The Stablty Regon of the Two-User Broadcast Channel Nkolaos appas *, Maros Kountours, * Department of Scence and Technology, Lnköpng Unversty, Campus Norrköpng, 60 74, Sweden Mathematcal and Algorthmc Scences Lab, France Research Center, Huawe Technologes Co. Ltd. Boulogne-Bllancourt, 900, France Emals: nkolaos.pappas@lu.se, maros.kountours@huawe.com arxv:50.0667v3 [cs.it] 3 Feb 06 Abstract In ths paper, we characterze the stablty regon of the two-user broadcast channel. Frst, we obtan the stablty regon n the general case. Second, we consder the partcular case where each recever treats the nterferng sgnal as nose, as well as the case n whch the packets are transmtted usng superposton codng and successve decodng s employed at the strong recever. I. INTRODUCTION A fundamental queston to whch Informaton Theory ams to provde an answer s how to maxmze the use of a communcaton channel between transmtters and recevers. In other words, ts major objectve s to characterze the maxmum achevable rate of nformaton that can be relably transmtted over a communcaton channel, whch s called the channel capacty. In contrast to pont-to-pont channels, f the channel s shared among multple nodes multuser channel, the goal s to fnd the capacty regon,.e. the set of all smultaneously achevable rates. One of the man assumptons n the nformaton-theoretc formulaton of the capacty regon s that the maxmum achevable rate s obtaned under nfntely backlogged users. However, the bursty nature of the sources n communcaton networks gave rse to the development of a dfferent concept of capacty regon, whch s the maxmum stable throughput regon or the stablty regon []. Understandng the relatonshp between the nformaton-theoretc capacty regon and the stablty regon has receved consderable attenton n recent years and some progress has been made prmarly for multple access channels. Interestngly, the aforementoned regons capacty and stablty are not n general dentcal and general condtons under whch they concde are known only n very few cases []. In ths work, we consder the two-user broadcast channel [3], whch models the smultaneous communcaton of nformaton dfferent messages from one source to multple destnatons. Marton n [4] derved an nner bound, whch s the best known achevable nformaton-theoretc capacty regon for a general dscrete memoryless broadcast channel. Fayolle et al. [5] provded a theoretcal treatment of some basc problems related to the packet swtchng broadcast Ths work has been supported by the eople ogramme Mare Cure Actons of the European Unon s Seventh Framework ogramme F7/007-03/ under REA grant agreement no.[636] SOrBet. channel. The work n [6] provded a partal characterzaton of the capacty regon of the two-user Gaussan fadng broadcast channel. Care and Shama n [7] nvestgated the achevable throughput of a mult-antenna Gaussan broadcast channel. In [8], schedulng polces n a broadcast system were consdered and general condtons coverng a class of throughput optmal schedulng polces were obtaned. In [9], the authors characterzed the stablty regons of twouser Gaussan fadng multple access and broadcast networks wth centralzed schedulng under the assumpton of nfnte backlogged users. In [0], the capacty regon of the two-user broadcast erasure channel was characterzed and algorthms based on lnear network codng and ther stablty regon were also provded. Superposton Codng SC [3] s one of the fundamental buldng blocks n network nformaton theory. The objectve of SC s to smultaneously transmt two messages by encodng them nto a sngle sgnal n two layers. The recever wth the better less nosy channel, also named stronger recever, can recover the sgnal on both layers by applyng successve nterference cancelaton, whle the other weaker or worse can decode the message on the coarse layer treatng the message on the fne layer nterference as nose. In [], SC wth conventonal frequency dvson n a osson feld of nterferers was analyzed. Furthermore, n [], the authors provded a software-rado based desgn and mplementaton of SC. Ther results show that SC can provde substantal spectral effcency gans compared to orthogonal schemes, such as tme dvson multplexng. The stablty regon of the two-user nterference channel was derved n [3], where the case of successve nterference cancelaton was also consdered. In ths paper, the stablty regon of the two-user broadcast channel s obtaned. We frst provde the stablty regon for the general case as a functon of success probabltes and afterwards we specalze our study consderng two partcular cases. The frst case s when both recevers treat nterference as nose. The second case s when superposton codng s employed and the user erencng better channel uses a successve decodng scheme. Two smple transmt power allocaton schemes are consdered n the latter case: the assgned power remans fxed, and the transmt power s adapted to the state of the queues.
Queue D h λ λ S Fg. : The two-user broadcast channel wth bursty arrvals. Queue h D II. SYSTEM MODEL We consder a two-user broadcast channel, as depcted n Fg., n whch a sngle transmtter havng two dfferent queues ntends to communcate wth two recevers. The frst resp. second queue contans the packets messages that are destned to recever D resp. D. Tme s assumed to be slotted, the packet arrval processes at the frst and the second queue are assumed to be ndependent and statonary wth mean rates λ and λ n packets per slot, respectvely. Both queues have nfnte capacty to store ncomng packets and Q denotes the sze n number of packets of the -th queue. The source transmts packets n a tmeslot f at least one of ts queue s not empty. The transmsson of one packet requres one tmeslot and we assume that receve acknowledgements ACKs are nstantaneous and error-free. If only the -th queue at the source s non-empty durng a certan tmeslot, then the transmtter sends nformaton to the -th recever only. When both queues have packets, the source transmts a packet that contans the messages of both recevers, whereas whenever both queues at the source are empty, the transmtter remans slent. Let D /T denote the event that D s able to decode the packet transmtted from the -th queue of the transmtter gven a set of non-empty queues denoted by T, e.g. D /, denotes the event that the frst recever can decode the packet from the frst queue when both queues are not empty T = {, }. It s evdent that D / D /,. The average servce rate seen by the frst queue s µ = Q > 0 D /, + Q = 0 D /. Respectvely, the average servce rate of the second queue s µ = Q > 0 D /, + Q = 0 D /. If a packet from the -th queue fals to reach D, t remans n queue and s retransmtted n the next tmeslot. The sgnal y receved at user D at a tmeslot t s gven by y t = h t x t + n t 3 where n t s the addtve whte Gaussan nose at tmeslot t wth zero mean and unt varance. The channel gan from the transmtter to D at tme nstant t s denoted by h t, and the transmtted sgnal s x t. A block fadng channel model wth Raylegh fadng s consdered here,.e. the fadng coeffcents h t reman constant durng one tmeslot, but change ndependently from one tmeslot to another. In the transmsson phase downlnk, the transmtter assgns power for messages packets from queue. The event D / s defned as the probablty that the uncoded receved sgnal-to-nose rato SNR s above a certan threshold,.e. D / = { SNR }. The dstance between the transmtter and D s denoted by d and α s the pathloss onent. The SNR threshold for recever s. Then SNR h assumng a physcal layer model. The probablty that the lnk between the transmtter and D s not n outage when only the -th queue s non-empty s gven by Ch. 5.4 n [4] D / = {SNR } = d α. 4 Note that ths s an approxmaton on the success probablty under specfc assumptons on the underlyng physcal layer model and s done n order to relate the success probabltes wth a physcal layer and channel model. Actually, the above resson on the success probablty comes from the rates for arbtrarly relable communcaton, whch mples that the channel uses go to nfnty. Ths asymptotc resson s an approxmaton of the nstantaneous rate when nformaton s transmtted n packets. Nevertheless, the man result of ths paper,.e. the stablty regon derved n the followng secton, s general as t s ressed n terms of success probabltes, whch can n turn take on dfferent ressons dependng on the adopted physcal channel or the nformaton-theoretc model. When both queues at the source are non-empty at tmeslot t then the source transmts the sgnal x t = x t + x t where x t s the sgnal for the user D. Then, the sgnal y receved at user D at a tmeslot t s gven by y t = ht xt + n t, =,, where n t s the addtve whte Gaussan nose at tmeslot t wth zero mean and unt varance. The channel gan from the transmtter to D s denoted by h t at nstant t and a Raylegh block fadng model s consdered. In the transmsson phase downlnk, the transmtter assgns power for messages packets of queue wth + =. We assume that each recever D knows each channel h t perfect CSIR and that the transmtter has perfect channel state nformaton CSIT,.e. t knows h t,. Each recever decodes separately ts message usng the receved sgnal y. The success probabltes n the case that both queues are nonempty depend on the nterference handlng technque and for that we study certan dfferent cases n the followng sectons. A. Stablty Crtera We use the followng defnton of queue stablty [5]: Defnton. Denote by Q t the length of queue at the begnnng of tme slot t. The queue s sad to be stable f
lm t r[q t < x] = F x and lm x F x =. If lm x lm t nf r[q t < x] =, the queue s substable. If a queue s stable, then t s also substable. If a queue s not substable, then we say t s unstable. Loynes theorem [6] states that f the arrval and servce processes of a queue are strctly jontly statonary and the average arrval rate s less than the average servce rate, then the queue s stable. If the average arrval rate s greater than the average servce rate, then the queue s unstable and the value of Q t approaches nfnty almost surely. The stablty regon of the system s defned as the set of arrval rate vectors λ = λ, λ for whch the queues n the system are stable. III. THE STABILITY REGION THE GENERAL CASE In ths secton, we provde the stablty regon as a functon of the success probabltes n the general case wthout consderng specfc nterference handlng technques. The average servce rates of the frst and second queue are gven by and, respectvely. Snce the average servce rate of each queue depends on the queue sze of the other queue, t cannot be computed drectly. Therefore, we apply the stochastc domnance technque [],.e. we construct hypothetcal domnant systems, n whch one of the sources transmts dummy packets when ts packet queue s empty, whle the other transmts accordng to ts traffc. A. Frst Domnant System: the frst queue transmts dummy packets In the frst domnant system, when the frst queue emptes, then the source transmts a dummy packet for the D, whle the second queue behaves n the same way as n the orgnal system. All other assumptons reman unaltered n the domnant system. Thus, n ths domnant system, the frst queue never emptes, thus the servce rate for the second queue s gven by µ = D /,. Then, we can obtan stablty condtons for the second queue by applyng Loyne s crteron [6]. The queue at the second source s stable f and only f λ < µ, thus λ < D /,. Then we can obtan the probablty that the second queue s empty by applyng Lttle s theorem and s gven by Q = 0 =. 5 D /, After replacng 5 nto, we obtan that the servce rate for the frst queue n the frst domnant system s D / D /, µ = D / λ. 6 D /, The frst queue s stable f and only f λ < µ. The stablty regon R obtaned from the frst domnant system s gven n 7. λ D / D /, D /, D / + D /, D / D /, D / > < Fg. : The stablty regon for the two-user broadcast channel n the general case. B. Second Domnant System: the second queue transmts dummy packets In the second domnant system, when the second queue emptes then the source transmts a dummy packet for the D whle the frst queue behaves n the same way as n the orgnal system. In ths domnant system, the second queue never emptes, so the servce rate for the frst queue s µ = D /,. 9 The frst queue s stable f and only f λ < µ. The probablty that Q s empty s Q = 0 =. 0 D /, The servce rate of the second queue, after substtutng 0 nto s D / D /, µ = D / λ. D /, The stablty regon R obtaned from the second domnant system s gven n 8. The stablty regon of the system s gven by R = R R, where R and R are gven by 7 and 8 respectvely and s depcted by Fg.. An mportant observaton made n [] s that the stablty condtons obtaned by the stochastc domnance technque are not only suffcent but also necessary condtons for the stablty of the orgnal system. The ndstngushablty argument [] apples to our problem as well. Based on the constructon of the domnant system, t s easy to see that the queues of the domnant system are always larger n sze than those of the orgnal system, provded they are both ntalzed to the same value. Therefore, gven λ < µ, f for some λ, the queue at S s stable n the domnant system, then the correspondng queue n the orgnal system must be stable. Conversely, f for some λ n the domnant system, the queue at node S saturates, then t wll not transmt dummy packets, λ
R = λ λ, λ : + D / R = λ λ, λ : + D / D / D / D / D / D /, D /, λ <, λ < D /, D /, λ <, λ < D /, D /, 7 8 and as long as S has a packet to transmt, the behavor of the domnant system s dentcal to that of the orgnal system because dummy packet transmssons are elmnated as we approach the stablty boundary. Therefore, the orgnal and the domnant system are ndstngushable at the boundary ponts. The obtaned stablty regon for the two-user broadcast channel n the general case has the same resson wth the stablty regon of the two-user nterference channel obtaned n [3]. IV. TREATING INTERFERENCE AS NOISE In ths secton, we consder the case where the users treat the nterferng sgnal as nose. When the -th queue s empty at the source, whle the j-th queue s not, then the success probablty for the -th user s gven by 4. When both queues are non-empty then the transmtted sgnal at tmeslot t from the source to the recevers s denoted by x t = x t + x t. The receved sgnal y t by the user D s y t = ht xt + n t. The event D /,j denotes that user D s able to decode ts ntended packet. Ths s feasble when the receved SINR s above a threshold and s ressed by { } h D /,j = + j h. The success probablty of the second user can be obtaned smlarly to the frst. The transmsson from the source to D s successful when h + h h h. Note that + =. Thus, f < 0 then, the success probablty s zero because the ntal nequalty s not feasble. Thus, f > + then h. Assumng Raylegh block fadng, we have h and the success probablty can be ressed as [ ] D /, = h = [ ] = 0 x f h x dx. 3 Thus, 0 F h x D /, = f h x dx. 4 Note that f x = x and F h h x = x. To summarze, the success probablty for the second user when both queues at the source are non-empty s gven by 5, where { } s the ndcator functon. Thus f > and > then D /, = dα and D /, = dα after replacng n 7 and 8 we obtan the stablty regon R = R R. Smlarly we can obtan the regon for the other cases. V. SUCCESSIVE DECODING In ths secton we consder the case where the channel from the transmtter to D s better than that to D ;.e. h > h. When only one queue at the transmtter s non-empty, then the procedure of a successful transmsson s descrbed n Secton II. When both queues at the source are non-empty, the procedure of decodng a packet by a recever s as follows. We refer to the two packets used n a sngle superpostonbased transmsson as two levels layers. The packet ntended for the weaker recever.e. D s referred to as the frst level. We refer to the other level as the second level. A transmtter usng superposton codng splts the avalable transmsson power between the two level, selects the transmsson rate for each of the levels, then encodes and modulates each of the packets separately at the selected rate. The modulated symbols are scaled approprately to match the chosen power splt and summed to obtan the transmtted sgnal. More detals about mplementaton of superposton codng at the medum access layer can be found n [7]. At the recever sde, D treats the message of D as nose and decodes ts data from y. Recever D, whch has a better channel, performs successve decodng,.e. t decodes frst the message of D, then t subtracts t from the receved sgnal, and afterwards decodes ts message wth a sngle-user decoder. Note that n the broadcast channel wth superposton codng, the decodng order s dfferent from SIC n the nterference channel, n whch the sgnal wth the strongest channel s decoded frst. The successve decodng s feasble at the frst recever f
{ D /, = > } d α + = { > } d α 5 { } h + h, h. 6 D s able to decode ts ntended packet f and only f the receved sgnal-to-nterference-plus-nose rato SINR s greater than. The success probablty seen by the frst user, D when both queues are non-empty s gven by 7. The proof s omtted due to space lmtatons. The probablty that the lnk between the transmtter and D s not n outage when both queues are non-empty s gven by 5 whch s obtaned n the prevous secton. In the remander, we consder two smple schemes regardng the transmsson power for each recever s packets. The frst scheme s the case where we have fxed transmt power for the -th recever, such that + =. The second scheme comes naturally whenever a user s nactve,.e. has no packets to receve. We consder that the transmtter adapts the power consderng the queue state of each recever,.e. f the queue Q s empty, then all power s allocated to the j-th queue, j. A. Fxed ower Scheme We assume here that the transmtter assgns fxed power resp. at the D resp. D on every tmeslot. The case where > + : The servce rate seen by the frst queue s gven by. Snce constant transmttng power s used and D has better channel than D, from 7 we have that D /, = D /. Thus, we have µ = D /. 8 From Loyne s crteron for stablty [6], the frst queue s stable f and only f λ < µ. From Lttle s theorem Ch. 3. n [8], we have that λ Q > 0 =. 9 D / The servce rate for the second queue s gven by. After substtutng 9 nto we obtan D /, D / µ = D / + λ. 0 D / From Loyne s crteron we have that the second queue s stable f and only f λ < µ. The stablty regon for the degraded broadcast channel s gven by s depcted by Fg. 3. Recall that the success probablty D /, s gven by 5. Note that n ths case we do not face the problem of coupled queues as mentoned n the general case descrbed n Secton III. D / D /, D / Fg. 3: The stablty regon for fxed transmt powers when > + and D apples successve decodng and D treats nterference as nose. The case where < + : In ths case clearly D /, D /. We obtaned that D /, = d α. In ths case the queues are coupled so, we have to use the results obtaned n Secton III derved by the stochastc domnance technque. After replacng and 5 nto 7 and 8 we obtan the stablty regon. B. Varable ower Scheme based on Queue State In ths part, we consder a smple adaptve scheme regardng the power allocaton for each recever s packets. The power allocaton s performed as follows: when both queues are not empty, the transmt power for the frst and second queue s and, respectvely, satsfyng + =. However, when the queue of -th recever s empty, the total transmt power s used for transmttng the packets from for the j-th where j recever. The average servce rates of the frst and the second queue, µ, and µ are gven by and respectvely. The success probabltes D / for =, are gven by D / = d α, 5 snce when a queue s empty, the transmtter assgns all power to the other queue, and can be obtaned from 4. The success probablty D /, s gven by 7. In the above scheme, t s evdent that D / D /,, and as a result, there s couplng between the queues. Thus we can use drectly the stablty regon obtaned n Secton III by replacng the success probabltes.
{ } + D /, = < d α { } + + > d α R = λ λ, λ : + D / D / D / D /, D / λ <, λ < D / 7 R = λ λ, λ : + dα R = λ λ, λ : + dα dα dα dα d λ α d α <, λ < + 3 + dα dα d α + dα λ <, λ < d α 4 The stablty regon R has two parts, R and R where R = R R. If > +, then R s gven n 3 after replacng 5, 7 and 5 nto 7, smlarly R gven by 4. The stablty regon can be obtaned smlarly for the case < +. The ndstngushablty argument mentoned n Secton III apples to ths case as well. VI. CONCLUSIONS In ths work, we derved the stablty regon for the twouser broadcast channel. We consdered two decodng schemes at the recever sde, namely treatng nterference as nose by both recevers and successve decodng by the strong recever. For the latter, two smple power allocaton polces were studed, a fxed power allocaton and an adaptve power scheme based on the queues states. REFERENCES [] R. Rao and A. Ephremdes, On the stablty of nteractng queues n a mult-access system, IEEE Transactons on Informaton Theory, vol. 34, no. 5, pp. 98 930, Sep. 988. [] A. Ephremdes and B. Hajek, Informaton theory and communcaton networks: an unconsummated unon, IEEE Transactons on Informaton Theory, vol. 44, no. 6, pp. 46 434, Oct. 998. [3] T. Cover, An achevable rate regon for the broadcast channel, IEEE Transactons on Informaton Theory, vol., no. 4, pp. 399 404, July 975. [4] K. Marton, A codng theorem for the dscrete memoryless broadcast channel, IEEE Transactons on Informaton Theory, vol. 5, no. 3, pp. 306 3, May 979. [5] G. Fayolle, E. Gelenbe, and J. Labetoulle, Stablty and optmal control of the packet swtchng broadcast channel, J. ACM, vol. 4, no. 3, pp. 375 386, July 977. [6] A. Jafaran and S. Vshwanath, The two-user Gaussan fadng broadcast channel, n IEEE Internatonal Symposum on Informaton Theory oceedngs ISIT, July 0, pp. 964 968. [7] G. Care and S. Shama, On the achevable throughput of a multantenna Gaussan broadcast channel, IEEE Transactons on Informaton Theory, vol. 49, no. 7, pp. 69 706, July 003. [8] C. Zhou and G. Wunder, General stablty condtons n wreless broadcast channels, n 46th Annual Allerton Conference on Communcaton, Control, and Computng, Sep. 008, pp. 675 68. [9] V. Cadambe and S. Jafar, Dualty and stablty regons of multrate broadcast and multple access networks, n IEEE Internatonal Symposum on Informaton Theory ISIT, July 008, pp. 76 766. [0] Y. Sagduyu, L. Georgads, L. Tassulas, and A. Ephremdes, Capacty and stable throughput regons for the broadcast erasure channel wth feedback: An unusual unon, IEEE Transactons on Informaton Theory, vol. 59, no. 5, pp. 84 86, May 03. [] S. Vanka and M. Haengg, Analyss of the benefts of superposton codng n random wreless networks, n IEEE Internatonal Symposum on Informaton Theory oceedngs ISIT, June 00, pp. 708 7. [] S. Vanka, S. Srnvasa, Z. Gong,. Vz, K. Stamatou, and M. Haengg, Superposton codng strateges: Desgn and ermental evaluaton, IEEE Transactons on Wreless Communcatons, vol., no. 7, pp. 68 639, July 0. [3] N. appas, M. Kountours, and A. Ephremdes, The stablty regon of the two-user nterference channel, n IEEE Informaton Theory Workshop ITW, Sep. 03. [4] D. Tse and. Vswanath, Fundamentals of wreless communcaton. New York, NY, USA: Cambrdge Unversty ess, 005. [5] W. Szpankowsk, Stablty condtons for some dstrbuted systems: Buffered random access systems, Adv. n App. ob., vol. 6, no., pp. 498 55, June 994. [6] R. Loynes, The stablty of a queue wth non-ndependent nter-arrval and servce tmes, oc. Camb. hlos. Soc, vol. 58, no. 3, pp. 497 50, 96. [7] L. E. L, R. Alm, R. Ramjee, J. Sh, Y. Sun, H. Vswanathan, and Y. R. Yang, Superposton codng for wreless mesh networks, n 3th Annual ACM Internatonal Conference on Moble Computng and Networkng, ser. MobCom 07. ACM, 007, pp. 330 333. [8] D. Bertsekas and R. Gallager, Data networks nd ed.. Upper Saddle Rver, NJ, USA: entce-hall, Inc., 99.