CCCT 05: INTERNATIONAL CONFERENCE ON COMPUTING, COMMUNICATIONS, AND CONTROL TECHNOLOGIES 1 Approxmatng User Dstrbutons n CDMA Networks Usng 2-D Gaussan Son NGUYEN and Robert AKL Department of Computer Scence and Engneerng Unversty of North Texas Denton, TX, 76203 ABSTRACT In ths paper, we present an analytcal model for approxmatng the user dstrbutons n mult-cell thrd generaton CDMA networks usng 2-dmensonal Gaussan dstrbutons by determnng the means and the standard devatons of the dstrbutons for every cell. Ths allows us to calculate the nter-cell nterference and the reverselnk capacty of the network. e compare our model wth smulaton results and show that t s fast and accurate enough to be used effcently n the plannng process of large CDMA networks. Keywords: Inter-cell nterference, Capacty, CDMA, User dstrbuton, 2-D Gaussan dstrbuton. 1. INTRODUCTION deband Code Dvson Multple Access (CDMA) s an ar nterface that s proposed for thrd generaton wreless networks that provdes a vast range of data servces wth bt rates of up to 2Mbps wth varyng qualty of servce requrements. Snce CDMA was frst ntroduced n 1989 by QUALCOMM, the number of subscrbers has grown to more than 240 mllon globally. The ablty to offer greater capacty, mult-rate transmsson wth backward compatblty, effortless ntegraton, and easy mgratng path to 3G cellular systems, has created the wdespread deployment of CDMA systems all over the world [1]. It has been shown n [2], [3], [4], [5] that the capacty of a CDMA network s reverse lnk lmted, and therefore our study s focused on reverse lnk capacty. One of the prncpal characterstcs of a CDMA network s that the capacty of the system s a functon of the total nterference experenced by the network, and s upper bounded by the cell experencng the most nterference. Thus, t s mmnent to characterze the total nter-cell nterference seen by a sngle cell n terms of the user dstrbuton n all other cell for determnng the capacty n that sngle cell. Tradtonally, the total nterference contrbuted by a cell has been vewed as an approxmaton, determned by smply multplyng the number of users n that cell by the average nterference offered by that cell [6]. In other words, a user placed anywhere wthn a cell generated the same amount of nterference. Clearly, a more realstc approach wll use peruser nterference as a functon of ts actual dstance to the pont of nterest. There s a dearth of lterature where actual dstance was used n the nterference model. In [7], even though nterference was calculated usng actual dstance, the capacty calculatons were done usng mean value of nterference. User postons were vared over tme, but the number of users was kept constant. In ths paper, we present an analytcal model for the approxmaton of the user dstrbuton n mult-cell CDMA networks usng 2-dmensonal Gaussan dstrbutons by determnng the means and the standard devatons of the dstrbutons for every cell. Once the user dstrbutons are approxmated, the average nter-cell nterferences can be determned smlar to what was done n [8]. e compare our model wth smulaton results presented n [9] and show that t s fast and accurate enough to be used effcently n the plannng process of large CDMA networks. The remander of ths paper s organzed as follows. In secton 2, we use the 2-D Gaussan functon for modelng user dstrbutons and calculatng the average nter-cell nterference. In secton 3, we compute the capacty of a CDMA network. Numercal results are presented n secton 4. Fnally, our conclusons are gven n secton 5. 2. AVERAGE INTER-CELL INTERFERENCE MODEL USING 2-D GAUSSIAN DISTRIBUTION It s assumed that each user s always communcatng and s power controlled by the base staton (BS) that has the hghest receved power at the user. Let r (x, y) and r j (x, y) be the dstance from a user to BS and BS j, respectvely. Ths user s power controlled by BS j n the cell or regon C j wth area A j, whch BS j servces. It s assumed that both large scale path loss and shadow fadng are compensated by the perfect power control mechansm. Let I j,t be the average nter-cell nterference that all users n j,t usng servces t wth actvty factor v t and receved sgnal S t at BS j mpose on BS. Modfyng the average nter-cell nterference gven by [8], we have e (γσs)2 rj m (x, y) I j,t = S t v t n j,t w(x, y) da(x, y), (x, y) A j C j (1) where γ = ln(10)/10, σ s s the standard devaton of the attenuaton for the shadow fadng, m s the path loss exponent, and w(x, y) s the user dstrbuton densty at (x, y). r m
CCCT 05: INTERNATIONAL CONFERENCE ON COMPUTING, COMMUNICATIONS, AND CONTROL TECHNOLOGIES 2 e defne κ j,t to be the per-user (wth servce t) relatve nter-cell nterference factor from cell j to BS,.e., r κ j,t = e(γσs)2 j m (x, y) w(x, y) da(x, y). (2) A j (x, y) C j The nter-cell nterference densty Ij nter from all servces T becomes I nter j r m = 1 from cell j to BS I j,t, (3) where s the bandwdth of the system. Eq. (3) can be rewrtten as Ij nter = 1 S t v t n j,t κ j,t. (4) Thus, the total nter-cell nterference densty I nter all other cells to BS s I nter = 1 j=1,j from S t v t n j,t κ j,t, (5) where M s the total number of cells n the network. If the user dstrbuton densty can be approxmated, then, κ j,t needs to be calculated only once. e model the user dstrbuton by a 2-dmensonal Gaussan dstrbuton as follows η w(x, y) = e 1 2 ( x µ 1 σ ) 2 1 e 1 2 ( x µ 2 σ ) 2 2, (6) 2πσ 1 σ 2 where η s a user densty normalzng parameter. e show that by specfyng the means µ 1 and µ 2 and the varances σ 1 and σ 2 of the dstrbuton for every cell, we can approxmate a wde range of user dstrbutons rangng from unform to hot spot clusters. e compare these results wth smulatons and determne the value of η expermentally. 3. CDMA CAPACITY In CDMA, the energy per bt to total nterference densty at BS for a servce t s gven by [10] ( E b I 0 ),t = S t N 0 + I nter + I own, (7) S t v t where N 0 s the thermal nose densty, s the bt rate for servce t. I own s the total ntra-cell nterference densty s gven by = 1 S t v t n,t. (8) caused by all users n cell. Thus I own I own Let τ t be the mnmum sgnal-to-nose rato, whch must receved at a BS to decode the sgnal of a user wth servce t, and S t be the maxmum sgnal power, whch the user can transmt. Substtutng (5) and (8) nto (7), we have for every cell n the CDMA network, the number of users n,t n BS for a gven servce t needs to meet the followng nequalty constrant τ t S t (9) N 0 + S t [X(, t)], where X(, t) = n,t v t + j=1,j n j,t v t κ j,t v t. (10) After rearrangng terms, (9) can be rewrtten as where n,t v t + j=1,j c (t) eff = [ 1 τ t n j,t v t κ j,t v t c (t) eff, (11) S t /N 0 ]. (12) The capacty n a CDMA network s defned as the maxmum number of smultaneous users (n 1,t, n 2,t,..., n M,t ) for all servces t = 1,..., T that satsfy (11). 4. NUMERICAL ANALYSIS The results shown are for a twenty-seven cell network topology used n [9]. The COST-231 propagaton model wth a carrer frequency of 1800 MHz, average base staton heght of 30 meters and average moble heght of 1.5 meters s used to determne the coverage regon. The path loss coeffcent m s 4. The shadow fadng standard devaton σ s s 6 db. e assume only one servce,.e., T = 1. The processng gan R s 21.1 db. The bt energy to nterference rato threshold, τ, s 9.2 db. The actvty factor, v, s 0.375. These parameters yeld c eff of 38.25. The smulator used for comparson s an extenson of the software tools CDMA Capacty Allocaton and Plannng (CCAP) [11]. CCAP, wrtten n MATLAB, was developed at ashngton Unversty n St. Lous for numercal analyss of optmzaton technques developed n [8] to compute the capacty of CDMA networks. e extended CCAP for CDMA networks and used the 2-dmensonal Gaussan functon for w(x, y). In what follows, we show that by usng 2-D Gaussan dstrbuton, we can model many dfferent scenaros ncludng users unformly dstrbuted, users clustered at the center of the cells, and users at the cells boundares. e verfed the results n [9], where actual dstances were used to smulate real-tme users enterng the network for the calculaton of nterference. e analyzed the network wth dfferent values of σ 1 and σ 2, whle keepng µ 1 and µ 2 equal to zero n (6). Table I shows the maxmum number of users n every cell for the 27 cell CDMA network as the values of σ 1 and σ 2 are ncreased from 5000 to 15000 whle µ 1 =0 and µ 2 =0. Ths results n users spread out (almost unformly) n the cells. Fg. 1 shows the 2-D Gaussan approxmaton of users unformly dstrbuted n the cells wth σ 1 =σ 2 =12000. The total number of users s 548. Ths compares well wth smulaton results presented n Fg. 2, whch yelds a total number of users equal to 554 when they are placed unformly n the cells. Table II shows the maxmum number of users n every cell for the 27 cell CDMA network as the values of σ 1 and σ 2 are ncreased from 100 to 400 whle µ 1 =0 and µ 2 =0. Ths results n users densely clustered around the
CCCT 05: INTERNATIONAL CONFERENCE ON COMPUTING, COMMUNICATIONS, AND CONTROL TECHNOLOGIES 3 TABLE I THE MAXIMUM NUMBER OF USERS IN EVERY CELL FOR THE 27 CELL CDMA NETORK AS THE VALUES OF σ 1 AND σ 2 ARE INCREASED FROM 5000 TO 15000 HILE µ 1 =0 AND µ 2 =0. THIS RESULTS IN USERS SPREAD OUT (ALMOST UNIFORMLY) IN THE CELLS. σ=σ 1, σ 2 5000 7000 10000 12000 15000 Un Dst Cell 1 18 18 18 18 18 18 Cell 2 18 18 18 18 18 18 Cell 3 18 18 18 17 17 17 Cell 4 18 18 18 17 17 17 Cell 5 18 18 18 18 18 18 Cell 6 18 18 18 17 17 17 Cell 7 18 18 18 17 17 17 Cell 8 18 18 18 18 18 18 Cell 9 18 17 17 17 17 17 Cell 10 22 21 21 21 21 21 Cell 11 22 22 22 21 21 21 Cell 12 22 21 21 21 21 21 Cell 13 17 17 17 17 17 17 Cell 14 18 18 18 18 18 18 Cell 15 18 17 17 17 17 17 Cell 16 22 21 21 21 21 21 Cell 17 22 22 21 21 21 21 Cell 18 22 21 21 21 21 21 Cell 19 18 17 17 17 17 17 Cell 20 25 25 25 25 25 25 Cell 21 25 25 24 24 24 24 Cell 22 25 25 24 24 24 24 Cell 23 25 25 25 25 25 25 Cell 24 25 25 25 25 25 25 Cell 25 25 25 24 24 24 24 Cell 26 25 25 24 24 24 24 Cell 27 25 25 25 25 25 25 Total Users 565 558 553 548 548 548 Fg. 2. Smulated network capacty where users are unformly dstrbuted n the cells. The maxmum number of users s 554. Fg. 3. 2-D Gaussan approxmaton of users densely clustered around the BSs. σ 1 =σ 2 =100, µ 1 =µ 2 =0. The maxmum number of users s 1026. Fg. 1. 2-D Gaussan approxmaton of users unformly dstrbuted n the cells. σ 1 =σ 2 =12000, µ 1 =µ 2 =0. The maxmum number of users s 548. BSs. Fg. 3 shows the 2-D Gaussan approxmaton wth σ 1 =σ 2 =100. The maxmum number of users s 1026. Ths compares exactly wth smulaton results presented n Fg. 4, whch yelds a total number of users equal also to 1026. In ths confguraton, the users cause the least amount of nterference to the network, by reducng the power gan requred to mantan a desred sgnal-to-nose rato. Fg. 5 shows the 2-D Gaussan approxmaton of users clustered at the boundares of the cells. The values of σ 1, σ 2, µ 1, and µ 2 may be dfferent n the dfferent cells and are gven n Table III. The maxmum number of users s 133. Ths result s close to what was attaned through smulaton. The maxmum network capacty was made low by havng the smulator place the users such that they cause maxmum nterference to the network. The smulaton yelded a total capacty of 108 users, wth only 4 users n each cell. The
CCCT 05: INTERNATIONAL CONFERENCE ON COMPUTING, COMMUNICATIONS, AND CONTROL TECHNOLOGIES 4 TABLE II THE MAXIMUM NUMBER OF USERS IN EVERY CELL FOR THE 27 CELL CDMA NETORK AS THE VALUES OF σ 1 AND σ 2 ARE INCREASED FROM 100 TO 400 HILE µ 1 =0 AND µ 2 =0. THIS RESULTS IN USERS DENSELY CLUSTERS AROUND THE BSS. σ=σ 1, σ 2 σ = 100 σ = 200 σ = 300 σ = 400 Cell 1 38 38 37 34 Cell 2 38 38 37 34 Cell 3 38 38 37 35 Cell 4 38 38 37 35 Cell 5 38 38 37 34 Cell 6 38 38 37 35 Cell 7 38 38 37 35 Cell 8 38 38 37 35 Cell 9 38 38 37 35 Cell 10 38 38 37 36 Cell 11 38 38 37 36 Cell 12 38 38 37 36 Cell 13 38 38 37 35 Cell 14 38 38 37 35 Cell 15 38 38 37 35 Cell 16 38 38 37 35 Cell 17 38 38 37 35 Cell 18 38 38 37 35 Cell 19 38 38 37 35 Cell 20 38 38 37 36 Cell 21 38 38 38 36 Cell 22 38 38 38 37 Cell 23 38 38 38 36 Cell 24 38 38 38 36 Cell 25 38 38 37 36 Cell 26 38 38 37 36 Cell 27 38 38 37 36 Total Users 1026 1026 1003 954 TABLE III THE VALUES OF σ 1, σ 2, µ 1, AND µ 2 FOR THE 2-D GAUSSIAN APPROXIMATION OF USERS CLUSTERED AT THE BOUNDARIES OF THE CELLS AS SHON IN FIG. 5. THE MAXIMUM NUMBER OF USERS IS 133. µ 1 σ 1 µ 2 σ 2 Cell 1-1400 300-900 300 Cell 2-1400 300 800 300 Cell 3-1400 300 800 300 Cell 4 0 300-1700 300 Cell 5 0 300-1600 300 Cell 6 1300 300-800 300 Cell 7-1400 300 900 300 Cell 8-1300 300 900 300 Cell 9 0 300 1500 300 Cell 10 0 300 1600 300 Cell 11 0 300 1550 300 Cell 12-1400 300 900 300 Cell 13 0 300 1500 300 Cell 14 1300 300 900 300 Cell 15 1300 300-800 300 Cell 16-1350 300-850 300 Cell 17-1400 300-900 300 Cell 18 0 300-1600 300 Cell 19-1400 300-800 300 Cell 20-1400 300-800 300 Cell 21-1350 300 800 300 Cell 22 0 300 1600 300 Cell 23 1350 300 800 300 Cell 24 1400 300-800 300 Cell 25 0 300-1700 300 Cell 26 0 300-1600 300 Cell 27-1350 300-850 300 pattern seen n Fg. 6 shows that the smulator placed the users at the extreme corners of ther respectve cells. The placement at extremtes would requre users to ncrease ther power gan causng a lot more nterference to other users. 5. CONCLUSIONS e presented an analytcal model for approxmatng the user dstrbutons n mult-cell CDMA networks usng 2-dmensonal Gaussan dstrbutons by determnng the means and the standard devatons of the dstrbutons for every cell. Ths allowed for the calculaton of the nter-cell nterference and the reverse-lnk capacty of the network. e compared our model wth smulaton results and showed that t s fast and accurate enough to be used effcently n the plannng process of large CDMA networks. REFERENCES Fg. 4. Smulated network capacty where users are densely clustered around the BSs causng the least amount of nter-cell nterference. The maxmum number of users s 1026 n the network. [1] CDMA Development Group, CDG : orldwde : CDMA orldwde, http://www.cdg.org/worldwde/ndex.asp?h area=0. [2] J. Evans and D. Evertt, On the teletraffc capacty of CDMA cellular networks, IEEE Trans. Veh. Technol., vol. 48, no. 1, pp. 153 165, January 1999. [3] R. Padovan, Reverse lnk performance of IS-95 based cellular systems, IEEE Personal Commun. Mag., vol. 1, no. 3, pp. 28 34, Thrd Quarter 1994. [4] K. Takeo and S. Sato, Evaluaton of a CDMA cell desgn algorthm consderng non-unformty of traffc and base staton locatons, IEICE Transactons Fundamentals, vol. E81-A, no. 7, pp. 1367 1377, July 1998.
CCCT 05: INTERNATIONAL CONFERENCE ON COMPUTING, COMMUNICATIONS, AND CONTROL TECHNOLOGIES 5 [5] A. Vterb, A. Vterb, K. Glhousen, and E. Zehav, Soft handoff extends CDMA cell coverage and ncreases reverse lnk capacty, IEEE J. Select. Areas Commun., vol. 12, no. 8, pp. 1281 1288, October 1994. [6] K. Glhousen, I. Jacobs, R. Padovan, A. Vterb, L. eaver, and C. heatley, On the capacty of a cellular CDMA system, IEEE Trans. Veh. Technol., vol. 40, no. 2, pp. 303 312, May 1991. [7] D.. Matolak and A. Thakur, Outsde cell nterference dynamcs n cellular CDMA, Proceedngs of the 35th Southeastern Symposum, pp. 418 152, March 2003. [8] R. Akl, M. Hegde, M. Naragh-Pour, and P. Mn, Mult-cell CDMA network desgn, IEEE Trans. Veh. Technol., vol. 50, no. 3, pp. 711 722, May 2001. [9] R. Akl and A. Parvez, Impact of Interference Model on Capacty n CDMA Cellular Networks, Proceedngs of SCI 04: Communcaton and Network Systems, Technologes and Applcatons, vol. 3, pp. 404 408, July 2004. [10] D. Staehle, et. al., Approxmatng the Othercell Interference Dstrbuton n Inhomogenous UMTS Networks, IEEE Vehcular Technology Conference, vol. 4, pp. 6 9, May 2002. [11] R. Akl, M. Hegde, A. Chandra, and P. Mn, CCAP: CDMA Capacty Allocaton and Plannng, ashngton Unversty, Tech. Rep., Aprl 1998. Fg. 5. 2-D Gaussan approxmaton of users clustered at the boundares of the cells. The values of σ 1, σ 2, µ 1, and µ 2 may be dfferent n the dfferent cells and are gven n Table III. The maxmum number of users s 133. Fg. 6. Smulated network capacty where users are clustered at the boundares of the cells causng the most amount of nter-cell nterference. The maxmum number of users s only 108 n the network.