UMTS Radio Network Evaluation and Optimization Beyond Snapshots

Transcription

1 Konrad-Zuse-Zentrum für Informatonstechnk Berln Takustraße 7 D Berln-Dahlem Germany ANDREAS EISENBLÄTTER AND HANS-FLORIAN GEERDES AND THORSTEN KOCH AND ALEXANDER MARTIN AND ROLAND WESSÄLY UMTS Rado Network Evaluaton and Optmzaton Beyond Snapshots ZIB-Report (October 2004)

2 Andreas Esenblätter Hans-Floran Geerdes Thorsten Koch Alexander Martn Roland Wessäly UMTS Rado Network Evaluaton and Optmzaton Beyond Snapshots Abstract. Ths paper s concerned wth UMTS rado network desgn. The task s to reconfgure antennas and the related cells as to mprove network qualty. In contrast to second generaton GSM networks, nterference plays a paramount role n the desgn of thrd generaton rado networks. A recent compact characterzaton of a rado network through a lnear equaton system allows to assess cell load and nterference. Ths characterzaton s based on user snapshots and s generalzed here to average traffc ntenstes. Ths allows to overcome the notorous dffcultes of snapshot-based network optmzaton approaches. A mxed-nteger programmng model for the network desgn problem s recalled that s based on user snapshots and t s contrasted wth a new network desgn model based on the average couplng formulaton. Exemplarly focusng on the partcularly mportant problem of optmzng antenna tlts, computatonal results are presented for a fast local search algorthm and for the applcaton of a MIP solver to both models. These results demonstrate that the new average-based approaches outperform state-of-the-art snapshot models for UMTS rado network optmzaton. 1. Introducton The Unversal Moble Telecommuncatons System (UMTS) s a 3rd generaton (3G) cellular system for moble telecommuncatons. UMTS supports all servces of the worldwde predomnant GSM and GPRS networks and s more powerful, more flexble, and more rado spectrum effcent than ts predecessors. UMTS s a Wdeband Code Dvson Multple Access (WCDMA) system. Rado transmssons are generally not separated n the tme or the frequency doman. Complex codng schemes are nstead used to dstngush dfferent rado transmssons at a recever. The capablty to properly detect the desred carrer sgnal, however, requres that ths s not too deeply bured among nterferng rado sgnals. Techncally speakng, the carrer sgnal to nterference rato shall not drop below some threshold value. Interference, thus, needs to be carefully controlled durng network plannng and operaton, because t s a lmtng factor for network capacty. Hans-Floran Geerdes, Thorsten Koch: Zuse Insttute Berln (ZIB) e-mal: {geerdes,koch}@zb.de Andreas Esenblätter, Roland Wessäly: Ateso GmbH and ZIB e-mal: {esenblaetter,wessaely}@ateso.de Alexander Martn: Darmstadt Unversty of Technology (TUD) e-mal: martn@mathematk.tu-darmstadt.de Ths work has been supported by the DFG Research Center Matheon, Mathematcs for key technologes.

3 2 Andreas Esenblätter et al Overvew Ths paper s organzed as follows. The rest of ths secton ntroduces the task of UMTS rado network optmzaton and addresses related work. Secton 2 ntroduces the techncal background n parallel wth the notaton used throughout the paper. Secton 3 ntroduces a method to perform fast approxmate analyss of network capacty. Ths depends on systems of lnear equatons descrbng the nterference couplng among cells. In Secton 4, we refne a mxed nteger programmng model for network optmzaton based on traffc snapshots [8, 14]. A novel, alternatve mathematcal program, whch overcomes some drawbacks of the prevous model, s presented n Secton 5. Secton 6 contans computatonal results for realstc plannng scenaros. These results are obtaned by mplementatons based on the two mathematcal models and from a local search procedure. We draw conclusons n Secton Network Plannng Network operators are currently deployng UMTS n many countres across Europe and Asa. Unlke durng the ntroducton of GSM n the 1990s, the new moble telecommuncaton technology faces a well-developed market wth GSM as a competng system. GSM already offers large areal coverage, hgh relablty, and acceptable prces. An ncentve for the user to change technology towards UMTS s the avalablty of more servces and hgher communcaton data-rates n large areas wth a compettve prcng. Even more complex plannng tools than those for GSM are ndspensable. Ths s partcularly true for the rado network plannng due to the change of the rado access technology from frequency and tme multplexng to code multplexng. As n the case of frequency plannng for GSM networks, t s to be expected that the qualty of the plannng results can be largely mproved f the rado network planner s asssted by automatc optmzaton functons. Frst commercal products n ths doman are avalable, but t wll certanly take a few more years to reach the maturty of GSM frequency plannng. The paper contrbutes to ths development wth respect to network evaluaton and network optmzaton. A central part n the ntal deployment and the subsequent expansons of a UMTS rado network s to decde about the locaton and confguraton of the base statons ncludng ther antennas. Among others, the type of an antenna (and thus ts radaton pattern), the mountng heght, and ts man radaton drecton, the azmuth n the horzontal plane and the tlt n the vertcal plane, have to be decded. Ths may look lke a faclty locaton problem at frst sght, but t s not! The coverage area and the capacty of a base staton may shrnk sgnfcantly due to rado nterference Tlt Optmzaton All three optmzaton approaches proposed n ths paper can be used to optmze all aspects of an antenna nstallaton mentoned above. From the practcal pont

4 UMTS Rado Network Evaluaton and Optmzaton Beyond Snapshots 3 of vew, however, tlt optmzaton s of prmary nterest. Operators often deploy an UMTS networks n addton to exstng 2G nfrastructure. Snce t s costly and ncreasngly dffcult to acqure new stes, UMTS antennas are often nstalled n places where there s already a GSM antenna. The two types of antennas are sometmes assembled on the same mechancal structure, dual band antennas that operate for both GSM and UMTS are also n use. Ths bascally fxes the antenna s heght and azmuth. In addton, operators typcally have standard antenna types. The one parameter that operators favor for optmzaton s tlt. We focus on optmzng ths parameter n our computatonal studes n Secton 6. Antenna tlt comes n two flavors, mechancal and electrcal tlt, whch can be vared ndependently. The mechancal tlt s the angle by whch the antenna s tlted out of the horzontal plane. For many UMTS antennas, the man radaton drecton n the vertcal plane can also be changed by electrcal means. The resultng change s called electrcal tlt. Tlt optmzaton alone has a consderable leverage on network coverage as well as capacty, because t nfluences cell ranges and nterference. In general, f an antenna s tlted down, the sze of the correspondng cell and thereby the load that ths antenna has to serve decrease. At the same tme, the amount of nterference n neghborng cells mght be reduced Related Work Ths work s largely based on the authors partcpaton n the EU-funded project Momentum. The nterdscplnary project had mathematcans, engneers, and telecommuncaton operators n the consortum, the scope was developng models and algorthms for smulaton and automatc plannng of UMTS rado networks. The frst verson of the snapshot model presented n Secton 4 s formulated n [10, 14], refnements and detaled techncal descrptons of the model parameters s gven n [8]; a summary and frst computatonal results are gven n [11]. Due to the practcal relevance of our topc we menton both works from the mathematcal as well as from the engneerng sde. In [39] a survey chartng the evoluton of centralzed plannng for cellular systems s gven, ncludng the authors contrbutons. There s a varety of mathematcal work on UMTS rado network plannng. Optmzaton models smlar to our snapshot model are suggested n [1, 2, 3] wth computatonal results. The problems nclude ste selecton and base staton confguraton. Heurstc methods such as tabu or greedy search are used to solve nstances. Integer programmng methods for 3G network plannng are presented n [29]. Some papers deal wth subproblems: plot power optmzaton under coverage constrants usng mathematcal programmng s performed n [35, 38], power control and capacty ssues wth partcular emphass on network plannng are treated n [6,27]. To our knowledge, however, there s no work focusng on the network tunng aspect under techncal constrants; tlt optmzaton falls nto ths category.

5 4 Andreas Esenblätter et al. On the other hand, there s a vast body of work by engneers on problems n network plannng. The frst and exemplary landmark monography coverng many techncal aspects of UMTS networks and some practce-drven optmzaton and tunng rules s [26]. Optmzaton of network qualty aspects not nvolvng ste selecton and usng meta-heurstcs s treated n [17]. Network desgn wth a specal emphass on optmzng cell geometry usng evolutonary algorthms s presented n [22, 23]. A mxed nteger program for optmzng qualty-of-servce for users wthn a snapshot s developed n [36]. The latter paper also uses the dmenson reducton technque that we present n Secton 3. The dmenson reducton had earler been proposed for monoservce traffc n [31], and was extended to a servce mx n [6,7]. We contrbute a generalzaton to average user densty maps and an optmzaton model based on these premses. Earler papers addressng power control by consderng systems of lnear equatons wth postve solutons are [18, 19, 20]. 2. Rado Network Desgn for WCDMA We gve an ntroducton to the propertes of UMTS technology nsofar as they are relevant to our paper. In moble telecommuncatons, two transmsson drectons are dstngushed: the uplnk and the downlnk. An uplnk (UL) transmsson takes place when a moble transmts to a base staton. A downlnk (DL) transmsson takes place from the base staton to moble users. The type of rado network we consder wth UMTS s called a cellular network: there s an nfrastructure of base statons. One or more antennas are nstalled at each base staton. Each antenna serves users n a certan area (typcally the area where ths antenna provdes the strongest sgnal), ths area s called cell. In a UMTS network, every antenna emts a specal constant power plot sgnal, whch s used by the mobles to measure whch antenna s receved best and for decdng whch cell they assocate to. In ths paper, we consder up- and downlnk transmssons on dedcated channels, whch can be seen as pont-topont connectons between a moble and a base staton. There are some other channels that are shared between the mobles n a cell, but apart from the plot channel, we do not model them explctly. In UMTS networks rolled out today, transmssons n uplnk and downlnk do not nterfere, because separate frequency bands are reserved for each drecton (frequency doman duplex, FDD). Examples for well-known moble communcaton servces are speech telephony or text messagng, some of the addtonal servces that wll be offered on the bass of UMTS are vdeo telephony, downlnk data streamng or web access. Ths dversfcaton mples an ncreasng heterogenety of the traffc n the network. An mportant dfference between servces s the requested data rate, especally as some of the new servces requre a substantally hgher data rate than speech telephony, for example.

6 UMTS Rado Network Evaluaton and Optmzaton Beyond Snapshots Prelmnares and Notaton For a complete account of the notaton used n ths paper please refer to Table 1. We consder an UMTS rado network wth a set N of antennas (or cells) and aset M of moble users n a traffc snapshot. We assume that each moble s connected to exactly one antenna, namely, the one whch has the strongest plot sgnal. The practcal stuaton s more complcated, snce a moble can be lnked to more than one antenna at a tme. We gnore ths feature of UMTS called soft-handover for our presentaton. We denote a vector wth components v j n bold, v. The notaton dag (v) stands for a dagonal matrx of matchng dmenson wth the elements of v on the man dagonal. Table 1. Notaton N Set of antenna nstallatons (cells) n the network I I N Set of all possble nstallatons for the network N, I Installaton M Set of mobles M Mobles served by antenna nstallaton M {m M α m > 0} M {m M α m > 0} m m M Moble S Set of servces s s S Servce A Plannng area p p A Locaton n the plannng area A A A Cell area (best server area) of nstallaton T s A R + Average spatal traffc dstrbuton of servce s p m R + Uplnk transmt power from moble m M p m R + Downlnk transmt power from nstallaton to moble m ˆp R + (Downlnk) plot transmt power from nstallaton ˇp R + (Downlnk) common channels transmt power from nstallaton p R + Total transmt power of nstallaton p R + Total receved power at nstallaton Π max R + Maxmum total transmt power for nstallaton γ m [0, 1] Uplnk attenuaton factor between moble m and nstallaton γ m [0, 1] Downlnk attenuaton factor between nstallaton and moble m η, η m 0 Nose at nstallaton /moble m α m, α m [0, 1] Uplnk/downlnk actvty factor of moble m ω m [0, 1] Orthogonalty factor for moble m µ m 0 Uplnk CIR target for moble m µ m 0 Downlnk CIR target for moble m ˆµ m 0 Plot E c/i 0 requrement for moble m Antenna nstallatons. The degrees of freedom (antenna type, heght, azmuth, tlt) when confgurng an antenna and thereby determnng the propertes of the related cell have been mentoned above. For the antenna of a certan cell, we call a fxng of the parameter values n these dmensons an nstallaton.

7 6 Andreas Esenblätter et al. The set of all consdered nstallatons wll be denoted by I. An actual network or network desgn s a choce of all possble nstallatons wth the property that exactly one of the potental nstallatons for each cell s chosen. In partcular, t always holds that N I. We wll use the ndces and j for elements of both N and I CIR nequalty for a transmsson. For a sgnal to be successfully decoded at the recever wth WCDMA technology, the rato between the receved strength of the desred sgnal and all nterferng sgnals ncludng background nose exteror to the system must exceed a specfc threshold. Ths rato s also called carrer-to-nterference rato (CIR), the threshold s called CIR target. For a transmsson to take place, the followng nequalty must be satsfed: Strength of Desred Sgnal Nose + CIR target (1) Strength of Interferng Sgnals Code multplexng technology ntroduces the possblty that the rght hand sde of ths nequalty s smaller than one, that s, the desred sgnal strength may be much weaker than the nterference. We wll now ntroduce the varous parameters that play a role n (1) CIR targets and actvty. The CIR target s servce specfc. In order to successfully decode a hgher data rate a hgher CIR target must be satsfed. There s also a specfc CIR target for the plot channel, manly dependng on user equpment. We denote the CIR targets for a user m M for uplnk, downlnk, and plot by µ m, µ m, and ˆµ m. The way that users access a servce over tme also vares: n a speech conversaton, each of the two users nvolved speaks 50 % of the tme on average, and no data s transmtted n slence perods (dscontnuous transmssons). Ths s taken nto account n the form of actvty factors. For each user m M we have two actvty factors, α m for the uplnk and α m for the downlnk. The actvty factor can be used to compute the average power that s sent over a lnk. There are servces that cause traffc n only one drecton, e. g. downlnk data streamng. In ths case the correspondng actvty factor for the other drecton s zero. (There s actually always some control traffc, but t s neglgble.) In (1), we take ths averaged value for all nterferng sgnals, snce we cannot tell whether the nterferng lnks are currently n an actve perod or not. For the lnk n queston, however, we need to reach the CIR target n actve perods only, snce there s no transmsson n nactve perods. The actvty s thus only taken nto account n the denomnator of (1), but not n the numerator Attenuaton. The strength of a sgnal transmtted over a rado lnk s attenuated on ts way to the recever. The receved power depends lnearly on the output power at the sender. The attenuaton on the rado channel (excludng transmsson and recepton equpment) s called path loss. Predctng the path loss n real-world settngs s dffcult and many methods rangng from rule-ofthumb formulas up to sophstcated raytracng methods usng 3D buldng data

8 UMTS Rado Network Evaluaton and Optmzaton Beyond Snapshots 7 and vegetaton are avalable. We refer the nterested reader to [16, 24, 5]. Besdes ths loss on the rado channel, some further losses and gans due to the cablng, hardware, and user equpment have to be consdered. All ths nformaton s then summed up nto attenuaton factors. We have two factors for each par N, m M, an attenuaton factor γ m for the uplnk and γ m for the downlnk Complete CIR constrants. We denote the transmsson power of a moble m M n uplnk by p m. The receved sgnal strength at nstallaton N s then γ m p m. If we denote the receved background nose at nstallaton N by η, the complete verson of (1) for the uplnk and the transmsson from m to reads γ m p m η + n m γ n α np n µ m. Wrtng p := η + γ m α m p m (2) m M for the average total receved power at nstallaton N, ths smplfes to γ mj p m p j γ mj α m p m µ m. (3) In the downlnk, the stuaton s more complcated. Frst of all, we have to take the plot and common channels nto account, whose transmsson power we denote by ˆp and ˇp for nstallaton. There s another UMTS feature to be consdered here: each cell selects orthogonal transmsson codes for the mobles n ts cell, whch n theory do not nterfere mutually (we neglect the use of secondary scramblng codes). However, due to reflectons on the way the sgnals partly lose ths property. Sgnals from other antennas do not have t at all. Hence, when summng up the nterference, the nterference from the same cell s reduced by an envronment dependent orthogonalty factor ω m [0, 1], wth ω m = 0 meanng perfect orthogonalty and ω m = 1 no orthogonalty. We defne the total average output power of nstallaton as p := m M α m p jm + ˆp + ˇp. (4) Wrtng η m for the nose value at moble m, we obtan the downlnk verson of (1) related to transmsson from to m: γ m ω m γ jm p jm ( ) p α m p m + j γ jm p j + η m µ m. (5)

9 8 Andreas Esenblätter et al. Note on downlnk orthogonalty. Strctly speakng, there s one control channel for whch orthogonalty to other channels n the cell does not apply, the synchronzaton channel (SYNC). We do not consder ths detal for clarty of presentaton. Ths s acceptable from the engneerng pont of vew, because ts transmsson power s farly low n comparson to the other common channels. For the plot, the stuaton s smpler. Techncally speakng, the receved chp energy of the plot sgnal called the plot channel s E c or CPICH E c relatve to the total power spectral densty I 0 has to le above a certan threshold. When computng the spectral densty I 0 as the denomnator n the plot verson of (1), no beneft due to orthogonalty apples and even the plot s own contrbuton appears as nterferer: γ m ˆp η m + ˆµ N γ m p m. (6) The left hand fracton n ths nequalty s called the plot channel s E c /I 0 (CPICH E c /I 0 ) Power control. The qualty of a rado channel can actually not be pnned to a constant factor γ. It vares strongly over tme due to a collecton of dynamcal phenomena grouped under the terms shadowng and fadng. In an UMTS system, these effects are largely made up for by power control. The recever measures the sgnal strength n very short tme ntervals and notfes the sender of any changes, whch then adjusts the transmsson power accordngly to meet the CIR requrements (3) and (5) most of the tme. We do not take ths mechansm nto account but assume perfect power control, that s, we suppose that the sender can adjust ts transmsson power exactly to the mnmum level requred to fulfll the CIR requrement Performance Metrcs for UMTS Rado Networks The qualty or performance of a UMTS rado network s measured by varous scales, see [21,26,25], for example. There s no sngle objectve for optmzng a rado network, but a varety of factors need to be taken nto account Coverage. The user assocates to the network va the plot channel. The qualty of the plot sgnal therefore determnes the network s coverage. The frst condton for coverage s that the plot s absolute sgnal strength E c, the numerator n (6), s suffcently large for the user s moble equpment to relably detect t. The areas n whch ths s the case are sad to have E c coverage. At a locaton wth suffcent E c coverage, the plot sgnal can only be decoded f the plot s E c /I 0 s suffcent, that s, f (6) holds true; the locatons where ths s the case are sad to have E c /I 0 coverage. Unlke E c coverage, E c /I 0 coverage n a certan locaton depends on nterference and, thus, on the traffc load wthn the network. Whle E c problems are lkely to occur at places that are too dstant from any antenna, falure to reach the requred E c /I 0 rato s often a sgn of too much nterference from other cells.

10 UMTS Rado Network Evaluaton and Optmzaton Beyond Snapshots Plot Polluton. Mobles contnuously track the strength not only of the plot sgnal of ther own cell, but also the plot sgnals from neghborng cells f these are receved wth a strength wthn a certan wndow (say, 5 db) below the own cell s sgnal. Ths serves to allow a seamless hand-off n case the user moves and crosses cell borders. Due to hardware constrants, however, the number of plots that can be tracked smultaneously s lmted to only a few. Thus, f too many plot sgnals are audble n a certan locaton, the locaton s sad to suffer from plot polluton Network load. Once that network coverage s gven by suffcently strong and clear plot sgnals, users are capable of establshng a connecton. The most mportant performance measure and the one that s most nvolved to compute essentally ams at determnng how many users, or how much load the network can carry. Servces dffer n ther demand for transmsson capacty. Whether there s suffcent transmsson capacty for all user requests and whch lnk capactes are lkely to be observed are evaluated by network smulatons. Monte-Carlo smulatons are typcally used for ths analyss: User demand realzatons called traffc snapshots are drawn, and each snapshot s analyzed. The analyss of ndependent snapshots s deally contnued untl network performance ndcators are statstcally relable [36]. In practce, however, the number of snapshots to be evaluated s often fxed n advance [26]. The detals of analyzng a sngle snapshot dffer. Each approach n essence has to decde at least whch of the users are served by whch cell(s), the power levels of each actve lnk, and whch users are out of coverage or unserved due to capacty or nterference reasons. Durng ths step, the lmts on user and base staton equpment power are consdered, ncludng the varatons n CIR targets dependng on servce, equpment type, and speed, the effects of varous forms of soft hand-over, etc. As a result, the transmt and receve sgnal power s determned for all cells. These power levels are n turn the bass for dervng E c /I 0 or servce coverage maps, snce they compose the I 0. The downlnk load of an ndvdual cell s measured relatve to a full cell: t s the fracton of the maxmum output power that the cell uses for transmsson. The uplnk load, on the other hand, s measured by the nose rse compared to an empty cell. The nose rse s the rato of total receved power at an antenna to the nose, whch s always present. A nose rse of two (whch means that the sgnals generated wthn the network reach the antenna wth a strength equal to the nose) corresponds to 50 % uplnk load; the load s at 100 % f the nose rse reaches nfnty. The average cell load s actually lmted to values sgnfcantly below 100% to leave room for compensaton of dynamc effects by the power control mechansms and to ensure the system s stablty. Typcally the maxmum average load should not exceed 50% n the uplnk and 70% n the downlnk. We denote the practcal lmt on the total transmt power n the downlnk by Π max.

11 10 Andreas Esenblätter et al. 3. Computng Network Load Ths secton ntroduces systems of lnear equatons that descrbe the up- and downlnk load per cell and the couplng among the cells. The descrpton s dealzed snce no transmt power or nose rse lmtatons are taken nto account. For the sake of smplcty, we also assume that no moble s n soft hand-over. Wth the necessary precautons, solvng these systems nevertheless allows to get an estmate of network load wthout extensve smulatons. We wll use ths method to quckly assess network load durng local mprovement methods. Moreover, these systems are at the core of an alternatve network optmzaton model proposed n Secton 5. The systems were ntroduced and extended n [31,37,7,6] to speed up an central operaton n Monte-Carlo smulaton for the performance assessment of UMTS rado network (see Secton 2.2). The orgnal dervaton for the equaton systems s therefore based on user traffc n the form of traffc snapshots. We follow ths approach to ntroduce the systems, but we then extend t to derve the systems based on average traffc load dstrbuton or to combnatons of average load and a traffc snapshot. The central assumptons are that all users are served and that all CIR targets n the uplnk (3) and downlnk (5) are met at equalty. We start from a network desgn wth nstallatons N and a traffc snapshot wth mobles m M, usng the notaton from Table Uplnk. Concernng the uplnk at antenna nstallaton j, recall that p j s the total amount of receved power ncludng thermal and other nose. Under the above assumptons elementary transformaton of the equalty verson of (3) allow to derve two quanttes for every moble m served by nstallaton j: Frst, the transmsson power p m of moble m gven the total receved power p j at the servng nstallaton j. Second, the fracton of the total receved power at the nstallaton j orgnatng n moble m: p m = 1 γ mj µ m 1 + α mµ m p j (7) α m γ mj p m p j = α m µ m 1 + α m µ m (8) It s convenent to defne the uplnk user load l m of a moble m as the rght hand sde of (8): α m µ m lm := 1 + α m µ m We break down the contrbutons to the total receved power p j at nstallaton j n dependence of all uplnk connectons (not just the ones served by j). Let (9)

12 UMTS Rado Network Evaluaton and Optmzaton Beyond Snapshots 11 M M denote the set of all users served by nstallaton. Then (2) reads as p j = γ mj α m p m + γ mj α m p m + η j. m M j j m M Defnng the nstallaton uplnk couplng factors C j as C j := γmj m M γ lm, m and substtutng (7), the ndvdual uplnk transmsson powers can be wrtten as p j = C jj p j + j C j p + η j. (10) Thus, the total power receved at the nstallaton s composed of three contrbutons, those from the own cell, those from other cells, and nose. The quantty C jj measures the contrbuton from the own users, and C j scales the contrbuton from nstallaton. The matrx C := ( C ) j 1,j N s called the uplnk cell load couplng matrx. Collectng (10) for all nstallatons and wrtng η for the vector of nose values, we obtan the desred system of lnear equatons governng the uplnk cell recepton powers: p = C p + η. (11) Under the assumptons stated n the begnnng of ths secton, the soluton of (11) s the uplnk receved powers at each nstallaton. Necessary and suffcent condtons on C for the exstence of postve and bounded solutons to (11) are dscussed n [6, 7] Downlnk. In the downlnk case, we bascally repeat what has just been done for the uplnk. The startng pont s the CIR constrant (5). Assumng that the constrant s met sharply for all mobles t can be rewrtten as 1 + ω m α m µ m α m µ α m p jm = ω m p j + m j γ m γ jm p + η m γ jm We defne the downlnk user load of servng moble m as: lm α m µ m := 1 + ω m α m µ. (12) m

13 12 Andreas Esenblätter et al. Recall that the break-down of the downlnk transmsson power at nstallaton j s p j, as defned n (4). Smlar to the uplnk case, further notaton s helpful to express the dependency of p j on the downlnk transmsson power at all nstallatons. We ntroduce the downlnk couplng factors C jj := m M j ω m l m and C j := m M j γm γ lm jm ( j) for nstallatons and j as well as the nstallaton s traffc nose power p (η) j := η m m M j γ jm l m. The transmt powers at the nstallatons satsfy the expresson p j = C jj p j + j C j p + p(η) j + ˆp j + ˇp j. (13) The nterpretaton of C jj, C j, and p(η) j are as follows. The frst term, C jj, captures the effects due to ntra-cell nterference. The second term, C j, quantfes the fracton of transmsson power spent on overcomng nterference from other nstallatons. The thrd quantty, p (η) j, states how much transmsson power s needed to overcome the nose at the recevers f no ntra-system nterference were present. We call C := ( C ) j 1,j N the downlnk load couplng matrx. Fgure 1 contans a graphcal representaton of the downlnk couplng matrx. The connectons between each par of antennas are shaded accordng to the absolute value of the sum of the two correspondng off-dagonal elements. Darker shades ndcatng hgher values. The dfference between Fgs. 1(a) and 1(b) shows the decouplng effect that can be acheved by tltng. Agan, equaton (13) for all antenna nstallatons n the network form a lnear equaton system that descrbes the downlnk transmt power n each cell: p = C p + p (η) + ˆp + ˇp (14) The same qualfcatons as dscussed n the uplnk case hold Generalzed Couplng Matrces. We have to assess network load as part of network optmzaton, for example, n order to fnd out whether a tentatve network desgn s capable of supportng the offered load. In ths context, the above equaton systems have two shortcomngs:

14 UMTS Rado Network Evaluaton and Optmzaton Beyond Snapshots 13 (a) Orgnal network (b) Optmzed electrcal tlts Fg. 1. Effect of tlt adjustng on mutual nterference, Berln scenaro The equaton systems allow to determne the network load for one snapshot. In order to obtan statstcally relable results many snapshots, hundreds or even thousands, have to be evaluated. Ths s computatonally prohbtve f the load evaluaton shall be used wthn a local search approach and has to be executed durng most steps. The results from the load calculaton become meanngless once the network s n overload. We address the frst ssue here. In order to obtan an estmaton of the network load based on the analyss of only one equaton system, we derve the couplng matrces and the downlnk traffc nose power vector on the bass of average traffc load dstrbutons. Ths approach s appealng because the computatonal complexty s radcally dmnshed. The speed-up comes, however, at the expense of an estmaton error. To our knowledge, there s no extensve research or upper bound for ths error. It s reported [30] that for a mono-servce stuaton the cell load calculaton on the bass of the average load dstrbuton leads to a systematc underestmaton of user blockng, but the correspondng error s estmated as less than 10%. A more rgorous error analyss s n demand to provde the cell load estmates wth the necessary accuracy estmates. We denote by A the total plannng area and by A A the best server area of cell. Let S be the set of servces under consderaton, let T s be the servcespecfc spatal traffc dstrbutons, and let T s (p) denote the average traffc ntensty of servce s at locaton p (for some specfc pont n tme, e. g., durng the busy hour). We assume that the traffc ntensty measures the quantty of concurrent (packet) calls at locaton p. The poston p specfes a two-dmensonal

15 14 Andreas Esenblätter et al. coordnate at some reference heght, e. g., 1.5 m, or a three-dmensonal coordnate. For the sake of a smpler exposton, we assume that each servce s has reprasentatve CIR targets µ s, µ s and actvty factors α s, α s n up- and downlnk; we also assume a locaton-specfc nose η p at a moble n poston p. Instead of usng γ mj, γ jm, ω m for the connecton between nstallaton j and a moble at locaton p, we smply wrte γ jp, γ jp, ω p. Wth these conventons, the snapshotbased defntons of couplng matrces and downlnk traffc nose power can be easly generalzed to traffc load dstrbutons. The basc load defntons (9) and (12) are replaced by lp := α s µ s s S 1 + ω p α s µ T s (p). s The uplnk couplng matrx s gven by C jj p A := lp dp, C j := j γjp p A γ p l p dp, the downlnk couplng matrx and the traffc nose power are gven by C jj p A := ω p lp dp, C j := γp lp dp, η p j p A j γ jp p(η) j := p A j γ jp lp dp. (15) The above defntons can be generalzed further n two drectons. Frst, the defntons also make sense f some part of the traffc s taken accordng to average load dstrbutons and another part from a snapshot. Ths can be helpful for Monte-Carlo smulatons: servce usage well approxmated by average behavor can be treated through average load maps, whle hgh data-rate users wth bursty traffc can be analyzed at snapshot-level. Second, the best server nformaton can be replaced by assgnment probablty nformaton, ndcatng the probablty wth whch some pont s served from an nstallaton. 4. Mxed-Integer Programmng Model Based on Snapshots In ths secton we sketch how to model a snapshot based MIP for the UMTS plannng problem. Detals on ths and a full account of the model can be found n [10,14]. Besdes the set of potental antenna nstallatons for each cell of the rado network under consderaton, the nput to the model conssts of a set D of snapshots, each snapshot contanng many users. Every user belongs to a specfc snapshot d D. The model allows to select a rado network desgn by pckng one antenna confguraton for each cell. Dependng on ths desgn choce, the qualty of the current soluton s assessed n each snapshot. To ths end, a decson has to be made for each user whether t s dropped (left wthout servce) or served, and f so, by whch cell. For the served users power levels for each lnk have to be determned such that all CIR requrements are met. Our objectve s to maxmze the number of served users.

16 UMTS Rado Network Evaluaton and Optmzaton Beyond Snapshots Outlne of the Model Bnary varables z, I are used to ndcate whch nstallatons are used n the network. It s requred that exactly one potental nstallaton s chosen for each cell. Bnary varables x m, m M, I, for each par of moble m and nstallaton ndcate whether or not m s served by. We requre x m z to ensure that only selected nstallatons serve mobles. Bounded contnuous varables are used for the uplnk power level p m of moble m, for the total receved power p d at nstallaton n snapshot d, for the downlnk transmsson power level p m, and for the for the total downlnk transmsson power p d of nstallaton n snapshot d. The transmsson powers of the plot ˆp and the other common channels ˇp are fxed. If an nstallaton serves a moble, the requred carrer-to-nterference ratos for up- and downlnk have to be fulflled. To ensure ths, we multply the rght hand sde of the CIR nequaltes n the uplnk (3) and n the downlnk (5) by x m. The resultng quadratc nequaltes s lnearzed usng a bg-m formulaton. We show ths exemplarly for the downlnk. Defnng the denomnator of the modfed downlnk CIR nequalty (5) as φ(m, ) = ω m γ m ( p ρ(m) α m p m ) + j I j γ jm p jρ(m) we can wrte Settng bg-m to γ m p m µ mφ(m, ) x m + µ mη m x m. Θ m = ω m γ m Πmax + j I j γ jm Πmax j + η m we obtan a lnearzed verson of (5) sutable for the MIP model γ m µ m p m φ(m, ) Θ m x m η m Θ m. (16) The η m on the rght-hand sde cancels out when Θ s expanded. Mxed nteger roundng (MIR) cuts, see for example [33,28], can be added to the model n order to tghten the lnear relaxaton of the model. The followng vald nequalty s derved from the modfed downlnk CIR constrant: ( ) γ x m 1/µ m + ω mα m m p m η 0. m Note that p s dropped from the constrant. Ths s feasble, because ts value has to be non-negatve and ts coeffcent s one. The uplnk case can be dealt wth n a smlar manner.

17 16 Andreas Esenblätter et al Assessment We conducted extensve computatonal experments [9, 11] wth snapshot-based models usng several dfferent objectves. In the cted publcatons the goal was to mnmze network cost whle mantanng acceptable qualty. For the computatonal results n ths paper, we essentally fxed the network cost and tred to optmze the qualty of the network. All our computatonal studes, ncludng the ones presented n Secton 6, reveal some shortcomngs of the snapshot model. The most serous ssue s ther poor solvablty due to the very hgh dynamc range of the nput data and excessve symmetres. The hgh dynamc range s due to the necessty to deal wth attenuaton values n lnear scale. Relevant attenuaton coeffcents γ vary between 60 db and 160 db. They apply to output powers p m of mobles n the range of 50 dbm up to 21 dbm. Ths results n a dynamc range of receved powers of 171 db; these are 17 orders of magntude. Wth these dynamcs, we are on the very border of precson for double precson IEEE floatng pont arthmetc. The symmetres are caused by alternatve but relatvely smlar nstallatons and mobles located close to each other. A sgnfcant and bad consequence of the MIP s poor solvablty n our studes s that only very few snapshots could be handled n an optmzaton run; often only a sngle snapshot. Ths s smply not enough to obtan sgnfcant optmzaton results, snce the whole dea of snapshot evaluatons s based on the hope of convergence for a hgh number of snapshots. Furthermore, the model has no noton of a coverage area. Thus, coverage s often neglected n very low traffc areas. A possble remedy would be the ntroducton of artfcal snapshots to cover these areas. Last but not least, the number of varables n the model grows wth the number of potental nstallatons tmes the number of mobles. Even though we apply elaborate preprocessng to remove mpossble combnatons, ths severely lmts the scalablty of the model. To overcome these problems of poor numercs and scalablty, we present a new optmzaton model n the next secton. That model s based on the compact evaluaton method presented n Secton 3 and works on the bass of average load rather than snapshots. 5. Optmzng the Couplng Matrx The performance propertes of a partcular network desgn on a gven snapshot are to a large extent descrbed by the couplng matrces C and C derved n Secton 3. The generalzed average couplng matrces descrbed n Secton 3.3 provde an estmaton of the network propertes that s ndependent of sngle snapshots. Ths leads to the dea of lookng at rado network desgn as a problem of desgnng a good average couplng matrx. At the core of the model developed n ths secton resdes a lnear descrpton of the couplng matrx for the potental network desgns. In addton, we mpose coverage constrants nspred by the performance metrcs descrbed n Secton 2.2. Our presentaton wll focus only on the downlnk and on the average-

18 UMTS Rado Network Evaluaton and Optmzaton Beyond Snapshots 17 load based descrpton wth best server areas as presented n Secton 3.3, an extenson to the uplnk drecton s obvous. Agan, we descrbe any possble network desgn by an ncdence vector z {0, 1} I. For a feasble network desgn, we have to choose exactly one nstallaton for each cell. We call the set of network desgns that fulfll ths condton F Determnng the Entres of the Couplng Matrx Recall the entres of the average couplng matrx as gven n (15). All entres n row of the couplng matrx are computed by ntegratng over the area A served by cell. For determnng the elements of the couplng matrx, t s therefore crucal to determne these best server areas Server of Sngle Ponts. Consder a pont p n the plannng area. We ntroduce a decson varable c (p) that expresses whether or not p s served by nstallaton. Ths s the case f and only f a) nstallaton s selected and b) no nstallaton wth a stronger sgnal at p s selected. In the example n Fg.2, p s served by f and only f nstallaton j (and not k) s selected for the rght-hand cell. Assumng that the plot powers of all antennas are set to the same value, ths depends only on the attenuaton values γ. We denote by D (p) the set of all nstallatons that domnate at p: { } D (p) = j I : γ jp > γ p In the example, only k domnates at p, so D (p) = {k}. We have the relaton c (p) = 1 z = 1 z j = 0 j D (p). Ths relaton can be expressed by lnear nequaltes as follows: c (p) z z g c (p) z g D (p) c (p) 1 z g g D (p) (17) Parttonng the Plannng Area. The above constructon can be carred out for all ponts n the plannng area. The area served by nstallaton s then the set of all ponts p A wth c (p) = 1. However, ths lnear descrpton of the area served by may be very large On the other hand, the descrpton for a sngle pont p depends only on the set D (p) of nstallatons that domnate at p: all ponts wth the same set of nstallatons domnatng wll lead to the same nequaltes (17). We aggregate these ponts nto one set. Ths yelds a partton of the plannng area A accordng to the nstallatons that domnate. For a gven

19 18 Andreas Esenblätter et al. D (p) = {k} p j k A ( ) A ({k}) A ({j, k}) Fg. 2. Example set D I, / D of nstallatons, we defne A (D) as the set of ponts n whch exactly the nstallatons n D domnate : { } A (D) := p A : D (p) = D } = {p A : γ jp > γ p for j D, γ jp γ p for j / D These sets do not actually for a parttonng of the area, snce there mght be many ponts recevng the same sgnal strength from dfferent nstallatons. Ths can be resolved by breakng tes arbtrarly (whch s on all accounts vald f the set of such ponts s very small). However, a more refned verson of ths model should nclude soft handover and thereby deal wth regons of smlar sgnal strength from several nstallatons n a dfferent way. Assumng that there are no tes, we have A = A (D). D I The example parttonng generated by nstallaton s shown n Fg.2. Ether all ponts n A (D) are served by or none, dependng on whether or not any nstallaton n D s selected. We thus defne a bnary varable c (D) {0, 1} statng wether or not ths s the case, and couple t to the decson varables z n analogy to (17). For a gven network desgn z F we then know that A (z) = D I : c (D) =1 A (D). (18) We obtan a mathematcal programmng model that descrbes the best server area of any network desgn. Ths can be used to calculate both the man dagonal entres of the downlnk couplng matrx and the (nterference dependent) offdagonal entres.

20 UMTS Rado Network Evaluaton and Optmzaton Beyond Snapshots Calculatng Man Dagonal Entres and Nose Load. The man dagonal entres of C are gven as C jj := p A j ω p l p dp. Usng the parttonng (18) of A, we can calculate ths value for any network desgn z F by summng the contrbutons to C jj on each set: C (z) = ( ) ω p lp dp c (D) (19) D I p A (D) The nose load p (η) can analogously be determned: p (η) (z) = ( ) η p D I p A (D) γ lp dp p Calculatng Off-Dagonal Entres. The same prncple can be used to determne the off-dagonal elements of C. However, there s one more dffculty. The man dagonal elements only depend on the servce area A of nstallaton, whereas the off-dagonal element C j are also nfluenced by the settngs for nstallaton j. We thus ntroduce another dependent bnary varable c (D) j that specfes whether or not any contrbuton to C j s generated on A (D). Ths s the case f and only f A (D) belongs to the servce area of nstallaton and nstallaton j s selected c (D) j = c (D) Ths product of bnary varables can be transformed to three lnear nequaltes n the canoncal way: The entry C j to (19): c (D) j c (D) j c (D) j z j c (D) z j c (D) + z j 1 c (D) s now determned for any network desgn z F n analogy C j (z) = D I ( γ jp p A (D) γ p l p dp ) Techncally speakng the matrx C (z) as defned here has dmenson I I. The network s actual couplng matrx C of dmenson N N can of course be obtaned by deletng the all-zero rows and columns of nstallatons that have not been chosen. The analogue apples to p (η) (z) and p (η). c (D) j

21 20 Andreas Esenblätter et al Coverage. There s another condton for a pont to be served by nstallaton : the plot sgnal has to be receved wth suffcent absolute strength (E c coverage) and sgnal qualty (E c /I 0 coverage) for the user to regster wth the network. We only treat the frst pont here. Assumng a fxed plot power level, ths means that the attenuaton γ p to any pont p served by n queston must not fall below a threshold value γ. Ths condton can easly be added to the defnton of A (D). For mposng addtonal coverage constrants (e. g. that 99% of the area A must be covered by the network), we can then use the area covered by the cell of nstallaton : A (z) = D I A (D) c (D) The total area covered by the network s A(z) = I A (z). The load covered by the ndvdual cells can be read off the man dagonal elements of the matrx Objectve and Addtonal Constrants The model presented n the prevous paragraphs can be used to desgn the downlnk couplng matrx C n a varety of settngs. For the followng, mnmzaton of the total downlnk load s emphaszed, that s, mnmzng the sum 1 T p of the components of the downlnk power vector. Ths vector cannot smply be determned by desgnng the matrx, snce t s the soluton of the (scaled) downlnk couplng equaton (14). Solvng (14) when desgnng the matrx falls out of the scope of a lnear optmzaton model. However, at the same tme t should be ensured that other performance measures are not compromsed when mnmzng downlnk load. We present one way to formulate a full optmzaton model for network desgn based on the matrx entres, others are lkely to be nvestgated n the future. In the case of network tunng, there s an orgnal network desgn z 0 on whch we want to mprove. The assocated power vector p (z 0 ) can be computed for ths network desgn as descrbed n Secton 3. We propose to use the followng optmzaton model for network desgn: mn z F ( ) 1T C (z) p (z 0 ) + p (η) (z) s. t. A(z) A(z 0 ) (20) 1 T dag ( C (z) ) 1 T dag ( C (z 0 ) ) (21) The basc dea s to approxmate the quanttes that cannot be computed n a lnear model by the referrng values of the reference network. Ths s vald as long as the optmzaton result s smlar to the reference network. In the objectve functon, we use the downlnk power vector p (z 0 ) n the rght hand sde of the fx pont form (14). Constrant (20) ensures that E c coverage does not dmnsh compared to the reference network. An analogue to ths condton s stated n (21), but the covered area s here weghted wth the user load. Ths

22 UMTS Rado Network Evaluaton and Optmzaton Beyond Snapshots 21 makes sure that coverage n hghly frequented areas s not traded for coverage n areas wth lttle traffc. When producng good solutons n the sense of ths objectve functon, dfferent ssues play a role: Interference reducton s rewarded: cells try to mnmze ther nfluence on the neghbors Nose load s reduced: cells try to mprove the attenuaton to ther own mobles Load s shfted away from overloaded cells to empter cells These ponts (whch are to some extent contradctory) suggest that network desgns that mprove on the objectve functon also have better performance propertes. Computatonal experments (cf. Secton 6) confrm ths. However, there are some dsadvantages of ths approach. Frst, ths optmzaton model depends largely on the nput network. Dfferent nput networks wll lead to dfferent optmzaton results. When usng ths model teratvely, no stablzaton of the result could be observed as of yet. Second, f the used approxmaton s not vald on the entre set F, the objectve functon mght be msleadng and results of less qualty than the reference network desgn are produced. Ths has also been observed n computatonal experments Model Sze The full model as descrbed above s of exponental sze n the nput length, snce the contrbuton varables are ndexed wth all subsets D I. However, the relevant sets of domnatng nstallatons are comparatvely small n practce, and t s crucal to determne these sets beforehand. Ths s done n a preprocessng step. The area A s usually descrbed n the fnte form of a pxel grd. We determne for each element p A and for all I wth γ p γ the domnator set D (p). There are at most A I such sets, leadng to a maxmum of A I varables c (D). However, ths number s n practce consderably reduced by the E c -coverage condton, snce t typcally admts only cells n a regonal vcnty of p. The constrant that one nstallaton per cell must be selected can be used to further reduce the number of relevant domnator sets n an obvous way. The potental nterference sources per pxel s at most I ( I + 1)/2. The number s thus O( A I 3 ). At the expense of accuracy n determn- and c (D) j wth very small contrbutons of varables c (D) j ng the matrx entres, varables c (D) p A ω (D) p lp dp and p A (γ (D) jp /γ p ) l p dp, respectvely, can be deleted from the model. Example model szes for the plannng scenaros used for computaton are lsted n Table 2. In all cases, three potental nstallaton confguratons are allowed for each central cell; cells at the border of the scenaros were not vared. The nstallaton confguratons dffer only n tlt (mechancal and electrcal). All domnator sets and hence all varables c (D) are ncluded n the model. Varables

23 22 Andreas Esenblätter et al. Table 2. Sze of Matrx Desgn Models Scenaro Cells Optons Bnary Vars Constrants Nonzeroes The Hague , , ,921 Berln , , ,409 Lsbon , , ,842 are deleted f ther potental contrbuton to the referrng off-dagonal element s less than Ths ensures that man dagonal elements and nose load are computed wth the MIP solver s precson. The off-dagonal elements have n all tests been computed wth an absolute error no larger than In general, the model szes are acceptable and stll tractable wth standard MIP solvers. A lttle surprsng s the large sze of the model for The Hague, whch has the smallest number of cells. Ths s due to some peculartes n the data; rado sgnals propagate wth lttle attenuaton across the scenaro, so mutual couplng s very hgh. c (D) j 6. Computatonal Results Ths secton descrbes the computatonal results from the models descrbed n Sectons 4 and 5. The results are analyzed wth the methods outlned n Secton 3. We focus on adjustng the tlts of the antennas n a gven rado network startng from reference networks based on GSM settngs. The potental mpact of adjustng tlts s descrbed n Secton 1.3, a graphcal llustraton can be found n Fg. 1 on page The Scenaros Three real-world scenaros developed wthn the Momentum project [32] are used as test cases. We gve only a bref descrpton of the scenaros, the detals can be found n [34,15,12,13] General descrpton. The scenaros cover the downtown areas of The Hague, Berln, and Lsbon; some key nformaton can be found n Table 3. The total (downlnk) load n the scenaro as gven n the table s A l p dp. The locatons of base statons are called stes, there are commonly three antennas and thereby three cells assocated to each base staton. Table 3. The scenaros Avg. Total Area Name Stes Cells Users DL-Load km 2 The Hague Berln Lsbon