A model for mobility-dependent large-scale. propagation characteristics of wireless channel. D. Moltchanov, Y. Koucheryavy, J.

Transcription

1 A moel for mobility-epenent large-scale propagation characteristics of wireless channels D. Moltchanov, Y. Koucheryavy, J. Harju Abstract In this paper we propose an extension to existing Markovian wireless channel moeling techniques introucing a notion of mobility behavior of the user. Particularly, we represent a large-scale propagation characteristics of wireless channels as a mobilityepenent stochastic process that explicitly tracks the movement of the user between areas with ifferent receive local average signal strength (RLASS). Basically, our moel consists of two ifferent parts: mobility moel an large-scale propagation moel. Mobility of the user is moele by a Markov chain with finite state space. Large-scale propagation characteristics of wireless channel is represente by a function of mobility moel. Propose moel can be parameterize using either real measurements or classic large-scale propagation moels. Base on the available information regaring a given environment we evelope three ifferent parametrization methos for our moel. Propose approacllows to capture large-scale propagation characteristics of wireless channels as a function of mobility of the user an can be use in performance evaluation of user s applications running over the air interface. I. Introuction Thir generation (3G) mobile systems, which were recently given a lot of attention, are nowaays seen as an intermeiate step between conventional secon generation (2G) mobile systems an next generation (NG) mobile networks. While NG mobile systems are not well-efine, there is a common agreement that these networks will rely on IP protocol as en-to-en transport technology. The motivation is to buil a common service platform for future mobile Internet, known as NG All-IP networks. In aition to broaban wireless access to the Internet NG All-IP networks shoul also provie quality of service (QoS) to their applications. Provision of QoS is an inherent problem for many service types even in fixe networks an, as expecte, will get worse when user services have to be extene to the air interface. Inherent characteristics of mobile users like their mobility, traffic emans an unstable nature of the air interface have to be aresse before the require quality of user services will be achieve. These new challenges require evelopment of novel methos of teletraffic theory, optimization an esign. Among others, the special attention shoul be pai to wireless channel moeling. Since the error rate of fixe transmission meium is negligibly low, in orer to evaluate performance of user applications in fixe networks it is sufficient to Authors are with the Network an Protocol group, Institute of Communication Engineering, Tampere University of Technology, Tampere, Finlan. WWW: {moltchan,yk,harju}@cs.tut.fi. estimate packet losses, elays an elay jitters cause by buffer overflows. The istinguishing property of wireless links is that their performance is affecte by atmospheric conitions an other physical factors incluing reflection, iffraction an scattering resulting in hign correlate bit errors at the physical layer. Therefore, ealing with wireless networks we cannot neglect packet level QoS egraation cause by bit errors of wireless transmission meium. Inee, they have completely ifferent nature compare to what we ealt in fixe networks an may contribute a lot in en-to-en performance egraation. Survey of research papers has shown that wireless channel moels esigne for mobile systems o not explicitly take into account mobility behavior of a single user. Most of them capture propagation characteristics of wireless channels at a given istance from the transmitter an neglect those changes cause by a movement of a user between areas with ifferent signal strengths. However, it is clear that the receive signal strength epens on istance between transmitter an receiver, that, in turn, epens on mobility of the user. Novel state-of-the-art channel wireless channel moel must capture such epenence. In this paper we propose an extension for conventional large-scale propagation moels of wireless channels to the case of mobility-epenent behavior. Particularly, we propose a moel that explicitly captures mobility of a single user an integrates our moel consists of two ifferent parts: mobility moel an largescale propagation moel. Mobility of the user is moele by a Markov chain with finite state space. Largescale propagation characteristics of wireless channel is represente by a rate process associate with this Markov chain. So that the whole moel is actually a oubly stochastic process. Propose approacllows to capture large-scale propagation characteristics of wireless channels as a function of mobility of the user an can be use in performance evaluation of user s applications running over the air interface. Our paper is organize as follows. In Section II we make necessary remarks on current wireless channel moeling techniques an provie reasons why we have to exten those moels to capture mobility behavior of the user. Then, in Section III, we outline the structure of our moel. Parameters estimation methos are given in Section IV. Some examples are in Section IV. Conclusions an further work are outline in the last section.

2 A moel for mobility-epenent large-scale propagation characteristics of wireless channels II. Propagation characteristics of wireless channels The propagation path between transmitter an receiver may vary from simple line-of-sight (LOS) to very complex one ue to iffraction, reflection an scattering. To estimate performance of wireless channels, propagation moels are often use. Basically, we istinguish between two types of propagation moels [1]. These are large-scale propagation moels an small-scale propagation moels. A. Large-scale propagation moels. When a mobile user moves away from the transmitter over large istances the RLASS graually ecreases. This signal strength can be preicte using large-scale propagation moels. Usually, we istinguish between analytical an empirical large-scale propagation moels. The former ones capture large-scale propagation base on analytical representation of physical phenomenons incluing iffraction, reflection an scattering. Empirical moels are base on fitting analytical expressions to a set of measure ata. The main avantage of these moels is that they implicitly take into account all propagation factors, both known an unknown. Most analytical an empirical large-scale propagation moels assume that the RLASS ecays as the power law function of istance between the transmitter an a receiver. There are a number of large-scale propagation moels available in literature. However, neither outoor [2], [3], [4], [5] nor inoor [6], [], [8], [9] moels o not take into account mobility behavior of mobile users between areas with ifferent RLASS. B. Small-scale propagation moels. When a mobile use moves over short istances the instantaneous signal strength may vary rapily. The reason is that the receive signal is a sum of many components coming from ifferent irections ue to reflection, iffraction an scattering. Since phases, amplitues an arriving times of components are ranom, the resulting signal varies significantly. Depening on the relation between signal parameters, channel characteristics an velocity of the user, ifferent signals experience ifferent types of faing. Base on multipath time elay sprea we istinguish between flat an frequency-selective faing, base on Doppler sprea we istinguish between fast an slow faing. Due to implicit incorporation of small-scale mobility in small-scale propagation moels [10], [11], [12], most performance evaluation stuies of information transmission over wireless channels performe so far [13], [14], [15], [16], were limite to small-scale propagation phenomenon. C. Integrate cross-layer wireless channel moels. It is well-known that a mobile user may change its location many times uring an active session an these changes are not always limite to short travel istances. Wireless channel moels evelope to capture large-scale propagation phenomenon cannot be effectively use in performance evaluation of user s applications running over the air interface, since both the mobility of the user between areas with ifferent RLASS an rapi fluctuations of the signal strengtre not taken into account. Moels of small-scale propagation implicitly inclue the mobility behavior of the user. However, they fail to preict signal strengtttenuation cause by movements over large istances. Taking into account nomaic behavior of mobile user, novel wireless channel moels must capture both mobility of the user an propagation characteristics of wireless channels. Large-scale an small-scale propagation characteristics must be consiere as functions of user s mobility an represente by stochastic processes. One shoul also note that moels of receive signal strengtre not appropriate for performance evaluation purposes an must be further extene to higher layers proviing, for example, IP packet error probabilities. Thus, we have to take into account characteristics of unerlying layers incluing ata-link error concealment techniques like forwar error correction (FEC), automatic repeat request (ARQ) or combination of them, an moulation schemes at the physical layer. Finally, an aequate wireless channel moel for NG All-IP mobile systems must be cross-layer integrate one capturing propagation characteristics of wireless channels by a mobility-epenent stochastic process an important properties of unerlying layers. Such moels along with traffic moels have to be further applie to preict QoS expectations experience by applications running over the wireless channels. III. Wireless channel moel A. Structure of the moel Assume that a given cell is somehow ivie into a finite number of areas M such that these areas are non-overlappe an the sum of their areas equals to the cell area. A simple example of ivision of the cell into regions satisfying abovementione assumptions is shown in the left part of Fig. 1. We assume that a mobile user may move between these areas an eacrea i =1, 2,...,M is associate wit certain RLASS. Let us assume a iscrete-time environment, i.e. time axis is slotte wit certain granularity, the slot uration is constant an given by t =(t i+1 t i ), i =0, 1,... Assume also that changes of areas are only allowe at slot bounaries. Consiering the mobility behavior of the user within an between areas one may expect some type of positive autocorrelation in this process. Roughly speaking, if the user is in a certain area in the slot n it is more likely it will continue to be in the same area in the slot (n + 1). In this paper we propose to capture such type of positive autocorrelation an represent user s mobility between ifferent areas using a iscrete-time homogenous Markov chain {S L (n),n=0, 1,..} efine at the

3 EUNICE 2004 state space S L (n) {1, 2,...,M}, where M is the number of areas. Accoring to this moel, time a particular user stays in eacrea is geometrically istribute an can only be change at slot bounaries. However, one shoul note that this restriction can be relaxe to capture more general istributions of sojourn times in eacrea, incluing sum of geometrical, hypergeometrical, etc. This can be one allowing more than one state of the Markov chain to enote eacrea. Anyway, the minimum time the user stays in any mobility state is still t, an therefore, the careful choice of t is of paramount importance. In our moel transitions are only allowe between ajacent areas which is a natural assumption regaring user s movement within any given area. It is also possible to represent the irectional movement of the user between areas introucing a perioicity in the Markov chain. For example, ealing wit highway scenario the sequence of areas visite by a mobile user may be known in avance. As we have seen previously, the RLASS is the function of both istance between transmitter an receiver an user s mobility. To stochastically represent it, let us now associate a certain value of RLASS with eactate of mobility moel. To o so let {L(n),n= 0, 1,..}, L(n) {L 1,L 2,...,L M } be the RLASS process whose unerlying Markov chain is {S L (n),n = 0, 1,...}. So that the value of RLASS is moulate by a unerlying Markov chain an the whole moel is a oubly-stochastic process. Let L =(L 1,L 2,...,L M ) be the RLASS vector. The value of RLASS in the slot is a ranom variable L that takes on values of the vector L =(L 1,L 2,...,L M ) with respective steaystate probabilities π = (π 1,π 2,...,π M ) of unerlying Markov chain {S L (n),n =0, 1,...}. Therefore, the RLASS process {L(n),n =0, 1,...} is efine as L(n) =L i, i {1, 2,...,M}, while the Markov chain is in the state i at the time slot n. Due to incorporation of moulating process the propose moel is characterize by the following structure of autocorrelation function of RLASS process: K L (m) λ m l, m =1, 2,..., (1) l, l 1 λ l, l =2, 3,...,M are eigenvalues of transition probability matrix of Markovian process {S L (n),n = 0, 1,...} given that λ 1 = 1. So that the autocorrelation function of the RLASS process is the sum of M 1 geometrically istribute terms whicre given by non-unit eigenvalues of moulating Markov chain. A simple example is shown in Fig. 1 where the cell of a circular type is partitione into several areas with ifferent RLASS. In general, these areas may be of an arbitrary configuration. Inee, the RLASS epens not only on the istance between the transmitter an the receiver but on the environmental characteristics too. RLASS in every area is actually a range of RLASS an must be estimate using either real measurements or large-scale propagation moels available in literature. Fig. 1. L1... LM L1 L M Tampere, Finlan A moel for large-scale propagation characteristics. B. Parametrization of the moel Our moel is just a oubly stochastic process. To completely efine it we have to etermine the transition probability matrix P L of the unerlying Markov chain {S L (n),n =0, 1,...} an RLASS vector L = (L 1,L 2,...,L M ). Assume that all information about areas witppropriate values of RLASS is available. In this case parameters of unerlying Markov chain {S L (n),n = 0, 1,...} are easy to estimate. Let us consier a cell of a circular configuration wit number of areas as shown in the left part of Fig Fig. 2. to state 1: p 10,1 to state 8: p 10,8 to state 9: p 10,9 to state 10: p 10,10 10 LM to state 11: p 10,11 to state 12: p 10,12 to state 13: p 10,13 Example of the cell of a circular type. In accorance with our moel every area correspons to a state of the Markov chain. It is easy to notice that the sojourn time in a certain area must epen on its size. Inee, with the increasing of the size of area, time a user stays in this area is increasing. One shoul also note that the area sojourn time also epens on the velocity of the mobile user. Hence, we shoul istinguish between mobile users with ifferent velocities. Aitionally, in certain cases it is necessary to take into account a irectional movement of users. For example, consiering a highway scenario if a mobile user is in a certain area in the time slot n, itis more likely it will continue to the next area along the highway than move to any other areas. Consier a mobile user which is in the state i, i = 1, 2,..., M, in the slot n. In accorance with propose moel, mobile user in the next slot (n +1) may either stay in the area i or move to other areas. The only areas to whic user can move in one slot are neighboring areas enote by Ω i, i =1, 2,...,M.We propose to compute transition probabilities between area i, i =1, 2,...,M an those areas from sets Ω i, i =1, 2,...,M as follows: p ij = S ip ij w ij v i S R P i, i,j =1, 2,...,M, (2) where S i is the area of i, P i is the perimeter of i, S R is the area of the cell, P ij, i, j Ω i is the length of the borer between areas i an j. Parameter v i,is

4 A moel for mobility-epenent large-scale propagation characteristics of wireless channels the factor proportional to velocity of the user. Parameters w ij, i, j Ω i are intene to represent irectional movement of the user in a highway or urban environments. Note that: w ij =1, i =1, 2,...,M. (3) j Ω i Parameters w ij, i, j Ω i, i =1, 2,...,M,arespecific for a given environment, an therefore, no general expression can be provie. In Section 6 we show how to estimate w ij, i, j Ω i, i =1, 2,...,M,for a highway scenario. To complete parametrization, in the following section we consier how to etermine configuration, bounaries an areas of regions with ifferent values of RLASS using either measurements of RLASS or classic large-scale propagation moels. IV. Estimation of parameters A. Estimation base on measurements In practise we cannot measure RLASS in any point of the cell. Instea, measurements of RLASS are often represente by a three-imensional vector: P =,i=1, 2,...,M, (4) where M is the whole number of measurements, x i, y i are coorinates of the i th measurement an P i is the RLASS value of respective measurement. Information given by, i =1, 2,...,M are insufficient for our purposes. Inee, to estimate parameters of our moel using (2) we have to etermine areas to which these measurements belong to. To etermine areas with nearly the same RLASS we propose to use a suitable ivision of the cell into areas whose vertexes are measurement points (x i,y i ), i =1, 2,...,M. In our case an appropriate ivision of the cell is achieve using so-calle Voronoi tesselation that separate a certain region D of space R n into cells D i, i = 1, 2,..., M, using a certain point process. The polygon D i is the intersection of half planes H ij boune by the bisectors of the segments ((x i,y i ), (x j,y j )), i, j =1, 2,...,M an containing (x i,y i ) as a vertex. The system of all polygons forms a tesselation of the plane. In our case the space is R 2 an coorinates of measurement points can be consiere as a point process on the plane. Practically, the Voronoi tesselation is constructe as follows: for each measurement point (x i,y i ), i = 1, 2,...,M,letD i, i =1, 2,...,M, be the area consisting of all locations in the space whicre closer to (x i,y i ) than to any other measurement point. Aitionally, to ensure that the Voronoi tesselation is well efine the only requirement we have to impose on measurement points is that they must compose a non-egenerate realization of the point process on the plane. It means that there must be at least two measurement points, they are istinct, an there are only finitely many of these points in any boune region. All these requirements are satisfie within our assumptions. There are a number of approaches to compute Voronoi tesselation ([1] an references therein). To etermine an appropriate value of RLASS it is not strictly require to istinguish between every value of RLASS in every area. Instea, it is possible to consier ranges of RLASS. In this case the range (max i P i min i P i ) must be ivie into K nonoverlapping ranges of the following length: P = (max i P i min i P i ). (5) K Using (5), all measurements must be classifie to appropriate ranges of the RLASS. Areas corresponing to this ranges can be etermine using the the proceure outline below. Let l i,j, i, j =1, 2,...,M, to enote the length between i tn j th measurements on the plane. Assume that i th measurement falls into k th range of RLASS. Consier now the measurement point such that the conition min j,i j l ij, is satisfie. If this point falls into the the same range of the RLASS the bounary of the current area must be extene to that area. Thus, the resulting area consists two areas an correspons to k th range of RLASS. The same proceure must be performe for the next point with next minimal istance from i th point until a certain point will be classifie to another range of RLASS. Note that with the increasing of the range of RLASS the accuracy of the moel ecreases. An example of analysis of the cell configuration is shown in Fig. 3. Left figure shows measurement points on the plane. In the next figure Voronoi tesselation is shown. Finally, in the right figure an example of areas are shown. Fig Example of analysis of the cell configuration. B. Estimation base on propagation moels Unfortunately, often, measurements of RLASS are unavailable while characteristics of a given environment is known. In this case it is also possible to parameterize our moel base on classic large-scale propagation moels evelope to ate. There are a number of large-scale propagation moels available in literature. Most of them assume that with the increasing of the istance between transmitter an receiver, RLASS is ecreasing accoring to a power law of given a stanar istance 0. For example, free space moel assumes that there is only one unobstructe LOS path between transmitter an

5 EUNICE 2004 receiver. The free space propagation loss is given by: ( ) L() =L s ( 0 ) + 20 lg (6) However, it rarely occurs in practice that the LOS is unobstructe. So that, the estimation given by (6) fails is most cases. It was shown by Okumura et. al. [4] using real measurements that with the increasing of the istance between transmitter an receiver, propagation loss is rarely ecreasing at a rate when n = 2. Hata [18] fits those ata to empirical formulas. In general, propagation loss is proportional to separation istance accoring to the following expression: ( ) E[L()] = L s ( 0 )+10nlg. () Distance 0 is often assume to be equal to 1000 meters for macrocells, 100 meters for microcells an 1 meter for inoor wireless channels [1]. L s ( 0 )canbe either measure or approximate [1]. Parameter n epens on wavelength, antenna height an a given propagation environment. For example, if we take an assumption of free space propagation n = 2 an () egenerates to (6). When the LOS is shaowe n>2. Consiering urban areas, in certain circumstances n<2 (see [1] for a range of n for specific environments). Measurements carrie out by Seiel et. al. [19] that the actual propagation loss may vary significantly epening on the propagation environment. It was foun that the actual value of propagation loss L() is log-normally istribute with mean E[L()]. Finally, the expression for propagation loss L() in a particular environment is given by: ( ) L() =L s ( 0 )+10nlg + X σ, (8) where X σ N(0,σ 2 ) is zero-mean Gaussian istribute ranom variable expresse in B. The value of X σ can be estimate from real measurement for ifferent locations of transmitter an receiver. Usually, X σ {6 10} B. Propagation loss can be easily relate to RLASS [1] or use instea of RLASS. Using abovementione consierations we assume that areas with ifferent RLASS are given by a circular forms on the plane as shown in the left part of Fig. 1. The value of RLASS in every area can be estimate using either () or (8), while (6) an can be seen as rougpproximation. Finally, transition probabilities can be foun using (2). C. Estimation base on area configurations From previous subsection we have seen that the parameter n of large-scale propagation moels may vary significantly epening on the presence of shaowers in a given propagation environment. We took it into account introucing a eviation parameter X σ that epens on a given environment an must be estimate from measurements. However, in those environments where shaowers are rare this may lea to Tampere, Finlan significant errors ue large eviation of n. Thus, the assumption of circular configuration of areas with the same RLASS (actually, the range of RLASS) may no longer hol. To take it into account, in what follows we propose an approximate moel of the lanscape configuration base on estimation of shaowe areas. Let us assume that shaows are istribute on the plane accoring to a stationary Poisson process wit certain finite intensity λ, λ<. The parameter λ etermines the mean ensity of the points an epens on the lanscape. For example, if we consier an urban area λ must be high while ealing with highway or country-sie scenarios λ must be low. Poissonian assumption allow to avoi a etaile geographical escription of the network while improve the accuracy of the moel explicitly taking into account the presence of shaowers. One shoul note that the Poisson assumption can be relaxe when necessary. For example, Markov ranom fiel can be use instea. Let us take the following assumptions regaring shaowers configuration: eachaower is of rectangular from with zero with; withs of shaowers are arbitrary istribute; heights of shaowers are arbitrary istribute; height of transmitter antenna is given. Let us now consier three possible shaow placements that may be cause by shaowers with ifferent relation between parameters as shown in Fig. 4, where is the maximum shaow length, R is the raius of the cell. >, s <R- 90 o R Fig. 4. >, s >R- < 90 o R R Three ifferent shaow configurations. One may notice from the figure that in accorance with our assumptions regaring cell environment an parameters of the shaowers have have to istinguish between three ifferent cases: >, s R : in this case the shaow is fully within the cell; >, s >R : in this case we assume that the shaow continues up to the borer of the cell an is limite outsie the cell; < : in this case we assume that the shaow continues up to the borer of the cell an more up to infinity. Since we restricte our attention to only one cell, from abovementione consierations one may notice that that the latter two case are similar to us an can be consiere simultaneously. Let us firstly etermine the area of the shaow shown in the left part of Fig. 4. To to it consier

6 A moel for mobility-epenent large-scale propagation characteristics of wireless channels ifferent view of the similar configuration presente in Fig. 5. Fig. 5. R. w s s An example of shaow configuration: >, s One may note from this schematic illustration that the length of the shaow s is expresses via initial parameters as follows: s =, >, s R, (9) 1 ha hs where is the minimal istance between the center of the cell an shaower. Taking into account two other cases of the shaow placement the final expression for s is given by: ( ) s = min,r. (10) 1 ha hs Using (10) the area the shaow is given by: The same estimation proceure must be performe for all other shaowers obtaine via realization of the Poisson process with intensity λ. We also have to note that the overlapping of shaows are also allowe. In this case the estimation of the shaows is slightly more complicate ue to the fact that every shaow may be overlappe with more than one ajacent shaows. For every shaow the ranges of values of RLASS can now be estimate using Hata-Seiel moel given by (8) with n = n s > 2. However, consiering a worst case scenario there can be shaows whose length s is only insignificantly less raius of the cell R. So that the range of RLASS corresponing to this shaow can be very large resulting in significant moeling errors. To eal with this situation, in aition to shaow estimation we propose to consier ifferent areas of a circular form to whichaows can be further classifie as shown in the left part of Fig.. New shaowe an non-shaowe areas are now given by borers of these circles an borers of shaows as shown in the right part of Fig.. Accoring to it we have several zones with ifferent RLASS. Eachaow may completely or partially be within ifferent circles enoting areas with ifferent RLASS as consiere without obstacles. For non-shaowe areas RLASS can be estimate using (8) with n = n ns 2, n ns <n s. S s = min(s s,s R ), (11) where S s is the area of the segment with raius ( + s ) minus the area of triangle with height, S R is the area of the segment with raius (R ) minus the area of triangle with height. To etermine the area of the shaow let us now consier the Fig. 6. L1... LM Fig.. An example of a) ifferent circular areas, b) final areas. w s Fig. 6. Another example of shaow configuration: >, s R. The area of the shaow presente in Fig. 6 is given as follows: S s =4π( + s ) 2 γ 360 w s 2. (12) For those cases when >, s >R or < the area is given by: S R =4πR 2 γ 360 w s 2. (13) Finally, the area of the shaow can be expresse as follows: S s = min(4πr 2 γ 360, 4π( + s) 2 γ 360 ) w s 2. (14) s V. Conclusions In this paper we propose an extension to existing Markovian wireless channel moeling techniques introucing a notion of mobility behavior of the user. Particularly, we represent a large-scale propagation characteristics of wireless channels as a mobilityepenent stochastic process that explicitly tracks the movement of the user between areas with ifferent RLASS. Basically, our moel consists of two ifferent parts: mobility moel an large-scale propagation moel. Mobility of the user is moele by a Markov chain with finite state space. Large-scale propagation characteristics of wireless channel is represente as a function of this Markov chain. Propose approach allows to capture large-scale propagation characteristics of wireless channels as a function of mobility of the user an can be use in performance evaluation of user s applications running over the air interface. We provie three parametrization methos for our moel. Metho base on real measurements is base

7 EUNICE 2004 of ivision of the space to areas with ifferent RLASS using Voronoi tesselation. In those cases when only a limite information regaring a given lanscape is available we propose to use parametrization metho base on empirical large-scale propagation moels. To improve the accuracy of the moel we evelope a technique that explicitly takes into account the presence of shaowers in a given environment. Poissonian assumption regaring the istribution of shaowers on the plane allows to avoi a etaile geographical escription of the network. It is also allowe to use an arbitrary point process on the plane instea of Poisson one. Tampere, Finlan crocellular raiotelephone. IEEE Trans. on Veh. Tech., 40(4):21 30, November References [1] T. Rappaport. Wireless communications: principles an practice. Communications engineering an emerging technologies. Prentice Hall, 2n eition, [2] A. Longley an P. Rice. Preiction of tropospheric raio transmission loss over irregular terrain: as computer metho. Technical report: Erl 9-its 6, ESSA, [3] R. Ewars an J. Durkin. Computer preictions for service area for VHF mobile raio networks. Proceeings of the IEE, 116(9): , [4] T. Okumura, E. Omori, an Fakua K. Fiel strengtn its variability in VHF an UHF lan mobile service. Review of electrical communication laboratory, 16(9/10):825 83, September/October [5] M. Feuerestein, K. Blackar, T. Rappaport, S. Seiel, an H. Xia. Path loss, elay sprea, an outage moels as functions of antena height for microcellular system esign. IEEE Tran. on Veh. Tech., 43(3):48 498, August [6] S. Seiel an T. Rappaport. 914 MHz pass loss preiction moels for inoor wireless communications in multifloore builings. IEEE Trans. on Ant. an Propag., 50(2):20 21, February [] J. Anersen, T. Rappaport, an S. Yoshia. Propagation measurements an moels for wireless communications channels. IEEE Comm. Mag., 33(1):42 49, November [8] R. Valenzuela. A ray tracing approach to preicting inoor wireless transmission. In In Proc. of IEEE VTC, pages , May [9] T. Seiel, S. a Rappaport. Site-specific propagation preiction for wireless in-builing personal communication system esign. IEEE Trans, on Veh. Tech., 43(4):89 891, November [10] A. Salen R. Valenzuela. A statistical moel for inoor multipath propagation. IEEE JSAC, 5(2):128 13, February 198. [11] T. Rappaport. Statistical channel impulse response moels for factory an open plan builing raio communication system esign. IEEE Trans. on Comm., 39(5):94 806, May [12] D. Durgin an T. Rappaport. Theory of mltipathape factors for small-scale faing wireless channels. IEEE Trans. on Ant. an Propag., 48(5): , May [13] M. Krunz an J.-G. Kim. Flui analysis of elay an packet iscar performance for QoS support in wireless networks. IEEE JSAC, 19(2): , February [14] M. Krunz an J.-G. Kim. Delay analysis of selective repeat ARQ for a markovian source over a wireless channel. IEEE Trans. on Veh. Tech., 49(5): , September [15] M. Zorzi, R. Rao, an L. Milstein. ARQ error control for faing mobile raio channels. IEEE Trans. on Veh. Tech., 46(2): , May 199. [16] M. Zorzi an R. Rao. Perspectives of the impact of error statistics on protocols for wireless networks. IEEE Pers. Comm., 6(5):32 40, October [1] P. Green an R. Sibson. Computing irichlet tesselations in the plane. Computer Journal, 21:168 13, 198. [18] M. Hata. Empirical formula for propagation loss in lan mobile raio services. IEEE Trans. on Veh. Tech., VT- 29(3):31 325, August [19] S. Seiel. Path loss, scattering an multipath elay statistics in four european cities of igital cellular an mi-