Throughput Performance of Quantized Proportional Fair Scheduling with Adaptive Modulation and Coding

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1 Throughput Performnce of Quntized Proportion Fir Scheduing with Adptive Modution nd Coding Fumio Ishizki nd Gng Uk Hwng Deprtment of Systems Design nd Engineering, Nnzn University, Seto, Jpn Deprtment of Mthemtic Sciences nd Teecommuniction Engineering Progrm Kore Advnced Institute of Science nd Technoogy (KAIST, Deeon, South Kore Abstrct Assuming ogrithmic rte mode for SISO (Singe-Input Singe-Output systems, the informtion theoretic cpcity under scheduing gorithms expoiting mutiuser diversity in wireess networks hs been studied so fr. However, in mny cses, the throughput chieved in re wireess networks is very different from (nd is much ess thn the informtion theoretic cpcity obtined under the ssumption of the ogrithmic rte mode, nd the rte functions in re wireess networks re very different from the ogrithmic rte mode. Hence, to exmine the usefuness of the scheduing gorithms expoiting mutiuser diversity in re wireess networks, we shoud study the throughput performnce under more reistic rte function mode rther thn the informtion theoretic cpcity under the ogrithmic rte mode. In this pper, we consider wireess network where (Quntized Proportion Fir scheduing nd AMC (dptive modution nd coding scheme re empoyed. Assuming reistic rte function of AMC, we then nyze the throughput performnce under the scheduing with the AMC scheme. We so provide numeric resuts to investigte the usefuness of the scheduing with the AMC scheme. I. INTRODUCTION Since the utiiztion of mutiuser diversity 8] in wireess networks cn increse the informtion theoretic cpcity, much ttention hs been pid to scheduing gorithms expoiting mutiuser diversity (see, e.g., ], 2], 3], 4], 6], 7], ], ] nd references therein. Mutiuser diversity is diversity existing between the wireess chnne sttes of different users, nd this diversity comes from the fct tht the wireess chnne stte processes of different users re usuy independent for the sme shred medium. To chieve the efficient use of the bndwidth, the mutiuser diversity cn be expoited, for instnce, in such wy tht the scheduer in the BS (Bse Sttion seects the MS (Mobie Sttion whose received SNR (Sign-to-Noise Rtio is the best, nd trnsmits pckets to the seected MS. This scheduing scheme mximizes the informtion theoretic cpcity of the over system, but it is highy unfir when n MS hs very disprte chnne condition ]. To sove this unfir probem, the proportion fir ( scheduing ws proposed where the scheduer considers the normized SNR vues of MSs, defined by the received SNR vues divided by the corresponding verge received SNR vues, nd seects the MS whose normized SNR vue is the rgest. The /9/$2. 29 IEEE scheduing provides strict firness mong MSs becuse the normized SNR vues re i.i.d. (independent nd identicy distributed mong MSs ]. Here, the strict firness mens tht the ccess probbiities of MSs to the wireess chnne re equ. In prctice, probby the normized SNR vues re quntized, the quntized normized SNR vues re reported to the BS by MSs, nd the scheduing is performed bsed on the quntized normized SNR vues. This scheduing is ced the (Quntized scheduing. In gener, the informtion theoretic cpcity chieved by the scheduing increses nd pproches to tht chieved by the scheduing with the increse of the number of its quntiztion eves. On the other hnd, from view of reducing the mount of feedbck informtion from MSs to the BS, sm number of quntiztion eves is desirbe. It is reported tht with firy sm number of quntiztion eves, the scheduing cn chieve retivey good cpcity, if the quntiztion threshods re ppropritey determined 2], 4]. Assuming ogrithmic rte mode for SISO (Singe-Input Singe-Output systems, the informtion theoretic cpcity under scheduing gorithms expoiting mutiuser diversity hs been studied so fr. The informtion theoretic cpcity is the highest rte in bits per chnne use t which informtion cn be sent with rbitrriy ow probbiity of error. In mny cses, the throughput chieved in re wireess networks is very different from (nd is much ess thn the informtion theoretic cpcity obtined under the ssumption of the ogrithmic rte mode. For instnce, mny re wireess networks empoy dptive modution nd coding (AMC scheme where mutipe modution nd coding modes cn be used nd one of them is seected depending on the received SNR. The rte function of AMC is very different from the ogrithmic rte mode. Hence we shoud study the throughput performnce under more reistic rte function mode rther thn the informtion theoretic cpcity under the ogrithmic rte mode in order to exmine the usefuness of scheduing in re wireess networks. In this pper, we consider wireess network consisting of BS nd MSs where the BS empoys the scheduing nd n AMC scheme. We focus on the downink trnsmission nd nyze the throughput performnce t the downink. We

2 then obtin the expression of the normized throughput under the scheduing with AMC scheme. For comprison, we further obtin the expression of the normized throughput under the scheduing with AMC scheme nd tht under the round-robin ( scheduing with AMC scheme. We so provide numeric resuts to investigte the usefuness of the scheduing with AMC scheme. The reminder of this pper is orgnized s foows. In Section II, we describe system mode for the wireess network considered in this pper. Section III presents the nysis of the normized throughputs. Numeric resuts re provided in Section IV. Concusions re drwn in Section V. II. SYSTEM MODEL In this pper, we consider wireess network consisting of BS nd K MSs. We suppose tht the BS empoys pcket scheduer nd n AMC scheme for downink trnsmission from the BS to the MSs. We focus on the downink trnsmission nd nyze the throughput performnce t the downink. We ssume tht the downink chnne of MS i (i,...,k is described by ft Ryeigh fding chnne mode. Time xis is divided into physic (PHY frmes of equ size T f (sec nd time index is given by t,, 2,. The PHY frme durtion T f is considered to be unit time in our mode. Then, the received SNR process {z (i (t} (t,,... of MS i (i,...,k is sttionry stochstic process nd z (i (t for ny t is ccording to the foowing exponenti distribution: P{z (i (t x} exp( x/ z (i, ( where z (i denotes the verge received SNR of MS i nd is defined by z (i Ez (i (t]. We ssume tht the received SNR processes of the K MSs re independent with ech other. We suppose tht the BS empoys the scheduer s the pcket scheduer. Under the scheduing, the normized SNR processes of MSs re considered, where the normized SNR process mens the process {z (i (t/ z (i }. To reduce the mount of feedbck informtion from MSs to the BS, ech MS quntizes or prtitions the entire normized SNR rnge into L grdes with boundry points denoted by {γ } L with γ, γ <γ + (,...,L nd γ L. The normized SNR process of MS i is in the th (,...,L grde t t if γ z (i (t/ z (i <γ +. The scheduing then works s foows: Ony the MSs with the normized SNR vues rger thn γ re owed to feed bck informtion to the BS. The MSs with the normized SNR vues rger thn γ report their SNR grdes to the BS. The scheduer considers MSs whose SNR grdes re highest s cndidtes for trnsmission. If more thn two MSs hve the highest grde, the scheduer rndomy seects one of them for trnsmission. The scheduing is performed PHY frme-by-frme. We suppose tht the AMC scheme studied in this pper prtitions the entire SNR rnge into M trnsmission modes with boundry points denoted by {ξ m } M+ m with ξ, ξ m < ξ m+ (m,...,m nd ξ M+. Ifξ m z (i (t < ξ m+, the AMC controer of MS i uses the trnsmission mode m of the AMC t time t. We ssume tht no pcket is trnsmitted when the mode is. At the PHY yer, trnsmissions re performed PHY frmeby-frme. We here ssume tht the chnne condition is sowy vrying nd remins invrint per PHY frme. We further ssume tht ech MS reports the AMC mode corresponding to its current SNR vue to the BS every PHY frme. So the BS knows which AMC mode shoud be used for ech MS. As consequence, the trnsmission mode in the AMC scheme is dusted on PHY frme-by-frme bsis. When trnsmission mode m is used, d m MAC frmes in the queue of the MAC yer re mpped into PHY frme nd trnsmitted simutneousy in the corresponding PHY frme. We ssume tht d <d < <d M. An exmpe set of {d m } M+ m is found in 9]. For the service process for MAC frmes in the queue t the MAC yer, we ssume the foowing: If MAC frme is received incorrecty t the receiver fter error detection, this informtion is immeditey fed bck to the trnsmitter of the BS nd the trnsmitter retrnsmits the MAC frme in the next PHY frme. On the other hnd, if MAC frme is received correcty t the receiver fter error detection, this informtion is immeditey fed bck to the trnsmitter of the BS nd the trnsmitter removes the MAC frme from the queue. For convenience, MAC frme is referred to s pcket from now on. To mode the pcket service process t the MAC yer, we first consider the pcket error process t the PHY yer in our mode. From the ssumptions nd the settings we mde so fr, the pcket error rte (PER t the PHY yer is expressed s function of the trnsmission mode seected by the AMC controer. Let PER n (γ denote the PER when the mode n is used nd the received SNR is equ to γ. For the AMC modes in Tbe I, when the pcket ength is 8 bits, Liu et. 9] showed tht PER n (γ cn be pproximted s { ( <γ<γpn, PER n (γ (2 n exp( g n γ (γ γ pn, where n, g n, nd γ pn re the mode-dependent prmeters nd re given in Tbe I. In prctice, we hve γ n >γ pn. Tbe I shows n exmpe set of { n }, {g n }, nd {γ pn } for n AMC scheme with 7 trnsmission modes 9]. III. ANALYSIS OF THE NORMALIZED THROUGHPUT In this section, we nyze the normized throughput under the scheduing with AMC. The normized throughput is defined s the verge number of pckets which cn be successfuy sent to n MS during unit time T f. Without oss of generity, we herefter focus on the normized throughput of MS i. First, for,...,l nd i,...,k, et α be α P{z (i (t/ z (i <γ }.

3 TABLE I THE AMC SCHEME WITH 7 MODES 9] Mode Modution rte n g n γ pn BPSK QPSK QAM QAM QAM QAM QAM rte bits/symbo In other words, α denotes the probbiity tht the grde of the normized SNR process is worse thn the th grde. Note here tht α is independent of i, becuse z (i (t/ z (i hs the sme distribution for i,...,k. Since we ssumed the ft Ryeigh fding mode, the distribution is n exponenti distribution nd thus α is given by α exp( γ. Then, the probbiity tht MS i is seected for trnsmission by the scheduer given tht the normized SNR process of MS i is in the th (,...,L grde is expressed s 2] k k + ( K k (α + α k α k αk + αk K(α + α. The normized throughput C (i of MS i under the scheduing with AMC scheme is thus expressed s L C (i α K + αk γ+ f( z (i xe x dx, (3 K(α + α γ where for ξ m x<ξ m+ nd m,...,m, f(x is defined by f(x d m m exp( g m x] with nd g. Note tht f(x for ξ m x<ξ m+ is the condition normized throughput (or the condition rte function when AMC mode m is used. Aso note tht this rte function of AMC is very different from the ogrithmic rte function og 2 ( + x ssumed in mny studies on the informtion theoretic cpcity of SISO systems. In wht foows, to compute C (i, we focus on the integr prt γ + γ f( z (i xe x dx incuded in (3. We begin with the introduction of some nottions. For n,...,l, we define M (i n by M (i n {m z(i γ n <ξ m < z (i γ n+,m,...,m }. (4 For n,...,l, we define m n by m (i n sup{m ξ m z (i γ n,m,...,m }. For nottion convenience, insted of m (i n, we herefter use m n, though m (i n depends on the superscript (i. To compute the integr prt γ + γ f( z (i xe x dx incuded in (3, we seprtey consider the two cses: ( M (i nd (b M (i. We first consider the cse ( M (i. In this cse, we obtin γ+ γ+ f( z (i xe x dx d m (e x m e η (i x dx γ γ d m e γ e γ + where η (i m η (i (e η(i γ e η(i γ + (,...,L ; i,...,k is defined by η (i +g m z (i. ], (5 We next consider the cse (b M (i.for,...,l nd i,...,k, we define ζ (i by In this cse, we obtin γ+ ζ (i γ f( z (i xe x dx + ξ z (i. (i m + γ (i (i ζ + min M (i γ+ + f( z (i xe x dx ζ (i f( z (i xe x dx ζ (i m + f( z (i xe x dx. (6 We further seprtey consider the three terms incuded in the right hnd side of (6. The first term cn be rewritten to (i m + γ f( z (i xe x dx d m (i m + γ e x m e η d m e γ e ζ(i m + m η (i (e η(i The second term cn be rewritten to (i (i ζ + min M (i ζ (i f( z (i xe x dx (i γ e η(i ζ (i m + (i (i ζ + min M (i ζ (i d x dx ]. (7 (e x e (+g z(i x dx (i min M (i d e ζ(i e ζ(i + z {e (+g +g z (i (i ζ (i e (+g z(i ζ (i + } ]. (8

4 The third term cn be rewritten to γ+ ζ (i m + f( z (i xe x dx d m+ γ+ ζ (i m + e x m+ e η d m+ e ζ(i m + e γ + (i m + (e η(i + ζ(i m + e η (i η (i + γ + + x dx ]. (9 Combining (3 with (5 (9, we cn compute the normized throughput C (i under scheduing with AMC scheme. For comprison, we here obtin the normized throughput C (i under round-robin ( scheduing with AMC scheme. It is given by C (i K K M f( z (i xe x dx (i + ζ (i d (e x e (+g z(i x dx M d e ζ(i e ζ(i + K +g z (i (e (+g z(i ζ (i e (+g z(i ζ (i + ]. ( Aso, for comprison, we obtin the normized throughput C (i under the (not quntized scheduing with AMC scheme. It is given by C (i M f( z (i xe x ( e x dx (i ζ + d ζ (i ( e x dx M ( K d k k (e (k+ζ(i k + ( k ++g z (i (e x e (+g z(i x ( k e (k+ζ(i + e (k++g z(i ζ (i e (k++g z(i ζ (i + ].( Here, the foowing identity with constnts, b nd h is used in the st eqution of (. b e hx ( e x dx b k ( K e hx k k ( K ( k k k + ( k e kx dx ( e (k+h e (k+hb. Normized throughput Normized throughput Number of quntiztion eves Fig.. Normized throughput for z (i 5dB, K Number of quntiztion eves Fig. 2. Normized throughput for z (i 2dB, K 2 IV. NUMERICAL RESULTS In this section, we investigte the normized throughputs under the scheduing gorithm with AMC nd compre them with those under the or scheduing gorithms with AMC. In the numeric resuts, we suppose tht the AMC scheme with 7 modes shown in Tbe I is empoyed nd the boundry points {ξ m } 8 m of trnsmission modes for the AMC scheme re determined by the method presented in 5]. First, for the scheduing, we observe the effect of the number of quntiztion eves, i.e., L on the normized throughput performnce. Fig. 3 dispy the normized throughput C (i under the scheduing with AMC s function of the number of its quntiztion eves. For comprison, Fig. 3 so show the normized throughput C (i under the scheduing with AMC nd the normized throughput under the scheduing with AMC. In Fig. 3, the C (i verge SNRs nd the number of MSs re set s z (i 5dB, 2dB, 2dB nd K 3, 2, 3, respectivey. Note here tht the normized throughput C (i is func-

5 Normized throughput..5 Normized throughput.5..5 (2dB,2 (2dB,3 (3dB,2 (3dB, Number of quntiztion eves Fig. 3. Normized throughput for z (i 2dB, K Averge SNR (db Fig. 4. Normized throughput for K 3 tion of the quntiztion threshods {γ } nd it is difficut to determine the optim quntiztion threshods (excuding the cse of L 2. The reson of the difficuty is tht the normized throughput C (i is not continuous function of {γ } due to the discontinuity in the AMC. This requires us to sove muti-dimension optimiztion probems of discontinuous obective functions under the constrint <γ <γ 2 < <γ L <. Aso note tht the optim quntiztion threshods {γ } depend on both the number K of MSs nd the verge SNR z (i. In this study, we use the Neder-Med Simpex gorithm with tri nd error for the initi vues of quntiztion threshods, nd find quntiztion threshods which mximize the normized throughput C (i. The resuting mxim normized throughput C (i is potted in Fig. 3. We observe the foowing in Fig. 3. The normized throughput C (i under the scheduing with AMC increses with the number L of quntiztion eves. We see tht compred to the scheduing with AMC, the scheduing with AMC cn enhnce the throughput performnce. We so see tht with retivey sm number of quntiztion eves, sy, L 3, the scheduing with AMC cn chieve most sme throughput performnce tht cn be chieved by the scheduing with AMC. Even when the number L of quntiztion eves is equ to two, the scheduing with AMC yieds retivey good throughput performnce. From view of reducing the mount of feedbck informtion from MSs to the BS, smer number of quntiztion eves is desirbe. We thus concude tht the scheduing with sm number of quntiztion eve, sy, L 2, 3, is quite usefu in prctice, if the quntiztion threshods {γ } re ppropritey set. Next, for the scheduing, we observe the effect of the quntiztion threshods {γ } on throughput performnce. For the simpicity of discussion, we imit ourseves to the cse of L 2. We here consider the foowing scheduing with AMC: (zdb, k where the quntiztion threshod γ is Normized throughput (2dB,2 (2dB,3 (3dB,2 (3dB, Number of MSs Fig. 5. Normized throughput for z (i 2dB optimized for the condition tht the verge SNR z (i nd the number K of MSs re equ to zdb nd k, respectivey. Fig. 4 dispys the normized throughputs C (i for (2dB,2, (2dB,3, (3dB,2 nd (3dB,3 s function of the verge SNR z (i. The number K of MSs re set to 3 in Fig. 4. For comprison, Fig. 4 so shows the normized throughputs C (i nd C(i. In Fig. 4, we observe tht the normized throughputs C (i under the scheduing increse in discontinuous fshion with the increse of the verge SNR z (i. This occurs due to the combintion of the discontinuity in the AMC nd the quntiztion of the received SNR. More specificy, when (not normized quntiztion threshod z (i γ n (n,...,l moves from one AMC mode to nother AMC mode with the chnge of the verge SNR z (i, the discontinuous chnge of C (i occurs. We so observe tht C (i grety depends on the vue of the quntiztion threshod γ. If the ctu verge SNR z (i is quite different from the verge SNR where the qun-

6 tiztion threshod is optimized, the throughput performnce is considerby degrded. In prticur, if the ctu verge SNR z (i is somewht smer thn the verge SNR where the quntiztion threshod is optimized, the enhncement of the throughput performnce by using the scheduing is very imited. Fig. 5 dispys the normized throughputs C (i for (2dB,2, (2dB,3, (3dB,2 nd (3dB,3 s function of the number K of MSs. The verge SNR z (i is set to 2dB. For comprison, Fig. 5 so shows the normized throughputs C (i nd C (i. In Fig. 5, we see tht the number of MSs where the quntiztion threshod is optimized so ffects the throughput performnce chieved by the scheduing. The normized throughput chieved by (2dB,2 is greter thn tht chieved by (2dB,3 when the number of MSs is ess thn 24, whie the former is smer thn the tter when the number of MSs is greter thn or equ to 24. However, the difference is not so significnt in the rnge from K 5to K 35. Aso, the normized throughput chieved by (3dB,2 is most equ to the normized throughput chieved by (3dB,3 (these two ines re not distinguishbe in Fig. 5. They re smer thn both the normized throughput chieved by (2dB,2 nd tht chieved by (2dB,3 in the whoe rnge from K 5 to K 35. Thus, to utiize the potenti biity of the scheduing, we my need to more crefuy seect the verge SNR where the quntiztion threshod is optimized thn the number of MSs where the quntiztion threshod is optimized. REFERENCES ] R. Ferrús, L. Aonso, A. Umbert, X. Revés, J. Pérez-Romero, nd F. Csdev, Cross-yer scheduing strtegy for UMTS downink enhncement, IEEE Rdio Communictions, vo.2, no.2, pp.24 28, 25. 2] F. Forén, O. Edfors nd B.-A. Moin, The effect of feedbck quntiztion on the throughput of mutiuser diversity scheme, Proc. of IEEE GLOBECOM 23, pp.497-5, 23. 3] D. Gesbert nd M.-S. Aouini, How much feedbck is muti-user diversity rey worth?, Proc. of IEEE ICC 4, pp , 24. 4] G. U. Hwng nd F. Ishizki, Design of fir scheduer expoiting mutiuser diversity with feedbck informtion reduction, IEEE Communictions Letters, Vo.2, No.2, pp.24 26, 28. 5] F. Ishizki nd G. U. Hwng, Cross-yer design nd nysis of wireess networks using the effective bndwidth function, IEEE Trns. on Wireess Communictions, Vo.6, No.9, pp , 27. 6] F. Ishizki nd G. U. Hwng, Queuing dey nysis for pcket scheduers with/without mutiuser diversity over fding chnne, IEEE Trns. on Vehicur Technoogy, Vo.56, No.5, pp , 27. 7] H. Kim nd Y. Hn, An opportunistic chnne quity feedbck scheme for proportion fir scheduing, IEEE Communictions Letters, Vo., No.6, pp.5 53, 27. 8] R. Knopp nd P. A. Humbet, Informtion cpcity nd power contro in singe-ce mutiuser communictions, Proc. of IEEE ICC 95, pp , ] Q. Liu, S. Zhou, nd G. B. Ginnkis, Cross-yer combining of dptive modution nd coding with truncted ARQ over wireess inks, IEEE Trns. on Wireess Communictions, Vo.3, No.5, pp , 24. ] X. Qin nd R. Berry, Expoiting mutiuser diversity for medium ccess contro in wireess networks, Proc. of IEEE INFOCOM 3, pp.84 94, 23. ] L. Yng, M. Kng, nd M.-S. Aouini, On the cpcity-firness trdeoff in mutiuser diversity systems, IEEE Trns. on Vehicur Technoogy, Vo.56, No.4, pp.9 97, 27. V. CONCLUSIONS In this pper, we consider wireess network consisting of BS nd MSs where the BS empoys the scheduing nd n AMC scheme. We then nyze the throughput performnce under the scheduing with AMC t the downink. Numeric resuts show tht the scheduing with AMC cn enhnce the throughput performnce, compred to the scheduing with AMC. In the numeric resuts, we see tht the scheduing with sm number of quntiztion eve, sy, L 2, 3, is quite usefu in prctice, if the quntiztion threshods {γ } re ppropritey set. However, if the ctu verge SNR z (i is somewht smer thn the verge SNR where the quntiztion threshods re optimized, the enhncement of the throughput performnce by using the scheduing is very imited. Therefore we shoud notice tht to utiize the potenti biity of the scheduing, we need to crefuy determine the quntiztion threshods by supposing the ctu verge SNR. ACKNOWLEDGMENT This reserch ws supported by the MKE (Ministry of Knowedge Economy, Kore, under the ITRC (Informtion Technoogy Reserch Center support progrm supervised by the IITA (Institute for Informtion Technoogy Advncement (IITA-28- C9-8-3.