Mobility-dependent small-scale propagation model for applied simulation studies

Transcription

1 RESEARCH Open Access Moblty-dependent small-scale propagaton model for appled smulaton studes Dmtr Moltchanov Abstract We propose an extenson to exstng wreless channel modelng technques ntroducng a noton of moblty behavor of the user. We represent large-scale propagaton characterstcs of wreless channels as a mobltydependent stochastc process that explctly tracks the movement of the user between areas wth dfferent receved local average sgnal strength (RLASS). Our model conssts of two dfferent parts: moblty model and propagaton model. Moblty of the user s modeled by a Markov chan wth fnte state space. Large-scale propagaton characterstcs of wreless channel are represented as a functon of moblty model. Small-scale propagaton characterstcs are then obtaned takng nto account shadowng of the lne-of-sght propagaton path. Based on the amount of avalable nformaton regardng a gven landscape envronment we develop two dfferent parametrzaton methods () RLASS values are avalable for lmted number of ponts n a gven landscape and () RLASS nformaton s not avalable. The model s sutable for smulaton studes of applcatons performance n presence of RLASS changes caused by movement of the user. 1 Introducton To estmate performance of wreless channels propagaton models are often used. We dstngush between large-scale and small-scale models. The former models capture propagaton characterstcs on a coarse granularty usng the noton of the receved local average sgnal strength (RLASS), see e.g., [1-3]. Models characterzng rapd changes of the receved sgnal strength are called small-scale propagaton models, see e.g., [4-]. These models capture propagaton characterstcs on a fner granularty. Nether large-scale nor small-scale models take nto account the sgnal strength attenuaton caused by movements of a user. To be precse small-scale propagaton models do take nto account the so-called small-scale moblty of the user over short dstances [7]. Such models descrbe rapd fluctuatons around a constant mean whch are called fadng. Such processes are mplctly assumed to be statonary at least n the second-order sense and mean equals to the receved local average sgnal strength (RLASS). However, f we would consder larger travel dstances (.e., more than just few meters) RLASS starts to vary and the most mportant factors Correspondence: moltchan@cs.tut.f Department of Communcaton Engneerng, Tampere Unversty of Technology, P.O. Box 553, Tampere, Fnland affectng t are terran, speed of a moble and dstance from between transmtter and the recever. In ths artcle, contrarly to most small-scale propagaton models proposed so far, we explctly take nto account two of these three factors terran and moblty of a user over t. In a moble envronment a user s allowed to change ts poston at any nstant of tme and these movements are not restrcted to short travel dstances. A recever durng a sngle sesson experences dfferent propagaton characterstcs. To predct these changes, an adequate model must capture both movement of a user between areas wth dfferent propagaton characterstcs and small-scale propagaton characterstcs n each area. Propagaton characterstcs must be represented as a probablstc functon of user s movements. Such model would be useful to estmate performance provded to applcatons whle user s n travelng mode. The rest of the artcle s organzed as follows. In Secton 2, we descrbe the structure of the model and ntroduce our modelng assumptons. Then, n Secton 3, we propose three dfferent parametrzaton methods for our model. Extensons and refnements of the model are dscussed n Secton 4. Fnally, n Secton 5 we test some of our assumptons. Conclusons are provded n the last secton. 2 Moltchanov; lcensee Sprnger. Ths s an Open Access artcle dstrbuted under the terms of the Creatve Commons Attrbuton Lcense ( whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted.

2 Page 2 of 9 2 Structure of the model 2.1 Prelmnary notes The receved sgnal strength n wreless networks s bascally affected by three factors: () type of the terran we consder () speed of mobles, and () dstance between the transmtter and recever. Most small-scale propagaton models proposed so far ether mplctly or explctly assumed that all these factors are fxed,.e., a certan type of terran s consdered, speed of mobles as well as dstance between transmtter and recever are fxed. The man reasons s that even wth these assumptons the resultng receved sgnal strength s a covarance statonary stochastc process wth complcated dstrbutonal and correlatonal propertes. The reason for such reduced complexty of models s that smallscale propagaton models were manly used to desgn transcevers and assocated lower layer technques. For such tasks these models provde suffcent accuracy and should not be extended to more complex cases. When performance of nformaton transmsson at hgher layers s concerned we are nterestng n addng those addtonal factors to the small-scale propagaton models. Indeed, durng the actve sesson a moble user may move nsde and n-between cells experencng not only qualtatvely dfferent propagaton characterstcs (e. g., dfferent dstrbutons but the same mean) but quanttatvely dfferent too (e.g., dfferent RLASS). Thus, n addton to the type of small-scale propagaton at a certan separaton dstance from the transmtter we also have to take nto account movement of a user between areas wth dfferent RLASS. We would lke to note that allowng at least one factor affectng the receved sgnal strength to be varable may result n complex stochastc process makng the model applcable for smulaton studes only. 2.2 The moblty model Assume that a gven cell of a crcular shape s dvded nto a fnte number of areas, M, such that these areas are non-overlapped and the sum of ther areas equals to the area of the cell. Each area s assocated wth a certan, not necessarly dstnctve value of RLASS. We assume that a moble user may probablstcally move between these areas. Consder a dscrete-tme envronment,.e., tme axs s slotted, the slot duraton s constant and gven by Δt. Changes of areas are only allowed at the slot boundares. To capture movement of a user between areas we use a homogenous dscrete-tme Markov chan (DTMC) {S P (n), n=,1,...}, S P (n) Î {1, 2,...,M}, whose states correspond to areas. Let D P be ts transton probablty matrx. To parameterze ths model M, D P, and Δt shall be provded. 2.3 The large-scale propagaton model RLASS s manly a functon of the dstance between the transmtter and the recever and the type of propagaton envronment. To represent t, we assocate a value of RLASS wth each state of the moblty model. Let {P (n), n =,1,...}, P(n) Î {P 1,P 2,...,P M }betherlass process whose underlyng Markov chan s {S P (n), n =, 1,...},.e., the value of RLASS s modulated by the underlyng Markov chan. To parameterze ths model we have to provde the RLASS vector P =(P 1, P 2,..., P M ). 2.4 The small-scale propagaton model For performance evaluaton studes, small-scale propagaton characterstcs n a certan area can be modeled by doubly-stochastc process {R (n), n =,1,...}modulated by homogenous DTMC {S R, (n), n =,1,...}, S R, (n) Î {1, 2,..., H }, each state of whch s assocated wth condtonal probablty dstrbuton functon of the receved sgnal strength F R, (kδf j) (Δf) =Pr{R (n) =kδf S R, (n) =j}, k =1,2,..., N, j =1,2,..., H, where N s the number of bns to whch the sgnal strength s parttoned, Δf s the dscretzaton nterval. The slot duraton Δt of the model equals to the tme to transmt a sngle bt at the wreless channel. Slot duratons of moblty and small-scale propagaton models are equal and synchronzed. To capture small-scale propagaton characterstcs we dstngush between LOS and NLOS envronments. In the former case the small-scale propagaton envelop dstrbuton s Rcan. As the domnant component fades away the small-scale propagaton envelop dstrbuton degenerates to Raylegh dstrbuton. Thus, small-scale propagaton characterstcs n dfferent areas are, at least, qualtatvely (dstrbuton) or quanttatvely (RLASS) dfferent. Therefore, areas must be determned such that each of them s unquely characterzed by mean/dstrbuton par. Then, each area must be assocated wth unque small-scale propagaton model {R (n), n =,1,...}, =1, 2,..., M such that E[R ]=P,.e., the RLASS n the area s the mean of {R (n), n =,1,...}. We requre that the state space of all small-scale propagaton models s the same and gven by {1, 2,..., H}. Snce propagaton characterstcs are a probablstc functon of the movement between areas, the choce of the actve propagaton model {R (n), n =,1,...}, = 1, 2,..., M, ntheslotn depends on the area n the

3 Page 3 of 9 slot n,.e., {R(n), n=,1,...}={r (n), n =,1,...}, S P (n) =. Recallng that all small-scale propagaton models were assumed to have the same number of states, H, the state-space of the resultng model s S R (n) {(1, 1),..., (1, H),..., (M, 1),..., (M, H)}. (1) An approprate small-scale propagaton model s only assocated wth the state of the moblty model correspondng to the approprate area. Hence, transton probabltes of DTMC of moblty-dependent model s gven by d R,j = d R,,kj, k, j {1, 2,..., H}, S P (n) =. (2) To parameterze moblty-dependent small-scale propagaton model we shall provde H, D R,, F R, (kδf j)(δf), =1,2,..., M, k =1,2,..., N, j =1,2,..., H. 3 Parametrzaton of the model 3.1 The moblty model Assume that the number of areas M and assocated RLASS vector P are known. It s far to expect MP that the mean sojourn tme n a certan area depends on ts sze. Consder a user that s n the area n the slot n. In the next slot ths user may stay n the area or move to another area. The only areas to whch a user can move n a slot are adjacent areas denoted by Ω.Consder areas from Ω as a sngle area. We compute transton probablty between area and Ω as d L, = S j S + S, d L, = S S + S, (3) where S s the area of area, S Ω stheareaofω, d L,, = 1, 2,..., M, are transton probabltes between area and Ω. Dependng on the length of the border between area and areas from Ω transton probablty d L, s dstrbuted among transton probabltes d L, j, j Î Ω,as d L,j = d V j w j j V j, j, (4) where V j, j Î Ω are lengths of the border between and j. Parameters w j, j Î Ω are ntroduced to represent drectonal movement of the user n a certan applcaton scenaro, e.g., hghway, cty center. Snce ths movement s specfc for a gven envronment, no general expresson can be provded. Note that j w j =1, =1,2,..., M. 3.2 Large-scale propagaton model Parametrzaton based on measurements Measurements of RLASS are often gven by the 3D vector P XY =(x, y, P ), =1,2,..., M, wherem s the number of measurements, (x, y ) s the coordnate of th measurement and P s the RLASS value of respectve measurement. Practcally, t s not feasble to measure RLASS n each and every pont of the landscape. Thus, nformaton gven by P XY s often nsuffcent to parameterze our model. To estmate P and M we have to determne areas to whch these measurements belong to, such that the approxmaton error s mnmzed. To determne areas we propose to use a dvson of the cell nto areas whose vertexes are measurement ponts (x, y ), =1,2,..., M. An approprate dvson of the cell s gven by Vorono (Drchlet) tesselaton separatng a regon E of the space R 2 nto polygons E,=1,2,..., M, usng M ponts drawn from a certan pont process. In our case the space s R 2 and coordnates of measurements are consdered as a realzaton of the pont process. For each measurement pont (x,y ),E s the area consstng of all locatons that are closer to (x,y )thantoany other pont. There are a number of algorthms to compute Vorono tesselaton [8]. To determne areas wth dfferent RLASS t s not strctly requred to dstngush between values of RLASS n each area. Instead, t s possble to consder ranges of RLASS. In ths case the range (max P - mn P ) must be dvded nto K non-overlappng ranges of length ΔP =(max P - mn P )/K. Then, all measurements are classfed to these ranges. Adjacent areas havng RLASS values fallng n the same range can be grouped. An example s shown n Fgure 1, where measurements ponts, the Vorono tesselaton, and resultng areas are shown Theoretcal parametrzaton Measurements of RLASS are often unavalable. In ths case, we can parameterze our model based on approxmate estmaton of shadowed areas and subsequent applcaton of large-scale propagaton models. Assume that centers of shadowers are dstrbuted accordng to the statonary Posson pont process wth mean l. The parameter l depends on a gven landscape. Note that whenever possble actual placements of shadowers can be used. We also assume that (1) shadowers are of rectangular form, (2) thckness of shadowers s zero, (3) ther wdths and heghts are arbtrary dstrbuted, (4) heght of the transmtter antenna h a s known. Threepossbleshadowplacementscausedbyshadowers wth dfferent relaton between ntal parameters are shown n Fgure 2, where d s s the shadow length, R s the radus of a cell, d s mnmal dstance between the center of the cell and a shadower, and h s s the heght of shadower. We should dstngush between three cases: (1) h a >h s, d s R-d, the shadow s wthn the cell, (2) h a >h s, d s >R-d, the shadow s bounded

4 Page 4 of Fgure 1 Example of dvson of a cell based on measurements. outsde, (3) h a <h s, the shadow vrtually contnues up to nfnty. Snce we consder a sngle cell, the latter two cases can be treated smlarly. The area of an arbtrary shadow s gven by ( S s = mn 4π R 2 γ,4π(d + d s) 2 γ ) dw s 2,(5) where d s = mn ( ) hs cot β, R d, β =9 arctan d h a h s. To estmate RLASS vector P we use large-scale propagaton models. It was shown n [9] that the mean value of propagaton loss, L(d), can be approxmated by E [ L (d) ] ( ) d = L s (d ) +1n lg + X σ, () where d s the separaton dstance, n s the path loss exponent, d s the standard dstance, L s (d )sthepropagaton loss at d, X s s the factor unque for a gven envronment. For every shadowed area the propagaton loss s estmated usng () wth n>2, for unshad- owed areas n 2. Parameters for dfferent frequency bands are gven n [2,1]. Gven a certan transmsson power of the base staton, propagaton loss can be related to RLASS [7]. Observe that there can be shadows whose length s only slghtly less than the radus of the cell. The range of RLASS correspondng to these shadows s large d leadng to modelng errors. To avod t, n addton to dvson of the cell nto shadowed and non-shadowed areas, we propose to dvde the cell nto crcles wth dfferent rad. Shadows are then classfed to these crcles as shown n Fgure 3. Fnally, we need to menton that performng theoretcal estmaton of shadowed areas accordng to the proceduredescrbedabovetherecouldbethecasewhen two or more shadows overlap as shown n Fgure 4, where dfferent fllngs denote doubly and trply overlapped shadowed areas. In ths case RLASS n the area where shadows overlap could be dfferent compared to RLASS n non-overlappng regons of these areas. In ths case the propagaton loss n regons where two or more shadows overlap needs to be computed usng dfferent pass loss exponents. 3.3 The small-scale propagaton model We parameterze small-scale propagaton models usng the hstogram matchng method. Dependng on presenceoflosnthearea, process {R (n), n =,1,...} has ether Rcan or Raylegh dstrbuton wth mean E [R ]=L. Settng the number of states of the Markov chan n each state of the moblty model to one (H =1) and choosng Δf small enough we capture these dstrbutons as F R, ( k f )( f ) = p k x k+1 x k. (7) h a >h s,d s <R-d d R 9 o h a >h s,d s >R-d h a <h s d R d 9 o 9 o R d s d s d s Fgure 2 Three possble placements of a shadow.

5 Page 5 of 9 where p k = x k+1 x k f X (x)dx, f X (x) s ether Raylegh or Rcan dstrbuton, x k and x k+1 are lower and upper values for nterval k. Alternatvely, settng H and solvng the nverse-egenvalue problem one can capture autocorrelaton 1 and solvng the nverse-egenvalue problem one can capture autocorrelaton propertes of the receved sgnal strength. 4 Extensons and notes 4.1 Sgnal-to-nterference rato For a model to be useful, n addton to the receved sgnalstrengthwehavetotakentoaccountthelevelof nose and consder the sgnal-to-nterference (SI) rato. The nterference at the recever s gven by I = W + U, where W the thermal nose, U = L l=1 U (l) s receved sgnal strength from L nterferng base statons. W s usually assumed to be constant. U s stochastc, depends on the dstance and propagaton path between the recever and transmtters of nterferng cells. Assume that U s constant for any state of the moblty model. The nterference process {U(n), n =,1,...}, U(n) {U 1, U 2,..., U M }, s modulated by {S P (n), n =,1,...}. We determne the mean propagaton loss E[L(m j )] for a gven area and nterferng transmtter j as a functon of dstance them, m j,usng().e[l(m )] s then related to RLASS and mean other cells nterference s computed for areas as U = L l=1 U (l). Addng a constant W to {U(n), n =,1,...} affects the mean only allowng to consder the nterference process {I(n), n =,1,...}, =1,2,..., M. The resultng sgnal-to-nterference process s gven by (R 1 I 1 )(n), S P (n) =1, Y (n) = (8) (R M I M )(n), S P (n) = M, where I = W + L l=1 U (l), = 1, 2,..., M. Fgure 4 Doubly and trply overlapped shadowed areas. 4.2 Attracton ponts Notce that n our model movement of the user s homogenous n tme and depends on the sze of areas only, see (3). However, n addton to the sze of the areas other factors may affect movement of the user of the landscape. One of the most mportant s the socalled attracton ponts n a cell,.e., places (areas n our termnology) where a user may spent sgnfcant amount of tme compared to other areas. Takng nto account the effect of such attracton ponts s equvalent to ntroducng non-homogenety to the process of user movement between areas, that s, makng movement to depend on some addtonal factors. Formally, t can be done ntroducng a new multplers g, =1,2,..., M, each of whch corresponds to a certan area nto (3). 4.3 The scope of applcaton We would lke to note that when mcro or pcocells scenaros are consdered the areas we ntroduced n ths artcle are almost mpossble to defne. The reason s that n such cells attracton ponts seem to play much more mportant role compared to the sze of areas. Ths s clearly seen consderng offce scenaro, where employees move between pre-fxed locatons such as workng places, meetng rooms, etc. In ths case nstead of parameterzng our model based on szes of areas we need to consder attractveness of varous locatons and Fgure 3 Example of drect estmaton of shadowed areas.

6 Page of 9 nter-locaton movement of users developng new expressons for estmatng nter-locaton movements. Sncetherecouldbeanumberoflocaton-specfcfactors nvolved nto estmaton of these probabltes we do not provde formulas for mcro and pcocells. Our model was orgnally proposed for macrocell envronments where t s possble to dstngush between dfferent areas wth quanttatvely and/or qualtatvely dfferent receved sgnal strength. The theoretcal parametrzaton of the model s sutable for suburbs and country-sde areas. When measurements of RLASS are avalable the model can be appled n cty-center scenaros. The only dfference s that n addton to szes of areas we also need to take nto account possble tunnelng effects. Notce that even n ths case we can use theoretcal parametrzaton. However, nstead of propagaton models specfed n [1-3,1] we need to use specal models sutable for such envronment. Fnally, as mentoned above wth some modfcatons ths model can be appled to mcrocell and pcocell scenaros where a user moves between a number of attracton ponts. Applyng ths model n such scenaros addtonal care should be taken as even small travel dstances may lead to drastc changes n small-scaled propagaton characterstcs due to complcated reflecton and scatterng observed n the buldngs. 5 Testng assumptons Unfortunately due to unavalablty of data regardng moblty of users on the landscape we were not able to test closeness of the proposed moblty model and emprcal data. Asde from ths, one of the most mportant assumpton we take n ths artcle s that the receved sgnal strength at a certan separaton dstance from the transmtter s ndeed statonary whle when changng the locaton dfferent characterstcs of the receved sgnal strength dstrbuton changes. To test ths assumpton we carred out experments wth wreless local area network (WLAN) networks operatng n nfrastructure DCF mode accordng to IEEE 82.11b standard at 11 Mbps (DSSS) n laboratory envronments. The access pont s nstalled n the corrdors wth rooms along both sdes of t. The man ponts of nterest were characterstcs of the receved sgnal strength process at a certan separaton dstance from the transmtter and ts changes as users move between dfferent attracton ponts. Sgnal strength observatons were measures wth 1 ms granularty and then averaged over.5 ms tme ntervals. To test our assumpton regardng statonary nature f the receved sgnal strength two rooms were arbtrarly chosen and the sgnal was measured for a long duraton of tme. Two samples gather as a result of these experments are shown n Fgure 5. Just observng these traces no defntve conclusons regardng statonarty of the SNR process at a certan separaton dstance from the transmtter can be made. For ths reason we performed detaled statstcal analyss as dscussed below. Hstograms of relatve frequences servce as estmates of emprcal dstrbutons and ther approxmatons by Normal dstrbuton are show n Fgure. As one may observe approxmatons are qute good. Testng the null hypothess about Normal dstrbuton of data usng c 2 test confrmed t wth the level of sgnfcance, a, setto.1. Notce that for the second trace the null hypothess s accepted even when a =.5. Tests performed for other rooms (not shown here due to space constrants) also confrm ths approxmaton. We would lke to note that one should not be surprsed wth normalty of observatons. Recall that we averaged the receved sgnal strength over.5 s. ntervals provdng suffcent aggregaton level for central lmt theorem to hold. In addton to dstrbutonal propertes dscussed above the process s found to be autocorrelated as hghlghted as normalzed autocorrelaton functons (NACF) shownnfgurec,d.theljung-boxtest(aspecal type of portmanteau test) performed wth the level of sgnfcance a =.1 showed that frst three lags of the frst sample and the frst lag of the second one are statstcally dfferent from zero wth level of confdence a =.1. Observng the fgures we see that the both functons decay geometrcally fast. Note that the above mentoned analyss does not allow us to conclude that the process at a certan separaton dstance s covarance statonary. Unfortunately, there are no well-behaved tests to access statonarty of stochastc processes. We use the followng two-steps procedure. Frst of all, observe that explctly assumng covarance statonarty the above mentoned analyss allows to model SNR data at a certan separaton dstance from the transmtter usng autoregressve process of order 1, AR(1), n the form Y = j + j 1 Y -1 + ε, =, 1,...,wherej and j 1 are some constants, ε are ndependently and dentcally dstrbuted random varables havng the same normal dstrbuton wth zero mean and varance s 2 [ε]. Thus, we used aposteror change-pont exponental weghted movng average test for AR(1) (see [11,]) to detect change n the mean of SNR data. For our observatons no changes have been observed n both samples. However, ths does not necessarly mply that there are no changes n other parameters of the process. To address ths stuaton we dvded each sample nto three dfferent subsamples and compare ther dstrbutonal and correlatonal propertes. It appeared that usng c 2 test wth the level of sgnfcance set to a =.1 all the subsamples drawn from a certan sample are homogenous,.e., have the same 1D dstrbutons. Usage of rather bg value of a s explaned

7 Page 7 of 9 Y() (a) Sample 1 Fgure 5 Samples of SNR observatons (locaton s fxed). Y() (b) Sample 2 by presence of correlaton n subsamples. To conceal the effect of correlaton we also used every second observaton n each subsample to compute emprcal dstrbutons.theturnngponttestdemonstratedthatthe hypothess about whte nose should be rejected wth the level of sgnfcant a =.5 ndcatng that there s memory n each of those sample. Fnally, n each subsample NACF decayed geometrcally fast. The reason for performng the turnng pont test nstead of, e.g., Ljung-Box portmanteau test s that t s unreasonable to expect the same number of lags havng statstcally sgnfcant correlaton n samples of small szes. Although these observatons do not strctly prove that SNR process at a certan separaton dstance from the transmtter s covarance statonary they provde enough evdence that frst and second-order characterstcs lkely reman unchanged. Recall that n addton to covarance statonarty of SNR process as a certan separaton dstance from the f,e ( ) K Y () (a) Sample (c) Sample 1,lag f,e ( ) K Y () (b) Sample (d) Sample 2 Fgure Hstograms and NACFs of SNR observatons and ther approxmatons.,lag transmtter we also assumed that the mean (and possbly dstrbuton) changes as usermovesbetweendfferent locatons. To back up our assumpton we performed the followng experments. A moble staton was n statonary poston for some notceable amount of tme. Then, a user moved nto another offce room and the staton agan remaned n statonary state for some amount of tme. Two arbtrarly chosen SNR samples obtaned by changng the locaton of the user durng the measurng process are shown n Fgure 7. Observng these traces one may notce that the change n the receved sgnal strength happens almost nstantaneously. The reason s that the dstances between attracton ponts n offce envronment are rather short (recall that the granularty of SNR measurements s.5 s). Results of EWMA change pont test are shown n Fgure 8 for two values of smoothng parameters g. Inall demonstrated fgures the change n SNR statstcs s detected. Moreover, our results show that changes are detected for all values of smoothng parameters g that are less than.1 mplyng that there s no need to access accuracy of the test usng average run length (ARL) statstcs. Note that change-pont statstcal tests are rather coarse n general. To ensure that the change n some parameters of the dstrbuton happened we compared dstrbutonal characterstcs of two subsamples drawn from each sample. To get these subsamples we used results of EWMA test to exclude those observatons occurrng at the change pont. c 2 test performed wth the level of sgnfcance set to a =.5showedthatthe null hypothess about the same emprcal dstrbuton should be rejected. Note that the above mentoned statstcal analyss demonstrate that () SNR process at a certan separaton dstance from the transmtter s close to covarance statonary () SNR process may quanttatvely change when user changes ts locaton. Supplementng the above mentoned analyss wth moblty model between attracton ponts we a partcular case of the moblty-

8 Page 8 of 9 Y() (a) Sample 1 Fgure 7 Tme seres of SNR observatons (locaton s changed). Y() (b) Sample 2 dependent propagaton model ntroduced n ths artcle. Although these test have been performed n offce envronment usng IEEE 82.11b WLAN we can expect the same statstcal behavor for publc wreless networks. Stll there are mportant dfferences. Frst of all, due to larger spaces covered by these networks n addton to attracton ponts we need to consder areas as descrbed n Secton 2. Secondly, changes n characterstcs of the receved sgnal strength are not expected to happen drastcally as user moved between areas and/or attracton ponts. Conclusons We developed an extenson for wreless channel modelng technques explctly takng nto account the movement of the user between areas wth dfferent smallscale propagaton characterstcs. Although a sngle cell envronment has been consdered, results can be extended to multple cells scenaro. The model s sutable for performance evaluaton of applcatons n presence of sgnal strength changes caused by movement of the user. Although the Markov structure of the model allows for analytcal performance L Y (), C mn (), C max () (a) Sample 1: γ =.1 L Y (), C mn (), C max () (c) Sample 2: γ =.1 Fgure 8 Tme seres of SNR observatons (locaton s changed). L Y (), C mn (), C max () (b) Sample 1: γ =.1 L Y (), C mn (), C max () (d) Sample 2: γ =.1

9 Page 9 of 9 modelng, large state space may restrct ts usage to smulaton studes. We note that the model can be extended to capture propagaton-dependent behavor. For example, the current modulaton and codng scheme as a functon of moblty of the user can be modeled. Competng nterests The authors declare that they have no competng nterests. Receved: February 211 Accepted: 2 Aprl 2 Publshed: 2 Aprl 2 References 1. T Okumura, E Omor, K Fakuda, Feld strength and ts varablty n VHF and UHF land moble servce. Rev Electr Commun Lab. 1(9/1): (198) 2. M Hata, Emprcal formula for propagaton loss n land moble rado servces. IEEE Trans Veh Tech. VT-29(3): (198) 3. M Feueresten, K Blackard, T Rappaport, S Sedel, H Xa, Path loss, delay spread, and outage models as functons of antena heght for mcrocellular system desgn. IEEE Trans Veh Tech. 43(3): (1994). do:1.119/ A Saleh, R Valenzuela, A statstcal model for ndoor multpath propagaton. IEEE JSAC. 5(2):8 137 (1987) 5. T Rappaport, Statstcal channel mpulse response models for factory and open plan buldng rado communcaton system desgn. IEEE Trans Commun. 39(5):794 8 (1991). do:1.119/ D Durgn, T Rappaport, Theory of multpath shape factors for small-scale fadng wreless channels. IEEE Trans Ant Propag. 48, (2). do:1.119/ T Rappaport, Wreless Communcatons: Prncples and Practce, 2nd edn. (Prentce Hall, Upper Saddle Rver, 22) 8. P Green, R Sbson, Computng Drchlet tesselatons n the plane. Comput J. 21, 173 (1978) 9. S Sedel, Path loss, scatterng and multpath delay statstcs n four european ctes of dgtal cellular and mcrocellular radotelephone. IEEE Trans Veh Tech. 4(4):721 7 (1991). do:1.119/ Dgtal moble rado, towards future generaton systems. COST 231 Fnal Report, EUR957. (1999) 11. D Moltchanov, State descrpton of wreless channels usng change-pont statstcal tests. n Proc WWIC 2, vol (Bern, Swtzerland, 2), pp W Schmd, A Schone, Some propertes of the EWMA control chart n the presence of auto-correlaton. Ann Stat. 25(3):77 83 (1997). do:1.14/ aos/ do:1.1/ Cte ths artcle as: Moltchanov: Moblty-dependent small-scale propagaton model for appled smulaton studes. EURASIP Journal on Wreless Communcatons and Networkng 2 2:1. Submt your manuscrpt to a journal and beneft from: 7 Convenent onlne submsson 7 Rgorous peer revew 7 Immedate publcaton on acceptance 7 Open access: artcles freely avalable onlne 7 Hgh vsblty wthn the feld 7 Retanng the copyrght to your artcle Submt your next manuscrpt at 7 sprngeropen.com