Approximating Blocking Rates in UMTS Radio Networks

Transcription

1 Approxmatng s n UMTS Rado Networks Dplomarbet vorgelegt von Franzska Ryll Matrkel-Nr.: be Prof. Dr. Martn Grötschel Technsche Unverstät Berln Fakultät II: Mathematk und Naturwssenschaften Insttut für Mathematk Studengang Wrtschaftsmathematk Aprl 2006

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3 De selbständge und egenhändge Anfertgung verschere ch an Edes statt. Berln, den

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5 Acknowledgements I am deeply grateful to Hans-Floran Geerdes for the productve support throughout my whole tme at ZIB. Moreover, I thank Dr. Andreas Esenblätter for hs constructve advces. Both were good revewers and gave helpful suggestons. I acknowledge the great workng condtons at ZIB. In ths connecton, I am ndebted to Prof. Dr. Martn Grötschel gvng me the opportunty to develop my dploma thess n such a professonal surroundng. I thank Holger van Bargen for supportng me n topcs concernng the Central Lmt Theorem. Furthermore, I am grateful to my famly and Sören for ther encouragement and support.

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7 Contents 1 Introducton 1 2 Prelmnares Bascs of Wreless Communcatons Cellular Moble Rado Networks Rado Network Plannng Specfcs of UMTS CDMA Interference n CDMA Systems Blockng n UMTS General Mathematcal Model Statc Vew Input Data and Assumptons CIR Constrants and Blockng Exstng Methods System of Equatons Snapshot Based Dervaton Expected Couplng Approxmatng Blockng Monte Carlo Smulaton Shortcomngs The as Expected Value Constant User Load Prelmnares Uplnk Downlnk Extenson Varable User Load Prelmnares

8 v Contents Uplnk Downlnk The Effect of Couplng The Assumpton of a Normal Dstrbuton The Central Lmt Theorem Transformaton Proof Dscusson Power Knapsack Snapshot Based Dervaton Downlnk Uplnk The Expected Power Knapsack Downlnk Uplnk Comparson of both Methods Uplnk Downlnk Computatonal Results Implementaton Test Cases Valdaton One Cell Two Cells Results Runnng Tme Conclusons Summary and Outlook 85 A Notaton 87 B Further Results 91 C Zusammenfassung 111 Lst of Fgures 115 Bblography 118

9 1 Introducton The frst natonwde moble telecommuncatons system was nstalled n Germany n Untl then, regonal systems covered only small areas, lke ctes or communtes, and a customer could not use hs moble rado unt n another regon than hs servng one [5]. Stll n the begnnng of the 1980s, moble communcatons were not wdespread because the fees and prces for termnal equpment were too hgh for many people. Wth the lberalzaton of the telecommuncatons market and the ntroducton of a unfed standard for dgtal cellular moble rado systems n 1992 [2], moble communcatons developed to a mass market n the 1990s [22]. Ths ntroduced standard s called the Global System for Moble Communcatons (GSM). GSM s sad to belong to the second generaton of moble phone technology followng the frst generaton of analog rado networks [5]. Although the transmsson of data s possble n GSM besdes speech telephony, the system s nadequate for varous applcatons that requre hgher bt rates [22]. Ths s one reason why the Unversal Moble Telecommuncatons System (UMTS) was developed. UMTS s a thrd generaton cellular moble phone technology, whch s deployed commercally n Germany snce 2004 [24]. Wth UMTS, t s possble to transmt at varable data rates of up to 384kbt/s [2]. Ths s 40 tmes faster than the connecton speed GSM offers [23]. Therefore, a varety of new servces s avalable lke multmeda applcatons or vdeo transmssons. Furthermore, UMTS systems are more resstant aganst falure than second generaton rado networks. For example, connectons break off less often when a moble user moves. The provders of moble communcatons want to offer a system that covers a large area wth hgh qualty servces and acceptable prces to ther customers. For ths reason, t s mportant to desgn effcent rado networks. One fundamental topc n rado network plannng s the capacty of a rado network. Ideally, a suffcent amount of rado resources has to be provded for all users to establsh a connecton. However, n practce ths s not always possble. Due to the lmtaton of rado resources a moble user mght not be served. The rejecton of a customer who wants to establsh a new connecton

10 2 1 Introducton to the rado network s called blockng. One of the goals when desgnng a rado network s that the rato of rejected mobles called the blockng rate does not exceed a certan threshold. In UMTS, the mult user access scheme Code Dvson Multple Access (CDMA) s employed n the rado nterface, the nterface between the user and the rado network. Because of ths technology, the capacty of the rado network s not fxed and therefore not known exactly durng the plannng phase. The capacty depends on the current nterference stuaton n the rado network whch n turn depends among others on the number and poston of smultaneous mobles and the knds of servces they use. For ths reason, n the case of UMTS, we speak of soft capacty [5]. Ths specal characterstc of UMTS systems makes t dffcult to determne the maxmum number of users the rado network s able to carry. Consequently, t turns out to be complcated to predct the average blockng rate of the rado network relably. Varous methods have been proposed to assess the consumpton of resources n a UMTS rado network, whch must be known n order to determne the blockng rate of the system. There s a trade-off between accuracy and effcency n all of these models. Ther nadequaces led to the necessty of an mproved method for the calculaton of the blockng rate of a UMTS rado network. One such method s ntroduced n ths dploma thess. The thess develops and analyzes a model to effcently approxmate the average blockng rate of a UMTS rado network. In dong so, we consder a moment durng the perods, when the average expected amount of traffc s hghest. One such perod s called the busy hour. The presented method s based on a stochastcal estmaton of the average nterference n the system. Wth ths model, t s possble to predct the average blockng rate of a confgured rado network quckly durng the plannng phase. Shortcomngs n the rado network desgn can thus be detected easly. Outlne Chapter 2 presents a short survey of UMTS and ts rado technology. Among others, cellular rado networks are ntroduced, as well as the mult user access technque CDMA. Furthermore, blockng s dscussed n more detal and the dffcultes of assessng the blockng rate of UMTS rado networks are descrbed. A common mathematcal model s gven that represents a UMTS rado network n a statc way. All these topcs are summarzed from comprehensve lterature studes. In Chapter 3, establshed methods to determne the average blockng rate of a UMTS rado network are ntroduced. In one approach, a system of equatons s set up to compute the average blockng rate. Ths approach can

11 1 Introducton 3 be used n two ways. One way of usng ths method s very tme consumng whle the other one leads to unacceptable estmaton errors. The extensve verson s a numercal method known as Monte Carlo smulaton. The basc prncple of ths popular method s explaned brefly. The nadequaces of these approaches are the motvaton to propose another model that reduces ther shortcomngs. The next three chapters cover a new method to approxmate the blockng rate of a UMTS rado network. Ths method s based on the expected couplng approach presented n the precedng chapter. The new model s ntroduced n two dfferent ways n Chapter 4 and n Chapter 5. In both cases, the expected value of the average blockng rate s computed. In dong so, the nterference of other rado cells s estmated by constant values whle the nterference of the own rado cell s modeled stochastcally. Chapter 6 shows analytcally that the results of both approaches are equal. The results of extensve computatonal tests are presented and analyzed n Chapter 7. These results are compared to outputs of Monte Carlo smulaton. Fnally, n Chapter 8, a summary of the entre thess and an outlook are gven.

12 4 1 Introducton

13 2 Prelmnares Ths chapter ntroduces necessary prelmnares for ths thess brought together from several lterary sources. A short overvew of the techncal bascs of general cellular rado networks s gven as well as the specfcs of the UMTS technology. Moreover, a common mathematcal model for the explaned features s shown. The chapter s organzed as follows. Frst, some bascs of wreless communcatons are mentoned. Then, characterstcs of cellular rado networks are ponted out. At the same tme, basc notaton s ntroduced. Afterwards, the tasks and purposes of rado network plannng are hghlghted. In the next secton, the partculartes of UMTS are dscussed. We deal wth the access scheme CDMA and the consequences for the system caused by ths technology. Furthermore, the resultng dffcultes for the computaton of the blockng rate are revealed. Fnally, a statc mathematcal model whch s the bass for the consderatons n ths thess s gven. Whenever we use the term network throughout ths thess, a rado network s actually meant. 2.1 Bascs of Wreless Communcatons A communcatons system conveys messages. These messages orgnate n an nformaton source and are transmtted to a destnaton. Bascally, there are three components n the communcatons ppe: the transmtter, the channel and the recever [12], as shown n Fgure 2.1. The transmtter and the recever are dstant of each other. The physcal manfestaton of a message s a sgnal [14]. The transmtter adapts the sgnal of the nformaton source such that t can be transmtted over the channel. In wreless communcatons systems, the transmsson medum delverng the sgnal from the transmtter to the recever s a rado channel. Durng transmsson n space, the channel s mpared by nterference and nose. Interference orgnates from other sources occupyng the same frequency band. Nose s generated by electronc devces 5

14 6 2 Prelmnares Informaton Transmtter Channel Recever Informaton Source Snk Fgure 2.1: Block dagram of a communcatons system, based on [12, p. 4] at the recever. Fnally, the recever creates an estmate of the orgnal sgnal out of the receved nformaton-bearng sgnal. An exact reproducton s not possble because of the mentoned nfluences on the rado channel (cf. [12]). The frequency range occuped by the energy of a sgnal s denoted by the bandwdth of the sgnal. The bandwdth of the communcatons system s the frequency range the rado channel s able to transmt. The power spectrum of a sgnal descrbes the dstrbuton of the sgnal s power along the dgtal frequency range [14]. The power spectrum can be understood as a functon of frequency. Then the ntegral over the entre bandwdth represents the average power of the sgnal [12]. The power of narrowband sgnals s concentrated on a relatve small bandwdth. In contrast, wdeband sgnals have a wde bandwdth. 2.2 Cellular Moble Rado Networks A cellular rado network conssts of a set of base statons whch are set up n the terran. At each base staton, one or more antennas are nstalled. The electromagnetc sgnals they emt are conveyed n space and are attenuated on ther way to the recever. The complete attenuaton on the rado channel s called path loss. The hgher the dstance from the transmtter s, the weaker s the power of the recevng sgnal at a specfc poston. However, ths power has to be suffcently hgh n order to establsh a connecton to the rado network. Because the maxmum power of an antenna s restrcted ths leads to a regonal confnement of the rado sgnal range. The complete area s dvded nto so-called cells or sectors (cf. [2]). Users (moble statons) n one sector are served by a certan base staton antenna. Ths s usually the one that provdes the strongest sgnal n ths regon. Ths antenna s called the best server. Mostly, cells are coherent areas whch overlap partly [22]. In Fgure 2.2, the cellular structure of a UMTS rado network s shown. Besdes the cells, the fgure depcts the locatons where the antennas could be set up (red ponts) and the nstalled antennas (black arrows). Cellular systems for moble telecommuncatons are organzed centrally.

15 2.3 Rado Network Plannng 7 Fgure 2.2: Cells n a UMTS rado network That s, users are not lnked drectly to each other but to central rado statons [5]. These connectons can be seen as pont-to-pont connectons [12]. Lnks of a base staton antenna to exactly one moble are denoted by dedcated channels. In contrast, common channels are used by all mobles of one cell [2]. There are two drectons of communcaton n rado networks. In the downlnk drecton, the base staton antenna transmts sgnals to the moble staton. The reverse drecton s called uplnk (cf. [23]). The capacty of a cell denotes the maxmum number of users the assocated antenna can serve wthout excessng ts avalable resources. If the capacty of an antenna s exhausted, users tryng to establsh a new connecton are left unserved. They are blocked. 2.3 Rado Network Plannng The purpose of rado network plannng s to create a rado network wth good performance for the expected demand at low costs. There are varous ndcators to estmate the performance of a confgured rado network. One such ndcator s the qualty of a servce. Another mportant crteron s the sze of the coverage area. Ths s the area where the sgnal can be receved wth suffcent strength to establsh a connecton. The blockng rate of a network also represents a measure for the performance of a network desgn. Usually, ths value should le below 2% n an acceptable rado network [23] such that hgh avalablty of good qualty servce s ensured (cf. [16]).

16 8 2 Prelmnares The problem of rado network plannng occurs n dfferent forms. In the so-called Green Feld Plannng, a complete rado network shall be desgned from scratch. Nowadays, ths s not relevant practcally snce base statons and antennas are already nstalled n large regons. More nterestng are the problems of Ste Selecton and Network Tunng. In the frst one, a subset of exstng stes s chosen where UMTS antennas are set up. In the second task, the qualty of an exstng rado network shall be mproved by changng the confguraton of the antennas, e. g. ther heght, ther azmuth angle n the horzontal plane or ther tlt angle n the vertcal plane. As much users as possble shall be provded wth hgh qualty servces. At the same tme, the arsng expenses for the deployment and mantenance of the network shall be as low as possble. That s, wth a mnmum amount of rado resources the network desgn whch handles the expected traffc best n terms of the gven requrements shall be conceved. In order to solve ths problem several optmzaton models are proposed. In [6] for example, an approach for optmzng antenna tlts s ntroduced. Wth the method presented n ths thess for quckly assessng the blockng rate of a UMTS rado network, t s possble to refne such models. One could, e.g. nsert an addtonal constrant concernng the maxmum allowed blockng rate of the cells n order to mprove the qualty of the results. 2.4 Specfcs of UMTS Ths secton descrbes the specfc partculartes of UMTS rado networks. Due to ts new access scheme, called CDMA, nterference plays a major role n the desgn of UMTS rado networks. Ths n turn leads to problems when assessng the blockng rates of the cells n the network. These topcs are handled successvely n the followng CDMA In moble communcatons systems, all mobles n a sector use a common physcal resource to transmt and receve sgnals. Ths transmttng medum s a frequency band n the rado spectrum [16]. The smultaneous access of all users to t (mult-user access) has to be controlled n order to avod a loss of nformaton [5]. In the technology GSM, users are separated by Frequency Dvson Multple Access (FDMA) and Tme Dvson Multple Access (TDMA). In FDMA systems, the avalable spectrum s subdvded nto several frequency bands whch are used smultaneously. Each band can be nterpreted as a physcal

17 2.4 Specfcs of UMTS 9 channel whch s assgned to exactly one user. TDMA means that a frequency channel s splt up nto dsjont tme slots. In dong so, every moble conveys sgnals n dfferent perods of tme (cf. [22]). The access technque used n UMTS s Code Dvson Multple Access (CDMA). In contrast to FDMA and TDMA, the complete frequency band s avalable for the total duraton of the connecton to every moble. Due to the smultaneous use of the rado spectrum by all mobles, varous sgnals arrve at the recever. From those, t has to separate the desred one. Ths s done by assgnng a unque code sequence to each lnk (cf. [22, 23]). Fgure 2.3 vsualzes the operatng mode of ths access technque. Code 1 Code 1 Sgnal 1 Nose x Flter 1 x Sgnal Code n Code n Sgnal n x x Flter n Sgnal n Transmtter Rado Channel Recever Fgure 2.3: Operatng mode of Code Dvson Multplex Besdes separaton between the lnks, the code sequences are used to spread the narrowband rado sgnals to wdeband sgnals for the transmsson across the wreless channel. That s, the energy whch was concentrated on a narrow frequency range s then spread to a wder bandwdth. For ths reason, CDMA systems are commonly called spread spectrum systems. The spread spectrum technque deployed n UMTS s Drect Sequence-CDMA (DS-CDMA). That s, the user data stream s multpled by a specfc code sequence whose bt rate s by a multple hgher than the user bt rate. In dong so, the resultng bt sequence has a hgher bandwdth and a lower power spectrum than the orgnal stream. The sgnal s sad to be spread. Fgure 2.4 llustrates the spreadng operaton. At the recever, the arrvng data stream contans addtonally spread bt sequences from other users and other nterferng sgnals. Ths stream s multpled wth exactly the same code sequence used n the spreadng operaton. Ths despreadng process restores the lower bandwdth and the hgher power

18 10 2 Prelmnares User Data Stream User Data Stream 1 1 Tme Power Spectrum Frequency Code Sequence Code Sequence 1 1 Tme Power Spectrum Frequency Resultng Sequence Resultng Sequence 1 1 Tme Power Spectrum Frequency Behavor n Tme Behavor n Power Spectrum Fgure 2.4: Wdeband spreadng, based on [5, p. 221] spectrum of the orgnal user bt stream. At the same tme, the power spectrum of nterferng narrowband sgnals, such as thermal nose, s decreased because they are spread now. Narrowband means that the bandwdth s sgnfcantly smaller than that of the spread user sgnal. The wdeband sgnals from other users reman wdeband, and thus ther power spectrum remans low. Hence, the power spectrum of the desred sgnal s ncreased relatve to the power spectrum of the nterferng sgnals. Afterwards, the resultng product s fltered wth a flter adapted to the current sgnal [2]. The whole operaton at the recever n case of narrowband nterference s depcted n Fgure 2.5. The property of CDMA to reduce nterferences, especally those orgnatng from other smultaneously proceedng calls, s fundamental n order to reuse the avalable frequences over geographcally close dstances. Ideally, the codes of the dfferent users are perfectly orthogonal such that they are ndependent and the dfferent physcal channels do not dsturb each other. A more detaled descrpton of code spreadng can be found n [5, 23, 13].

19 2.4 Specfcs of UMTS 11 Power Spectrum Interference Informaton Frequency Before Despreadng Despreadng Power Spectrum Informaton Spread Interference Frequency After Despreadng Power Spectrum Improved CIR Frequency Flterng After Flterng Fgure 2.5: Interference attenuaton n CDMA, based on [5, p. 222] WCDMA The most wdely adopted rado nterface for thrd generaton systems s WCDMA (Wdeband-DS-CDMA). Ths rado nterface s used n UMTS n Europe and Asa. In WCDMA, the bandwdth s around 5 MHz (cf. [13]). In the uplnk drecton, the spreadng codes of dfferent mobles are quasorthogonal. That s, the dsturbances from other physcal channels do not dsappear but are very small. In the downlnk, the code sequences that a base staton antenna uses to convey messages to ts assocated mobles are perfectly orthogonal f the sequences belong to the same code famly. However, ths property s partly lost due to reflecton and scatterng of the rado waves on ther way to the recever. The codes of dfferent cells are quas-orthogonal (cf. [23]). In UMTS rado networks, each base staton antenna emts a specal sgnal wth constant power, called plot sgnal. A moble staton connects to that antenna from whch t receves the strongest plot sgnal [6]. Snce several base statons are usng the same frequency band n WCDMA systems, t s possble that one moble s connected to up to three servng antennas at a tme f the receved rado sgnals offer a comparable strength. The receved nformaton from each physcal channel are combned approprately. Ths usually happens when the user s located at the border or overlappng area between some cells. Then besdes the connecton to the best server, a connecton to one or more neghborng base staton antennas s establshed. The moble s sad to be n soft handover n ths moment. If ths feature was

20 12 2 Prelmnares not avalable, the moble staton at the cell border would have to transmt and receve at hgh power because of the large dstance to the base staton. Ths would cause a hgh amount of nterference to the assocated cell as well as to the neghborng ones. Thus, the lnk qualty n these sectors would be downgraded (cf. Secton 2.4.2). Due to the addtonal lnk(s), the transmt powers of both, the moble and the best server, can be decreased. Hence, the arsng nterference s weakened. Moreover, the probablty of the connecton beng nterrupted when the user moves between the cells s almost elmnated (cf. [23, 5]) Interference n CDMA Systems Interference s receved power from other transmtters than the desred one that radate energy n the same frequency band. That s, nterference s an unmeant contrbuton to the receved power that complcates the detecton of the desred sgnal (cf. [23]). The hgher the amount of nterference s, the more dffcult s t to flter out the desred rado sgnal properly. In CDMA systems, all mobles n one cell use the same frequency spectrum smultaneously as descrbed n Secton Hence, they cause nterference, denoted by ntra-cell nterference. Furthermore, the same frequency channels are avalable to several base staton antennas n the network [5]. Therefore, all mobles from those cells use the same frequency band at the same tme, too. These mparments orgnatng from other sectors are called nter-cell nterference. Consequently, there s a hgh amount of nterference n rado networks usng the CDMA technology. In the uplnk drecton, the sgnals from other moble statons overlay the own rado waves. In the downlnk, nterference s produced by other base statons [23]. Both drectons do not nterfere because ether two dfferent frequency bands are used (FDD: Frequency Dvson Duplex) or recevng and transmttng happen at dfferent moments n tme (TDD: Tme Dvson Duplex) [22]. The strength of the nterferng sgnals depends among others on the dstance between recever and dsturbng transmtter due to the propagaton characterstcs of rado waves. That s, the spatal constellaton between the users nfluences the amount of nterference each lnk receves. Typcally, the nterferers n the own cell are located much closer than those of other sectors. For ths reason, the power of the ntra-cell nterference s usually hgher than that of the nter-cell nterference. Another nfluence on the strength of the nterferng sgnal s the power wth whch the dsturber transmts data (cf. [23]). Every knd of nterference causes a modfcaton of the rado sgnal durng propagaton. Possbly, ths could lead to an ncorrect detecton at the

21 2.4 Specfcs of UMTS 13 recever. The stronger the wanted sgnal C (carrer) and the smaller the nterference power I, the lower s the error rate. Therefore, the Carrer to Interference Rato (CIR) C/I must exceed a specfc threshold, called the CIR target. The followng nequalty must hold: Strength of Desred Sgnal Strength of Interferng Sgnals + Nose CIR target. (2.1) Besdes the nterference caused by the system, there are natural mparments lke the thermal nose at the recever, whch s always present (cf. [23]). Also emssons of other external sources lke radars or ndustral equpment have to be consdered [16]. The nfluence of such factors s ncluded n the term Nose n the nequalty. For the requred CIR target to be mantaned n spte of the hgh amount of dsturbance, nterference control s crucal n UMTS rado networks. A recever s able to tolerate a specfc maxmum level of nterference power to whch each user contrbutes [2]. If ths level s exceeded the desred sgnal s bured among the nterferng sgnals after despreadng. For ths reason, a complex power control s appled to dedcated lnks n UMTS rado networks. The power control mnmzes the nterference n the system by adjustng the transmsson powers as low as possble. On the other hand, t ensures an adequate sgnal qualty at the recever accordng to the CIR target (cf. [16]). If the nterference stuaton n the network changes the CIR target has to be adapted to the actual crcumstances by the power control mechansms [5]. Furthermore, the power control equalzes sgnal varaton due to dynamcal phenomena called shadowng and fadng. In UMTS systems, many users share the same frequency spectrum smultaneously. Therefore, the value of the CIR at the recever s smaller than one snce the power of the desred sgnal s usually weaker than the sum of the powers of the other sgnals [23]. The CIR target s also much smaller than one. Due to the ablty of CDMA systems to apprecably reduce the nterference power proportonal to the power of the desred sgnal (cf. Secton 2.4.1), the requred power densty s hgher than the nterference power densty after despreadng. In UMTS rado networks, the sgnal power can thus be lower than the power of the nterference and the recever can stll detect the desred sgnal. The CIR target depends manly on the requested servce. A hgher threshold has to be acheved when t s transmtted at a hgher data rate [23]. Moreover, the user s velocty nfluences the CIR target. The faster he moves the faster changes the fadng stuaton of ts lnk. For hgh speeds, the varances are too fast to be made up by power control. In order to guarantee the qualty of a connecton even n ths case, the CIR target to meet s hgher.

22 14 2 Prelmnares Snce uplnk and downlnk are usually asymmetrcally loaded [16], the target values for uplnk and downlnk dffer Blockng n UMTS If all channels n the rado network are occuped t s mpossble to establsh a connecton. In ths stuaton, a new arrvng call would be refused or blocked. In UMTS rado networks, the admsson control handles all new ncomng traffc. Ths control admts a new request to the system only f ths would not overload the network and f the necessary resources are avalable. The admsson control belongs to a varety of functons whch ensure that the rado nterface load does not exceed predefned thresholds. They are grouped under the so-called congeston control whch n turn belongs to the Rado Resource Management. Besdes admsson control, the congeston control contans the load control whch s responsble to brng the system nto a feasble stuaton when t s overloaded. The Rado Resource Management ncludes among others the power control as well as the handover control (cf. [16]). The capacty of a CDMA cell manly depends on the orthogonalty and number of the used spreadng codes. When havng perfectly orthogonal code sequences the dfferent dedcated channels do not nfluence each other. In ths case, the capacty of a sector s determned by the number of orthogonal codes. However, as ponted out n Secton 2.4.1, the codes are not perfectly orthogonal n UMTS rado networks. For ths reason, nterference s the factor determnng the capacty of a UMTS cell. UMTS networks are sad to be nterference lmted (cf. [23]). Every new accepted lnk n the whole network as well as n one arbtrary cell causes a degradaton of the qualty of all other exstent connectons n the same frequency band snce each CIR decreases. In the case that one CIR drops below the accordng CIR target, power control trggers the approprate transmtter to rase ts emtted energy. Ths n turn ncreases the nterference power on all other connectons n the frequency band whch possbly causes other transmtters to emt wth more power and so on (cf. [5]). The transmsson and recepton powers of the base staton antennas are lmted due to the nstalled hardware. If the avalable rado resources are exhausted, no more users can be served. Then new requests are rejected. When a user tres to establsh a completely new connecton to the rado network and s refused by a base staton antenna due to the explaned reasons, he s blocked. A smlar stuaton appears f an actve moble moves from one cell to another one havng no rado resources avalable. Then t may happen that the connecton s broken off. We speak of a dropped call. Snce often users estmate such an experence more negatve than a blocked request some

23 2.5 General Mathematcal Model 15 channels are reserved especally for handover by the rado network operators (cf. [23]). Therefore, we wll not consder dropped calls n ths dploma thess. Besdes rejectng new arrvals t s possble that the lnk qualty for some mobles of crcut-swtched servces s downgraded. Crcut-swtched servces are real-tme traffc servces lke speech telephony or vdeo transmssons. In contrast, packet-swtched servces are servces whch can be carred out delayed such as sendng e-mals. Furthermore, t may happen that a desred lnk s blocked even though there are rado resources avalable n order to guarantee the qualty of the entre system [5]. In ths dploma thess, we address blockng only n the case of exceeded cell powers leadng to a rejected user request. In second generaton moble communcatons systems lke GSM, the capacty of a cell can be specfed durng the plannng phase. The common use of the frequency band s controlled by assgnng a specfc frequency channel and tme slot to each user (cf. Secton 2.4.1). To every base staton antenna, a certan number of channels and slots s assocated. From that, the maxmum number of smultaneous lnks can be derved. If a new arrval fnds them all occuped, then t s blocked (cf. [5]). In UMTS rado networks, the number of smultaneous users s restrcted by ther mutual nterference at the recever [22]. In contrast to second generaton cellular systems, each cell has a varyng capacty whch manly depends on the current nterference stuaton. Therefore, t s called soft capacty [5]. The dffculty s that t s not known exactly beforehand but can only be estmated. Thus, the capacty n CDMA systems s not determnstc but a stochastc value. 2.5 General Mathematcal Model In the current secton, we set up a mathematcal model of a UMTS rado network. The presented approach s the bass for the further consderatons n ths thess. Frst, we brefly explan the essental smplfcatons of the generated model compared to a UMTS rado network n realty. Afterwards, the nput data s explaned as well as the central assumptons. Fnally, formulas for the CIR targets and powers of the antennas are derved Statc Vew The proposed model s an abstracton of the real processes n a UMTS rado network. That s, the propertes of the modeled system are covered whch are essental for our purpose whle other features are gnored. In ths manner, the complexty of the orgnal system s reduced n order to be able to better

24 16 2 Prelmnares understand and analyze t. Nevertheless, the represented propertes have to be modeled as precse as necessary to obtan reasonable study results whch can be appled to the orgnal system. Hence, a trade-off between accuracy and smplcty has to be found. Actually, a UMTS rado network s a dynamc system. That s, the state of the network changes steadly. Due to movng mobles and successvely ncomng servce requests the nterference stuaton vares n the complete network. The power control effects the transmt powers to change accordng to the new CIR targets, whch can be updated every 10ms [2]. Other dynamc features are, e.g. the handovers of the exstent lnks from one cell to another [16] or blockng as explaned n Secton However, we consder ths dynamc rado network n a statc way. That s, users are located at fxed postons nstead of movng n the area. The dfferent arrvng tmes of the requests are not taken nto account, rather the whole traffc demand s present at once. Furthermore, the changng CIR targets at a recever are modeled each by one constant, average value. The same apples to the nterference n the rado network and the transmsson and recepton powers. Consequently, we just consder the UMTS rado network at one nstance n tme. Moreover, we gnore the possblty of soft handover and assume, that each moble staton s lnked to exactly one antenna, namely the one wth the strongest plot sgnal Input Data and Assumptons We consder a plannng area A. Ths regon s embedded nto the two dmensonal plane for a fxed heght or nto the three dmensonal space wth varable heghts for each pont. The dmenson of the area s denoted by d {2, 3}. In order to dscretze the plannng regon, t s subdvded nto a fnte set of pxels. Each pxel marks a d-dmensonal locaton n the area. In the plannng area A, a UMTS rado network wth a set N of antennas s nstalled. The best server area of an antenna N s denoted by A A. The users n the network are represented by a set M of moble statons. The set M M denotes the users served by antenna N. Furthermore, a set S of avalable servces s consdered. All these sets are fnte. Ther cardnalty s a natural number, that s, A N, N N, M N and S N. The moble users M are gven by a traffc snapshot. Ths s a statc realzaton of the average user demand obtaned on bass of spatal average traffc load dstrbutons. A traffc snapshot gves detaled nformaton on the poston, moblty, and servce of each user. The average spatal traffc dstrbuton of a servce s S s denoted by T s : A R +. For a poston

25 2.5 General Mathematcal Model 17 p A, T s (p) s the average traffc ntensty of the servce s at one nstance n tme. Fgure 2.6 llustrates the average spatal traffc dstrbuton for one servce. Fgure 2.6: Average traffc dstrbuton for one servce The number of users and ther locatons s a random varable. It s a common assumpton that the average user dstrbuton n one pxel follows a Posson dstrbuton. In general, the Posson dstrbuton s a dscrete probablty dstrbuton whch arses n a varety of stuatons n whch t s desred to count the number of occurrences of some phenomenon n an nterval of tme or space [20, p. 199]. Usually, the number of possble successes s large whle the probablty for one success s small [15]. Both features apply n our case. The pxels n the plannng area are small compared to the sze of the entre regon. For ths reason, the number of pxels s hgh whle the average traffc ntensty n one pxel s very low. The expected number of users n a pxel s always much smaller than one. Thus, the probablty for one user beng located at a pxel s low. Consequently, the Posson dstrbuton s an adequate characterzaton of the spatal user dstrbuton. The user ntenstes n non-overlappng areas are assumed to be ndependent. The sum of ndependent Posson dstrbuted random varables s agan a Posson dstrbuted random varable whose parameter s the sum of the parameters of the orgnal random varables [15]. Hence, for each sequence

26 18 2 Prelmnares (A n ) n N, A n A, of parwse dsjont sets followng equaton apples: ( ) T s A n = n n T s (A n ). (2.2) Furthermore, T s ( ) = 0 holds. These propertes show that T s s a measure on A [9]. Actually, t s a countng measure whch maps to a regon the expected number of users n t. Ths measure s fnte, that s, T s (A) < snce we only consder stuatons n whch the traffc ntensty n the entre plannng area s fnte. We assume that the number of users for each servce s S n a certan regon à A s proportonal to the sze of the regon λd (Ã). The measure λd s the d-dmensonal Lebesgue-Measure. We assume that there exsts a user densty f s : A R + for each servce s S. The expected number of users of servce s n area à s thus expressed by T s (Ã) = f s (p) dp. (2.3) p à CIR Constrants and Blockng At frst, we derve the complete CIR constrants for the uplnk and downlnk drecton. The average CIR targets for a moble staton m M are denoted by µ m for uplnk and µ m for downlnk. Furthermore, there are transmt actvty factors αm and αm for every moble ndcatng the average rato of tme t s transmttng data on the rado channel. In speech conversatons, for example, every user speaks on average 50% of the tme. The CIR nequalty has to be satsfed n actve perods only. At other nstances n tme, there s no data transmsson. For ths reason, we assgn a transmt actvty factor of one to the desred moble. We do not know f the sgnals of other users are currently n an actve perod or not. Therefore, we apply the transmt actvty factors to the other sgnals n (2.1) n order to consder the average nterference power. Fnally, γ m n uplnk and γ m n downlnk are the attenuaton factors for moble staton m M and base staton antenna N. Apart from the path loss between the moble and the antenna, addtonal losses and gans are ncluded dependent on the cablng, hardware, and user equpment. In the uplnk drecton, the transmsson power of a moble m M s denoted by p m. Then the strength of the desred sgnal at base staton antenna N s γ m p m. The receved background nose at antenna s marked by η. Wth these conventons, the basc CIR target nequalty (2.1)

27 2.5 General Mathematcal Model 19 for the uplnk transmsson from moble m to antenna reads as: γ m p m n m γ n α n p n + η µ m. (2.4) Several base statons use the same frequency band (cf. Secton 2.4.1). In ths model, we assume that all base statons use the same frequency spectrum. Hence, all users n the area convey nformaton n the same frequency band at the same tme. All these transmssons are receved wth varyng strength by each base staton antenna. For moble statons usng another frequency band n realty, the attenuaton factor s approprately low. The average total recepton power at antenna N s thus gven by p := m M γ m α m p m + η. (2.5) As mentoned prevously, t s not known for a lnk whether t s actve or not. For ths reason, we take the transmt actvty factors of the mobles nto account and obtan the average power. Usng the last equaton, (2.4) smplfes to γ m p m p γ m α m p m µ m. (2.6) In the downlnk drecton, the plot and common channels are ncluded, whose power we denote by p (c) at base staton antenna N. Ths value s assumed to be constant. Furthermore, p m s the strength of the sgnal from antenna to moble m and ω m [0, 1] s an envronment dependent orthogonalty factor. The sgnals an antenna transmts to ts assocated mobles partly lose ther orthogonalty due to multpath propagaton (cf. Secton 2.4.1). If ω m = 0 holds, the sgnals are perfectly orthogonal and ω m = 1 means no orthogonalty. The average total transmsson power of antenna s defned by p := αm p m + p(c). (2.7) m M We denote by η m the nose at moble m. Then the CIR constrant n downlnk satsfes followng nequalty: γ m ω m γ m p m ( ) p α m p m + j γ jm p j + η m µ m. (2.8) The transmsson power of a base staton antenna s restrcted. Typcally, a UMTS antenna cannot emt more than 20W. In addton, there are lmts

28 20 2 Prelmnares on the average load of a cell. These load lmts le sgnfcantly below 100% snce t s mportant to have a buffer to compensate for dynamc effects. The downlnk load s defned as the rato of the current transmsson power to the maxmum possble output power. Usually, the lmt of the downlnk load les 1 at 70%. The uplnk load s gven by 1. The nose rse s the rato nose rse of the total receved power at a base staton antenna to the nose power. The uplnk load should not rse above 50%. We denote by Π max the maxmum possble transmsson power and by p max and p max the maxmum allowed recepton and transmsson power of a base staton antenna N. The latter can be derved by resolvng 1 η p max = load lmt and p max Π max = load lmt. Throughout ths thess, we mean wth maxmum total power the maxmum allowed total power p max and p max, respectvely. The followng nequaltes express that on average all users n cell are served and thus no blockng occurs: m M p pmax and p pmax. (2.9) Usng equatons (2.5) and (2.7), ths can be transformed nto γ m α m p m pmax η and αm p m pmax m M p (c). (2.10)

29 3 Exstng Methods In ths chapter, we dscuss establshed methods to assess the average blockng rate of a base staton antenna n a UMTS rado network. Frst, we ntroduce an approach to approxmate the blockng rate based on a system of equatons. Ths system can be set up for one traffc snapshot as descrbed n the frst part of the followng secton. In order to obtan statstcally relable results, such equaton systems have to be solved for a large number of snapshots. Snce the computatonal complexty of ths procedure s too hgh to be applcable n some stuatons, the basc dea of ths approach s generalzed on the bass of average traffc load dstrbutons. Ths dea s explaned afterwards. Ths method speeds up calculaton radcally. In exchange, t causes a sgnfcant underestmaton of the blockng rate of a cell n a regon under around 5%. In the next secton, the so-called Monte Carlo smulaton on traffc snapshots s descrbed brefly. Ths s a popular method but ths approach s very extensve and tme consumng. The snapshot based system of equatons s also a Monte Carlo smulaton. The last secton summarzes the shortcomngs of the formerly presented methods. The notaton can be found n Appendx A. Throughout ths thess, we assume perfect power control on dedcated channels. That s, the CIR targets are met at equalty. Moreover, no user s n soft handover, and effects of shadow fadng are neglected. Uplnk and downlnk are consdered ndependently. 3.1 System of Equatons In ths secton, the deas from [6] are ntroduced brefly. A system of equatons s set up for uplnk and downlnk respectvely descrbng the average transmsson and recepton powers of the antennas n the rado network. These results are then used to assess the blockng rate of each cell. Frst, the equatons are derved based on a traffc snapshot and then generalzed on the bass of stochastcal average load. Afterwards, we pont 21

30 22 3 Exstng Methods out how the blockng rate s calculated n both cases. Ths model s the basc prncple of the method we develop n the next chapters. Throughout ths thess, the ndces and j wll be used for base staton antennas. The subscrpt denotes the cell whose blockng rate we wsh to determne. A vector wth elements v j s denoted n bold font v. Moreover, dag (v) marks a dagonal matrx havng the same dmenson as v and the components of v on the man dagonal Snapshot Based Dervaton We consder a set M of moble statons gven by a traffc snapshot. Followng assumptons are made n ths model for the tme beng: () lmtatons of transmsson powers and nose rse are neglected and () all users are served. These restrctons are mportant for the dervaton of the equatons. Later, they wll be abolshed when blockng s modeled. Uplnk In the uplnk drecton, we start from equaton (2.5), whch descrbes the average recepton power of antenna, wrtten as p = γ m α m p m + γ m α m p m + η. (3.1) m M j m M j In ths way, t can be recognzed that the total recepton power at an antenna conssts of three parts: one porton for the nterference from the own and from the other cells respectvely and the nose exteror to the system. For the uplnk CIR target to be mantaned by transmsson from moble staton m M to antenna N, nequalty (2.6) must hold. As stated n the begnnng, we assume that equalty apples. When convertng ths equaton properly and defnng the uplnk user load of a moble m as l m := α m µ m, (3.2) 1 + αm µ m the uplnk couplng factors result n C j := γm m M j γ mj l m. (3.3)

31 3.1 System of Equatons 23 Consequently, wth (3.1) and (3.3), the uplnk transmsson power of base staton antenna reads as p = C p + j C j p j + η. (3.4) We call the matrx C := ( ) C j 1,j N the uplnk cell load couplng matrx (uplnk couplng matrx). The components of C can be nterpreted n followng way. The dagonal entry C measures the contrbuton from the ntra-cell nterference to the total receved power. The value C j scales the nter-cell nterference contrbuton from antenna j. The desred system of equatons arses from (3.4): p = C p + η. (3.5) The soluton of ths system s the vector wth the uplnk recepton powers at each base staton antenna. Downlnk The same approach s appled n the downlnk case. The total average output power of base staton antenna N s defned by (2.7). The CIR constrant s gven by (2.8). Agan, the assumpton of perfect power control holds and the constrant s an equaton. The downlnk user load reads as l m := α m µ m. (3.6) 1 + ω m αm µ m We use t to ntroduce the downlnk couplng factors C := m M ω m l m and C j := γjm l m M γ m m (j ) (3.7) for antennas and j, as well as the traffc nose power of sector := lm. (3.8) p (η) η m m M γ m The meanng of the couplng factors C j s the followng. The dagonal entry C represents the contrbuton from the ntra-cell nterference to the total transmsson power. The value C j specfes the porton of transmsson power

32 24 3 Exstng Methods allocated on overcomng the nter-cell nterference from antenna j. The tem p (η) expresses the fracton of transmsson power spent on overcomng the nose at the mobles f there was no ntra-system nterference. For the transmsson power at antenna we obtan p = C p + j C j p j + p(η) + p (c). (3.9) The matrx C := ( C ) j 1,j N s called the downlnk cell load couplng matrx (downlnk couplng matrx). Equaton (3.9) for each base staton antenna yelds the followng system of equatons p = C p + p (η) + p (c). (3.10) The soluton of ths system s the downlnk transmsson power for every cell Expected Couplng The couplng matrces C and C are stochastcal. They depend on the postons and servces of the actve mobles. We assume the user dstrbuton n the plannng area to be known (cf. Secton 2.5.2). The matrx entres defned n (3.3) and (3.7) are lnear compostons. For ths reasons, t s possble to determne the expected values of the load couplng matrces, denoted by C and C. Then, the equaton systems (3.5) and (3.10) can be set up wth these expected values. For a clearer presentaton, t s mpled, that we have representatve CIR targets µ s, µ s and transmt actvty factors α s, α s n both drectons for each servce s S. Furthermore, η p s the nose and ω p the orthogonalty factor at a moble n poston p. The attenuaton factors between a base staton antenna and a user located n p are denoted by γ p n uplnk and γ p n downlnk. The defntons of the user load (3.2) and (3.6) are substtuted by l p := s S αs µ s T 1 + αs µ s (p) and lp := s s S α s µ s 1 + ω p α s µ s T s (p). (3.11) Remember, that T s (p) s the expected value of the traffc ntensty for servce s at locaton p. The other factors n the above defntons are constants. Thus, lp and l p are the expected values of l m and l m at locaton p. We derve the

33 3.1 System of Equatons 25 entres of the expected uplnk couplng matrx by C := lp dp, C j := p A p A j γ p l γ p jp dp. (3.12) The components of the expected downlnk couplng matrx and traffc nose power read as C := γ jp η p p A ω p l p dp, C j := p A γ p l p dp, Approxmatng Blockng p (η) := p A γ p l p dp. (3.13) Untl now, the system of lnear equatons ntroduced n the former sectons gnores the effects due to load control whch s trggered f the power of a cell would excess ts lmt (cf. Secton 2.4.3). To mmc load control, the approach from [7] s adopted to reduce the load n saturated cells. In ths proposed model, t s not necessary to dstngush whether a user s rejected or whether the servce qualty of other users s downgraded. Followng two propertes characterze a proper load control: () Admssblty: After load control has been appled, all antenna power values are feasble, that s, (2.10) holds. () Greedness: Users are only rejected by a cell f t cannot serve them wthout rsng above ts own capacty. That s, an antenna does not reject users to ease the stuaton of ts neghborng cell. Furthermore, we assume that a base staton antenna s able to serve all ts users up to a certan fracton of ther resource demands. Ths s realzed by scalng the relatve user load ( (3.2) and (3.6) or (3.11) ) n the accordng cell by a value λ between 0 and 1. In dong so, only as lttle load as necessary s wthdrawn. The blockng rate s then 1 λ. However, n realstc settngs, the assumpton of compressble user demand s not vald. The obtaned scalng vectors can be used as a gudelne for determnng how many mobles need to be refused. A complementarty condton has to hold for the resultng power and scalng vectors n order to acheve the above two propertes. In general, n a complementarty condton one or several subgroups of nequaltes are comprsed. In each group at least one of these nequaltes should be met at equalty [4]. In our case, t clams that f user demand n a cell s reduced, then the cell power s equal to ts maxmum allowed value.