Distance dependent Call Blocking Probability, and Area Erlang Efficiency of Cellular Networks

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1 Distnce dependent Cll Blocking Probbility, nd Are Erlng Efficiency of Cellulr Networks Subhendu Btbyl 1, Suvr Sekhr Ds 1,2 1 G.S.Snyl School of Telecommuniction, Indin Institute of Technology Khrgpur, Indi. 2 Dept. of Electronics & Electricl Communiction Engineering, Indin Institute of Technology Khrgpur, Indi Abstrct In this pper the cpcity nd coverge of cellulr network is nlyzed in the perspective of Grde of Service (GoS) constrint. The cellulr networks considered, unlike GSM systems, llocte vrible rdio resource sizes to users even for constnt bit-rte (CBR) trffic. The Grde of Service is the cll blocking probbility. It is shown tht the blocking probbility (P b ) vries with user distnce from the Bse Sttion (BS). It is lso shown tht the effective cell-rdius vries for different services (bit-rtes) given certin GoS. Furthermore the vrition of useful re of cell long with offered lod is shown for different services. If mximizing energy efficiency is one of the considertions in deploying cellulr network, then it is shown tht the optimum trnsmit power for cell lyout is different when GoS is used s the criterion s compred to minimum throughput or minimum Signl to Interference plus Noise Rtio (SINR) t cell edge. I. INTRODUCTION The cellulr systems such s GSM, cdmone etc. llocte fixed rdio resource to users. Such networks re designed to crry trditionl voice trffic. Modern mobile wireless networks primrily crry dt. One of the importnt improvements introduced in these systems is the cpbility to llocte multiple rdio resources to one user s session (cll). Exmples of such networks re 3G nd 4G (Wimx, 3GPP LTE). In cse of 4G networks even voice is crried s VoIP. The cpcity of such networks is given in terms of throughput nd mximum no. of VoIP users per MHz. The method used to estimte mx VoIP cpcity is to find the no. of users when present in the system tht would llow t lest 95% of the users to be stisfied [1] [2]. A stisfied user is one who successfully receives t lest 98% of the trnsmitted pckets. Wheres the mesure of Erlngs is used to define the cpcity of wireline telephone exchnges nd erlier cellulr systems, where fixed resources re llocted to users mking voice cll. The cpcity of the systems re defined s the mximum offered trffic intensity in Erlngs for which the the GoS, which is the blocking probbility, is less thn trget threshold. The definition of Erlng cpcity ws developed originlly for circuit switched telephone networks. It ws ssumed tht in such networks resource is llocted to the user for the entire cll durtion nd is not shred with other users even when idle. On the other hnd wireless systems re bndwidth (resource) limited due to the brodcst nture of the chnnel. Furthermore due to the bursty nture of the trffic the protocols re being designed which mximize system cpcity by shring the rdio resource using dynmic Rdio Resource Alloction (RRA) mechnism. The RRA tkes into ccount the rte of informtion t the source nd the chnnel condition of the user nd then lloctes s much resource s required to support the requested bit rte. Thus vrible mount of rdio resource is llocted to users which my further vry with time. It is not strightforwrd to derive the Erlng cpcity of such systems. We build our work upon the frmework in [3] [6] to develop the Erlng cpcity for Orthogonl Frequency Division Multiple Access (OFDMA) wireless cellulr networks. These works show cellulr cpcity from Grde of Service point of view, nd study the cellverge blocking probbility. Our objective is to nlyze the GoS s function of user distnce from Bse Sttion (BS). We lso investigte the percentge (%) re coverge in the perspective of GoS, nd cll it the useful service re of the cell. This prmeter is defined in Section II. The clssicl definition of cell coverge re [7] is in terms of the received signl strength (nd hence the Signl to Interference plus Noise Rtio (SINR)) being bove certin threshold. The vrition of the useful service re of the cell s function of the offered trffic lod in Erlngs, the streming rte nd the cell rdius is lso presented in this work. Energy efficiency of cellulr networks hs been studied s [8], [9] the cell spectrl efficiency (SE) divided by the totl power consumed by the BS. The cell SE is verged over the cell nd unnormlized by the re of the cell. While the bove pproch my optimize the trnsmitted power in cell where only best effort trffic is present, it need not necessrily be optiml in cell where streming trffic predomintes. Hence we propose to study Energy Efficiency from n Erlng Cpcity (nd ssocited GoS) viewpoint. The rest of the pper is orgnized s follows. Erlng cpcity is introduced nd studies of blocking probbility vrition within cell nd useful service re vrition with offered lod etc. re presented in Section II. Are Erlng efficiency 1 (AEE) is introduced nd Energy Efficiency from Erlng cpcity point of view is presented in Section III. Finlly the conclusions re presented in section IV. II. BLOCKING PROBABILITY The nlysis frmework nd system prmeters (Fig. 1)used re s given in [3]. Selected prmeters re shown in Tble I. Typicl dt rte for voice is r =12.2kbps [3], 64kbps is the rte for 3G Video Telephony, nd 256kbps for good 1 This term is defined in Section III /12/$ IEEE

2 Symbol Mening ρ SINR Πi i th Trffic Clss Pb() Pb s function of Erlng Lod In Section generte Men nd Outge Cell SE Erlng cpcity from Pb( ) vs Consider Hexgonl Network Model, BSs plced on regulr hexgonl grid torus comprising 5x5 cells Estimte SINR t uniformly distributed rndom loctions within desired cell 1) Mp ρ to SE 2) Compute Bndwith Requirement. 3) Mp to Subcrriers Find distribution of No. of subcrriers required by rndom cll within the cell Kufmn Roberts Algorithm gives Πi specific Pb( ) weighted verge of which gives Pb( ) R P = 52 dbm (mx trnsmit power) Pn = 3 dbm (noise power) d ε = 12 % (control chnnel power) Frequency Reuse Fctor = 1 (Other Cell) Interference is white No mobility Blocked clls re clered η Propgtion: L(d) = (Kd) Pth Loss Only Κ = 8667 η = 3.38 No Shdowing No smll scle fding SE = b log (1 + ρ ) b=.83 2 (ρ) Πi : Ech region S i of the cell (nnulr) requiring different no. of subcrriers comprises trffic clss i ε {1,2,... P} w i : In rndom distribution of lrge no. of users (U) within cell, the proportion of users (U i) belonging to S i Anlyze vrition of Pb vs d. Mesure useful service re where Pb <.2 Fig. 1: Flow-chrt nd key prmeters for Erlng Cpcity, Useful Service Are of Cell nd lter Cell Spectrl Efficiency Evlution. qulity streming video. Fding is ccounted for by using constnt mrgin [] which is detiled lter. The Kufmn Roberts Algorithm (KRA) is used to evlute the blocking probbility in cellulr network employing OFDMA [3]. We use modifictions of Shnnon formul in order to get closer to rel world system (eg. LTE) performnce. Typiclly this sort of modifiction looks like [11] ρ C = b log 2 (1 + ), (1) ((ρ) fding ) where C is the spectrl efficiency, b ccounts for system overheds, ρ is the verge SNR nd (ρ) ccounts for the gp between Shnnon limit nd performnce of rel world coding schemes which vries with ρ. Furthermore, to ccount for fding [] we bring in nother extr fctor fding =3dB. With these modifictions, we do not need to use the exggerted bndwidth requirement of 18kbps (insted we use r =12.2kbps) per cll [3], thereby enhncing the ccurcy of our simultions. We lso study the effect of vrition in the no. of subcrriers (N subc ) (hence in Δ f - the bndwidth per subcrrier) for given system bndwidth W sys. For most of the simultion results (unless otherwise specified) N subc = 128 corresponding to Δ f =39.625KHz. Some results, such s the mximum voice trffic per cell (Erlngs) nd the Are Spectrl Efficiency s function of cell rdius re evluted for both N subc = 128, nd 512 (Δ f =9.7656KHz). The Energy Efficiency study uses N subc = 512 for evluting the Erlng cpcity for voice clls, nd mps it to n equivlent spectrl efficiency s described in section III. Streming clls whose inter-rrivl times to the network re i.i.d exponentil rndom vribles with rte λ (men 1 λ ) re hndled by the BSs. Durtion of cll is exponentilly distributed with men 1 μ. Ech cll request origintes from point uniformly distributed within the geogrphicl re of the cell (sy S C ). Ech cll must be gurnteed its intrinsic bitrte (r) for the durtion of the cll. The mesure = λ μ is TABLE I: Simultion Prmeters Prmeter Choice b, (1).83, TU SISO [11] r 12.2kbps [3], 64, 256kbps P b.2 W sys 5Mz N subc 128, 512 clled trffic demnd density (Erlng/m 2 ). Depending on its distnce from the BS, the SINR nd hence SE chievble for MS vries. In our model, since only signl ttenution due to propgtion loss is considered, the no. of subcrriers required by cll is constnt over ring surrounding the BS. Hence we cn clssify the cell into nnulr regions (S i, see Fig. 1) within which the blocking probbility of cll is constnt [3]. Ech nnulr region comprises trffic clss (Π i ) with specific GoS (P Πi b ) for given lod tht cn be evluted by KRA. The verge blocking probbility P b is defined s the weighted verge of the ring-wise GoS, i.e. P b = P i=1 w ip Πi, where w i = Ui U is the frction of users belonging to clss Π i, nd P is the no. of distinct trffic clsses. The useful cell re from GoS perspective is defined s F u (P bmx )=Pr(P bx,y <P bmx P b = P bmx ) x, y S C, (2) where x, y is n rbitrry point within the cell nd P bmx is typiclly.2. TheRHSof(2) is given by the probbility tht rndomly selected point in the cell flls within the circle composed of ll the rings for which P b <P bmx.the Admission Probbility is defined s P =1 P b.thep% outge P is vlue P o,p such tht Pr(P x,y <P o,p )= p. (3) P o,p is thus the vlue of P bmx for which F u is ( p)%. b

3 Fig. 2 shows P b s function of offered lod ( = λ μ where λ is cll rrivl rte, cll rrivl process being ssumed Poisson, nd 1 μ is the men cll holding time) for vrious streming dt rtes (r =12.2, 64, 256kbps). The mximum lod on the system for which P b < P bmx (which we denote s mx for P bmx =.2) is defined s the Erlng Cpcity which vries with bit-rte nd cell rdius (R). The vrition of mx vs R for vrious rtes is shown in Fig. 3. The Erlng cpcity for 12.2kbps voice is improved by mximum of 5% by incresing N subc from 128 to 512 becuse of the incresed resource lloction grnulrity, wheres tht for streming video shows much smller chnge so in Fig. 3 we plot mx for voice (only) with both 128 s well s 512 subcrriers. P b (%) r = 256 kbps r = R=525m, r = 12.2kbps 3m 5m 525m, r=64kbps 3m 5m 525m, r = 3m 5m r = 12.2 kbps (Erlngs) Fig. 2: Blocking probbility s function of offered lod in Erlngs for streming trffic (r =12.2, 64, nd256kbps). It is observed tht mx stys more or less flt in the interference limited region (R <2km) [3] nd flls shrply for R > 3km. Also mx decreses shrply with increse in r, hving vlues < 1 for r = 256kbps. For supporting these rtes Multiple Input Multiple Output (MIMO) nd other spectrl efficiency enhncement techniques become invluble. For vlues of R<2km, mx 66 with 128 subcrriers nd mx 93 with 512 subcrriers. This is significntly higher thn the vlue 24 in [3] becuse of the constnt SNR penlty of 11.7dB 2 ssumed t ll verge SNRs insted of penlty tht vries with ρ between 4 to 11dB s in LTE spectrl efficiency [11]. Mximum Trffic Per Cell (Erlng) kbps (128 subcrriers) 12.2 kbps (512 sucbcrriers) Cell Rdius (km) 1 Fig. 3: Mximum trffic lod (Erlngs) per Cell vs Cell Rdius for vrious streming trffic rtes (r =12.2, 64, nd 256kbps) e3 =W sys log 2 (1 + ρ 12 ); 18e3 =W sys log 2 (1 + ρ 18 ) SNR penlty = ρ 18 ρ dB Fig. 4 shows the no. of subcrriers N subcreq needed for vrious r s function of d BS MS, the distnce of the MS from the BS within cell of rdius R = 525m (Figs. 5, 6, 7 lso ssume this vlue of R). N subcreq vries between 1 4 for 12.2kbps becuse of its low bndwidth requirement. For the higher vlues of r such s 256kbps, N subcreq increses steeply with d BS MS especilly for higher vlues of d BS MS where the fll in SINR is ccelerted by the combined effect of incresing distnce from the desired BS nd decresing distnce reltive to (dominnt) interfering BSs. Figure 5 shows P b (evluted by KRA, which ssigns different P b to ech clss of clls with different N subcreq, P b being their weighted verge) s function of d BS MS for = mx. To study the impct of higher P b t nd ner the cell-edge, two pproches re possible: 1) Study the Jin Firness mesure [12] for the Cll Admission Probbility metric (P =1 P b ) defined s: F (P )= 1 1+ V (P) E 2 (P ), (4) where V (P ) is the vrince of P, nd E(P ) is its expecttion. 2) Compute the p% outge P (p =5, ) nd lso F u. No. of reqd. subcrriers r = 12.2 kbps r = r = Distnce of MS from BS (m) Fig. 4: Subcrrier Requirement (N subcreq ) for Cell Rdius of 525m nd vrious Streming Dt Rtes s function of distnce from BS. Pb (%) r = 12.2 ( mx = 66.6) r = ( mx = 11.7) r = ( mx =.9) Distnce of MS from BS (m) 45 5 Fig. 5: Blocking Probbility (P b ) for Cell Rdius of 525m (P b =.2) nd vrious Streming Dt Rtes s function of distnce from BS. Figure 6 shows the CDF of P when P b =.2. It is seen tht for ll rtes, F u is less thn 8%, i.e.intlest% of the possible loctions within the cell, P < 98%. F (P ),F u nd

4 P o,p for vrious streming rtes re listed in Tble II for R = 5m, nd 3m. It is observed from these tbles tht though F (P ) decreses with incresing r, it is lwys firly close to vlues, however, re much lower thn 98% with the minimum being 91.82% for p =5(r = 256kbps, R =3km) which corresponds to P b of 8.18% nd 94% for p = (r = 256kbps, R =3km) - which corresponds to P b of unity. P o,p decreses with incresing dt rte. F u on the other hnd increses with incresing dt rte, being the lowest for 12.2kbps (.55 for R =3km) nd the highest for 256kbps (.72 for R =3km). The increse of F u with incresing rtes is becuse the finl rise of P b vs. d BS MS (Fig. 5) is steeper for higher rtes, hence to mintin the sme P b, P b must cross P b lter (i.e. t higher vlue of d BS MS ). Note tht the mx is significntly lower for higher dt rtes. 6%. P o,p CDF: Pr(P (%) < bsciss) kbps P : Admission Probbility (%) Fig. 6: CDF of Probbility of Cll Admission (P ) for Cell Rdius of 525m (P b =.2) nd vrious Streming Dt Rtes. Useful Cell Are from Pb perspective kbps / mx Fig. 7: Useful Cell Are (in P b sense) for Cell Rdius of 525m vs. rtio of nd mx - cell diltion t low lod (P b =.2 when mx =1). An interesting question is wht hppens to F u when the lod in the cell is less thn mx. To nswer this, Fig 7 contins plot of F u vs mx (for < mx ) nd it is seen tht F u increses s mx decreses - phenomenon tht cn be visulized s the useful cell expnding s the Erlng lod decreses, similr to the concept of cell size brething [13]. This lso leds to possible solution for incresing F u nd P o,p whichistouselowervlueofp bmx while nlyzing network cpcity for dimensioning nd ccordingly incresing surfce density of rolled out BSs so s to reduce per cell. Overll, Jin Firness of P seems to be indequte in nlyzing P b vritions over the cell. TABLE II: Cll Admission Probbility Firness (F(P )), Useful Cell Are from GoS perspective (F u =Pr(P b <.2 P b =.2)), nd Outge (P b >.2) Rtes for Streming Clls originting from Rndom Loctions within Cell (Rdius R 1 =.525km, R 2 = 3km) when = mx r F(P ) F u p% Outge P (%) (kbps) ( <F() < 1) (p =5, ) R 1 R 2 R 1 R 2 R 1 R , , , , , , 94.4 Are Spectrl Efficiency (bits/sec/hz/m 2 ) ASE(b/s/Hz/m 2 ) R(km) 9 Men ASE Outge ASE AEE (r = 12.2 kbps) Cell Rdius (km) Fig. 8: Men nd Outge Are Spectrl Efficiency nd Are Erlng (r = 12.2kbps) Efficiency s function of Cell Rdius (512 subcrriers, inset subcrriers). Energy Efficiency (bits/sec/hz/w) x 3 III. ENERGY EFFICIENCY ESE for R =.5km R = 1km R = 2km R = 5km R = km EEE for R =.5km R = 1km R = 2km R = 5km R = km Trnsmit Power (including ntenn gin nd loss) [dbm] Fig. 9: Energy (Spectrl, voice-erlng) Efficiency s function of Cell Rdius (512 subcrriers). TABLE III: Trnsmit Power to chieve Pek Network (Energy) Efficiency - Men Spectrl (ESE) nd Erlng (EEE) Efficiency terms P R tx m Ptx e % EEE loss Extr Power t ESE mx t EEE mx t Ptx m for ESE mx (km) (dbm) (dbm) (db) The nlysis frmework nd system prmeters (see Fig. 1) used for the study of energy efficiency re sme s tht of [9]. The clssicl Are Spectrl Efficiency (ASE) considers the cell-verge of the Shnnon spectrl efficiency nd ignores

5 the impct of QoS constrined trnsmissions; we extend it by directly incorporting Erlng cpcity into the definition [14]. Are Erlng efficiency (Ω) for streming rte r is defined s: Ω(r) = mx(r)r W sys A cell (5) where nd mx re s defined before, r is the bit rte of ech streming user (bits/sec), nd A cell is the cell re = R2, R being the cell rdius of the hexgonl cell. Are Erlng efficiency (AEE) is thus (rte-specific) mesure of spectrl efficiency per unit re inferred from the Erlng cpcity. Fig. 8 shows the vrition of men (A e ) nd outge (A o e) re spectrl efficiency nd re Erlng efficiency Ω s function of cell rdius (inset considers AEE with 128 subcrriers while min plot hs AEE with 512 subcrriers). The Outge ASE in the plot implies vlue of SE (A o e) such tht Pr(ω(x, y) <A o e)=.1; (6) where ω(x, y) is the LTE spectrl efficiency [11] t loction defined by coordintes (x, y) in the Argnd plne (within the cell of interest). Men ASE (A e ) is defined s usul [15] but with modified Shnnon formul [11]. Fig. 8 (inset) indictes tht for N subc = 128, Ω is much closer to the A o e thn to A e. The rte of fll of A e, A o e nd tht of Ω re similr. At high vlues of cell rdius, A o e nd Ω re lmost identicl. Fig. 8 (min) shows tht for N subc = 512, Ω is closer to A e for lower cell rdii, nd to A o e for higher cell rdii, decresing t rte fster thn both s R cell increses. The plot of Energy Efficiency (Fig 9) contins on its y-xis the rtio Energy Spectrl Efficiency (ESE) = AeA cell 3 P BS nd the trnsmit power lever (P tx, including ntenn gin/loss) in the x-xis. Energy Erlng Efficiency (EEE) is defined similrly, but with Ω in the numertor in plce of A e.itis observed tht for low nd high vlues of R (eg. =.5km nd km), the peks of ESE nd EEE occur t different vlues of P tx. For higher vlues of R, the pek for ESE requires less trnsmit power level compred to EEE, wheres for lower vlues of R, EEE pek requires lower trnsmit power level. Thus operting t EEE pek hs the dvntge of high Erlng efficiency t lower (by db) trnsmit power level for smll cell rdii (.5kms), resulting in lower cpitl expenses. Moreover, from Tble III we observe tht if the Bse Sttion trnsmit power is set in order to chieve pek ESE (which might be strightforwrd conclusion from [9]), then penlty in terms of Erlng cpcity will be pid for R 5km vrying from 1% t R =.5km to 42% t R =km. Thus the network designer cnnot simply operte by trditionl energy efficiency if he needs to mximize it from Erlng cpcity perspective. 3 P BS = cp + d, c =21.45,d= W, p = P tx G; P BS being the totl power consumed t the Bse Sttion, G(= 12dB) being the ntenn gin - see [9]. A difference from the Energy Efficiency plot in [9] istht we hve not multiplied by W sys in the numertor, so ESE hs units of bits/sec/hz/w insted of bits/sec/w. IV. CONCLUSION We hve built upon the initil work on Erlng cpcity of wireless cellulr networks with spectrl efficiency models tht re closer to rel world system performnce. We hve shown the choice of subchnnel bndwidth strongly influences Erlng cpcity, nd needs to be smll enough to permit high resource lloction grnulrity nd hence high Erlng cpcity. We hve nlyzed the distnce dependence of cll blocking probbility nd shown tht P b my significntly exceed 2% t the cell-edge even when the cell-verge P b is less thn this limit. We hve extended the clssicl concept of cell coverge re from GoS viewpoint s useful service re (F u )ofthe cell. We hve shown tht F u decreses s the lod on the cell increses, concept useful in dimensioning, especilly if F u is significntly less thn %. We hve studied Energy Efficiency from Erlng Cpcity perspective. We hve shown tht use of Are Erlng Efficiency (AEE), which extends the clssicl concept of Are Spectrl Efficiency to ccount for QoS constrined trnsmissions, leds to different settings for optiml trnsmit power of BS. We hve observed tht for cells with rdii R>5km, the Energy Efficiency from Erlng Cpcity perspective my be scrificed by 7 4% if the BS trnsmit power is set to the optimum from ASE viewpoint. REFERENCES [1] S Sesi, I Toufik, M Bker, LTE The UMTS Long Term Evolution: From Theory to Prctice, 1st ed. John Wiley & Sons, 9. [2] ITU-R, Guidelines for evlution of rdio interfce technologies for imt-dvnced, ITU-R, Tech. Rep. M , 9. [Online]. Avilble: [3] M. K. Krry, Anlyticl Evlution of QoS in the Downlink of OFDMA Wireless Cellulr Networks Serving Streming nd Elstic Trfc, IEEE Trnsctions on Wireless Communictions,. 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