Diversity-Multiplexing-Delay Tradeoffs in MIMO Multihop Networks with ARQ

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1 ISIT, Austin, Texas, U.S.A., June - 8, Divesity-Multiplexing-Delay Tadeoffs in MIMO Multihop Netwos with ARQ Yao Xie, Andea Goldsmith Depatment of Electical Engineeing, Stanfod Univesity, Stanfod, CA. yaoxie@stanfod.edu, andea@wsl.stanfod.edu Abstact The tadeoff between divesity, multiplexing, and delay in multihop MIMO elay netwos with ARQ is studied, whee the andom delay is caused by queueing and ARQ etansmission. This leads to an optimal ARQ allocation poblem with a pe-hop delay o end-to-end delay constaint. The optimal ARQ allocation has to tade off between the ARQ eo that the eceive fails to decode in the allocated maximum ARQ ounds and the pacet loss due to queueing delay. These two pobability of eos ae chaacteized using the divesity-multiplexing-delay tadeoff (DMDT (without queueing and the tail pobability of andom delay deived using lage deviation techniques, espectively. Then the optimal ARQ allocation poblem can be fomulated as a convex optimization poblem. We show that the optimal ARQ allocation should balance each lin pefomance as well as avoid significant queue delay, which is also demonstated by numeical examples. I. INTRODUCTION In a multihop elaying system, each teminal eceives the signal only fom the pevious teminal in the oute and, hence, the elays ae used fo coveage extension. Multiple inputmultiple output (MIMO systems can povide inceased data ates by ceating multiple paallel channels and inceasing divesity by obustness against channel vaiations. Anothe degee of feedom can be intoduced by an automatic epeat equest (ARQ potocol fo etansmissions. With the multihop ARQ potocol, the eceive at each hop feeds bac to the tansmitte one-bit indicating on whethe the message can be decoded o not. In case of a failue the tansmitte sends additional paity bits until eithe successful eception o message expiation. The ARQ potocol povides impoved eliability but also causes tansmission delay of pacets. Hee we study a multihop MIMO elay system using the ARQ potocol. Ou goal is to chaacteize the tadeoff in speed vesus eliability fo this system. The ate and eliability tadeoff fo the point-to-point MIMO system, captued by the divesity-multiplexing tadeoff (DMT, was intoduced in []. Consideing delay as the thid dimension in this asymptotic analysis with infinite SNR, the divesitymultiplexing-delay tadeoff (DMDT analysis fo a point-topoint MIMO system with ARQ is studied in [], and the DMDT cuve is shown to be the scaled vesion of the coesponding DMT cuve without ARQ. The DMDT in elay netwos has eceived a lot of attention as well (see, e.g., []. In ou ecent wo [], we extended the point-to-point DMDT analysis to multihop MIMO systems with ARQ and poposed an ARQ potocol that achieves the optimal DMDT. The DMDT analysis assumes asymptotically infinite SNR. Howeve, in the moe ealistic scenaio of finite SNR, etansmission is not a negligible event and hence the queueing delay has to be bought into the pictue (see discussions in []. With finite SNR and queueing delay, the DMDT will be diffeent fom that unde the infinite SNR assumption. The DMDT with queueing delay is studied in [] and an optimal ARQ adapted to the instantaneous queue state fo the point-to-point MIMO system is pesented theein. In this wo, we extend the study [] of optimal ARQ assuming high but finite SNR and queueing delay in point-topoint MIMO systems to multihop MIMO netwos. This wo is also an extention ou pevious esults in [] to incopoate queueing delay. We use the same metic as that used in [], which captues the pobability of eo caused by both ARQ eo, and the pacet loss due to queueing delay. The ARQ eo is chaacteized by infomation outage pobability, which can be found though a divesity-multiplexing-delay tadeoff analysis [], []. The pacet loss is given by the limiting pobability of the event that a pacet delay exceeds a deadline. Unlie the standad queuing models fo netwos (e.g., [], [7] whee only the numbe of messages awaiting tansmission is studied, hee we also need to study the amount of time a message has to wait in the queue of each node. Ou appoach is slightly diffeent fom [], whee the optimal ARQ decision is adapted pe pacet; we study the queues afte they ente the stable condition, and hence we use the stationay pobability of a pacet missing a deadline. An immediate tadeoff in the choice of ARQ ound is: the lage the numbe of ARQ attempts we use fo a lin, the highe the divesity and multiplexing gain we can achieve, meaning a lowe ARQ eo. Howeve, this is at a pice of moe pacets missing thei deadline. Ou goal is to find an optimal ARQ allocation that balances these two conflicting goals and equalizes pefomance of each hop to minimize the oveall pobability of eo. The emainde of this pape is oganized as follows. Section II intoduces system models and the ARQ potocol. Section III pesents ou fomulation and main esults. Numeical examples ae shown in Section IV. Finally Section V concludes the pape. II. MODELS AND BACKGROUND A. Channel and ARQ Potocol Models Conside a multihop MIMO netwo consisting of N nodes: with the souce coesponding to i, the destination coesponding to i N, andi,,n coesponding to the intemediate elays, as shown in Fig.. Each node is equipped with M i antennas. The pacets ente the netwo fom the souce node, and exit fom the destination node, foming an open queue. The netwo uses a multihop automatic epeat equest (ARQ potocol fo etansmission. With the //$. IEEE 8 ISIT

2 ISIT, Austin, Texas, U.S.A., June - 8, multihop ARQ potocol, in each hop, the eceive feeds bac to the tansmitte one-bit indicating about whethe the message can be decoded o not. In case of a failue the tansmitte etansmits. Each channel bloc fo the same message is called an ARQ ound. We conside the fixed ARQ allocation, whee each lin i has a maximum of ARQ ounds, i, N. The pacet is discaded once the maximum numbe of ounds has been eached. The total numbe of ARQ ounds is limited to L: N i L.ThisfixedARQ potocol has been studied in ou ecent pape []. time t time t+ time t time t time t+ time t+ + - Fig. : Left: full duplex multihop elay netwo. Right: half duplex elay multihop MIMO elay netwo. M ARQ H M Relay H H N- M N- Relay ARQ H N- M N Fig. : Uppe: elay netwo with diect lin fom souce to destination. Lowe: multihop MIMO elay netwo without diect lin. Assume the pacets ae delay sensitive: the end-to-end tansmission delay cannot exceed. One stategy to achieve this goal is to set a deadline i fo each lin i with N i i. Once a pacet delays moe than i it is emoved fom the queue. This pe-hop delay constaint coesponds to the finite buffe at each node. Anothe stategy is to allow lage pe-hop delay while imposing an end-to-end delay constaint. Othe assumptions we have made fo the channel models ae (i The channel between the ith and (i +th nodes is given by: SNR Y i,l H i,l X i,l + W i,l, l. ( M i The message is encoded by a space-time encode into a sequence of L matices {X i,l C Mi T, : l,,l}, whee T is the bloc length, and Y i,l C Mi+ T, i,,n, is the eceived signal at the (i +th node, in the lth ARQ ound. The ate of the space-time code is R. Channels ae assumed to be fequency non-selective, bloc Rayleigh fading and independent of each othe, i.e., the enties of the channel matices H i,l C Mi+ Mi ae independent and identically distibuted (i.i.d. complex Gaussian with zeo mean and unit vaiance. The additive noise tems W i,l ae also i.i.d. complex Gaussian with zeo mean and unit vaiance. The fowad lins and ARQ feedbac lins only exist between neighboing nodes. (ii We conside both the full-duplex and half-duplex elays (see, e.g., [] whee the elays can o cannot tansmit and eceive at the same time, espectively, as shown in Fig.. We assume the elays use a decode-and-fowad potocol (see, e.g., []. (iii We assume a shot-tem powe constaint at each node fo each bloc code. Hence we do not conside powe contol. (iv We conside both the long-tem static channel, whee H i,l H i fo all l, i.e. the channel state emains constant duing all the ARQ ounds, and independent fo diffeent i. Ou esults can be extended to the shot-tem static channel using the DMDT analysis given in []. Log of Cost Function... Fig. : The logaithm of the cost function (7 fo the (,, multihop MIMO elay netwos. SNR is db. B. Queueing Netwo Model We use an M/M/ queue tandem to model the multihop elay netwos. The pacets aive at the souce as a Poisson pocess with mean inte-aival time μ, (i.e., the time between the aival of the nth pacet and (n th pacet. The andom sevice time depends on the channel state and is uppe bounded by the maximum ARQ ounds allocated.asan appoximation we assume the andom sevice time at Node i fo each message is i.i.d. with exponential distibution and mean. With this assumption we can teat each node as an M/M/ queue. This appoximation maes the poblem tactable and chaacteizes the qualitative behavio of MIMO multihop elay netwo. Node i has a finite buffe size. The pacets ente into the buffe and ae fist-come-fist-seved (FCFS. Assume μ so that the queues ae stable, i.e., the waiting time at a node does not go to infinity as time goes on. Bue s theoem (see, e.g., [7] says that the pacets depat fom the souce and aive at each elay as a Poisson pocess with ate p i /μ, wheep i is the pobability that a pacet can each the ith node. With high SNR, the pacet eaching the subsequent elays is high pobability: p i (equivalently the pobability of a pacet dopping is small because it uses up the maximum ARQ ound. Hence all nodes have pacets aive as a Poisson pocess with mean inte-aival time μ. C. Thoughput Denote by b the size of the infomation messages in bits, B[t] the numbe of bits emoved fom the tansmission buffe at the souce at time slot t. Define a enewal event as the event that the tansmitted message leaves the souce and eventually is eceived by the destination node possibly afte one o moe ARQ etansmissions. We assume that unde full-duplex elays the tansmitte cannot send a new message until the pevious message has been decoded by the elay at which point the elay can begin tansmission ove the next hop (Fig. a. Unde half duplex elays we assume tansmitte cannot send L 9

3 ISIT, Austin, Texas, U.S.A., June - 8, L + L L 8 7 Fig. : Allocations of optimal ARQ: L, L, L + L,,ina(,, MIMO elay netwo. SNR is db. (The optimal. a new message until the elay to the next hop completes its tansmission (Fig. b. The numbe of bits B tansmitted in each enewal event, fo full-duplexing B (N b, and fo half-duplexing B (N b/ when N is odd, and B Nb/ when N is even. The long-tem aveage thoughput of the ARQ potocol is defined as the tansmitted bits pe channel use (PCU [], which can be found using enewal theoy [8]: s η lim inf s Ts t L 8 B[t] B E(τ. B (N T R, Full duplex; R, Half duplex, N is odd; R ( + ( N, Half duplex, N is even. whee τ is the aveage duation fom the time a pacet aives at the souce until it eaches the destination node, and. denotes asymptotic equality. A simila agument as in [] shows E(τ. (N T fo high SNR. D. Divesity-Multiplexing-Delay Tadeoff The pobability of eo P e in the tansmission has two souces: fom the ARQ eo: the pacet is dopped because the eceive fails to decode the message within the allocated numbe of ARQ ounds, denoted as P ARQ, and the pobability that a message misses its deadline at any node due to lage queueing delay, denoted as P Queue. We will give P e fo vaious ARQ elay netwos. Following the famewo of [], we assume the size of infomation messages b(ρ depends on the opeating signal-to-noise atio (SNR ρ, and a family of space time codes {C ρ } with bloc ate R(ρ b(ρ/t log ρ. We use the effective ARQ multiplexing gain and the ARQ divesity gain [] η(ρ e lim ρ log ρ, d lim ρ log P e (ρ. ( log ρ We cannot assume infinite SNR because othewise the queueing delay will be zeo, as pointed out in []. Howeve we assume high SNR to use the DMDT esults in ou subsequent analysis. III. DIVERSITY, MULTIPLEXING, AND DELAY TRADEOFF VIA OPTIMAL ARQ ROUND ALLOCATION A. Full-Duplex Relay in Multihop Relay Netwo Pe-Hop Delay Constaint: The pobability of eo depends on the ARQ window length allocation, deadline constaint i, multiplexing ate, andsnrρ. Foagiven and ρ, wehave P e ({ }, { i } ρ, N P ARQ (ρ, { }+ P Queue (D i > i. ( Hee D i denotes the andom delay at the ith lin when the queue is stationay. This P e expession is simila to that given by Equation ( of []. Ou goal is to allocate pe-hop ARQ ound { } and delay constaint { i } to minimize the pobability of eo P e. Fo the long-tem static channel, using the DMDT analysis esults [] wehave: N ( P ARQ (ρ, { } ρ fi. ( i Hee f i ( is the divesity-multiplexing tadeoff (DMT fo a point-to-point MIMO system fomed by nodes i and i +. Assuming sufficient long bloc lengths, f i ( is given by Theoem in [] quoted in the following: Theoem : [] Fo sufficiently long bloc lengths, the divesity-multiplexing tadeoff (DMT f( fo a MIMO system with M t tansmit and M eceive antennas is given by the piece-wise linea function connecting the points (, (M t (M, fo,, min(m t,m. Denote the amount of time spent in the ith node by the nth message as Dn. i The pobability of pacet loss P Queue (D i > i can be found as the limiting distibution of lim n P (Dn i > i (adapted fom Theoem 7.. of [8]: Lemma : The limiting distibution of the event that the delay at node i exceeds its deadline i,fom/m/ queue models, is given by: i i ( μ. ( P Queue (D i > i lim P n (Di n > i μ e i Hee the diffeence in the sevice ate and pacet aival ate Li μ and utility facto μ both indicate how busy node m is. Using the above esults, ( can be witten as N ( P e ({ }, { i } ρ, [ρ Li fi + L ( i ] μ e i L i μ. (7 Note that the queueing delay message loss eo pobability is deceasing in, and the ARQ eo pobability is inceasing in. Hence an optimal ARQ ounds allocation at each node should tade off these two tems. Also, the optimal ARQ allocation should also equalize the pefomance of each lin, as the weaest lin detemines the system pefomance []. Hence the optimal ARQ allocation can be fomulated as the following optimization poblem: min P e({ }, { i } ρ, {},{ i} A (8

4 ISIT, Austin, Texas, U.S.A., June - 8, whee A N i L, μ, i,,n N i i. The following lemma (poof omitted due to the space limit shows that the total tansmission distotion function ( is convex in the inteio of A. Lemma : The tansmission distotion function ( is convex jointly in and i in the convex set { { }, { i } : i > ( μ, i, N. }, Lemma says that except fo the cones of A the cost function is convex. Howeve these cones have highe pobability of eo: i and tae exteme values and hence one lin may have a longe queueing delay then the othes. So we only need to seach inteio of A whee the cost function is convex. To gain some insights into whee the optimal solution esides in the feasible domain fo the above poblem, we pesent a maginal cost intepetation. Note that the pobability of eo can be decomposed as a sum of pobability of eos on the ith lin. The optimal ARQ ounds allocated on this lin should equalize the maginal cost of the ARQ eo and the pacet loss due to queueing delay. Fo node i, with fixed i, the maginal costs (patial diffeentials of the ARQ eo pobability, and the pacet loss pobability due to queueing delay, with espect to ae given by ( ( L f i ρ Li fi ln ρ<, ( i and P Queue (D i > i (+ Li ( e i L i μ >. ( μ Note that f i <. The optimal solution equalizes these two maginal costs by choosing [,μ]. Note that these maginal cost functions ae monotone in, hence the equalizing L i exists and <L i <μif the following two conditions ae tue fo and μ: ( (i : P Queue(D i > i < ρ fi L, ( ( L (ii : P Queue(D i > i > ρ fi Lμ, ( Lμ ( ρ fi These conditions involve nonlinea inequalities involving μ, ρ,, M i and M i+, which defines the case when the optimal solution is in the inteio of A. Analyzing these conditions eveals that these conditions tend to be satisfied at lowe multiplexing gain, smallm i o M i+,small i,andlageμ (light taffic. Note that with high SNR condition (ii is always tue fo modeate values. When (i and (ii ae violated, which means one eo dominates the othe, then the optimal solution lies at the bounday of A. With the total ARQ ounds constaint in (8, using the Lagangian multiplie an agument (9 simila to above still holds. End-to-End Delay Constaint: When the buffe pe node is lage enough the pe hop delay constaint is not needed, and we can instead impose an end-to-end delay constaint. The exact expession fo the tail pobability of the end-to-end delay is intactable. Howeve a lage deviation esult is available. The following theoem can be deived using the main theoem in [9]: Theoem : Fo a stationay M/M/ queue tandem (with full-duplex elays: ( N lim lim n log P Queue Dn i θ, whee θ min N i { μ i }. This theoem says that the bottlenec of the queueing netwo is the lin with longest mean sevice time. Hence the optimal ARQ ound allocation poblem can be fomulated as: whee min P e({ }, { i } ρ, ( {} B P e ({ }, { i } ρ, ( N P ARQ (ρ, { }+P Queue Dn i, N. i ( ρ fi i + e θ. ( { N B i L } i L, ( μ, i,,n Fo high SNR, this can be shown to be a convex optimization poblem. A simple agument can show that the pacet loss pobability of the pe-hop delay constaint is lage than that using flexible end-to-end constaint. B. Half-duplex Relay in a Multihop Netwo Half-duplex elay is not a standad queue tandem model. Howeve we can also deive a lage deviation esult fo the tail pobability fo the end-to-end delay of a multihop netwo with half-duplex elays (poof in the Appendix: Theoem : Fo a stationay M/M/ queue tandem (with half-duplex elays, when the numbe of node N is lage: ( N lim lim n log P Queue Dn i θ. (7 i Fom this theoem we conclude that the optimal ARQ allocation poblem with the end-to-end constaint and half-duplex elays can be fomulated similaly to that with full-duplex elays (. IV. NUMERICAL EXAMPLES Conside a MIMO elay netwo consists of a souce, a elay, and a destination node. The elay is full-duplex. The numbe of antennas on each node is (M,M,M, M, M,andM, whee the elay has a single antenna. Othe paametes ae: ρ db,, L 8,andthe

5 ISIT, Austin, Texas, U.S.A., June - 8, multiplexing gain is. The base logaithm of the cost function (7 is shown in Fig.. We have optimized the cost function with espect to L and so we can display it in thee dimensions. Note that the suface is convex in the inteio of the feasible egion. The optimal L, L, ae shown in Fig.. Also note that as inceases to the maximum possible, the total numbe of ARQ ounds allocated L + L gadually inceases to the uppe bound L 8as inceases. V. CONCLUSIONS AND FUTURE WORK We have studied the divesity-multiplexing-delay tadeoff in multihop MIMO netwos by consideing an optimal ARQ allocation poblem to minimize the pobability of eo, which consists of the ARQ eo and the pacet loss due to queueing delay. Ou contibution is two-fold: we combine the DMDT analysis with queueing netwo theoy, and we use the tail pobability of andom delay to find the pobability of pacet loss due to queueing delay. Numeical esults show that optimal ARQ should equalize the pefomance of each lin and avoid long sevice times that cause lage queueing delay. Wo in pogess uses esults theein to investigate the joint soucechannel coding in multihop MIMO elay netwos, extending the esults of []. APPENDIX Poof of Theoem Fo node i, i N, let the andom vaiable Sn i denotes the sevice time equied by the nth custome at the ith node (the numbe of ARQs used fo the nth pacet, and A i n be the inte aival time of the nth pacets (i.e., the time between the aival of the nth and (n th pacages to this node. The waiting time of the nth pacet at the ith node Wn i satisfies Lindley s ecusion (see [9]: Wn i (W n i + Si+ n Ai n +, i N, (8 whee (x + max(x,. The total time a message spent in a node is its waiting time plus its own sevice time, hence D i n W i n + Si n. (9 The aival pocess to the (i+th node is the depatue pocess fom the ith node, which satisfies the ecusion: A i n Ai n + Dn i Dn i, i N. ( with A i n a Poisson pocess with ate /μ. Also the waiting time at the souce satisfies: W n (W n + S n + S n A n +. ( A well-nown esult is that (see, e.g. [9], if the aival and sevice pocesses satisfy the stability condition, then the Lindley s ecusion has the solution: Wn i max j (σi,n τ i n i+,n, i, N, Wn max(σj j,j + σj,j τj +,j. ( j whee the patial sum τl,p i p l Ai and σ l,p p l Si. Hence D i n max n (σi,n + S i n τ i +,n, i, N. ( Fom ( wehaveτl,p i τ i l,p + Di p D i l fo l p +, and othewise. Plug this into ( wehave Dn i max j (σi+ j + i,n Si n τ i j i+,n Di n + D i. ( i n Hence the ecusive elation if we move Dn i to the left-handside: D i n + Di n S i max n (σi+,n + Si n τ i +,n + Di. ( Now fom ( wehaved i max ji (σ (i,j + i τ i j (m +,j m. Plug this in the above ( wehave Dn i + Dn i max j (i j (σi+ j + i,n Si n + σi j (i,j + i Si τ i n (i +,n Do this inductively, we have N i D i n max j j N n [ N (σ i+ m ],j (i+ + Si + τj +,n. If we also add Dn W n + S n to the above equation, afte eaanging tems we have: N i N Dn i i (σ i j (i, + S i j (i+ τ j +,n +Sj + σj +,j S j + σj N N,j (N. ( Note that σ (i,s independent of i Si j (i+. Fo long queue we can ignoed the last fou tems caused by edge effect (the souce and end queue of the multihop elay netwo. By stationaity of the sevice pocess σ (i,j + i Si j (i+ has the same distibution as σ, i j (i.then( educes to the case studied in [9] and we can boow the lage deviation agument theein to deive the exponent θ. REFERENCES [] L. Zheng and D. N. C. Tse, Divesity and multiplexing: A fundamental tadeoff in multiple-antenna channels, IEEE Tans. Infom. Theoy, vol. 9, pp. 7 9,. [] H. El Gamal, G. Caie, and M. O. Damen, The MIMO ARQ channel: Divesity-multiplexing-delay tadeoff, IEEE Tans. Infom. Theoy, vol., pp.,. [] T. Tabet, S. Dusad, and R. Knopp, Divesity-multiplexing-delay tadeoff in half-duplex ARQ elay channels, IEEE Tansactions on Infomation Theoy, vol., pp , Octobe 7. [] Y. Xie, D. Gunduz, and A. Goldsmith, Multihop MIMO elay netwos with ARQ, IEEE Globecom 9 Communication Theoy Symposium, Dec. 9. [] T. Holliday, A. J. Goldsmith, and H. V. Poo, Joint souce and channel coding fo MIMO systems: Is it bette to be obust o quic?, IEEE Tansactions on Infomation Theoy, vol., no., 8. [] N. Bisni and A. Abouzeid, Queueing netwo models fo delay analysis of multihop wieless Ad Hoc netwos, pp , Poceedings of the Intenational Confeence on Wieless Communications and Mobile Computing, July. [7] G. Bolch, S. Geine, and H. de Mee, Queueing Netwos and Maov Chains : Modeling and Pefomance Evaluation With Compute Science Applications. Spinge Seies in Statistics, Wiley-Intescience, ed., Aug.. [8] S. M. Ross, Stochastic Pocesses. John Wiley & Sons, ed., 99. [9] A. J. Ganesh, Lage deviations of the sojoun time fo queues in seies, Annals of Opeations Reseach, vol. 79, pp., Jan. 998.