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1、3.4 Traffic handling capacity:Erlang performance and cell sizing34 流量控制：爱尔兰厄兰性能和小流量控制：爱尔兰厄兰性能和小区尺寸区尺寸 With the appropriate cell cluster With the appropriate cell cluster size or reuse parameter C determined,size or reuse parameter C determined,by the SIR required for a given by the SIR required for
2、a given system,system,the number of channels available per the number of channels available per cell is immediately established.For cell is immediately established.For example,the first-generation AMPS example,the first-generation AMPS provides 832/C channels per cell,asprovides 832/C channels per c
3、ell,asalready noted.With C=7,as already noted.With C=7,as determined in the previous section,thedetermined in the previous section,thesystem can provide 118 channels persystem can provide 118 channels per cell.cell.一旦由系统给定的信干比决定的再用参数C或小区簇的大小确定，有效的信道数量也就确定了。例如，第一代AMPS系统每个小区提供832C个信道，这已经在前面提到过。当C=7时，系
4、统每个小区提供118个信道。Given the number of channels Given the number of channels available per cell,we can now available per cell,we can now determine the geographical determine the geographical extent or size of cells.This extent or size of cells.This determination is based on adetermination is based on aco
5、nsideration of the traffic consideration of the traffic expected in a given expected in a given regionregion and and the the system performancesystem performance,other,otherthan than SIRSIR,that is to be provided,that is to be provided to a typical user.to a typical user.给定每个小区的有效信道数量，能确定小区的地理范围和尺寸。
6、在一个给定区域和系统性能的情况下，主要考虑流量，而不是信干比，信干比主要是针对每个特定的用户而言的。In this introductory section on In this introductory section on system performancesystem performance we focus we focus exclusivelyexclusively（专有地，单独地）（专有地，单独地）on the on the probability of call blocking as a probability of call blocking as a measure o
7、f performance or user measure of performance or user satisfaction.satisfaction.The call blocking probability has,The call blocking probability has,historically,been the principalhistorically,been the principal（主要的）（主要的）measure of performance in telephonemeasure of performance in telephone systems.sy
8、stems.Simply put,Simply put,it describes the chanceit describes the chance（可能性）（可能性）that a user,attempting to that a user,attempting to place a call,receives a busy signalplace a call,receives a busy signal.在这部分，我们将介绍一下体现了系统性能和用户满意度的方法之一呼叫阻塞概率。历来，呼叫阻塞率一直是电话系统性能指标的主要测量方式。简单地说，它是表征一个用户打算通话但系统告知繁忙的可能性.
9、In the In the mobile wireless domainmobile wireless domain the calculation of the calculation of the call the call blocking probabilityblocking probability in a given in a given cell depends on cell depends on the number of the number of channels availablechannels available and and the trafficthe tr
10、affic expected to use that cell expected to use that cell.在无线通信领域，给定一个小区，呼叫阻塞概率的计算依赖于可以同时处理的有效信道数量以及此小区期望的流量.In the materialIn the material（素材，资料）（素材，资料）following,we first define what isfollowing,we first define what is meant by“meant by“amount of trafficamount of traffic（流（流量）量）,or,or traffic inten
11、sitytraffic intensity,or ,or traffic traffic loadload（负荷）（负荷）,using a simple,using a simple statistical model,then indicate howstatistical model,then indicate how the blocking probability is related the blocking probability is related to it.to it.在下面，定义了通信总流量，或者流量密度，或者流量负荷。使用了一个简单的模型，用其说明阻塞概率与它的关系。T
12、raffic intensity Traffic intensity is commonlyis commonly defined to be the product defined to be the product（产物，结（产物，结果）果）of the average number of call of the average number of call attempts per unit time,times the attempts per unit time,times the average call length.average call length.流量密度被定义为流量密
13、度被定义为每单位时间内每单位时间内呼叫的平均数量，乘以平均呼叫长度呼叫的平均数量，乘以平均呼叫长度.Let the symbolLet the symbol（符号（符号)represent)represent the average number of call attempts the average number of call attempts per unit time and 1/be the average per unit time and 1/be the average call length.The traffic intensity or call length.The
14、traffic intensity or load A is then load A is then/,in units of,in units of ErlangsErlangs.令为每单位时间内的平均呼叫数量，1是平均呼叫长度。流量密度或者负荷A为，用爱尔兰厄兰进行单位表示。As an example,say there is anAs an example,say there is an average of 50 call attempts per average of 50 call attempts perminute,with each call,if accepted,minu
15、te,with each call,if accepted,lasting 3 minutes.The traffic load lasting 3 minutes.The traffic load is then 150 is then 150 ErlangsErlangs.举个例子，假如平均每分钟有50次呼叫，每次呼叫如果成功的话，持续通话3分钟，流量负荷是150爱尔兰厄兰。Calculate the Calculate the ErlangErlang loads on a loads on a system for the following cases:system for the
16、following cases:1.An average call lasts 200 seconds;1.An average call lasts 200 seconds;there are 100 call attempts per minute.there are 100 call attempts per minute.2.There are 400 mobile users in a 2.There are 400 mobile users in a particular cell.Each user makes a call particular cell.Each user m
17、akes a call attempt every 15 minutes,on the attempt every 15 minutes,on the average.Each call lasts an average of 3 average.Each call lasts an average of 3 minutes.minutes.计算下面系统的爱尔兰厄兰负荷：1呼叫平均持续200 s，每分钟平均100次呼叫。2小区内有400个移动用户，每个用户平均每隔15 min呼叫一次。呼叫平均持续3 min。Call lengths are assumed in thisCall length
18、s are assumed in this simple model to be exponentially simple model to be exponentially distributed,with average value distributed,with average value 1/.A simple analysis then shows 1/.A simple analysis then shows that,with N channels available,thethat,with N channels available,the call blocking pro
19、bability P call blocking probability PB B is given is given by the so-called by the so-called ErlangErlang-B formula:-B formula:呼叫时间长度在单一模型中被假设为服从指数分布，平均值为1/。一个简单的分析表明，拥有N个有效信道，呼叫阻塞概率PB，也就是爱尔兰厄兰B式为：(3.7)A table or plot of PA table or plot of PB B vsvs A A indicates that N A for Pindicates that N A f
20、or PB B=10%.=10%.N N A channels are requiredA channels are requiredto have Pto have PB B=5%or less.=5%or less.一个PB为纵轴和A为横轴的表格或曲线图说明，P=10时，NA。当P5时，需要NA个信道。As an example,consider the As an example,consider the traffic load A=150 traffic load A=150 ErlangsErlangs calculated above.For Pcalculated above.
21、For PB B=5%,=5%,N=153 channels are found to beN=153 channels are found to berequired.For a blocking required.For a blocking probability of 1%,169 channels probability of 1%,169 channels are required.Small increases in are required.Small increases in the number of channels made the number of channels
22、 made available reduce Pavailable reduce PB B substantially substantially（实质上）（实质上）.举个例子，如上所计算的，如果通信负荷A=150爱尔兰厄兰PB=5时，需要N=153个信道。对于1的阻塞概率，需要169个信道。信道数量只要增加一点，PB的降低也是很明显的。Consider the converse now.Consider the converse now.Say the number of channels is Say the number of channels is fixed.How does the
23、 blocking fixed.How does the blocking probability vary with probability vary with ErlangErlang load?load?反过来，信道数量是固定的，阻塞反过来，信道数量是固定的，阻塞概率会随着爱尔兰厄兰负荷发生什么样概率会随着爱尔兰厄兰负荷发生什么样的改变呢的改变呢?Let N=100 channels,to beLet N=100 channels,to be concrete.To keep P concrete.To keep PB B at 1%or at 1%or less,a load of n
24、o more than 84 less,a load of no more than 84 ErlangsErlangs can be allowed.If A were can be allowed.If A were to increase by 13%,to 95 to increase by 13%,to 95 ErlangsErlangs,the blocking probability,the blocking probability would jump to 5%,a five-fold would jump to 5%,a five-fold increase!At low
25、blockingincrease!At low blocking probabilities,the blocking probabilities,the blocking probability is very sensitiveprobability is very sensitive（敏感（敏感的）的）to changes in load.to changes in load.取N=100，且不变。若保持PB1，则只允许至多84个爱尔兰厄兰的负荷。如果A增加13，到95爱尔兰厄兰，则阻塞概率将变为5，增加了5倍。在低的阻塞概率情况下，阻塞概率对于负荷的变化非常敏感。现在来考虑上面讨论的几
26、个量与给定信道灵活使用之间的关系。如果每个呼叫平均持续了200s。平均每隔15 min用户进行一次呼叫，有100个有效信道，呼叫阻塞概率为1，则可以容纳378个用户同时使用(这个数字是怎么得到的?)。如果平均每个呼叫持续了400 s，那么仅仅可以容纳189个用户同时使用。Consider now the significance of Consider now the significance of these numbers in relation to mobile these numbers in relation to mobile usage of a given channel.u
27、sage of a given channel.Say a call lasts 200 seconds on Say a call lasts 200 seconds on the average.Say a typical user the average.Say a typical user makes a call every 15 minutes,on makes a call every 15 minutes,on the average.If 100 channels are the average.If 100 channels are available and a bloc
28、king probability ofavailable and a blocking probability of 1%is desired,378 users can be 1%is desired,378 users can be accommodated.(How is this numberaccommodated.(How is this number obtained?)If the average call obtained?)If the average call durationduration（持续，持久）（持续，持久）increases to increases to
29、400 seconds,only 189 users can be 400 seconds,only 189 users can be accommodated.accommodated.如果每个呼叫平均持续了200 s，但是平均每隔10 min用户进行一次呼叫，那么小区中的用户数量只能有250个。所以，小区电话或终端的使用在决定同时容纳的用户数量时起到了非常重要的作用。If the typical call lasts 200 seconds If the typical call lasts 200 seconds,on the average,but a mobile user,on t
30、he average,but a mobile user attempts a call every ten minutes,onattempts a call every ten minutes,on the average,the number of users the average,the number of users allowed in the cell is now 250.allowed in the cell is now 250.So cell phone or terminal usage So cell phone or terminal usage clearly
31、plays a role in determining theclearly plays a role in determining the number of mobiles that may be number of mobiles that may be accommodated.accommodated.现在来考虑小区尺寸的计算所带来的影响。首先看一个AMPS的例子，回顾前面说到的例子，832个有效信道，7小区再用，每个小区118个信道。Now consider the impact of Now consider the impact of these calculations on
32、 the sizing these calculations on the sizing of a cell.we focus first on an of a cell.we focus first on an AMPS example.Recall that with aAMPS example.Recall that with a total of 832 frequency channels total of 832 frequency channels and C=7 reuse,118 channels and C=7 reuse,118 channels are availabl
33、e per cell.are available per cell.要保持阻塞概率PB小于1，通过使用式(3.7)和基于此等式的表格和图，爱尔兰厄兰负荷为A=101爱尔兰厄兰。假设每个用户平均每15min进行一次200 s长的通话，则总的平均呼叫到达速率为:=A/200=0.5次s，在每个小区有15(min)60(s)=450个用户可以同时通话。To keep the blocking probability To keep the blocking probability P PB B to less than 1%,the to less than 1%,the ErlangErlang
34、load load allowed turns out to be,using(3.7)allowed turns out to be,using(3.7)or tables or plots based on that or tables or plots based on that equation,A=101 equation,A=101 ErlangsErlangs.Say a.Say a typical user makes 200-second-longtypical user makes 200-second-long calls once every 15 minutes on
35、 the calls once every 15 minutes on the average,then the total call arrival average,then the total call arrival rate is =0.5 call/sec,and 15 x rate is =0.5 call/sec,and 15 x 60/2=450 users may be 60/2=450 users may be accommodated in each cell.accommodated in each cell.现在假设这些用户在小区内为均匀分布，半径为R的六边形小区面积
36、为：可接受的用户密度为:450/2.6R2=173R2个单位面积。Assume now that these users Assume now that these users are uniformly distributed over theare uniformly distributed over the cell.Since the area of a cell.Since the area of a hexagonal cell of radius R is hexagonal cell of radius R is ,the allowable user ,the allowab
37、le user density is 450/2.6Rdensity is 450/2.6R2=2=173173R R2 2mobiles/unit of area.mobiles/unit of area.例如，首先考虑农村地带，移动终端的密度为每平方千米两个移动台，小区半径：R=9.3 km（2（个）=173R2（个/单位面积）。郊区移动终端的密度为100个km2移动台，相应的半径为R=1.3 km（100（个）=173R2（个/单位面积）。城市中的密度为1000个km2移动台，相应的半径为R=0.26 km（1000（个）=173R2（个/单位面积）。Consider first a r
38、uralConsider first a rural（农村）（农村）region.region.Say the density of mobile terminals Say the density of mobile terminals is two terminals per kmis two terminals per km2 2.Then the.Then the cell radius should be set at about cell radius should be set at about R=9.3 km.R=9.3 km.Say now a suburban regio
39、n has a Say now a suburban region has a mobile terminal density of 100 mobile terminal density of 100 mobiles/kmmobiles/km2 2.The corresponding cell.The corresponding cell radius drops to 1.3 km.radius drops to 1.3 km.In an urban region with a mobile In an urban region with a mobile density of 1000
40、mobiles/kmdensity of 1000 mobiles/km2 2,the cell,the cell radius would reduce further to 0.26 radius would reduce further to 0.26 km.km.3.5 Probabilistic signal calculations35 信号的概率计算信号的概率计算 在一个小区内，更合理的性能测量应该为在这个点所处移动台（下行链路）或基站（上行链路）接收到的信号功率大于任何干扰信号功率的概率。A more appropriate measure of A more appropri
41、ate measure of performance in a cell might be the performance in a cell might be the probability that the signal power for probability that the signal power for either a mobile(downlink)or a base either a mobile(downlink)or a base station(uplink)was greater than any station(uplink)was greater than a
42、ny of the interfering signal powers of the interfering signal powers measured at that point.measured at that point.我们选择下行链路计算，可以计算在移动终端实际的信号功率与干扰功率的比值（而不是平均信干比）大于某个指定阈值的概率。这里重复式(2.4)，接受信号功率为PR：(3.9)AlternativelyAlternatively（做为选择（做为选择,二者择一地）二者择一地）one could calculate the probability one could calcula
43、te the probability that the ratio of actual signal power that the ratio of actual signal power received at a mobile terminal to the received at a mobile terminal to the interference power,the measured interference power,the measured SIR,rather than the average SIR,SIR,rather than the average SIR,was
44、 greater than a specifiedwas greater than a specified threshold.threshold.Repeating(2.4)here,we have the Repeating(2.4)here,we have the received signal power P received signal power PR R given by given by：(3.9)(3.9)是由于多径造成的小尺度瑞利衰落和莱斯衰落的因子，这里忽略多径效应。g(d)表示距离，形式为d-n，在这一章中主要用于计算信干比。其他的参数是发送端的发射功率和发送端及接收
45、端的天线增益。这里集中讨论函数10 x/10。is the small-scale Rayleigh or is the small-scale Rayleigh or RiceanRicean-fading term due to multipath-fading term due to multipath that we will neglect in this discussion.that we will neglect in this discussion.The term g(d)represents the distance-The term g(d)represents the
46、 distance-dependent ddependent d-n-n term on which we term on which we focused in calculating the SIR in this focused in calculating the SIR in this chapter.The other terms,are,of chapter.The other terms,are,of course,the transmitted power andcourse,the transmitted power and antenna gains at the tra
47、nsmitter and antenna gains at the transmitter and receiver,respectively.It is the receiver,respectively.It is the function 10 function 10 x/10 x/10 on which we focus on which we focus here.here.考虑用dB表示的接收功率。忽略小尺度衰落模型，设定2=1，PR,dB=x+PRav,dB其中，PRav,dB=10 lg(g(d)PTGTGR)。简化这个表达式，令P=PR,dB,Pav=PRav,dB。Cons
48、ider the received power Consider the received power measured in dB.Then we have,measured in dB.Then we have,neglecting the neglecting the small-scale fadingsmall-scale fading term term,setting,setting 2 2=1=1，P PR,dBR,dB=x+=x+P PRav,dBRav,dB,where where P Rav,dB=10 lg(g(d)PTGTGR).To To simplify the
49、notation,let simplify the notation,let P=P PR,dB R,dB,P av=P P RavRav,dB,dB.则用dB表示的随机变化的接收信号功率p，其平均值为pav，是服从高斯分布的，也用dB表示。它的概率密度分布函数为：(3.10)标准方差取值在610dB之间。以分贝表示的接收信号功率的高斯分布模型是如何使用的呢?Then the randomly varying Then the randomly varying received signal power p,measured inreceived signal power p,measured
50、 in dB,is dB,is gaussiangaussian distributed distributed about aboutits average value its average value P Pavav,also measured,also measured in dB.Its probability distribution in dB.Its probability distribution f f(p)(p)is given by,is given by,(3.10)(3.10)The standard deviationThe standard deviation（