Adaptive QoS for Mobile Multimedia Applications With Power ...

the DMZ x M matrix R is R(W) = ($:, . . . , $h), where eachDM x M submatrix $i is:and the asterisk indicates complex conjugate. And finally, the DM xDM2 central matrix M is:/ fl 0 ...I 0 fzM(F,E,A) = . . ,where each 1 x M vector, fi is:I : : .. .: I(9)B. SINR Fine-tuningThe previous section describes a fast method to get a reasonable rthat lets us obtain the power vector and the antenna weights; once wehave those, we can activate the new users. **With** the users now functioning,and we may devote our computing resources to improving andfine-tuning the SINR of those new users, and other existing users.Now, we do not have time constraints, and we can use more computationallyexpensive methods to get better results. We wish to increaseas many 7;'s as we can, and by as much as we can, but we must notdrive p(rF) to I. The tool that we use to make this choice is therate of change of p(rF) as we change each 7,. Let's first assume thatthese derivatives exist, and we have sorted them from the smallest tothe largest.If we have convergence with the present r **for** our algorithm, and wewish to increase some 7; while imposing the smallest possible effecton p(rF), we use the sorted derivatives to choose the smallest one.Specifically, we seekaPides = arg min -i=1, ... M a7,The prohibitively large dimensions of these matrices should not be adiscouragement, since these matrices are dealt with during this derivationonly. Working with these large matrices is not tedious either, sincemost elements are zero.Now we use these matrices to answer our convergence question:thenllrFll1 < 1. We can state that if llLll1 . llRll1 < h,p(rF) 5 IlrFlll IlLlll . IlMlll ' IlRlll < 1.(1 1)We can write llMll1 and llRll1 in terms of {w,}~EI. There**for**e,the problem reduces to finding a set of vectors, {wi}i~~ such thatw";, = 1,i = 1, ... M, andNote that we have no control over IIMII1. Except **for** the 7,. the righthand side must be estimated from the system. We wish to know if thereis any set of vectors that complies with these constraints. If there are,we have convergence, and we can continue with our algorithms. Wemust, there**for**e, find a set of vectors that minimize the left hand side of'the inequality.Theorem I: The collection of vectors, {wi};~~, that minimizes theleft hand expression of (12), with wFai; = 1,i = 1, ... M iswhere ai,maz = ai;(k) **for** the k that maximizes the argument. oThere**for**e, problem (4) has an element by element positive solution.as in (6). ifIt should be repeated that if the inequality does not hold, we may 01may not have convergence, it is not possible to know simply from that.inequality.Equation (14) tells us the fast way to determine if the system withthe new users will yield a positive power vector.We should only consider those i corresponding to users who have multimediaservices. It is of little use to increase the channel quality **for** auser who is simply using voice quality service, while ignoring anotheruser who is transmitting video.Once the ides is selected, we increase -(ides and test **for** convergence.Now, we assume that in the small span of time in which this takes place,the positions of the mobiles do not change significantly. This is importantbecause we assume that the antenna weights need not be recalculated.We only wish to find a new power vector that is more favorableto user ides, while not chastising the other users. To do this, we can useexisting techniques [13], [14] that quickly give us a new power vector,and this time, we can use the more exact test, p(rneWF) < 1.If the test still passes, we can find a new power vector, and implementit. If the test fails, we reduce the SINR **for** a different user:And we test **for** convergence. The sequence of increasing the 7i thatleast affects p(rF) and decreasing the 7j that most affects it is repeateduntil the system perceives new users to have been added to thegroup, or until a certain amount of time has passed. After some time,it is no longer possible to assume that the mobile units have not movedsignificantly, so the system must recalculate the weight vectors as wellas the power vector. The algorithm will be summarized in Section V.This suboptimal approach may not yield a unique answer. Considerwe have not made any statements regarding the convexity of the spectralradius of our matrix. Even further, we can count on the spectralradius to have expressions with high powers of the matrix elements.This will imply a number of local minima. We implement a steepestdescent algorithm that may lead ultimately to a local minimum. Furthermore,it does not give a unique answer, since our selection of thesize of increments and reductions may lead us through different paths.IV. CRITERIA FOR SlNR ADJUSTMENTThis section deals with the question of choosing which SINR to adjustto best fit our purpose, be it to quickly find a feasible r as seen inSection Ill-A, or to meticulously increase as many -(i by as much as wecan as seen in Section 111-B. For the first goal, we wish to choose the 7,to be altered so that a small decrease in 7; will have a large impact onthe reduction of p(rF). For the second goal we would like to increasesome SINR without driving p(I'F) above one. For this case, we wishto choose a 7j to be altered so that a large increase in 7j will have asmall impact on the increase of p(rF).62

1Check system **for** new cochannel users, assign thedesired Y to new users according to their service tvDeFig. 2. Both eigenvalues change continuously ils the elements of the matrix change, Butel suns as the spectral radius, and then at point (17.8,0.7) e2 becomes the specvalradius. Thus, the spectral radius is not smooth at that pint.For both of these, we wish to calculate the derivative of p(rF) withrespect to each yi. The question of the existence of the derivative ofthe spectral radius of such a matrix is addressed in [12], in conjunctionwith Frobenius’ Theorem,[l I]. We can now derive an expression **for**the derivative.Theorem 2: Let p be a simple eigenvalu? of I’F, with right and lefteigenvectors, x and y. respectively. Let F = rF + E. Then thereexists a unique b, eigenvalue of F such thatProof: [I21 oIn our case, E = Ay, . Fi, where F; is a matrix whose ith row isthe same as F, and all other elements are zero. Then, using 17, weconclude thatNote that the denominator is constant **for** all i = 1, ... M. We can sortthese derivatives to suit our needs **for** the Quick Activation and FineTuning schemes.We may choose to reduce several SINR, but our decision of whichand by how much should be influenced by the sorted indexes. The criteria**for** choosing the yi may still have some subjective contribution.For example, even if a certain user, j has the higher derivative, we maynot wish to reduce this users SINR, since he has been functioning witha high quality, and would be displeased with a reduction. It would perhapsmake more sense to take the subset of new users and only workwith their SINR. Another consideration is that we may have reduced acertain user’s SINR several times during the iterations. We may wish totag users so that they do not suffer repeated quality deprivation. Theseconsiderations have been included in the simulations, though not in Table2. They are provided as an indicator of what the designer may doto improve customer satisfaction, but they are subjective, and thus, tangentialto the material we present.It should be noted that the derivative is a measure of infinitesimalchange, there**for**e, we must remind ourselves to choose small perturbationsof the yi, so that these results may have some value. Anotherreason to keep the perturbations small is that the spectral radius maychange from one eigenvalue index to another, as was seen in Fig. 2.V. INTEGRATED MULTIMEDIA **QoS** SCHEMEIn this section, we put together the tools described in the previoussections and we **for**mulate a working algorithm, which is detailed inTable 2.3.a3.b456If success, then this r works. Go to Step 4.If failure, reduce the yi **for** i = arg maxj y.Return to Step 3.Obtain {wi},~~ and P.Assign antennas weights in bases and transmit powersettings to mobile units.Check to see if new mobiles have been added or if theclock has timed out.Increase the yi **for** i = arg minja7j .7Call this new set which we are testingII’,,,,,.8 Check ifp(r,,,F) < 1.8.a If success, mew, go to Step 9.8.b If failure, reduce the yi **for**i = arg minj v.I Got to Step 6.9 I Obtain Dower vector. P. Go to SteD 5. .Table 2Algorifhm descripfion.1 IVI. SIMULATION RESULTSWe assumed a wireless network with M = 40 cochannel bases usingFDMA, and a (2,l) reuse pattern. D = 4 **for** all base stations. Weassumed only two user types, voice quality and multimedia quality. Thealgorithm can be readily extended to several quality types. We assumedthat voice users would get a constant 9” SINR, and the other userswould try to have ;I , SINR, which was considerably higher than 9”.The simulations presented run along a time axis. M is the maximumnumber of cochannel calls that may be working at any given time. Weassume that about eighty percent of those M links would be on at anygiven time, and of the ones that are on, p fraction of them desire amultimedia quality channel. Call lengths are exponentially distributedwith an average of six minutes. The time-out **for** exiting Steps 7, 8,and 9 of the algorithm was set to be 16 time units times the number ofactive users.The values that are plotted are: the desired SlNR **for** the multimediausers, +m which is labeled “desired”, the effective SINR **for** those highquality users using the entire algorithm, ym, which can be seen in Fig. 3to be close to the desired level, the SINR **for** the high quality usersif they do not use the fine-tuning method, which can be seen to belower, and the SINR **for** voice users, 9”. For Fig. 3, T,,, = 30dB andTV = 16dB. Another example with larger difference between i,,, andqV can be seen in Fig. 4.Fig. 5 compares the results of our method with the possibility ofgiving a constant SINR to all users, and implementing the algorithmwithout the benefit of calculating the derivative of the spectral radiuswith respect to the yi.Fig. 6 compares the time that our scheme takes to calculate the mobilepowers and antenna weights when new users are being added to thesystem, to a trial and error scheme. In the trial and error scheme, thedesired 9s are assigned to the new users. The system then engages inan iterative algorithm to get the powers and antenna weights. If the algorithmdid not converge in 4 * M iterations, it would reduce the SINR63

Y26mP 222016'O't /116WE. qualnySlNR1450 IW 150 2W 250limFig. 3. p = .3; i; = 16dB; 4, = 30dB. The avenge difference between the melhodwithout steps 7 through 9 and bat with those steps was 10.2dB.Fig. 6. Comparison of the processing time **for** the fast activation method proposed in thispaper 10 a trial and error method.**for** a randomly chosen multimedia user and try again.28 -*e qualay SNR50 20 U) 60 60 1W 120 140 160Fig. 5. +, = 32dB. The average SlNR **for** our method using variable SINR was31.5dB. The average SINR using the same algorithm, but using a constant SINR **for**all users and rising and lowering them at the same time was 19.2dB.IlMVII. CONCLUSIONSWe can gain a significant increase in the SINR **for** certain users byestimating the spectral properties of the system matrix and using thoseproperties to adjust the power **for** the mobile units. Our simulationsshow over lOdB of improvement of the SINR by using the proposedSINR fine-tuning scheme.It should be mentioned that one advantage to our iterative method**for** improving the SINR **for** multimedia users is that the system is seekinga better quality off-line, so there is no latency associated to it. Itguarantees no detriment to voice quality users, and it seeks to drive theSINR closer to the desired levels.We've also managed to reduce the activation latency time **for** newusers significantly by using a coarser and faster method **for** finding feasibleSINR. Our simulations have shown an improvement of about twoorders of magnitude in the time required to find a feasible solution.REFERENCES[I1 E Ruhid-Fmokhi: K.J.R. Liu: and L. Tmiulu. 'Tnnsmit Beam**for**ming and **Power** Control **for**Cellular Wircless Systems". IEEE Journd on Selecred Areas in Communicarions. 16(8):pp. 1437-1449. Cctobcr 1998.121 E Rashid-Fmokhi: L. Tassiulu; and K.I.R. Liu. "Joint Optimal **Power** Control and Beam**for**mingin Wirclcss Nctworks Using Antenna Arrays". IEEE Transactions on Communicarions. 46( I0):pp.131 > 1323. Cctobcr I 998.131 1. Winters; C. Martin: and N. Sollcnbcrgcr. "Fnnvard Link Smart Antennas and **Power** Control **for**IS-136". 48rh IEEE Uhicular Technology Confermcc. pages pp. 60605. Miy 1998.141 Y. Liamg; F. Chin: and K.J.R. Liu. "Downlink Beam**for**ming **for** DS-CDMA **Mobile** Ridio with**Multimedia** SCN~CCS". Pmc. of IEEE VTS 50lh Vrhiculur Technology Confcrcncr. pages pp. 17-2 I,IW.151 A. Grandhi: R. Yates: D. Goodman. "Ruourcc Allocation **for** Cellular Radio Systems". IEEETmn.rmrions on Uhiculor Technology. 46(3):pp. 581-587. August 1997.161 S. Rappapon: C. Purzynski. "Priaritizcd Rcsourcc Assignment **for** **Mobile** Ccllulu CommunicationSystems with Mixed SCN~CCS and Plat**for**m Types". IEEE Tmnsactionr on Uhicular Technology.45(3):pp. 443458. August 1996.17) E Santucci: F. Graziosi. "**Power** Allocation in a MultimcdiaCDMA Wirc1e.s System with Impcrfcct**Power** Conuol". IEEE Inrrmaririnal Conference on Communicarians. June 1W.181 R. Saclc. **Mobile** Radio Communiwrims. IEEE Press. 1992.191 R. Monzingo and T. Miller. hrmduction IO Adoprivr Arruys. Wiley - Interscience. 1980.[IO] E Gantmuhcr. 7hhr lkmy ofMmricrs. volume 2. Chclxt Publishing Company. 195Y.11 I I R. Bellman. lnrnwlucrion ro Marrix Analysis, Srcond Edirion. SIAM, 1W7.I121 G. Stewan and 1. Sun. MmrixPcnurborion Thmry. Academic Presv. IW).I131 F. R;r