On the Delay-Throughput Tradeoff in Distributed Wireless Networks

Transcription

1 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER 2009 O the Delay-Throughput Tradeoff Dstrbuted Wreless Networks Jamshd Aboue, Alreza Bayesteh, ad Amr K. Khada Codg ad Sgal Trasmsso Laboratory Departmet of Electrcal ad Computer Egeerg, Uversty of Waterloo Waterloo, Otaro, Caada, N2L 3G arxv: v [cs.it] 20 Oct 2009 Tel: , Fax: Emals: {jaboue, alreza, khada}@cst.uwaterloo.ca Abstract Ths paper deals wth the delay-throughput aalyss of a sgle-hop wreless etwork wth trasmtter/recever pars. All chaels are assumed to be block Raylegh fadg wth shadowg, descrbed by parameters α, ), where α deotes the probablty of shadowg ad represets the average cross-lk gas. The aalyss reles o the dstrbuted o-off power allocato strategy.e., lks wth a drect chael ga above a certa threshold trasmt at full power ad the rest rema slet) for the determstc ad stochastc packet arrval processes. It s also assumed that each trasmtter has a buffer sze of oe packet ad droppg occurs oce a packet arrves the buffer whle the prevous packet has ot bee served. I the frst part of the paper, we defe a ew oto of performace the etwork, called effectve throughput, whch captures the effect of arrval process the etwork throughput, ad maxmze t for dfferet cases of packet arrval process. It s proved that the effectve throughput of the etwork asymptotcally scales as log ˆα, wth ˆα α, regardless of the packet arrval process. I the secod part of the paper, we preset the delay characterstcs of the uderlyg etwork terms of the packet droppg probablty. We derve the suffcet codtos the asymptotc case of such that the packet droppg probablty ted to zero, whle achevg the maxmum effectve throughput of the etwork. Fally, we study the trade-off betwee the effectve throughput, delay, ad packet droppg probablty of the etwork for dfferet packet arrval processes. I partcular, we determe how much degradato wll be eforced the throughput by troducg the aforemetoed costrats. Idex Terms Throughput maxmzato, delay-throughput tradeoff, droppg probablty, Posso arrval process. Ths work s facally supported by Nortel Networks ad the correspodg matchg fuds by the Natural Sceces ad Egeerg Research Coucl of Caada NSERC), ad Otaro Ceters of Excellece OCE). The materal ths paper was preseted part at the IEEE Iteratoal Symposum o Iformato Theory ISIT), Nce, Frace, Jue 24-29, 2007 [].

2 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER I. INTRODUCTION As the demad for hgher data rates creases, effectve resource allocato emerges as the prmary ssue wreless etworks order to satsfy Qualty of Servce QoS) requremets. Cetral to the study of resource allocato schemes, the dstrbuted power cotrol algorthms for maxmzg the etwork throughput have attracted sgfcat research atteto [2] [7]. Moreover, achevg a low trasmsso delay s a mportat QoS requremet wreless etworks [8]. I partcular, for buffer-lmted users wth real-tme servces e.g., teractve games, lve sport vdeos, etc), too much delay results droppg some packets. Therefore, the ma challege wreless etworks wth real-tme servces s to utlze a effcet power allocato scheme such that the delay s mmzed, whle achevg a hgh throughput. The throughput maxmzato problem cellular ad multhop wreless etworks has bee extesvely studed [9] [3]. I these works, delay aalyss s ot cosdered. However, t s show that the hgh throughput s acheved at the cost of a large delay [4]. Ths problem has motvated the researchers to study the relato betwee the delay characterstcs ad the throughput wreless etworks [5] [8]. I partcular, most recet lterature [4], [9] [26], the tradeoffs betwee delay ad throughput have bee vestgated as a key measure of the etwork s performace. The frst studes o achevg a hgh throughput alog wth a low-delay ad hoc wreless etworks are framed [7] ad [8]. Ths le of work s further expaded [4], [20] ad [2] by usg dfferet moblty models. El Gamal et al. [4] aalyze the optmal delay-throughput scalg for some wreless etwork topologes. For a statc radom etwork wth odes, they prove that the optmal tradeoff betwee throughput T ad delay D s gve by D = ΘT ). Referece [4] also shows that the same result s acheved radom moble etworks, whe T = O/ log ). Neely ad Modao [2] cosder the delay-throughput tradeoff for moble ad hoc etworks uder the assumpto of redudat packet trasmsso through multple paths. Sharf ad Hassb [22] aalyze the delay characterstcs ad the throughput a broadcast chael. They propose a algorthm to reduce the delay wthout too much degradato the throughput. Ths le of work s further exteded [23] by demostratg that t s possble to acheve the maxmum throughput ad short-term faress smultaeously a large-scale broadcast etwork. I [27], we addressed the throughput maxmzato of a dstrbuted sgle-hop wreless etwork wth K lks, where the lks are parttoed to a fxed umber M) of clusters each operatg a subchael wth badwdth W. We proposed a dstrbuted ad o-teratve power allocato strategy, where the M objectve for each user s to maxmze ts best estmate based o ts local formato,.e., drect chael ga) of the average sum-rate of the etwork. Uder the block Raylegh fadg chael model wth shadowg effect, t s establshed that the average sum-rate the etwork scales at most as Θlog K) the asymptotc case of K. Ths order s achevable by the dstrbuted threshold-based o-off scheme

3 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER e., lks wth a drect chael ga above certa threshold τ trasmt at full power ad the rest rema slet). I addto, the strog terferece scearo, the o-off power allocato scheme s show to be the optmal strategy. Moreover, the optmum threshold level that acheves the maxmum average sum-rate of the etwork s obtaed as τ = log 2 log log + O), where = K M s the umber of lks each cluster. We also optmzed the average etwork s throughput terms of the umber of the clusters, M. It s proved that the maxmum average sum-rate of the etwork, assumg o-off power allocato scheme, s acheved at M =. However, [27] oly focuses o the etwork throughput ad other ssues lke delay ad packet droppg probablty) were ot addressed ths work. I ths paper, we follow the dstrbuted sgle-hop wreless etwork model proposed [27] wth M = whch s the case wth the maxmum throughput) ad address the delay-throughput tradeoff of the etwork. The chaels are assumed to be block Raylegh fadg wth shadowg the same model as [27]), where the trasmsso block s assumed to be equal to the fadg block whch s assumed to be equal for all lks). Moreover, the lks are assumed to be sychroous. The assumpto of block Raylegh fadg wth sychroous users s used may works the lterature lke [28] for the pot-to-pot scearo, [29] for the multple-access chael, ad [22] ad [23] for the broadcast scearo). We cosder a buffer-lmted etwork, whch the users have a buffer sze of oe packet. Ths assumpto troduces droppg evet the etwork, whch s defed as the evet whe a packet s arrved the buffer whle the prevous packet has ot bee served yet. Although the assumpto of oe packet buffer sze s harsh for may practcal applcatos, t smplfes the aalyss whle gvg a good sght about the worst case performace the etwork. Notg the optmalty of o-off power allocato scheme terms of achevg the maxmum order of the sum-rate throughput [27], we use t ths work. Therefore, for ay lk, f the drect chael s above a pre-determed threshold ad there s ay packet the buffer, the trasmtter seds that packet durg a trasmsso block wth full power ad f ot, remas slet. I the frst part, we defe a ew oto of throughput, called effectve throughput, whch descrbes the actual amout of data trasmtted through each lks. Ths oto captures the effect of arrval process by takg to accout the full buffer probablty. We compute the optmum threshold level τ, ad the correspodg maxmum effectve throughput of the etwork, for each packet arrval process. It s proved that the effectve throughput of the etwork scales as log, wth ˆα α, regardg the packet arrval ˆα process. Ths throughput scalg s exactly the same as what we had derved [27],.e., the case of backlogged users. Moreover, we show that the maxmum throughput s acheved the strog terferece scearo, whch the terferece term domates the ose. As a terestg cosequece, the results of ths secto are vald eve wthout the assumpto of sychrozato betwee the users or equalty of ther fadg coherece tme fadg blocks).

4 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER I the secod part, we preset the delay characterstcs of the uderlyg etwork terms of the packet droppg probablty for determstc ad stochastc packet arrval processes. We derve the suffcet codtos the asymptotc case of such that the packet droppg probablty of the lks teds to zero, whle achevg the maxmum effectve throughput of the etwork, asymptotcally. The mportace of ths result s showg the fact that the loss the etwork performace due to the lmted buffer sze ca be made eglgble the asymptotc regme of. I the subsequet secto, we study the tradeoff betwee the effectve throughput of the etwork ad other performace measures,.e., packet droppg probablty ad delay for dfferet arrval processes. I partcular, we determe how much degradato wll be eforced the throughput by troducg the aforemetoed costrats, ad how much ths degradato depeds o the arrval process. The setup ths paper s qute dfferet from that of wth the o-off Beroull scheme [30]. I fact, we utlze a dstrbuted approach usg local formato,.e., drect chael gas, whle [30] reles o a cetral cotroller whch studes the chael codtos of all the lks ad decdes accordgly. Furthermore, we cosder a homogeeous etwork model wthout path loss. Ths dffers from the geometrc models cosdered [4], [20] ad [2], whch are based o the dstace betwee the source ad the destato.e., power decay-versus-dstace law). The rest of the paper s orgazed as follows. I Secto II, the etwork model ad objectves are descrbed. The throughput maxmzato of the uderlyg etwork s preseted Secto III. The delay characterstcs terms of the packet droppg probablty are aalyzed Secto IV. Secto V establshes the tradeoff betwee the throughput, delay, ad packet droppg probablty the uderlyg etwork. Smulato results are preseted secto VI. Fally, Secto VII, a overvew of the results ad coclusos are preseted. Notatos: For ay fuctos f) ad g) [3]: f) f) = Og)) meas that lm g) <. f) f) = og)) meas that lm = 0. g) f) = ωg)) meas that lm f) g) =. f) = Ωg)) meas that lm f) g) > 0. f) f) = Θg)) meas that lm = c, where 0 < c <. g) f) f) g) meas that lm =. g) f) g) meas that f) s approxmately equal to g),.e., f we replace f) by g) the equatos, the results stll hold. Throughout the paper, we use log.) as the atural logarthm fucto ad N for represetg the set {, 2,, }. Also, E[.] represets the expectato operator, ad P{.} deotes the probablty of the gve evet.

5 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER T T 4 R R 2 R 4 T 3 T 2 R 3 Fg.. A dstrbuted sgle-hop wreless etwork wth = 4. A. Network Model II. NETWORK MODEL AND PROBLEM DESCRIPTION I ths work, we cosder a dstrbuted sgle-hop wreless etwork, whch pars of odes, dexed by {,..., }, are located wth the etwork area Fg. ). We assume the umber of lks,, s kow formato for the users. All the odes the etwork are assumed to have a sgle atea. Also, t s assumed that all the trasmssos occur over the same badwdth. I addto, we assume that each recever kows ts drect chael ga wth the correspodg trasmtter, as well as the terferece power mposed by other users. However, each trasmtter s assumed to be oly aware of the drect chael ga to ts correspodg recever. The power of Addtve Whte Gaussa Nose AWGN) at each recever s assumed to be N 0. We assume that the tme axs s dvded to slots wth the durato of oe trasmsso block, whch s defed as the ut of tme. The chael model s assumed to be Raylegh flat-fadg wth the shadowg effect. The chael ga 2 betwee trasmtter j ad recever at tme slot t s represeted by the radom varable L t) j 3. For j =, the drect chael ga s defed as L t) j h t), where ht) s expoetally dstrbuted wth ut mea ad ut varace). For j, the cross chael gas are defed based o a The term par s used to descrbe the trasmtter ad the related recever, whle the term user s used oly for the trasmtter. 2 I ths paper, chael ga s defed as the square magtude of the chael coeffcet. 3 I the sequel, we use the superscrpt t) for some evets to show that the evets occur tme slot t.

6 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER shadowg model as follows 4 : L t) j β t) j ht) j, wth probablty α 0, wth probablty α, ) where h t) j s have the same dstrbuto as ht) s, 0 α s a fxed parameter, ad the radom varable β t) j, referred to as the shadowg factor, s depedet of ht) j ad satsfes the followg codtos: β m β t) j β max, where β m > 0 ad β max s fte, E [ β t) ] j. All the chaels the etwork are assumed to be quas-statc block fadg,.e., the chael gas rema costat durg oe block ad chage depedetly from block to block. I other words, L t) j s depedet of L t ) j for t t. Moreover, the fadg block of all chaels are assumed to be equal to each other ad ths value s equal to the durato of the trasmsso block for all users. Ths model s also used [22] ad [23]. Also, users are assumed to be sychroous to each other. However, as we wll see later, the results of the paper are stll vald eve the cases that the users are ot sychroous or the fadg block coherece tme) of the chaels are ot equal. B. O-Off Power Allocato Strategy I [27], we have show that a dstrbuted scheme, called threshold-based o-off scheme, acheves the maxmum order of the sum-rate throughput a sgle-hop wreless etwork wth lks, uder the block Raylegh fadg chael model possbly wth shadowg effect, the asymptotc regme of. Moreover, the strog terferece scearo, the o-off power allocato scheme s the optmal strategy, terms of the sum-rate throughput, assumg the avalablty of drect chael gas at the trasmtters. Motvated by the results of [27], we assume that all the lks utlze the threshold-based o-off power allocato strategy proposed [27] 5. Ulke most of the works the lterature that assume backlogged users, here we assume a practcal model for the packet arrvals whch the buffer of each lk s ot ecessarly full of packet) all the tme. Based o ths observato, we adopt the o-off power allocato scheme durg each tme slot t as follows: - Based o the drect chael ga, the trasmsso polcy s 6 p t) =, f h t) > τ ad the buffer of lk s full at tme slot t 0, Otherwse, 2) 4 For more detals, the reader s referred to [32] ad [33] ad refereces there. 5 We cosder a homogeeous etwork the sese that all the lks have the same cofgurato ad use the same protocol. Thus, the trasmsso strategy for all users are agreed advace. 6 I fact, f there s o packet the buffer, t does ot make sese for the user to be actve, eve f ts chael s good.

7 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER where p t) deotes the trasmsso power of user at tme slot t ad τ s a pre-specfed threshold level that s a fucto of ad also depeds o the chael model ad packet arrval process. 2- Kowg ts correspodg drect chael ga, each actve user trasmts a Gaussa sgal wth full power ad the rate equal to: where I t) = j= j R t) L t) j pt) j = E t) h,it) [ log )] + ht) pt) I t) + N 0 ats/chael use, 3) s the power of the terferece term see by recever N at tme slot t. The above rate s achevable by ecodg ad decodg over arbtrarly large umber M) of blocks. More precsely, assumg the umber of chael uses per each trasmsso block to be N, the th trasmtter maps the message m {m, m 2,, m L }, where L = 2 MNRt), to a Gaussa codeword of sze MN, C m {C, C 2,, C L }. I the k th block, f p t) =, the trasmtter seds the k th porto of C m, deoted by C m k). At the recever sde, the decoder cosders oly the blocks whch the trasmtter was trasmttg wth full power, deoted by {a,, a l }, ad s able to decode the message m, f L 2 NlR [ ], where R E t) h log + ht) p t),it) I t) +N =. Notg that as M, l MP{p t) = }, ad 0 R t) = P{p t) = }R, t s cocluded that the rate R t) s achevable. As we wll see later, the optmal performace regme, whch s the strog terferece regme, ecodg ad decodg over sgle blocks s suffcet to acheve 3). C. Packet Arrval Process Oe of the most mportat parameters the etwork aalyss s the model for the packet arrval process. The packet arrval process s a radom process whch s descrbed by ether the arrval tme of the packets or the terarrval tme betwee the subsequet packets. These quattes may be modeled by the determstc or stochastc processes Fg. 2). I ths paper, we cosder the followg packet arrval processes: Posso Arrval Process PAP): I ths process, the umber of arrved packets ay terval of ut legth s assumed to have a Posso dstrbuto wth the parameter. Ths process s a commoly used model for radom ad mutually depedet packet arrvals queueg theory [34]. Beroull Arrval Process BAP): I ths process, at ay gve tme slot, the probablty that a packet arrves s ρ 7. Moreover, the arrval of the packets dfferet slots occurs depedetly. Ths model has bee used may works the lterature such as [2] ad [35]. Costat Arrval Process CAP): I ths process, packets arrve cotuously wth a costat rate of packets per ut legth Fg. 2-b) [36]. 7 We choose the parameter ρ as to be cosstet wth other packet arrval processes.

8 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER x k +3 x k +2 x k + x k Trasmtter Buffer t Ak+ 4 t Ak+ 3 t Ak+ 2 t Ak+ t Ak Trasmtter Buffer t Ak+ 5 t Ak+ 4 t Ak+ 3 t Ak+ 2 t Ak+ t Ak b) Fg. 2. A schematc fgure for a) stochastc packet arrval process, b) costat packet arrval process. It s assumed that the packet arrval process for all lks s the same. Let us deote t ) A k as the tme stat of the k th packet arrval to the buffer of lk. It s observed from Fg. 2-a that t ) A k = k j= x) j + t ) 0 where t ) 0 s the startg tme for lk, ad the radom varable x ) j s the terarrval tme defed as x ) j t ) A j+ t ) A j, 4) wth E[x ) j ] =. For the CAP, x) j = ad t ) A k = k )+t ) 0 8, whle for the PAP, x ) j s are depedet samples of a expoetal radom varable x wth the probablty desty fucto pdf) f X x) = e x, x > 0. 5) Also for the BAP, x ) j s are depedet samples of a geometrc radom varable X wth the probablty mass fucto pmf) p X m) P{X = m} = ρ) m ρ, m =, 2,..., 6) wth ρ. We represet t ) D k as the tme stat at whch ether the k th arrvg packet departs the buffer of lk for the trasmsso or drops from the buffer. I such cofgurato, we have the followg defto: Defto Delay): The radom varable D ) k t ) D k t ) A k for each lk s defed as the delay betwee the departure ad the arrval tme of each packet k, expressed terms of the umber of tme slots. I ths work, we assume that the buffer sze for each trasmtter s oe packet. Due to the ths lmtato o the buffer sze ad the o-off power allocato strategy, the exstg buffered packet may be dropped f 8 For aalyss smplcty, we assume that s a teger umber.

9 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER t s ot served before the arrval of the ext packet. Mathematcally speakg, the evet that the droppg of packet k occurs lk N s defed as { B D ) k { D ) k t ) A k+ t ) x ) k 7) }. 8) Therefore, the packet droppg probablty each lk N, deoted by P {B }, ca be obtaed as { } P {B } = P D ) k x ) k 9) { } = P D ) k x ) k x ) k = x f X x)dx, for PAP, 0) 0 { } = P D ) k x ) k x ) k = m p X m), for BAP, ) m= { } = P D ) k, for CAP. 2) where f X x) ad p X m) are defed as 5) ad 6), respectvely. I Secto IV, we wll obta P {B } for dfferet packet arrval processes terms of ad τ. A k } D. Objectves Part I: Throughput Maxmzato: The ma objectve of the frst part of ths paper s to maxmze the throughput of the uderlyg etwork. To address ths problem, we frst defe a ew oto of throughput, called effectve throughput, whch deotes the actual amout of data trasmtted through the lks. I order to derve the effectve throughput, we obta the full buffer probablty of a lk for the determstc ad stochastc packet arrval processes. The, we compute the optmum threshold level τ, ad the maxmum effectve throughput of the etwork, for each packet arrval process. Part II: Delay Characterstcs: The ma objectve of the secod part s to formulate the packet droppg probablty of each lk the uderlyg etwork based o the aforemetoed packet arrval processes terms of the umber of lks ),, ad the parameter of the o-off power allocato scheme τ ). Ths aalyss eables us to derve the suffcet codtos the asymptotc case of such that the packet droppg probabltes ted to zero, whle achevg the maxmum effectve throughput of the etwork. Part III: Delay-Throughput-Droppg Probablty Tradeoff: The ma goal of the thrd part s to study the tradeoff betwee the effectve throughput of the etwork ad other performace measures,.e., the droppg probablty ad the delay-boud ) for dfferet packet arrval processes. I partcular, we are terested to determe how much degradato wll be eforced the throughput by troducg the other costrats, ad how much ths degradato depeds o the packet arrval process.

10 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER A. Effectve Throughput III. THROUGHPUT MAXIMIZATION I ths secto, we am to derve the maxmum throughput of the etwork wth a large umber ) of lks, based o usg the dstrbuted o-off power allocato strategy. We preset a ew performace metrc the etwork, called effectve throughput, whch s a fucto of the threshold level τ ad. Let us start wth the followg defto. Defto 2 Effectve Throughput): Uder the o-off power allocato strategy, the effectve throughput of each lk, N, s defed o a per-block bass) as L T lm R t) I t), 3) L L where R t) s defed as 3) ad I t) t= s a dcator varable whch s equal to, f user trasmts at tme slot t, ad 0 otherwse. Furthermore, the effectve throughput of the etwork s defed as T eff T. 4) = The quatty T represets the average amout of formato coveyed through lk a log perod of tme. Ths metrc s sutable for real-tme applcatos, where the packets have a certa amout of formato ad certa arrval rates. It should be oted that I t) = s equvalet to the case whch the buffer s full ad the chael ga h t) s greater tha the threshold level τ at tme slot t. Defg the full buffer evet as follows we have C t) {Buffer of lk s full at tme slot t}, 5) { } P I t) = { { where q P h t) > τ }, ad P C t) { = P = P } h t) > τ, C t) { } { h t) > τ P C t) } 6) 7) = q, 8) } s the full buffer probablty. I the above equatos, follows from the fact that the full buffer evet depeds o the packet arrval process as well as the drect chael gas h t ), for t chael model). Thus, < t, whch s depedet of the chael ga h t) I t) =, wth probablty q, 0, wth probablty q. due to the block fadg 9)

11 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER 2009 It s observed that I t) s a Beroull radom varable wth parameter q. I fact, q s the probablty of the lk actvato whch s a fucto of. I the sequel, we derve for the aforemetoed packet arrval processes. B. Full Buffer Probablty Let us deote t ) a same tme t. The evet C t) as the tme stat the last packet has arrved the buffer of lk before or at the mplctly dcates that durg X t) of lk s less tha the threshold level τ. Clearly, X t) t t ) a tme slots, the chael ga s a radom varable whch vares from zero to fty for the stochastc packet arrval processes ad s fte for the CAP 9. Uder the o-off power allocato scheme ad usg the block fadg model property, the full buffer probablty ca be obtaed as 0 = E where the expectato s computed wth respect to X t), ad q P ] [ q ) X t), 20) { h t) > τ } = e τ. Lemma Let us deote the full buffer probablty of a arbtrary lk N, for the Posso, Beroull ad costat arrval processes as PAP, BAP Proof: For the PAP, sce X t) Also for the BAP, X t) smplfed as PAP = BAP = CAP ad CAP, respectvely. The, + log q ), 2), + )q 22) = q ) q. 23) s a expoetal radom varable, 20) ca be smplfed as PAP = = 0 q ) x e x dx 24) + log q ). 25) s a geometrc radom varable wth parameter ρ =. Thus, 20) ca be BAP = = q ) m ρ ρ) m 26) m=0 + )q, 27) 9 Note that, here we assume that f a packet arrves at tme t ad the chael ga s greater tha τ at ths tme, the packet wll be trasmtted. 0 As we wll show Lemma, s depedet of dex.

12 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER where follows from the followg geometrc seres: x m =, x <. 28) x m=0 For the CAP, the full buffer probablty 20) ca be wrtte as CAP = b) = q ) m P{X t) = m} 29) m=0 q ) m m=0 30) c) = q ), 3) q where follows from Fg. 2-b, whch X t) vares from zero to ad b) follows from the fact that for the determstc process, X t) has a uform dstrbuto. I other words, for every value of m [0, ], P{X t) = m} =. Also, c) comes from the followg geometrc seres: s x m = xs+ x. 32) m=0 Havg derved the full buffer probablty, we obta the effectve throughput of the etwork the followg secto. C. Effectve Throughput of the Network Rewrtg 3), the effectve throughput of lk ca be obtaed as T = lm L L [ = E = E L t= R t) I t) [ R t) b) = q E c) = q E I t) [ [ ] R t) log R t) I t) ] { } I t) = P I t) = + E ] h t) > τ, C t) + ht) I t) + N 0 [ R t) I t) ] I t) = 0 { } P I t) = 0 33) 34) 35) 36) ] ht) > τ, 37) where the expectato s computed wth respect to h t) ad the terferece term I t). I the above equatos, follows from the ergodcty of the chaels due to the block fadg model), whch mples that the average over tme s equal to average over realzato. b) results from 6)-8) ad E [ R t) I t) t) I = 0 ] = 0. Fally, c) results from the fact that the term log + ht) s depedet of C t) I t) +N. 0

13 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER I order to derve the effectve throughput, we eed to fd the statstcal behavor of I t) performed the followg lemmas: whch s Lemma 2 Uder the o-off power scheme, we have [ ] E I t) = )ˆαq, 38) [ ] Var I t) )2ακq ), 39) [ ] 2 where ˆα α ad κ E β t) j. Proof: See Appedx I. Lemma 3 The maxmum effectve throughput s acheved at = o) ad the strog terferece regme whch s defed as E[I t) ] = ω), N. Proof: Suppose that o) whch mples that = Ω). Usg 37), we have [ ] T q E log + ht) N 0 ht) > τ 40) [ ] E h t) h t) > τ q log + 4) N 0 = q log + τ ) +, 42) N 0 where comes from the cocavty of log.) fucto ad Jese s equalty, E [log x] loge [x]), x > 0. Followg 2) - 23), t s revealed that m, q for all packet arrval processes. Substtutg 42), we have T log + log + ) log log, 43) whch follows from the fact that the maxmum value of q log + τ+ N 0 wth the codto of m, q s attaed at q =. Notg that = Ω), we have T Θ log log. Now, suppose that = o) but E[I t) ] ω), or equvaletly, E[I t) ] = O) for some. Sce E[I t) ] = )ˆαq, the codto E[I t) ] = O) mples that there exsts a costat c such that q c. Notg 2) - 23), t follows that ether q N 0 or = Θ). I the frst case, the

14 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER codto q c mples that c whch caot hold due to the assumpto of = o). Therefore, we must have q c, for some costat c. Substtutg 42) yelds T c log + τ ) + N 0 ) c log + log/c ) + N 0 log log = Θ, 44) where results from the fact that q log + τ+ N 0 s a creasg fucto of q ad reaches ts maxmum at the boudary whch s c. I the sequel, we preset a lower-boud o the effectve throughput of lk the rego = o) ad E[I t) ] = ω) ad show that ths lower-boud beats the upper-bouds derved the other regos, provg the desred result. For ths purpose, usg 37), we wrte T q log + [ ] E I t) h t) > τ + N o ) b) τ = q log + )ˆαq + N o c) τ q log + τ )ˆαq ), 45) where follows from the covexty of the fucto log+ b ) wth respect to x ad Jese s equalty, x+a b) results from the depedece of I t) from h t), ad c) follows from eglectg the term N 0 wth respect to )ˆαq due to the strog terferece assumpto. Settg q = log2 ad =, log 2 τ t s easy to check that τ )ˆαq = o) ad hece, log + τ )ˆαq )ˆαq whch gves τ the effectve throughput as = Θ log )ˆα whch s greater tha the throughput obtaed the other regmes. Due to the result of Lemma 3, we restrct ourselves to the case of = o) ad the strog terferece regme the rest of the paper. [ Lemma 4 Let us assume 0 < α s fxed ad we are the strog terferece regme.e., E ω)). The wth probablty oe w. p. ), we have I t) ] = I t) )ˆαq, 46) as. More precsely, substtutg I t) throughput of the etwork. by )ˆαq does ot chage the asymptotc effectve Proof: See Appedx II.

15 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER Lemma 5 The effectve throughput of the etwork for large values of ca be obtaed as τ T eff q log +. 47) ˆαq Proof: obtaed as Usg 37), the effectve throughput of the etwork the asymptotc case of s T eff = b) = T [ h t) q E log + )ˆαq + N 0 [ q E log + ht) ˆαq ) ] ht) > τ 48) 49) ] ht) > τ, 50) where results from the strog terferece assumpto ad Lemma 4, ad b) follows from approxmatg )ˆαq + N 0 by ˆαq due to the strog terferece assumpto ad large values of. A lower-boud o 50) ca be wrtte as T l eff = q log + τ ˆαq ). 5) Furthermore, due to the cocavty of log.) fucto ad Jese s equalty, a upper-boud o T eff ca be gve as T u eff [ E = q log + h t) ] > τ h t) ˆαq = q log + τ ) +. 52) ˆαq I order to prove that the above upper ad lower bouds have the same scalg, t s suffcet to show that the optmum threshold value τ ) s much larger tha oe. For ths purpose, we ote that f τ = O), the the effectve throughput of the etwork wll be upper-bouded by T eff τ + ˆα 53) = O), 54) where follows from log + x) x. I other words, the effectve throughput of the etwork does ot scale wth, whle the throughput of Θlog ), as wll be show later, s achevable. Ths suggests that the optmum threshold value must grow wth, ad hece, the bouds gve 5) ad 52) are asymptotcally equal to q log + τ ˆαq ad ths completes the proof of the lemma. Lemma 6 The maxmum effectve throughput of the etwork s obtaed the rego that τ = o ˆαq ).

16 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER Proof: Rewrtg the expresso of the effectve throughput of the etwork from 47) ad otg the fact that log + x) x, for x 0, we have τ T eff q log + ˆαq τ ˆα. 55) It ca be show that f the codto τ = o ˆαq ) s ot satsfed, the rato log+ less tha oe. Havg τ = o ˆαq ) results log + τ ˆαq τ ˆαq τ ) ˆαq τ ˆαq s strctly yeldg the upper-boud τ ṋα. Ths meas that to acheve the maxmum throughput, the terferece should ot oly be strog but also be much larger tha τ. Observato - A terestg observato of Lemmas 3-6 s that there s o eed to have sychrozato betwee the users or equalty of the fadg blocks coherece tme) of the chaels to obta these results. Ths s due to the fact that durg a trasmsso block whch s equal to the fadg block of the correspodg drect chael), the recever observes dfferet samples of terferece I due to asychroousy betwee the users). However, as the terferece s strog, from the result of Lemma 4, all samples of terferece asymptotcally almost surely scale as ˆαq, ad hece, the recever s stll capable of decodg the message correctly f the trasmsso rate s below q log + τ ˆαq ). Moreover, the ecodg ad decodg do ot eed to be performed over large umber of blocks. I fact, the blocks where h t) > τ, the trasmtter seds data wth the rate log + τ ˆαq ats/chael use ad the decoder wll be able to decode the packet formato correctly. Havg the expresso for the effectve throughput of the etwork 47), the ext theorem, we fd the optmum value of q or equvaletly τ ) terms of ad for the aforemetoed packet arrval processes,.e.: ˆq = arg max q T eff. 56) As show the proof of Lemma 5, sce the optmum threshold value s much larger tha oe, the optmzer ˆq s suffcetly small,.e., ˆq = o). Theorem Assumg the Posso packet arrval process ad large values of, the optmum soluto for 56) s obtaed as q PAP = δ log2 for some costat δ. Furthermore, the maxmum effectve throughput of the etwork asymptotcally scales as log, for = o. ˆα log Proof: See Appedx III. 57)

17 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER Theorem 2 Assumg the Beroull packet arrval process ad large values of, the optmum soluto for 56) s obtaed as q BAP = δ log2 for some costat δ. Furthermore, the maxmum effectve throughput of the etwork asymptotcally scales as log, for = o. ˆα log Proof: See Appedx IV. Theorem 3 Assumg a determstc packet arrval process, the optmum soluto of 56) ad the correspodg maxmum effectve throughput of the etwork are asymptotcally obtaed as ) q CAP = δ log2 ad T eff log, for = o, ˆα log 2 ) ) q CAP = δ log2 ad T eff log, for = Θ, «ˆα log 2 ) q CAP = log log 2 ˆα ad T eff log, for = ω ad = o, ˆα log 2 log for some costats δ ad δ. Proof: See Appedx V. The above theorems mply that the effectve throughput of the etwork scales as log, regardless of the ˆα packet arrval process. Note that ths value s the same as the sum-rate scalg of the same etwork wth backlogged users [27], whch s a upper-boud o the effectve throughput of the curret setup. I other words, the effect of the real-tme traffc the throughput whch s captured the full buffer probablty) s asymptotcally eglgble. However, we dd ot cosder the effect of droppg o the calculatos. I the subsequet secto, we clude the droppg probablty the aalyss ad fd the maxmum effectve throughput of the etwork such that the droppg probablty approaches zero. 58) IV. DELAY ANALYSIS I ths secto, we frst formulate the packet droppg probablty the uderlyg etwork terms of the umber of lks ) ad for the aforemetoed packet arrval processes. The, we derve the suffcet codtos for the delay-boud ) the asymptotc case of such that the packet droppg probabltes ted to zero, whle achevg the maxmum effectve throughput of the etwork. Lemma 7 Let us deote the packet droppg probablty of a lk, N, for the Posso, Beroull ad costat arrval processes as P { } { } { } B PAP, P B BAP ad P B CAP, respectvely. The, P { } B PAP = + log q ), 59) P { } B BAP q )q ) = + q )q ), 60) P { } B CAP = q ). 6)

18 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER Proof: Recallg t ) A k as the tme stat of the k th packet arrval to the buffer of lk, each user s actve at tme slot t t ) A k oly whe h t) > τ. I other words, assumg the buffer s full, o trasmsso or o servce) occurs each slot wth probablty q. From 4) ad 7)-), sce the tme durato betwee subsequet packet arrvals s x ) k, the packet droppg probablty for a lk s obtaed as P {B } = E ] [ q ) x) k, 62) where the expectato s computed wth respect to x ) k. For the PAP, sce x) k varable, 62) ca be smplfed as Also for the BAP, x ) k smplfed as s a expoetal radom P { } B PAP = 0 q ) x e x dx 63) = + log q ). 64) s a geometrc radom varable wth parameter ρ =. Thus, 62) ca be P { B BAP } = q ) m ρ ρ) m 65) = = m= ρ ρ [ q ) ρ)] m 66) m= q )q ) + q )q ), 67) where comes from the followg geometrc seres: x m = x, x x <. 68) Accordg to Fg. 2-a, x ) k m= for the CAP s a determstc quatty ad s equal to. Thus, we have P { } B CAP = q ). 69) It should be oted that 64), 67) ad 69) are vald for every value of q [0, ]. I partcular, the extreme case of q =, P { } { } { } B CAP = P B PAP = P B BAP = 0. We are ow ready to prove the ma result of ths secto. I the ext theorem, we derve the suffcet codtos o, such that the correspodg packet droppg probabltes ted to zero, whle achevg the maxmum effectve throughput of the etwork. Theorem 4 For the optmum q obtaed Theorems -3 resultg the maxmum effectve throughput of the etwork, ) lm P { B PAP } = 0, f PAP = ω log 2 ) ad PAP = o log ),

19 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER ) lm P { B BAP ) lm P { B CAP } = 0, f BAP = ω } = 0, f CAP = ω Proof: ) From 59), we have log 2 log 2 ) ad BAP = o ) ad CAP = o P { } B PAP = PAP log q PAP ). 70) It follows from 70) that achevg P { } B PAP = ǫ results PAP ǫ = ǫ log q PAP ) log log ), ). ǫ, 7) q PAP where comes from q PAP = o) ad the approxmato log z) z, z. Notg the fact that the optmum value of q PAP log scales as Θ 2, havg PAP = ω results lm log 2 P { } B PAP = 0. O the other had, from Theorem, the codto PAP = o s requred to acheve the maxmum log T eff, ad ths completes the proof of the frst part of the Theorem. ) It s realzed from 60) that achevg P { } B BAP = ǫ results BAP ǫ = q BAP ǫ q BAP [ q BAP )ǫ q BAP ) ], 72) for small eough ǫ. Notg the fact that the optmum value of q BAP log scales as Θ 2, havg BAP = ω results lm P { } B BAP = 0. O the other had, from Theorem 2, BAP = o log 2 guaratees achevg the maxmum effectve throughput of the etwork. ) From 6), we have where follows from log z) z, must have P { } B CAP = e CAP log q CAP ) log 73) e qcap CAP 74) z for q CAP CAP ǫ = q CAP = o). To acheve P { } B CAP = ǫ, we log ǫ. 75) It follows from 74) that settg q CAP CAP = ω) makes e qcap CAP 0. Usg part ) Theorem 3, t turs out that choosg CAP = ω satsfes q CAP log 2 CAP = ω) whch yelds lm P { } B CAP = 0. We also eed the codto CAP = o to esure achevg the maxmum effectve throughput of the etwork. log

20 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER Remark - It s worth metog that the delay-boud ) each lk for the CAP scales the same as that of for the PAP ad BAP. However, P { } { } { } B CAP decays faster tha P B PAP ad P B BAP terms of, whe teds to fty expoetally versus learly). A terestg cocluso of Theorem 4 s the possblty of achevg the maxmum effectve throughput of the etwork whle makg the droppg probablty approach zero. More precsely, there exsts some ǫ such that P {B } ǫ, N, whle achevg the maxmum T eff of log. Ths s true for all ˆα aforemetoed arrval processes. However, for arbtrary values of ǫ, there s a tradeoff betwee creasg the throughput, ad decreasg the droppg probablty ad the delay-boud ). Ths tradeoff s studed the ext secto. V. THROUGHPUT-DELAY-DROPPING PROBABILITY TRADEOFF I ths secto, we study the tradeoff betwee the effectve throughput of the etwork ad other performace measures,.e., the droppg probablty ad the delay-boud ) for dfferet packet arrval processes. I partcular, we would lke to kow how much degradato wll be eforced the throughput by troducg the other costrats, ad how much ths degradato depeds o the packet arrval process. A. Tradeoff Betwee Throughput ad Droppg Probablty I ths secto, we assume that a costrat P {B } ǫ must be satsfed for the droppg probablty. It ca be easly show that the costrat P {B } ǫ s equvalet to P {B } = ǫ. The am s to characterze the degradato o the effectve throughput of the etwork terms of ǫ for dfferet packet arrval processes. Frst, we cosder PAP. Lookg at the equatos 2) ad 59), t turs out that P { B PAP } = ǫ s traslated to PAP P { B PAP ca be wrtte as } = PAP. Hece, the codto = ǫ. Therefore, usg 47), the effectve throughput of the etwork T eff q ǫ log + τ ). 76) ˆαq ǫ From the above equato, t ca be realzed that the effectve throughput of the etwork s equal to the average sum-rate of the etwork wth ǫ users the case of backlogged users, whch s gve [27] as logǫ) ˆα for the case of ǫ or ǫ = ω ). Also, the optmum value of q s show to scale as δ log2 ǫ) ǫ for some costat δ ad hece, the optmum value of s gve as ǫ q = δ log 2 ǫ). Let us deote T eff as the degradato the effectve throughput of the etwork, whch s defed as the dfferece betwee the maxmum effectve throughput the case of o costrat o P {B } Theorem -3) ad the case

21 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER wth costrat o P {B }. Usg Theorem, T eff for the PAP ca be wrtte as T eff log ˆα logǫ) ˆα = logǫ ), 77) ˆα for ǫ = ω ). Moreover, for values of ǫ such that logǫ ) = olog ), t ca be show that the scalg of the effectve throughput of the etwork s ot chaged,.e., T eff log ˆα. For the BAP, ad usg 22) ad 60), we have P { } B BAP q = + )q + )q = BAP, 78) where follows from the fact that q = o). Therefore, smlar to the case of the PAP, we have P { } B BAP BAP = ǫ ad as a result, the rest of the argumets hold. I partcular, For the CAP, ad usg 23) ad 6), we have whch gves T eff logǫ ). 79) ˆα q ) = ǫ = q logǫ ), 80) CAP = q ) 8) q logǫ ). 82) Hece, usg 47), the effectve throughput of the etwork ca be wrtte as T eff logǫ ) q τ log + ˆαq, 83) logǫ ) whch s equal to the average sum-rate of a etwork wth equal to log logǫ ) ˆα logǫ ) backlogged users ad s asymptotcally, for values of ǫ satsfyg logǫ ) = o). Therefore, the degradato the effectve throughput of the etwork for the CAP ca be expressed as T eff log log ˆα logǫ ) ˆα ) = log logǫ ). 84) ˆα I the case of ǫ = O ), t s easy to see that the effectve throughput of the etwork does ot scale wth.

22 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER Comparg the expressos of T eff for the Posso, Beroull ad costat packet arrval processes, t follows that the degradato the effectve throughput of the etwork the cases of PAP ad BAP both grow logarthmcally wth ǫ, whle the case of CAP t grows double logarthmcally. I other words, the degradato the throughput the cases of the PAP ad BAP s much more substatal compared to the CAP. Ths fact s also observed the smulato results the ext secto. B. Tradeoff Betwee Throughput ad Delay I ths secto, we am to fd the tradeoff betwee the effectve throughput of the etwork ad the delay-boud ), for a gve costrat o the droppg probablty,.e., P {B } ǫ. ad ) PAP: Usg 2) ad 59), t follows that for a gve ad ǫ, we have q ǫ, = τ logǫ), 85) q. 86) Substtutg q ad τ from the above equatos 47) yelds T eff log + logǫ) ). 87) ˆα It ca be verfed that T eff has a global maxmum at PAP ˆα opt. I other words, for < log 2 ˆαǫ PAP ) opt, there s a tradeoff betwee the throughput ad delay, meag that creasg results creasg both the throughput ad delay. However, the crease the throughput s logarthmc whle the delay creases learly wth. It should be oted that the rego > PAP opt PAP opt s ot of terest, sce creasg from results decreasg the throughput ad creasg the delay whch s ot desred. 2) BAP: Due to the smlarty betwee the values of P {B } ad for the PAP ad the BAP, the results obtaed for the PAP are also vald for the BAP. ad 3) CAP: Usg 23) ad 6), t follows that for a gve ad ǫ, we have q logǫ ), = τ log logǫ ) ), 88) q. 89) As ca be observed, all the results for the cases of PAP ad BAP are extedable to the case of CAP by substtutg ǫ wth logǫ ). I partcular, the optmum value for ca be wrtte as CAP opt

23 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER ˆα log 2 ˆαlogǫ )) log ˆα logǫ ), ad for < CAP opt, the effectve throughput of the etwork ca be gve as T eff ). Ths meas that the rego < CAP opt, whch s the rego of terest, there s a tradeoff betwee the throughput ad delay such that by creasg, T eff creases logarthmcally, whle the delay creases learly wth. Furthermore, comparg the value of opt for the PAP ad BAP wth the CAP, t s realzed that CAP opt > PAP opt ad CAP opt > BAP opt. Ths fact s also observed the smulatos. VI. NUMERICAL RESULTS I ths secto, we preset some umercal results to evaluate the tradeoff betwee the effectve throughput of the etwork ad other performace measures,.e., droppg probablty ad the delayboud ) for dfferet packet arrval processes. For ths purpose, we assume that all users the etwork follow the threshold-based o-off power allocato polcy. I addto, the shadowg effect s assumed to be logormal dstrbuted wth mea = 0.5, varace ad α = 0.4. Furthermore, we assume that = 500 ad N 0 =. Fgures 3 ad 4 show the effectve throughput of the etwork versus ǫ for the PAP, BAP ad CAP ad dfferet values of ǫ. It s observed from these fgures that for a gve costrat o the droppg probablty e.g., ǫ = 0.05), ad for < opt, creasg results creasg both the throughput ad delay. However, the crease the throughput s logarthmc whle the delay creases learly wth as expected. Also, creasg from opt results decreasg the throughput ad creasg the delay whch s ot desred. Furthermore, comparg the value of opt for the PAP ad BAP wth the CAP, t s realzed that CAP opt > PAP opt ad CAP opt > BAP opt, as expected. To evaluate the degradato the effectve throughput of the etwork terms of droppg probablty, we plot T eff versus log ǫ for dfferet packet arrval processes Fg. 5. It ca be see that the degradato the throughput the cases of the PAP ad BAP s much more substatal compared to the CAP, as expected. Hece, the performace of the uderlyg etwork wth the CAP s better tha that of the PAP ad BAP from the delay-throughput ad delay-droppg probablty tradeoff pots of vew. VII. CONCLUSION I ths paper, the delay-throughput of a sgle-hop wreless etwork wth lks was studed. We cosdered a block Raylegh fadg model wth shadowg, descrbed by parameters α, ), for the chaels the etwork. The aalyss the paper reled o the dstrbuted o-off power allocato strategy for the determstc ad stochastc packet arrval processes. It was also assumed that each trasmtter has a buffer sze of oe packet ad droppg occurs oce a packet arrves the buffer whle the prevous packet has ot bee served. I the frst part of the paper, we defed a ew oto of performace the etwork, called effectve throughput, whch captures the effect of arrval process the etwork throughput,

24 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER ε=0. ε=0.05 ε=0.02 Network s Effectve Throughput ε 8 7 ε=0. ε=0.05 ε=0.02 Network s Effectve Throughput ε Fg. 3. Effectve throughput of the etwork versus ǫ for N 0 =, = 500, α = 0.4, ad dfferet values of ǫ a) PAP ad b) BAP. b)

25 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER ε=0. ε=0.05 ε=0.02 Network s Effectve Throughput ε Fg. 4. Effectve throughput of the etwork versus ǫ for the CAP ad N 0 =, = 500, α = 0.4, ad dfferet values of ǫ. ad maxmze t for dfferet cases of arrval process. It was proved that the effectve throughput of the etwork asymptotcally scales as log, wth ˆα α, regardless of the packet arrval process. I the ˆα secod part of the paper, we preseted the delay characterstcs of the uderlyg etwork terms of the packet droppg probablty. We derved the suffcet codtos the asymptotc case of such that the packet droppg probablty ted to zero, whle achevg the maxmum effectve throughput of the etwork. Fally, we studed the trade-off betwee the effectve throughput, delay, ad packet droppg probablty of the etwork for dfferet packet arrval processes. It was show from the umercal results that the performace of the determstc packet arrval process s better tha that of the Posso ad the Beroull packet arrval processes, from the delay-throughput ad throughput-droppg probablty tradeoff pots of vew. Let us defe χ t) j L t) j pt) j APPENDIX I PROOF OF LEMMA 2, where Lt) j s depedet of p t) j, for j. Note that { } { } P p t) j = = P h t) jj > τ, C t) j A-) = q, A-2)

26 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER CAP BAP PAP 4 Network s Effectve Throughput logε Fg. 5. Effectve throughput of the etwork versus log ǫ for dfferet packet arrval processes ad N 0 =, = 500, α = 0.4. where follows from 8). Thus for the o-off power scheme, we have [ ] E = q. A-3) p t) j Uder a quas-statc Raylegh fadg chael model, t s cocluded that χ t) j s are depedet ad detcally dstrbuted..d.) radom varables wth [ ] [ E χ t) j = E [ ] Var = E [ ] 2 where E h t) j = 2, E [ ] 2 [ Thus, E p t) j E [ ] 2 β t) j ] p t) j µ ad varace ϑ 2, where [ µ E [ Var ϑ 2 I t) χ t) j L t) j pt) j [ χ t) j ] = ˆαq, A-4) ) ] 2 [ ] E 2 A-5) χ t) j 2ακq ˆαq ) 2, A-6) 2 κ ad ˆα α. Also, follows from the fact that p t) t) j p j. = q. The terferece I t) = j= j χ t) j s a radom varable wth mea ] = )ˆαq, ] A-7) )2ακq ˆαq ) 2 ) )2ακq ). A-8) I t)

27 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER APPENDIX II PROOF OF LEMMA 4 Usg Lemma 2 ad the Cetral Lmt Theorem [37, p. 83], we obta { } P I t) ψ µ < ψ Q ϑ e ψ 2 2ϑ 2, B-2) for all ψ > 0 such that ψ = o 6ϑ ). I the above equatos, the Q.) fucto s defed as B-) Qx) 2π x e u2 /2 du, ad follows from the fact that Qx) e x2 2, x > 0. Selectg ψ = q ) 8 2ϑ, we obta P{ I t) µ < ψ } e q ) 4. B-3) Therefore, defg ε ψ µ, otg that as ϑ = Oq ) from A-8) Appedx I) ad µ = Θq ), we have ε = O q ) 3 8, t reveals that Notg that q, t follows that I t) P{µ ε) I t) µ + ε)} e q ) 4. B-4) µ, wth probablty oe. T eff τ [ = q APPENDIX III PROOF OF THEOREM Takg the frst-order dervatve of 47) wth respect to τ yelds ] τ log + b) τ [ ] q τ τ ˆαq ) + τ ) τ τ + q ˆαq + τ τ C-) + τ ) τ + q, C-2) ˆαq ˆαq + τ where comes from q = e τ ad q τ = q. Also, b) follows from Lemma 6 ad usg the approxmato log + x) x, for x. Settg C-2) equal to zero yelds ˆαq 2 = ) τ 2 τ. C-3) It should be oted that C-3) s vald for every packet arrval process. Recallg from 2), the full buffer probablty for the PAP s gve by PAP = + log q ) C-4), + q C-5)

28 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER where follows from the fact that for q = o), log q ) q. I ths case, PAP τ q +q ) 2, whch results PAP PAP τ Thus for the Posso arrval process, C-3) ca be smplfed as = PAP q q τ = + q ) 2 = PAP 2. C-6) ˆαq = τ 2. C-7) It ca be verfed that the soluto for C-7) s Usg q = e τ, we coclude that for some costat δ. τ PAP = log 2 log log + O). C-8) q PAP To satsfy the codto of lemma 6, we should have Usg C-5), C-8), ad C-9), t yelds τ ˆαq PAP = δ log2, C-9), C-0) PAP = o. C-) log Thus, the maxmum effectve throughput of the etwork obtaed 47) ca be wrtte as T eff τ ˆα. C-2) Usg 22), we have BAP τ BAP = BAP q APPENDIX IV PROOF OF THEOREM 2 q τ = q BAP BAP = τ q = q ) Thus for the Beroull arrval process, C-3) ca be smplfed as + )q ) 2. I ths case, + )q ) 2 = BAP 2. D-) ˆαq = τ 2. D-2) It ca be observed that D-2) s exactly equal to C-7) ad hece, ts soluto ca be wrtte as τ BAP = log 2 log log + O), D-3) ad q BAP = δ log2, D-4)

29 SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER for some costats δ. Smlarly, the maxmum effectve throughput of the etwork for the BAP s obtaed as whch s acheved uder the codto T eff τ ˆα, D-5) BAP = o. D-6) log APPENDIX V PROOF OF THEOREM 3 Usg 23), we have CAP τ = CAP q q τ CAP = q q E-) E-2) = q ) q ) q E-3) = CAP q ). E-4) Hece, CAP or CAP τ = q ). I ths case, C-3) ca be smplfes as ˆαq [ q ) ] 2 q ) 2 = q ) τ 2. E-5) ˆα = τ2 2 q q ) [ q ) ] 2. E-6) Sce q = o), we have q ) ) log q) = e e q, ad q ) b) e q. It should be oted that ad b) are vald uder the codto q2 2 = o) 2. Thus, E-6) ca be smplfed as or ˆα = τ2 2 q e q [ e q ] 2, E-7) ν log ν = Ψ, E-8) ν) 2 where ν e q ad Ψ ˆα For ths setup, we have the followg cases: τ 2. Case : Ψ 2 As we wll show the codto q2 2 = o) s satsfed for the optmum q ad the correspodg.