howeve the gan due to coopeaton s small at hgh n as compaed to the case of d el = 0.1R. Ths s because, due to small d el, thee ae fewe elay nodes pe pmay tansmtte-eceve pa and the netwok s close to the case wthout coopeaton. Fo ths choce of netwok paametes we obseve n Fg 7 that the maxmum achevable thoughput, wth o wthout coopeaton, fst nceases wth n and then deceases. Ths s 7 Ths choce of R and R s suffcent to ensue connectvty of the two netwoks [21].

29 29 because λ max deceases wth nceasng aveage numbe of ntefeng neghbos to a seconday node. The aveage numbe of pmay and seconday ntefeng neghbos decease and ncease espectvely wth nceasng n, when R log(n = ). The fome effect s domnant at low n n ; the latte effect s domnant at hgh n. VIII. SIMULATIONS We compae ou analytcal esults wth those obtaned though smulaton n C-pogammng language so as to vefy the valdty of ou assumptons. The smulaton settng conssts of n + 1 pmay and n + 1 seconday nodes that ae unfomly dstbuted ove a tous of unt aea. The length of a pmay o a seconday packet s 1KB. The tansmsson bandwdth (W) of the channel s 10 6 bts/sec. The back-off tmes fo both pmay and seconday nodes ae exponentally dstbuted wth mean back-off duaton of 0.01 seconds. We aveage smulaton esults ove 20 dffeent andomly geneated topologes. Each topology s selected such that the aveage numbe of ntefeng neghbos to evey pmay and seconday node s wthn a small numbe, selected as 0.7, fom the espectve theoetcal values. The smulaton uses pobablstc outng scheme as descbed n Secton II. Each packet tansmsson fom a pmay node to ts tansmsson neghbo s successful wth pobablty d s fowaded to one of the seconday elays wth pobablty el {1 (1 Φ) n +1 }(1 d ) ; othewse(n case of coopeaton) t el {1 (1 Φ) n +1 }(1. The tem d ) s the theoetcal aveage pobablty of successful tansmsson fom a pmay node to elay, gven dect tansmsson fom pmay node fals and at least one such elay exsts fo the coespondng pmay tansmsson neghbo pa. Each seconday packet tansmsson fom a seconday node to ts tansmsson neghbo s successful wth pobablty d. Each pmay packet tansmsson fom a seconday node s successful wth pobablty el. Each smulaton s un fo 1000 seconds. A. Smulatons fo Non-coopeatve potocol Fo Fg 8a and Fg 8b R and R ae chosen as 0.8 Fo both fgues q and q ae chosen as log(n ) n ae used whle ψ and ψ ae vaed such that d n and n. and = and 0.8 n log(n ) log(n ) n d log(n ) n espectvely. espectvely. N 0, σ 0 and β = 0.7 fo evey combnaton of In Fg. 8a, we plot the aveage end-to-end delay fo thee values of λ = λ : 0.3, 0.5 and

30 Theoy, λ =0.3 Smul, λ =0.3 Theoy, λ = Theoy n =n =100 Smul n =n =100 Theoy n =n =400 Smul n =n =400 D(n,n ) 1 Smul, λ =0.5 Theoy, λ =0.7 Smul, λ =0.7 D(n,n ) n (=n ) Aveage seconday (=pmay) packet geneaton ate (a) Aveage end-to-end delay of seconday packets vesus numbe of pmay (= numbe of seconday) nodes. (b) Aveage end-to-end delay of seconday packets vesus seconday (= pmay) packet geneaton ate. Fg. 8. Compason of the analytcal and smulaton esults wthout coopeaton. 0.7 espectvely whle n (= n ) s vaed fom 100 to 400 at steps of 100. Fg. 8b shows the aveage end-to-end delay when n = n = 100 and 400, whle λ (= λ ) s vaed fom 0.07 to 0.7. As expected the aveage end-to-end delay monotoncally nceases wth nceasng numbe of nodes and taffc geneaton ate n both fgues. We obseve that the netwok model s easonably accuate fo a wde ange of channel utlzaton 8 of a seconday node (an nceasng functon of λ and λ ) except fo hgh ange. In Fg. 8a, the channel utlzaton s about 0.63 when λ = 0.7 and n = 300. A smla esult s obseved fom Fg. 8b. Fo hghe channel utlzaton scenaos, ou appoxmaton of the aval pocess of class-1 jobs at any staton as sum of ndvdual tansmsson pocesses s no longe accuate as thee ae sgnfcant numbe of nstances when two o moe ntefeng neghbos of any node ae smultaneously tansmttng. In ths case, ou appoxmaton leads to ove-estmatng the numbe of events when the sevce of a class-2 job s nteupted due to aval of a class-1 job. As a esult, the theoetcal aveage end-to-end delay acts as an uppe bound to that obtaned fom smulaton esults.