PRT ellular verage ncepts Lecture. why cells verage fr a terrestrial zne Signal OK if P rx > -X dm P rx c P tx d -4 greater P tx greater d S d ase Statin N channels (e.g. channel frequency) N simultaneus calls
ellular cverage target: cver the same area with a larger number f Ss 9 ase Statin frequencies 4 frequencies/cell Wrst case: 4 calls (all users in same cell) est case: 76 calls (4 users per cell) verage case >> Lw transmit pwer Key advantages: Increased capacity (freq. reuse) ecreased tx pwer ellular cverage (micrcells) many S Very lw pwer!! Unlimited capacity!! Usage f same spectrum ( frequencies) (4 freq/cell) isadvantage: mbility management
ellular system architecture S per cell ell: Prtin f territry cvered by ne radi statin One r mre carriers (frequencies; channels) per cell bile users fullduplex cnnected with S S cntrls many Ss f4 f4 f5 f5 f3 f3 f f f6 Wired netwrk S S f6 f f f7 f7 f4 f4 f5 f5 f3 f3 f f f6 f6 f f f7 f7 S cnnected t PSTN S S PSTN ase Statin bile Switching entre Public Switching Telephne Netwrk ellular capacity Increased via frequency reuse Frequency reuse depends n interference need t sufficiently separate cells reuse pattern cluster size (7 4 3): discussed later ellular system capacity: depends n verall number f frequencies Larger spectrum ccupatin frequency reuse pattern ell size Smaller cell (cell micrcell piccell) greater capacity Smaller cell lwer transmissin pwer Smaller cell increased handver management burden 3
hexagnal cells Hexagn: Gd apprximatin fr circle Ideal cverage pattern n hles n cell superpsitin Example case: Reuse pattern 4 ells in real wrld Shaped by terrain, shadwing, etc ell brder: lcal threshld, beynd which neighbring S signal is received strnger than current ne 4
PRT ellular verage ncepts Lecture. lusters and I Reuse patterns Reuse distance: Key cncept In the real wrld depends n Territrial patterns (hills, etc) Transmitted pwer» and ther prpagatin issues such as antenna directivity, height f transmissin antenna, etc Simplified hexagnal cells mdel: reuse distance depends n reuse pattern (cluster size) Pssible clusters: 3,4,7,9,,3,6,9, 4 6 5 3 7 4 K 4 3 6 5 7 4 luster: K 7 4 4 3 3 4 3 3 R 5
Reuse distance R 3K General frmula Valid fr hexagnal gemetry reuse distance R cell radius q /R frequency reuse factr K q/r 3 3,00 4 3,46 7 4,58 9 5,0 6,00 3 6,4 v Prf istance between tw cell centers: (u,v ) (u,v ) (,) (3,) u 30 [( u u ) cs 30 ] + [( v v ) + ( u u )sin ] 30 Simplifies t: ( u u) + ( v v ) + ( u u)( v v ) istance f cell (i,j) frm (0,0): i + j + ij 3R R i + j + ij luster: easy t see that K R i + j + ij hence: R 3K 6
lusters: Number f Ss cmprised in a circle f diameter Number f Ss whse interdistance is lwer than lusters K(i3,j) K7(i,j) K4 (i,j0) Pssible clusters all integer i,j values i j Kii+jj+ij q/r 0,73 3 3,00 0 4 3,46 7 4,58 6,00 3 0 9 5,0 3 3 6,4 3 9 7,55 3 3 7 9,00 4 0 6 6,93 4 7,94 4 8 9,7 4 3 37 0,54 4 4 48,00 5 0 5 8,66 5 3 9,64 7
-hannel hannel Interference F G E E F G G E F E F G E F S N S I S N Frequency reuse implies that remte cells interfere with tagged ne -hannel Interference (I) sum f interference frm remte cells nise pwer (N signal pwer (S) interfering signal pwer (I) S I as N S small signal pwer (S) S ) + interfering signal pwer (I) I mputatin - assumptins ssumptins N I 6 interfering cells N I 6: first ring interferers nly we neglect secnd-ring interferers Key simplificatin Signal fr S at distance R Signal frm S interferers at distance Negligible Nise N S S/N ~ S/I d η prpagatin law η4 (in general) Same parameters fr all Ss Same P tx, antenna gains, etc Pwer P R int int ~ R Pwer P 8
S N N I S I R I cmputatin η cst R N I k η cst N I η η R N y using the assumptins f same cst and same : I q Results depend n rati q/r (qfrequency reuse factr) lternative expressin: recalling that R 3K η S S R η 3K ( 3K ) N I N I R 3K N I 6 S ( 3K ) 3 N I 6,µ4 K I 6 USGE: Given an S/I target, cluster size K is btained η ( ) η Examples target cnditins: S/I9 d η4 Slutin: target cnditins: S 8d η4. Slutin: S I S I 0 0.9 ( 3K ) K.3 6 7.94 8 η η 4 K K 3 S 3 I S I lg [ d] 5η lg( 3K ) ( 3K ) 0 K 3 8 + 7.78.3.3 5.63 0lg6 K 7 9
S/I cmputatin assuming 6 interferers nly (first ring) K q/r S/I S/I d 3 3,00 3,5,3 4 3,46 4,0 3,8 7 4,58 73,5 8,7 9 5,0,5 0,8 6,00 6,0 3,3 3 6,4 53,5 4,0 6 6,93 384,0 5,8 9 7,55 54,5 7,3 7,94 66,5 8, 5 8,66 937,5 9,7 dditinal interferers case K4 nte that fr each cluster there are always N I 6 first-ring interferers In I cmputatin, cntribute f additinal interferers is marginal 0
sectrizatin irectinal antennas ell divided int sectrs Each sectr uses different frequencies T avid interference at sectr brders PROS: I reductin ONS: Increased handver rate Less effective trunking leads t perfrmnce impairments Sectr 3 f fa, L + L a, 3L Sectr f fa, L + L a, L ELL a Sectr f a, L f a, L I reductin via sectrizatin three sectrs case Inferference frm cells, nly Instead f 6 cells With usual apprxs (specifically, int ~) S I S I 0 0 R η η S d I S 3 I mni mni d + 4.77 F G G E F F G E E F G E F G E F nclusin: 3 sectrs 4.77 d imprvement
6 sectrs 60 irectinal antennas I reductin: interfereer nly 6 x S/I in the mni case Imprvement: 7.78 d PRT ellular verage ncepts Lecture.3 teletraffic cnsideratins, teletraffic planning
USY ILE 0 Traffic generated by ne user (statistical ntin f traffic) user making phne calls TRFFI is a stchastic prcess Hw t characterize this prcess? statistical distributin f the USY perid statistical distributin f the ILE perid statistical characterizatin f the prcess memry time Traffic characterizatin suitable fr traffic engineering amunt f busy time in t traffic intensity i lim t t ( average number λ f calls per hur) ( average call duratin τ ) prbability that, at a randm time t, user is in USY state mean prcess value E.g. at a given time, des the prbability that a user starts a call result different depending n what happened in the past? ll equivalent (if statinary prcess) Traffic generated by mre than ne users U U U3 Traffic intensity (adimensinal, measured in Erlangs): 4 i 4i i U4 TOT P E 4 k k [ k active calls] ( ) [ active calls] 4 i i 4 k i 3
example 5 users Each user makes an average f 3 calls per hur Each call, in average, lasts fr 4 minutes calls i 3 hur 5 5 4 60 [ erl] [ erl] eaning: in average, there is active call; but the actual number f active calls varies frm 0 (n active user) t 5 (all users active), with given prbability 5 [ hurs] [ erl] number f active users prbability 0 0,37680 0,409600 0,04800 3 0,0500 4 0,006400 5 0,00030 Secnd example 30 users Each user makes an average f calls per hur Each call, in average, lasts fr 4 minutes 4 30 Erlangs 60 SOE NOTES: -In average, active calls (intensity ); -Frequently, we find up t 4 r 5 calls; -Prb(n.calls>8) 0.0% -re than calls nly nce ver TRFFI ENGINEERING: hw many channels t reserve fr these users! n. active users binm prbab cumulat 0,3E-0 0,63 30,7E-0 0,396669 435,8E-0 0,676784 3 4060,9E-0 0,86357 4 7405 9,0E-0 0,953564 5 4506 3,3E-0 0,987006 6 593775,0E-0 0,996960 7 035800,4E-03 0,999397 8 58595 5,0E-04 0,999898 9 430750 8,7E-05 0,999985 0 3004505,3E-05 0,999998 5467300,7E-06,000000 864935,9E-07,000000 3 9759850,9E-08,000000 4 454675,7E-09,000000 5 55750,3E-0,000000 6 454675 8,4E-,000000 7 9759850 5,0E-3,000000 8 864935,6E-4,000000 9 5467300,E-5,000000 0 3004505 4,5E-7,000000 430750,5E-8,000000 58595 4,5E-0,000000 3 035800,E-,000000 4 593775,3E-3,000000 5 4506 4,0E-5,000000 6 7405 5,5E-7,000000 7 4060 5,8E-9,000000 8 435 4,4E-3,000000 9 30,E-33,000000 30 5,E-36,000000 4
nte n binmial cefficient cmputatin 60 60!.39936e +!48! but 60! 8.3099e + 8 ( verflw prblems!! ) 60 60 exp lg exp exp 60 i lg ( lg( 60! ) lg(! ) lg( 48! )) () i lg() i lg() i (n verflw!! befre exp...) i 60 48 i ( i ) 60 exp lg i i (n verflw!! never! ) 48 i 48 () i lg() i lg() i + lg( i ) + 48lg( i ) i P Infinite Users ssume users, generating an verall traffic intensity (i.e. each user generates traffic at intensity i /). We have just fund that P k k [ k active calls, users] ( ) i k i! k ( k)! k! Let infinity, while maintaining the same verall traffic intensity [ k active calls, users] k lim k! lim! ( k) ( ) L( k + ) k! k! k k k k e k k k! 5
Pissn istributin 30% 5% erl pissn binmial (30) 0% 5% 0 erl 0% 5% 0% 0 4 6 8 0 4 6 8 0 P k ( ) e k k! Very gd matching with inmial (when large with respect t ) uch simpler t use than inmial (n annying queueing thery cmplicatins) Limited number f channels THE mst imprtant prblem in circuit switching The number f channels is less than the number f users (eventually infinite) Sme ffered calls will be blcked What is the blcking prbability? We have an expressin fr P[k ffered calls] We must find an expressin fr P[k accepted calls] s: [ ] P [ blck] P accepted calls U U U3 U4 TOT X X N. carried calls versus t N. ffered calls versus t 6
hannel utilizatin prbability channels available ssumptins: Pissn distributin (infin. users) lcked calls cleared It can be prven (frm Queueing thery) that: P[k calls in the system, k (0,)] P[ k ffered calls] P[ i ffered calls] i 0 Hence: (very simple result!) 35% 30% 5% 0% 5% 0% 5% 0% P[system full] P[ accepted calls] ffered traffic: erl - 3 ffered calls accepted calls 0 3 4 5 6 7 8 P[ ffered calls] P[ i ffered calls] i 0 lcking prbability: Erlang- Fundamental frmula fr telephne netwrks planning ffered traffic in Erlangs Π blck! E j j 0 j!, ( ) blcking prbability 00,00% 0,00%,00% 0,0% 0,0% Efficient recursive cmputatin available E, ( ) E + E, ( ) ( ),,,3,4,5,6,7 0 3 4 5 ffered lad (erlangs) 7
NOTE: finite users Erlang- btained fr the infinite users case It is easy (frm queueing thery) t btain an explicit blcking frmula fr the finite users case: ENGSET FORUL: Π blck i k i k 0 i i Erlang- can be re-btained as limit case infinity i 0 i Erlang- is a very gd apprximatin as lng as: / small (e.g. <0.) In any case, Erlang- is a cnservative frmula yields higher blcking prbability Gd feature fr planning apacity planning Target: supprt users with a given Grade Of Service (GOS) GOS expressed in terms f upper-bund fr the blcking prbability GOS example: subscribers shuld find a line available in the 99% f the cases, i.e. they shuld be blcked in n mre than % f the attempts Given: channels Offered lad Target GOS target target, E btained frm numerical inversin f ( ) 8
hannel usage efficiency Offered lad (erl) arried lad (erl) channels ( ) c lcked traffic efficiency: η c ( E ( )), if small blcking Fundamental prperty: fr same GOS, efficiency increases as grws!! example blcking prbability 00,0% 0,0%,0% 40 erl 60 erl 80 erl 00 erl 0,% 0 0 40 60 80 00 0 capacity GOS % maximum blcking. Resulting system dimensining and efficiency: 40 erl > 53 60 erl > 75 80 erl > 96 00 erl > 7 η 74.9% η 79.3% η 8.6% η 84.6% 9
Erlang calculatin - tables Erlang calculatin - sftware Erlang- frmula very easy t implement Even if sme tricks needed fr numerical accuracy Erlang- inversin nt s easy Sftware tls Online calculatr: http://mmc.et.tudelft.nl/~frits/erlang.htm Given tw parameter, calculates the third N number f circuits blcking prbability ffered lad 0
pplicatin t cellular netwrks ell size (radius R) may be determined n the basis f traffic cnsideratins First step: Given num channels and GOS 50 available channels in a cell lcking prbability<% Evaluate maximum cell (ffered) lad Frm Erlang- inversin(tables) 40.5 erl Secnd step Given traffic generated by each user Each user: 4 calls/busy-hur Each call: min in average i 4x/600.333 erl/user Evaluate max num f users in cell 30.87 ~ 30 Third step: Given density f users δ500 users/km Evaluate cell radius δ R πr πδ R~438m Other example Three service prviders are planning t prvide cellular service fr an urban area. The target GOS is % blcking. Users make 3 calls/busy-hur, each lasting 3 minutes in average ( i 3/00.5) Questin: hw many users can supprt each prvider? Prvider cnfiguratin: 0 cells, each with 40 channels Prvider cnfiguratin: 30 cells, each with 30 channels Prvider cnfiguratin: 40 cells, each with 0 channels Prvider : 40 channels/cell Prvider : 30 channels/cell Prvider : 0 channels/cell at %: 30.99 erl/cell 69.8 erl-ttal 43 verall users at %:.93 erl/cell 654.9 erl-ttal 4386 verall users at %: 3.8 erl/cell 57. erl-ttal 355 verall users mpare case with! The reasn is the lwer efficiency f 0 channels versus 40
Sectrizatin and traffic ssume cluster K7 Omnidirectinal antennas: 0 sectrs: 60 sectrs: I8.7 d I3.4 d I6.4 d Sectrizatin yields t better I UT: the price t pay is a much lwer trunking efficiency! With 60 channels/cell, GOS%, Omni: 60 channels x46.95 46.95 erl η77.5% 0 : 60/30 channels 3x.03 36.09erl η59.5% 60 : 60/60 channels 6x4.46 6.76erl η44.% cnclusin This mdule has given sme hints regarding: ell sizing via prpagatin cnsideratins Frequency reuse via prpagatin cnsideratins ell planning via teletraffic cnsideratin Very elementary mdels ut sufficient t understand what s inside planning N mbility! Teletraffic mdels need t be extended t manage handver rates! lcking requirement fr an handver call UST be much lwer than blcking fr a new incming call severe math cmplicatins Guard channels fr handver Out f the scpes f this class!