Optimal Operating Point for MIMO Multiple Access Channel with Bursty Traffic

Transcription

1 Optiml Operting Point for MIMO Multiple Access Chnnel with Bursty Trffic Somsk Kittipiykul nd Tr Jvidi Abstrct Multiple ntenns t the trnsmitters nd receivers in multiple ccess chnnel (MAC) cn provide simultneous diversity, sptil multiplexing, nd spce-division multiple ccess gins. The fundmentl trdeoff in the symptoticlly lrge SNR regime is shown by Tse et l (24). On the other hnd, MAC scheduling cn provide sttisticl-multiplexing gin to improve the dely performnce s shown by Bertsims et l (998) nd Stolyr et l (2). In this pper, we formulte nd nlyticlly derive bounds on the optiml operting point for MIMO-MAC chnnel for bursty sources with dely constrints. Our system model brings together the four types of gins: diversity, sptil multiplexing, spce-division multiple-ccess, nd sttisticl-multiplexing gins. Our objective is to minimize the end-to-end performnce s defined by the dely bound violtion probbility s well s the chnnel decoding error probbility. We find the optiml diversity gin nd rte region in which the system should operte. As n exmple, we illustrte our technique nd the optiml operting point for the cse of compound Poisson source. In ddition, we note n interesting interply between the intensity of the trffic nd resource pooling with regrd to both multiple-ccess nd sttisticl-multiplexing gins. Index Terms MIMO multiple ccess chnnel, diversitymultiplexing trdeoff, cross-lyer optimiztion, sttisticlmultiplexing. I. INTRODUCTION Multiple ntenns cn be used to enhnce the performnce of wireless systems. The multiple ntenns cn be used to simultneously boost the relibility (providing diversity gin) nd the dt rte (providing sptil multiplexing gin). In ddition, in multiple ccess scenrios where multiple users re trnsmitting to common receiver, multiple receive ntenns lso provide multiple-ccess gin by llowing for sptil seprtion of the signls of different users. Tse, Viswnth, nd Zheng [2] hve chrcterized the fundmentl trdeoff between these three types of gins t high SNR. Our gol in this pper is to nswer the question first posed by Hollidy nd Goldsmith in [5]: given the diversitymultiplexing region, where should one choose to operte?. Hollidy nd Goldsmith nswered this question in the context of the cross-lyer design of point-to-point system where source encoder is conctented with MIMO link. Their gol in this context ws to minimize the end-to-end distortion. In their lter ppers [6] nd [7], Hollidy et l generlized their formultion to include the distortion due to dely, where the dely is cused by rndom service time of the ARQ process. In this pper, we nswer the sme question in multi-user context. We consider cross-lyer queue-chnnel optimiztion problem for bursty nd dely-sensitive trffic sources. We consider system where ech user hs bursty source conctented with n infinite buffer nd MIMO multiple ccess chnnel (MIMO-MAC). The end-to-end performnce metric of interest is the totl bit loss probbility, where loss cn be due to either dely violtion or decoding errors in the MIMO-MAC chnnel. From user s perspective, we fce the following trdeoff: the higher the multiplexing gin the better the dely performnce, but the inevitble decrese in diversity results in n increse in MIMO chnnel errors. At the sme time, the sttisticl vrition in the trffic ptterns mong users provides us with flexibility in llocting the resources. The min contribution of this pper is the formultion of cross-lyer optiml operting point for MIMO-MAC chnnel with bursty sources nd dely constrints. In prticulr, we provide methodology for chrcterizing the optiml diversity gin nd rte region in which the system should operte in MIMO-MAC chnnel with given high SNR nd description of the bursty trffic sources. To chieve this, we ssume n optiml scheduler design which dynmiclly controls users trnsmission rtes (or equivlently, the multiplexing gins) s function of queue bcklogs. This dynmic dpttion of multiplexing gins ccounts for sttisticl-multiplexing while leverging the known trdeoff between diversity, sptil multiplexing, nd multiple-ccess gins given in [2]. From scheduling perspective, sttisticl-multiplexing is key mechnism by which the network resources re used to improve the dely performnce for bursty users. In prticulr, sttisticlmultiplexing cpitlizes on the fct tht peks in trffic of simultneously ongoing trffic strems rrely coincide. We believe tht our result cn be viewed s first step in integrting the known sptil diversity nd multiplexing nd multiple-ccess gins with tht of the sttisticl-multiplexing. In other words, for the first time, our model brings together the four types of gins offered t MIMO MAC. The reminder of the pper is orgnized s follows. In Section II, we provide detiled description of the system model s well s the problem formultion for generl number of users. In Section III, we provide the min nlyticl results nd bounds on the optiml chnnel diversity gin. We lso discuss the notion of sttisticl-multiplexing nd its benefits. In Section VI, we find the optiml operting diversity gin d or its bounds for prticulr clss of compound Poisson sources. Finlly, in Section V, we discuss the shortcomings of the present pper, possible extensions, nd future work. II. SYSTEM MODEL We consider the rchitecture shown in Figure. The system is time-slotted nd consists of three min components, ech shown with different number. The first component consists

2 2 Source Buffer r W t t Rte Scheduler Spce-Time Encoder rte r t 3 Tx MIMO-MAC (diversity d) Rx Joint-ML Decoder 2 Rx App. Bit loss (decoding error & dely violtion) Source K W K t r K t Spce-Time Encoder rte r K t Tx Rx App. K Fig.. System model nd the two cuses of bit loss: dely violtion nd chnnel decoding error. of K homogeneous users, ech of which hs n identicl but independent bursty source: ech source genertes informtion bits ccording to stochstic process. When pproprite, the bits re buffered prior to trnsmission over the chnnel. The second component of interest is MIMO multiple ccess chnnel without chnnel stte informtion (CSI) t the trnsmitters but with perfect CSI t the receiver. The receiver consists of joint mximum-likelihood decoder. In the bsence of CSI t the trnsmitters, we ssume tht the MIMO-MAC opertes t common diversity gin, which in turn specifies the corresponding cpcity region of the MIMO MAC chnnel s given in [2]. However, the individul rte of ech user is determined dynmiclly by the rte scheduler which is the third component in our system. This is centrlized rte scheduler tht dynmiclly determines the trnsmission rtes of the individul users given queue stte informtion (QSI) of ech user. In this pper we re interested in the following questions: given sttisticl description of the sources, lrge dely bound D, nd high SNR vlue of the chnnel, wht is the optiml design of the scheduler, nd wht is the optiml operting diversity gin of the MIMO MAC chnnel. The notion of optimlity needs to tke into ccount the chnnel decoding error s well s the dely bound violtion probbilities. We ssume no restrnsmission of the bits in error nd mp our objective to the sum of the probbility of dely violtion nd the probbility of chnnel error. In order to mthemticlly define this problem, we now model ech of the bove components precisely. A. Source Model We ssume tht the totl number of informtion bits generted by user i (i =,..., K) is given by sequence S i = St, i t =, 2,..., where St i is the totl number of bits of user i generted up to timeslot t nd S i. In ddition, we ssume tht the rrivl processes S i, i =,..., K, re identicl nd mutully independent. We lso ssume tht ech rrivl process S i hs sttionry increments nd stisfies Lrge Devitions Principle (LDP). In the ppendix, we discuss n dditionl smple pth LDP ssumption (Assumption B) on the rrivl processes. Here, to keep the flow of the pper, in this section we only provide the LDP ssumption nd the consequent chrcteriztion of the sources which is bsed on LDP. In generl, consider source process S generting sequence S t, t =, 2,... of rndom vribles, where S t is the totl number of bits generted up to timeslot t. Definition : A source S is sid to stisfy n LDP with decy function Λ : R [, ] if, for lrge enough t nd for smll ǫ >, Pr [ St t ] ( ǫ, + ǫ) e tλ () where Λ is lower semicontinuous function nd hs compct level sets 2 (see [3] nd [9] for more discussions in LDP). Fct : (Gärtner-Ellis theorem) Suppose source S stisfies the following: Assumption A: ) The limiting log-moment generting function Λ(θ) := lim t t log ] (2) E[eθSt exists for ll θ, where ± re llowed both s elements of the sequence nd s limit points. 2) The origin is in the interior of the domin D Λ := θ Λ(θ) < of Λ(θ). 3) Λ(θ) is differentible in the interior of D Λ nd the derivtive tends to infinity s θ pproches the boundry of D Λ. 4) Λ(θ) is lower semicontinuous, i.e. liminf θn θ Λ(θ n ) Λ(θ) for ll θ. Then the source S stisfies n LDP nd its decy function described by () is given s () Λ () = sup [θ Λ(θ)]. (3) θ Remrk : It cn be shown tht Λ is convex function tking vlues in [, ] such tht Λ (E[S ]) = where E[S ] is the verge rrivl rte of process S [9]. Remrk 2: Mny source models commonly used to model bursty trffic in communiction networks stisfy Assumptions A nd B. Such source models include renewl processes, Mrkov-modulted processes, nd more generlly sttionry processes with mild mixing conditions [3]. In lrge devitions literture, the Λ function is typiclly clled rte function. Here we use the nme decy function to void confusion with trnsmission rte. 2 The level set x : Λ (x) is compct for every rel

3 3 Remrk 3: From LDP, source is fully chrcterized by either Λ( ) or Λ ( ). An lterntive to the LDP chrcteriztion of source is the well-known effective bndwidth (see [8]) which ws used in our previous study for point-to-point scenrio [4]. r Multiplexing gin region R d= d=.5 d= d=2 d=3.8 B. MIMO-MAC Chnnel Model nd PHY Model We use the sme symmetric MIMO multiple ccess chnnel model s described in [2] which ssumes symmetric trnsmitters seeing i.i.d. fding chnnels, perfect symbol synchroniztion nd perfect CSI t the receiver but no CSI t ny trnsmitters. Ech trnsmitter hs M trnsmit ntenns, while the receiver hs N receive ntenns. Spce-time coding hppens over chnnel coherence time which is ssumed to contin T symbols 3. Since the trnsmitters re ssumed to know only the chnnel sttistics, including the verge received SNR, they lwys trnsmit t the mximum powers which re equl for ll trnsmitters. The chnnel fding processes of the trnsmitters re ssumed to be sttionry over time, mutully independent, nd identicl. For ech trnsmitter, the chnnel fding for different ntenn pths re ssumed to be slow block-fding with i.i.d. Ryleigh fding. We denote by ρ the verge received signl-to-noise rtio (SNR) t ech receive ntenn. From system perspective, t ech SNR level ρ, the PHY lyer for the MIMO-MAC chnnel provides trdeoff between the relibility of the trnsmissions nd the trnsmission rtes. Equivlently, we cn sy tht the PHY lyer provides trdeoff between common diversity d nd the multiplexing gin region, denoted by R, where d nd R re s defined in [] nd [2]. We stte these definitions below. Definition 2: (Definition in []) A code scheme C(ρ), which is fmily of codes (coding over one single coherence block) with one codebook for ech SNR level ρ nd provides dt rte R(ρ) nd verge error probbility P e (ρ), chieves multiplexing gin r nd diversity gin d if R(ρ) lim ρ logρ = r nd lim log P e (ρ) = d. (4) ρ log ρ For nottionl simplicity we shorten (4) s R(ρ). = r log ρ nd P e (ρ). = ρ d. We lso use. nd. if nd hold in the limit. 4 Definition 3: (Theorem 2 in [2]) Let r i t be the multiplexing gin of user i, i =,...,K, t time t. Given common diversity requirement d for ll users, i.e., P i e. SNR d, i =,...,K, (5) where Pe i is the verge error probbility for user i. Then the sptil multiplexing gins (rt,...,rk t ) t ny timeslot t must 3 We ssume either sufficiently lrge symbol rte or sufficiently smll number of ntenns such tht T KM + N. 4 Note tht we use the nturl log insted of log 2 nd hence use nts insted of bits. This is for convenience of the presenttion r Fig. 2. Exmple of the multiplexing gin region R for M = N = 2 cse. be within the (time-independent) multiplexing gin region R = (r,..., r K ) : s S r s r S M,N, S,...,K where rm,n for ny integers m nd n is the lrgest multiplexing gin chieved for n m n point-to-point MIMO link for given diversity d nd is defined s piecewise liner function joining the points ((m k)(n k), k) for k =,...,min(m, n). In this pper, we consider system which lwys opertes t common diversity gin d t ny time t. This d directly determines the multiplexing region R nd its shpe. In prticulr, d determines the sum of ll the rtes t ll time, which is independent of time. However, the individul rte t time t, rt, i i =,...,K, is determined dynmiclly by the rte scheduler discussed lter. In Figure 2, we illustrte the dependence of the shpe of R nd d for simple cse of K = 2 users nd M = N = 2. As seen in this figure, there exists diversity gin d (in this exmple, d = 2) such tht, for lrge d (d > d ), the shpe of R follows rectngulr shpe (single-user performnce regime), while, for smll d (d < d ), R is polymtroid shpe (ntenn-pooling regime). Furthermore, [2] shows tht d is the unique solution to t, i.e. K i= ri t r KM,N. (6) r KM,N (d ) = Kr M,N (d ). (7) Lter we will see the impct of this chnge of shpe on the working of the scheduler block. C. Rte Scheduler Given tht ech user opertes t fixed nd common diversity gin d nd given n verge SNR of ρ in the MIMO- MAC subsystem, the function of scheduler g d : R K + R is to llocte, t the beginning of every timeslot t, the set of fesible multiplexing gins to the users. This is done, equivlently, by selecting vector of sptil multiplexing gins (rt,..., rt K ) from the multiplexing gin region R. The decision is bsed on the dely of the hed-of-the-line bit in queue i, denoted by Wt i, i =,...,K, t the beginning of

4 4 timeslot t. Specificlly, we ssume tht (r t,..., r K t ) = g d (W t,..., W K t ). Without loss of optimlity, one cn ssume tht the rte scheduler lwys ssigns the highest possible sum rte. At timeslot t, n mount of rt i T log ρ bits re tken out from hed-of-the-line of the buffer of user i. We ssume tht if ny buffers do not hve enough dt to trnsmit, the null dt is used to fulfill the rtes. Remrk 4: Recll tht the shpe of the multiplexing gin R depends on d (e.g. see Figure 2). As result, the choice of diversity gin d determines the clss of fesible dynmic schedulers. In the single-user performnce regime (d d ), the users re decoupled nd independent from one nother, hence, reducing the scheduler to sttic (nd decoupled) choice of multiplexing gin rt i = r M,N for ll i =,...,K nd ll time t Z. For the ntenn-pooling regime (d < d ), on the other hnd, R is polymtroid. In other words, in this regime, the multiplexing gins of the users re dependent on one nother nd must be jointly llocted. Remrk 5: The model in this pper ssumes tht there is no CSI vilble t the trnsmitters nd the centrl scheduler. However, the scheduler hs perfect knowledge of the queue stte informtion (QSI). This is not unrelistic given the fct tht it is less bndwidth consuming nd more ccurte to send the QSI of ech buffer (n observble sclr number) to the centrlized scheduler thn to estimte CSI for MIMO chnnels (K mtrices, ech of dimension M N) t the receiver nd feed bck these mtrices to the trnsmitters. Remrk 6: Due to lck of CSI, the role of the rte scheduler in this pper is not to minimize the chnnel error performnce; insted, the scheduler improves the dely violtion probbility by tking dvntge of the sttisticl-multiplexing gin provided by the multiple bursty sources shring the multiple ccess chnnel. D. Arrivl Rte Scling nd Stbility Condition Since the rtes of trnsmission in the MIMO-MAC chnnel re scled s log ρ, we scle the rrivl rtes with log ρ s well. In other words, we ssume tht the verge bit rrivl rte λ of ech user is λ = λt log ρ (8) bits per timeslot for given constnt positive λ. In ddition, to gurntee system stbility, we require tht the totl verge rrivl rte to be no greter thn the (sum) cpcity of the MIMO-MAC chnnel [4]. In prticulr, we ssume tht or equivlently K λ < min(km, N)T log ρ, λ < min(m, N/K). (9) Since the system is stble, it reches stedy stte. We let W i nd L i denote the stedy-stte dely nd queue length, respectively, for queue i, i =,...,K. For the rest of the pper, we denote r v s the verge multiplexing gin t the common diversity d, defined s rm,n r v := if d d, K r KM,N if d < d (). nd denote C v := r v T log ρ (bits per timeslot) () s the verge per-queue chnnel cpcity t diversity d. E. Objective Our system objective is to find the optiml operting chnnel diversity gin d nd the corresponding multiplexing gin region R(d ) in which the system should operte. This diversity d minimizes the end-to-end totl bit loss probbility cused by two phenomens: ) dely violtion of the dely bound D nd 2) chnnel decoding error. 5 In prticulr, we define the following probbilities: Pe i := Pr[decoding error for user i] P e := Pr[decoding error for ny user] (2) Pq i := Pr[dely violtion of user i] = Pr[W i > D] P q := Pr[dely violtion for ny user] = Pr[ mx i=,...,k W i > D]. (3) With the bove definitions, the totl loss probbility P tot is expressed s P tot := mx Pr[bit loss for user i] i=,...,k = mx P i e + ( Pe i )P q i i=,...,k where Pe i+( P e i)p q i is the totl bit loss probbility of user i due to chnnel nd dely violtion. We will lter show tht Pe i.. = P e = ρ d nd Pq i.. = P q = ρ g where g is functionl tking positive vlues. Hence, the symptotic lrge-snr expression of P tot is given s P tot. = P e + P q. = ρ d + ρ g. (4) We note tht both probbilities P e nd P q re functions of the diversity gin d s well s the verge SNR ρ. However, there is trdeoff between the two probbilities s function of d: Intuitively, for fixed ρ, we expect tht high diversity gin, which trnsltes into smller trnsmission rte region, results in fster queue build-up nd lrger delys. On the other hnd, this higher diversity gin yields better chnnel performnce. In the reminder of the pper, we will derive nlyticlly lrge SNR pproximtions for P q nd P e nd show tht given fixed nd high ρ, P q is incresing on d while P e is decresing on d (confirming the bove intuition). Furthermore, we find the best PHY lyer operting point, 5 We ssume no retrnsmission for the lost bits due to chnnel decoding errors or dely violtion. Furthermore, the source processes re not effected by the lost bits.

5 5 i.e. diversity gin d, so s to minimize the totl bit loss probbility P tot in the high SNR regime. In other words, we will find d tht blnces the exponents of the two probbilities. III. PROBLEM ANALYSIS In this section, we nlyticlly derive the two loss probbilities, P e nd P q, for symptoticlly lrge SNR. As we will see, the two probbilities decy exponentilly with SNR. For the chnnel, the definition of diversity gin [2] gives direct symptotic pproximtion of P e for lrge SNR. Obtining the symptotic P q, however, requires more work. Depending on the vlue of d, we either directly compute the symptotic P q or provide lower nd upper bounds of the symptotic P q. A. Asymptotic P e The symptotic expression of P e for lrge SNR comes directly from the definition of diversity gin given in Definitions 2 nd 3. By the union bound nd the symmetry mong users, we hve the following bounds: P e P e KP e where Pe is the probbility of decoding error for user. Using Pe =. ρ d in Definition 2 nd the fct tht K is constnt independent of ρ, we hve B. Asymptotic P q P e. = ρ d. (5) Similrly, by the union bound nd the symmetry mong users, we hve the following bounds: Pr[W > D] P q K Pr[W > D], (6) where Pr[W > D] implicitly depends on d. Now, let us first focus on Pr[W > D]. To get n nlyticl expression for the symptotic Pr[W > D], we consider two cses depending on the vlue of d. Cse : Single-user performnce regime (d d < MN) As discussed in Section II-C, the multiplexing gin region R in this regime is squre nd the scheduler ssign decoupled rtes to the queues. Hence, the optiml scheduler simply ssigns fixed trnsmission rte of C v given in () to ech user (i.e. we cll this the symmetric sttic scheduler). Therefore, the symptotic pproximtion (when D is sufficiently lrge) of the dely violtion probbility is given s follows. Lemm : For d d MN nd sufficiently lrge D, the symptotic lrge-snr pproximtion of Pr[W > D] is such tht log Pr [ W > D ] lim = σ s DTr v (7) ρ log ρ where σ s is defined such tht Λ(σ s ) = σ s C v. (8) Equivlently, we cn write (7) s Pr[W > D]. = ρ σsdtrv. (9) The proof of this Lemm is given in Appendix II. Cse 2: Antenn-pooling regime ( < d < d ) In this cse, the multiplexing gin region R is polymtroid nd the trnsmission rtes of the users must be jointly llocted by scheduler. Since the optiml policy (with respect to the dely violtion probbility objective) is unknown, we provide the following lower bound nd upper bound to Pr[W > D]. ) Upper Bound on Pr[W > D]: An upper bound on Pr[W > D] is esily found since ny fesible scheduling policy cn provide n upper bound. In prticulr, to rrive t the upper bound P u, we consider the sme symmetric sttic scheduler s described in Cse : the symmetric sttic scheduler lwys ssigns the symmetric rte of C v to ech user t ll time. By Lemm, the symptotic pproximtion of P u for lrge D is given s P u. = ρ σsdtrv. (2) We note tht this upper bound becomes tighter s d increses to d since R pproches K-dimensionl hypercube. 2) Lower Bound on Pr[W > D]: The lower bound P l on Pr[W > D] is obtined from Fct 2, the construction of fictitious system, nd Fct 3, s follow. Fct 2: Consider two systems whose multiplexing gin regions re given by R nd R 2, respectively, where R R 2. The dely violtion probbility ssocited with the second system is no greter thn tht of the first system. For given d, consider fictitious system whose multiplexing gin region is given by K R fic := (r,...,r K ) : r i rkm,n. (2) i= Since R R fic, Fct 2 sttes tht the dely violtion probbility for this system is lower bound for Pr[W > D]. Stolyr nd Rmnn [4] hve shown tht the lrgest-delyfirst (LDF) policy chieves the minimum symptotic dely violtion probbility for this fictitious system. Fct 3: (Theorem 2.2 in [4]) Consider single-server queuing model with K users (illustrted in Figure 4(iii) for K = 2). For the sources considered in this pper, the lrgest-delyfirst (LDF) policy chieves the minimum dely violtion probbility Pr[mx i=,...,k W i > D] when D is lrge. 6 Hence, we compute the minimum symptotic dely violtion probbility for this fictitious system to rrive t lower bound, P l, for Pr[W > D], s given in the following Lemm: Lemm 2: For < d < d nd sufficiently lrge D, the symptotic lrge-snr pproximtion of P l is given by P l. = ρ KσsDTrv. (22) 6 See more detils of this fct in Fct 4. The result in [4] is much more generl thn this. It works with ny weighted delys, i.e. Pr[mx i=,...,k W i /α i > D], where α i is the weight for user i.

6 6 R r 3 r* 3M,N nd, in prticulr, when d d < MN, P q. = ρ σsdtrv (24) R fic nd, when < d d, ρ KσsDTrv. P q. ρ σsdtrv. (25) r* 3M,N r < d < d 2 r* 3M,N Fig. 3. The MIMO-MAC multiplexing gin region R nd the multiplexing gin region of the fictitious system R fic for K = 3 users. r 2 In summry, so fr, we hve seen tht P q nd P e exponentilly decy with ρ. The rte of decy of P e is known. When d > d, the rte of decy of P q is known vi (24). The rte of decy of P q when d < d is, however, unknown but is bounded s in (25). Next, with P q nd P e t hnd, we proceed with the minimiztion of the totl bit loss probbility. + b/2 + b/2 b 2+b (i) (ii) (iii) Fig. 4. Queuing models of the upper nd lower bounds of P[W > D] for the cse of K = 2 users nd the ntenn-pooling regime (r2m,n 2rM,N ). nd b re defined such tht + b = r M,N nd 2 + b = r2m,n. The proof is given in Appendix II. We lso note tht P l becomes tighter bound s d. Remrk 7: Compring the exponents in (2) nd (22), we see tht the LDF scheduler improves the exponent of the dely violtion probbility by K times of tht of the symmetric sttic scheduler. Tlking in the lnguge of chnnel diversity, the LDF scheduler improves the diversity gin by K folds by tking dvntge of sttisticl-multiplexing of the sources. However, we wnt to emphsize tht the lower bound in P l derived from the fictitious system with the multiplexing gin region R fic becomes more loose s the number of users K grows. This is expected becuse the ctul multiplexing gin region R in (6) is polymtroid (see n exmple of K = 3 users in Figure 3) while tht of the fictitious system is just the K-dimensionl simplex given by the constrint K i= r i rkm,n. Thus, the lower-bound becomes more optimistic s the number of users increses. Remrk 8: For n exmple of K = 2 users, Figure 4 summrizes the two bounds with the queuing models in mind. The upper bound P l is the til probbility of system (i) which lwys serves ech queue with multiplexing gin r2m,n /2. The lower bound P l is the til probbility of system (iii) which ssigns the single server of multiplexing gin r2m,n bsed on LDF scheduling. System (ii) is the queuing model given by the multiplexing gin region R. Now, using the bove two cses nd the bounds in (6), we rrive t n symptotic chrcteriztion of P q s follows P q. = Pr[W > D], (23) C. Minimizing Asymptotic Totl Loss Probbility From the symptotic expressions of P e given in (5) nd P q in (24) nd (25), the symptotic chrcteriztion of the totl loss probbility P tot. = P q + P e is immedite: For d d MN, For < d < d, P tot. = ρ σsdtrv + ρ d. (26) ρ KσsDTrv + ρ d. P tot. ρ σsdtrv + ρ d. (27) Since the term σ s DTr v is decresing in d while the term d is incresing on d, the minimum of P tot in (26) or its bounds in (27) hppen when the vlue of d mkes the exponents of the two terms re within o() of ech other (note tht if the exponents were not in the sme order, one term would dominte in the sum s ρ ). We now introduce n lgorithm which gurntees such choices of d: Algorithm : ) Solve for d which is solution of σ s DTr v = d. (28) If d d, then d = d u = d l = d nd stop. Otherwise, set d l = d. Go to Step 2. 2) Solve for d which is solution of nd set d u = min(d, d ). Kσ s DTr v = d (29) Theorem : Algorithm results in closed intervl [d l, d u ] in which the optiml common diversity gin d lies. The proof of this theorem is given in Appendix II. Notice tht the optiml diversity d nd its bounds depend on the sttisticl chrcteristics of the symmetric sources (Λ, µ, λ), the prmeters of the MIMO-MAC chnnel (e.g. T, M, N), nd the dely bound D.

7 7 D. Sttisticl-Multiplexing nd Optiml Diversity Gin From the bove nlysis, we obtin the following criticl observtion. Given dely constrint, the sttisticl property of the source hs significnt impct on the level of diversity well-designed system cn enjoy. In other words, the optiml scheduler which sttisticlly multiplexes the MIMO resources llows the combined bursty sources to perceive s smller ggregte trffic nd hence higher degree of diversity. Rigorously, this performnce improvement cn be ttributed to sttisticl-multiplexing gin s follows: Definition 4: An optiml dynmic scheduler with the totl bit loss probbility Ptot provides sttisticl-multiplexing gin of s over the sttic rte scheduler with Ptot, f where s := lim ρ log Ptot log P f tot log ρ From this definition nd the fct tht P f tot. (3). = ρ d l, the following proposition is immedite. Lemm 3: Consider the system model in Section II. The optiml sttisticl-multiplexing gin s is given by s = d d l. (3) Furthermore, it is bounded bove by d u d l. Remrk 9: The two concepts of sttisticl-multiplexing gin nd multi-user diversity gin re relted conceptully. The former tkes dvntge of the troughs (due to burstiness) of the trffic of different users while the ltter tkes dvntge of the peks (due to fdings) of the chnnels of different users. But their impcts on the design re sufficiently different, s multi-user diversity gin requires chnnel CSI t the trnsmitters while sttisticl-multiplexing requires QSI. E. Resource Pooling nd Sttisticl-Multiplexing Here we discuss the effect of the rrivl rte λ nd the verge dely bound D to the performnce region of the MIMO-MAC. The reltionship between (λ, D) nd the system performnce is summrized in Figure 5. The system performnce is divided into three min regions: the singleuser performnce region, the ntenn-pooling with significnt sttisticl-multiplexing region, nd the ntenn-pooling with insignificnt sttisticl-multiplexing region. In the single-user performnce region, the chieved optiml diversity gin d is equivlent to the cse when only one user is in the system, i.e. the cse d d. Specificlly, this cse hppens when λ is sufficiently smll nd D is sufficiently lrge. We denote this region s A. d A := (λ, D) : λ r, D σ s (d )Tr where r = r M,N (d ). 7 Since in this region the trnsmission rte of ech user is independent, there is no resource shring nd hence no sttisticl-multiplexing gin. On the other hnd, the significnce of sttisticlmultiplexing gin outside A is impcted by the rte of rrivls λ s well s the verge dely bound D. In prticulr, for 7 d Note tht is n incresing function on λ since σ σ s(d )Tr s(d ) which is the dely violtion exponent is itself decresing on the rrivl rte λ. Fig. 5. D Single User Performnce A Antenn Pooling, significnt stt. mux. Antenn Pooling, insignificnt stt. mux. min(m,n/k) λ The reltion between (λ, D) to the system performnce. (λ, D) in the neighborhood of A, the sttisticl-multiplexing gin is not significnt s the queues still behve in roughly independent mnner. Similrly, s λ increses to n overlod sitution or the dely bound D becomes very tight, the benefits of juggling resources diminishes. In contrst, for medium vlues of λ nd D nd under the optiml dynmic scheduler, ech queue perceives the whole (pooled) resource to itself, compred to /K of the resource s in cse of symmetric sttic scheduler. To illustrte the pproch shown in this pper nd the corresponding clcultion, we look t simple exmple of compound Poisson source with K = 2 users in the next section. IV. EXAMPLE: COMPOUND POISSON SOURCES AND K = 2 In this section, we illustrte the proposed pproch vi n exmple. We consider two independent but identicl source processes. For ech source i, rrivls re independent cross timeslots. The number of bits tht rrive in timeslot t, A i t, is n ggregtion of rndom number of pckets whose sizes re lso rndom, i.e. A i t = N n= Y n. Furthermore, we ssume tht the number of pckets t ech slot, N, is n independent Poisson rndom vrible with rte ν pckets per timeslot, while the length of the pckets, Y i, i =, 2,..., re i.i.d. rndom vribles with exponentil distribution of men /µ. The verge bit rrivl rte λ for ech source is equl to ν/µ nd scles with log ρ s in (8), i.e. λ = ν/µ = λt log ρ. (32) Proposition : For the compound Poisson source with exponentil pcket length, the σ s defined in Lemms is given s σ s = µ( λ ). (33) r v This proposition is proved in Appendix II. Note tht the rtio of the per-queue verge bit rrivl rte λ over the verge service rte, r v, cn be clled the trffic lod per queue. It is importnt to note tht the dely violtion exponent in (33) is decresing function of the verge pcket size /µ, for fixed pcket rrivl rte ν. A lrger pcket size in effectively cretes more burstiness in the rrivls, hence higher dely violtion probbility.

8 8 Exponents of P q nd P e d * l Upperbound on exponent of P q Diversity Gin d d * u Liner estimte of exponent of P q Lowerbound on exponent of P q Exponents of P q nd P e d * l d * u Diversity Gin d d * d* l.5.5 Exponents of P q nd P e () λ =.5, D = Diversity Gin d d * l = d * u = d* (c) λ =.5, D = 5 Exponents of P q nd P e (b) λ =, D = Diversity Gin d λ =, D = 5 Fig. 6. Plots of the exponents of P q nd P e for two different verge rrivl rtes (λ =.5 nd ) nd for two different dely bounds D = 2 nd 5. For d < d = 7.3, the upper nd lower bounds of the exponent of P q re shown. In ddition, we drw simple liner estimte (dotted line) of the exponent of P q between the two bounds. With Proposition in hnd, we re now redy to use Algorithm to obtin d (or its bounds). Figure 6 shows the optiml d nd its bounds when M = N = 4, the verge pcket size /µ is nts, nd the symbol rte such tht there re T = 2M + N = symbols per timeslot. In these figures, we plot the exponents of P q or its bounds, i.e. σ s DTr v nd 2σ s DTr v, nd the exponent of P e, i.e. d. To better illustrte the procedure followed by Algorithm, we plot the exponents of P q nd P e seprtely. Note tht when d d (d = 7.3 in this exmple), we only hve lower nd upper bounds for the exponents of P q. In this cse, in ddition to the bounds, we plot liner pproximtion (dotted line) to emphsize the tightness of the lower bound round d nd the upper bound round. The optiml diversity gin d or its bounds (d l nd d u) re shown in ech plot s the crossing of the exponents. As we discussed in Section III-E, the sttisticl-multiplexing gin s defined in (3) depends on the optiml choice of d which itself is function of the rrivl rte λ nd the verge dely bound D. Depending on λ nd D, we my or my not hve sttisticl-multiplexing gin. For exmple, Figure 6(c) shows tht, in the cse of sufficiently low rrivl rtes nd lrge dely bound, the optiml Ptot hppens when the users operte in the single-user performnce region. Hence, in this cse, there is no sttisticl-multiplexing gin to be chieved by dynmic scheduler. Here, the dominnt form of loss occurs on the chnnels. On the other hnd, Figure 6(b) corresponds to the cse of lrge rrivl rte nd smll dely bound, where the loss probbility due to dely violtion domintes tht of the chnnel. In this cse, the optiml diversity d necessittes resource shring in form of ntenn pooling. As result, the impct of n optiml dynmic scheduler, in form of sttisticlmultiplexing gin, becomes more significnt. Figure 7 illustrtes the performnce region discussed in Section III-E. In prticulr, the figure gives chrcteriztion d * l d * u 4 D D () d d l.5 λ vs. (λ, D).5.5 λ (b) Contour Plot d d l Fig. 7. 3D nd contour plots chrcterizing the sttisticl-multiplexing gin, pproximted by d d l, v.s. dely bound D nd rrivl rte λ. of the region shown in Figure 5 for the compound Poisson cse. We pproximte the sttisticl-multiplexing gin s = d d l chieved with n optiml dynmic scheduler with tht of simple liner pproximtion d d l, where d is the optiml diversity gin derived from the dotted line in Figure 6..5 V. SUMMARY AND FUTURE WORK In this pper, we considered system of bursty nd delysensitive symmetric sources conctented with symmetric MIMO-MAC chnnel. We ssumed no CSI informtion vilble to the trnsmitters nd block fding model with block coding whose block lengths re mtched to the coherence time of the chnnel. Furthermore, we ssumed fixed nd equl high trnsmission power t ech trnsmitter, i.e. high SNR regime. We ddressed the optiml choice of the sptil diversity gin d such tht it minimizes n endto-end loss performnce where loss cn occur due to dely violtion s well s chnnel decoding error. We showed how n optiml choice of diversity gin d depends on queuebsed scheduler module whose job is to sttisticlly multiplex the resources of the MIMO-MAC. In doing so, we integrted the notion of sttisticl-multiplexing gin with those of sptil diversity, multiplexing, nd multi-ccess gins provided by the MIMO-MAC. To strengthen the result presented in this pper, we will need to nlyze the performnce of the optiml dynmic scheduler, rther thn working with bounds. As emphsized in the introduction to this pper, we view the result of

9 9 this pper s first step to fully integrte the notions of scheduling nd sttisticl-multiplexing with other spects of MIMO technology. As such, strengthening the current result is not our most criticl concern. Some future works tht cn extend the utiliztion of the system resources in other spects re s follows: Time diversity: In the present work, we ssume coding whose block length is mtched to the coherence time of the chnnel. This is rther limiting ssumption, given the dely bound in the order of multiple coherence times we considered in this pper. We hope to extend our optimiztion of sptil diversity to the time-diversity, in form of coding over multiple coherence times (see [5]) or hybrid ARQ (see [3], [6], nd [7]). Controlling SNR: In the current formultion, the verge SNR is fixed cross ll users nd time. In the bsence of CSI t the trnsmitters nd of dely constrints (D = ), this fixed SNR ssumption incurs no loss in optimlity. But for smll nd medium D, it is nturl to expect tht n improvement in performnce is possible when the trnsmit powers for users re functions of the queue sttes. Dynmic control of PHY lyer operting point: In this pper, we ssumed tht the sole responsibility of the dynmic scheduler is to control the trnsmission rtes or multiplexing gins while the diversity gin nd the sum of the rtes of ll users re kept constnt. It is cler tht llowing for dynmic control of the diversity gin, s proposed by [7], will only improve the sttisticlmultiplexing gin nd the performnce of the system. This cn nturlly be extended to time-diversity, nd is roughly relted to dynmic control of the trdeoff mong vrious forms of diversity gin. In this context, our work cn be viewed s providing lower bound on the performnce of n optimlly designed system. ACKNOWLEDGMENT This reserch is supported in prt by the Center for Wireless Communictions, UCSD nd UC Discovery Grnt No. Com4-76, ARO-MURI grnt No. W9NF , NSF CAREER Awrd No. CNS-53335, nd AFOSR Grnt No. FA REFERENCES [] L. Zheng nd D. Tse, Diversity-multiplexing: fundmentl trdeoff in multiple-ntenn chnnels, IEEE Trns. Info. Theory, v. 49, no. 5, My 23. [2] D. Tse, P. Viswnth, nd L. Zheng, Diversity-multiplexing trdeoff in multiple-ccess chnnels, IEEE Trns. Info. Theory, v. 5, no. 9, Sept 24. [3] D. Bertsims, I. C. Pschlidis, nd J. N. Tsitsiklis, Asymptotic buffer overflow probbilities in multiclss multiplexers: n optiml control pproch, IEEE Trns. Automtic Control, v. 43, no. 3, Mrch 998. [4] A. L. Stolyr nd K. Rmnn, Lrgest weighted dely first scheduling: Lrge devitions nd optimlity. Ann. Applied Probbility, v., pp. -48, 2. [5] T. Hollidy nd A. Goldsmith, Joint source nd chnnel coding for MIMO systems, Proc. Allerton Conf., 24. [6] T. Hollidy nd A. Goldsmith, Optimizing End-to-End Distortion in MIMO Systems, IEEE ISIT 5, 25. [7] T. Hollidy, A. Goldsmith, nd H. V. Poor, The Impct of Dely on the Diversity Multiplexing ARQ Trdeoff, ICC, 26. [8] F. Kelly, Notes on effective bndwidths, Stochstic Networks: Theory nd Applictions, vol. 4, 996. [9] A. Dembo nd O. Zeitouni, Lrge Devition Techniques nd Applictions, Boston, MA: Jones nd Brtlett, 992. [] A. Dembo nd T. Zjic, Lrge devitions: from empiricl men nd mesure to prtil sum process, Stochstic Processes nd their Apps., v. 57, 995. [] A. Gnesh, N. O Connel, nd D. Wischik, Big Queues, Springer-Verlg, Berlin, 24. [2] J. Wlrnd nd P. Vriy, High-Performnce Communiction Networks, 2nd ed., Morgn Kufmnn Publishers, 2. [3] H. El Gml, G. Cire, nd M. O. Dmen, The MIMO ARQ chnnel: diversity-multiplexing-dely trdeoff, IEEE Trns. Info. Theory, v. 52, Aug 26. [4] S. Kittipiykul nd T. Jvidi, Optiml operting point in MIMO chnnel for dely-sensitive nd bursty trffic, IEEE ISIT 6, July 26. [5] P. Eli, S. Kittipiykul, nd T. Jvidi, On the Responsiveness-Diversity- Multiplexing trdeoff, WiOpt, 27. APPENDIX I ADDITIONAL ASSUMPTION ON SOURCE MODEL Here we give the dditionl ssumption B on the source we consider in this pper. Assumption B is required in the proof of Lemm 2. Assumption B: Smple pth LDP (see [3], [9] nd []) For n rrivl sequence S, S 2,..., for ll m N, for every ǫ, ǫ 2 >, nd for every sclr,..., m, there exists M > such tht for ll n M nd ll k,..., k m with = k k k m = n, m exp nǫ 2 (k i+ k i )Λ ( i ) i= Pr [ Ski+ S ki (k i+ k i ) i nǫ, i =,..., m ] m exp nǫ 2 (k i+ k i )Λ ( i ) i= APPENDIX II PROOFS OF LEMMAS AND PROPOSITION (34) Proof of Lemm : Proof: Since the symmetric sttic scheduler lwys ssigns the service rte C v to ech queue, we hve tht Pr[W > D] = Pr[L > DC v ]. (35) To be more specific, the sttement holds becuse ny bits delyed more thn D timeslots see t lest DC v bits before them, nd ny bits delyed less thn D timeslots must see less thn DC v bits before them. This is vlid becuse of the first-come-first-serve discipline ssumption. Hence, the two events W > D nd L > DC v re equivlent nd hve the sme probbility. Now, since Pr[L > DC v ] is equl to the til probbility for buffer which is served t fixed cpcity of C v nd whose rrivl process is described by Λ( ) nd stisfying LDP, one cn clculte the til probbility for single queue system with fixed service rte c s (see [8], [] nd [2]) lim B B log Pr[L > B] = θ

10 where θ is the lrgest positive root of eqution Λ(θ) θ = c. By replcing B with DC v nd c with C v nd using (), we hve log Pr[L > DC v ] lim = σ s DTr v ρ log ρ where σ s is given s the solution to Λ(σ s ) = σ s C v. Proof of Lemm 2: Before showing the proof of Lemm 2, we recll the following result on the symptotic til probbility of the mximl weighted dely under the longest-weighted-delyfirst (LWDF) scheduling discipline from [4] nd simplify the result to our specific ssumptions of symmetric users with LDF scheduling discipline. Fct 4: (Theorem 2.2 in [4]) Consider single server of fixed service rte nd K mutully independent source processes with sttionry increments. The totl number of informtion bits generted by source i (i =,...,K) is given by sequence Ŝi t, t =, 2,... where Ŝi t is the cumultive totl number of work rrived until time t from source i. We ssume Ŝi t, t =, 2,... stisfies LDP nd smple pth LDP (Assumptions A nd B) with the convex decy function ˆΛ i nd the convex log moment generting function ˆΛ i. Assume KE[Ŝ ] < for stbility. Let α i be the weight for user i (ssuming < α α 2 α K ). Consider the longestweighted-dely-first (LWDF) scheduling discipline, which lwys ssign the server to the longest witing (i.e. hed-ofthe-line)) customer of the source i which hs the mximl weighted dely. Then, the LWDF scheduling discipline mximizes the exponentil decy rte of the sttionry distribution of the mximl dely, mong ll cusl nd work-conserving scheduling disciplines. Furthermore, the probbility is given s lim sup n log Pr n [ n mx i,...,k ŵi > ] J (36) where nd ŵ i is the sttionry dely for user i, i =...,K, nd J is given s: subject to nd J = min j;x,...,x j γ j ( α i γ)ˆλ i (x i) (37) i= j,...,k,x i >, j x i > (38) i= j i= < γ = x i α j j+ i= α (39) ix i α j with α K+. Since in this pper we consider symmetric users nd LDF scheduling which is the LWDF discipline with equl weights, the following corollry gives specific expression of J which will be used to show Lemm 2. Corollry : Under the ssumptions of symmetric users with equl weights, i.e. ˆΛ i = ˆΛ, ˆΛ i = ˆΛ, nd α i = for ll i =,...,K, J in Fct 4 is reduced to J = sup θ : ˆΛ(θ) θ/k. (4) Proof: Under the ssumption of equl weights (i.e. α i = for ll i =,...,K), there re fesible vlues of γ in (39) only when j = K. Hence, the minimiztion in (37) is reduced to subject to nd J = γ min x,...,x K γ K ˆΛ (x i ) (4) i= K x i >, x i >, i =,...,K (42) i= K i= = < γ = x i α K K+ i= x. (43) i However, we notice tht condition (43) is stisfied with ny choices of x i stisfying condition (42). Hence, plugging the expression of γ into (4), we get J = min x,...,x K K i= x i K ˆΛ (x i ) (44) subject to (42). We cn simplify J further by using the convexity property of ˆΛ, i.e. K K i= i= ˆΛ (x i ) ˆΛ ( K i= x i K ) =: ˆΛ ( K + K ), where we let be such tht K = K i= x i > by the condition in (42). The equlity holds when x i = + /K for ll i =,..., K. Hence, we cn rewrite (44) nd its conditions concisely s ) J = min > ˆΛ ( + K. (45) To finish the proof, we expnd ˆΛ using its definition, s follows: J = min > ˆΛ ( + /K) = min > sup θ( + /K) ˆΛ(θ) θ R = sup min θ R > = sup θ R = sup, θ + θ/k ˆΛ(θ) if θ/k < ˆΛ(θ), θ, if θ/k ˆΛ(θ) θ : ˆΛ(θ) θ/k, where the third equlity holds becuse the function θ + is convex on nd concve on θ (since ˆΛ is convex). θ/k ˆΛ(θ) Proof: (Lemm 2) Consider scled version of the system in Fct 4 where the service rte is scled to C (which

11 is equl to KC v ) nd the rrivls re lso scled up by C. We cn think of this scling s chnge of mesurement units. We denote Ws i s the sttionry dely of rrivls of user i, i =,...,K, for this scled single-server system with LDF scheduling. Since scling of the service nd rrivls do not chnge the distribution of the delys, we hve from Fct 4 tht lim sup n [ ] n log P n mx W s i > J. (46) i,...,k Noticing tht the log moment generting function Λ of the scled system is given s Λ(θ) = lim t t log E[eθCŜ t ] = ˆΛ(θC), (47) we hve, by using Corollry, = sup θ : ˆΛ(θ) θ/k J = sup θ : Λ(θ/C) θ/k = C sup θ : Λ( θ) θc/k = θcv = Cσ s (48) where σ s > is defined s the unique solution to Λ(σ s ) = σ s C v. The second equlity in (48) follows by using (47); the third equility follows by letting θ = θ/c; nd the lst equlity by using tht fct tht Λ is strictly convex nd Λ () = λt log ρ < C v (the stbility condition in (9) nd the fct tht ˆΛ () is the verge rrivl rte per source [8]) nd hence the supremum is ttined with θ = σ s. Replcing n with D nd J = Cσ s = KC v σ s = Kσ s Tr v log ρ in (46), we hve [ ] P mx W s i.= > D ρ Kσ sdtr v, i,...,k for lrge vlue of D. From symmetry, on the other hnd, we hve [ ] P[Ws > D] P mx W s i > D KP[Ws > D]. i,...,k This provides the ssertion of the lemm: P l := P[W s > D]. = ρ KσsDTrv. Proof: The limiting log moment generting function Λ( ) for compound Poisson source with exponentil pcket length is derived in [4], which is Λ(θ) = lim n n log E[ exp(θst )] [ ] n = lim n n log E exp(θ A t) t= ] = log E [ exp θa = νθ µ θ if θ < µ, if θ µ. (49) From Lemm, σ s is the solution to Λ(σ s ) = σ s C v. From (49), this reduces to finding σ s such tht νσ s µ σ s = σ s C v µλt log ρ µ σ s = r v T log ρ where we hve replced ν nd C v from (32) nd (). Hence, σ s = µ( λ/r v ). Proof of Theorem : Proof: We first show the existence of solution d in (28) of Algorithm. The LHS term is decresing on d nd equl to for d d, for some d such tht the rrivl rte λt log ρ is equl to the service rte C v ( d) (in which cse, σ s ( d) = ). On nother hnd, the RHS term is incresing on d nd is equl to when d =. Hence, (28) must hold for some d (, d). Next, if d solving (28) is less thn d, this d is the lower bound d l (i.e. symptoticlly mximizing the RHS term in (27)) nd the upper bound d u is obtined from mximizing the LHS in (27). The existence of d solving (29) cn be shown similrly. Proof of Proposition I: