network03数据量特征随机过程介绍自相似

NetworkDesignand Wang©UniversityacademyofNetworkDesign©Universityacademyof

Data CharacterizationData©UniversityacademyofNetworkDesign©Universityacademyof

Data DatatrafficisthesequenceofmovementofdatathroughapointoraphysicalAtypicaldataitemisacontiguoussequenceofbitsformingadatapacket.Whenthesedataitemspassthroughaphysicaldevice,thereisusuallysomeimpedimentintheformofreception,processing,andforwarding.Suchanimpedimentresultsinqueuingandcausestimedelays.Ingeneral,queueshavesuccessivearrivalsofcustomersasinputs.Thesearrivalsexperiencepossiblewaitingandservicebeforebeingoutputassuccessivedepartures.©UniversityacademyofNetworkDesign©Universityacademyof

Data Thestatisticalnatureofarrivalscanbeexpresseddifferentifsuccessivein rrivaltimes(IATs)areindependent,aspecificationoftheinitialconditionintheformofthetimeinstantatwhichtheoperationofthequeuestartsandtheprobabilitydensityfunction(pdf)ofIATsaresufficienttocomple ydescribethenatureofarrivals.Thenumberofarrivalsoveratimeintervalisanotherimportantwayofcharacterizingthenatureofarrivals.Ingeneral,thisrequiresthespecificationoftheinitialconditionandthetimeinstantsofthestartandendoftheintervaloverwhichtherandomvariablenumberofarrivalsischaracterized.©University©UniversityacademyofNetworkDesign

Data 计算机网络性能分析需要概率 ©UniversityacademyofNetworkDesign©Universityacademyof

Data 量简常用 量及其数字特二维 量的数字特 ©UniversityacademyofNetworkDesign©Universityacademyof

Data 概率论与数理统计是研究随机现象随机现象是通过随机试验©UniversityacademyofNetworkDesign©Universityacademyof

Data 样本空间，记 样本点:试验的单个结果或样本空间的单元素称为样本点,为由样本点组成的单点集称为基 ,也记为 当且仅当所包含的一个样本点发生 ©UniversityacademyofNetworkDesign©Universityacademyof

Data 关于随量(及向量)的研究，是概率论的中心内随机是从静态的观点来研究随机现象，而随机 ©UniversityacademyofNetworkDesign©Universityacademyof

Data 定义：设X是随 ©UniversityacademyofNetworkDesign©Universityacademyof

Data ©UniversityacademyofNetworkDesign©Universityacademyof

Data x0F(x)F(x)单调不减，且F(0，F(0P(x1Xx2)F(x2)F©University©UniversityacademyofNetworkDesign

Data 期望和方差是 量取值的平均特n定义1（离散）若X~P{X=xk}=pkk=1,2,…nnE(X)xkpkk©UniversityacademyofNetworkDesign©Universityacademyof

Data

数学期望——描述 量取值的平均特2（连续）若X~f(x),-|x|fx)dx E(X)xf(x)dx.©UniversityacademyofNetworkDesign©Universityacademyof

Data E(cX)=cE(X),c为常数©UniversityacademyofNetworkDesign©Universityacademyof

Data 量取值波动程度的一个数字特征定义若E(X),E(X2E{[X-E(X)]2为r.vX的方差，记为D(X),或(X)

D(X E(X2P x},D(X)

[xE(X)]

fxdx ©UniversityacademyofNetworkDesign©Universityacademyof

Data D(X 若X，Y©UniversityacademyofNetworkDesign©Universityacademyof

Data 量简常用 量及其数字特二维 量的数字特 ©UniversityacademyofNetworkDesign©Universityacademyof

Data 量（ X～P{X＝k}＝pk(1－p)1－k,(0<p<1)

1©University©UniversityacademyofNetworkDesign

Data 设 P(Xi)p(1p)i,i0,1,2

E[X]ip(X i)ip(1p)ii0 i0

1pD[m]E[m2]{E[m]}2(1p)/p ©UniversityacademyofNetworkDesign©Universityacademyof

Data 几何分布具有 P{ nm/ m}P{ n}(1p)n证明 P( n / m

P( nm, mP( m

P( nmP( m (1 p)nm(1 p)

(1 p

n布函数中惟一具有 性的概率分©UniversityacademyofNetworkDesign©Universityacademyof

Data ©UniversityacademyofNetworkDesign©Universityacademyof

Data 设进行n次独立试验，每次试验结果有：p=P(A发生)，q=P(B发生)，且p+q=1。若在n次试验中，A发生m次，则B发生(n-m)次，则称随机试验为n次 p(m)Cmpm(1p)nm

D[m]E[m2]{E[m]}2npqnp(1pE[m]mpE[m]mpn(m)npmn©Universityacademyof

Data ©UniversityacademyofNetworkDesign©Universityacademyof

Data 泊松分布(Poisson分布) eP(Xk) k0,1,2,,k称服从参数为λ的泊松分布， X~DD[m]E[m2]{E[m]}2E[X]ipE[X]ip(iii)iiieii(i1)eeiee©Universityacademyof

Data ：：C 1nn ,其中©Universityacademyof

Data

f(x) 称在区间(a,b)上服从均匀分布，记为©UniversityacademyofNetworkDesign©Universityacademyof

Data

©University©Universityacademyof

ef(x)

xx其中λ>0为常数，则称服从参数为λ的指数分布1e xF(x)0 xP( t| )P( t0t 1F(t0t)e1F(t0

P( t0P( t在连续型 ©UniversityacademyofNetworkDesign©Universityacademyof

Data 瑞利分布(Rayleigh ef(x)e

22,0xx均值 2

42©Universityacademy©UniversityacademyofNetworkDesign

Data 量简常用 量及其数字特二维 量的数字特 ©UniversityacademyofNetworkDesign©Universityacademyof

Data 定义设(X,Y)是二维 量，则称EXEXYEYX与Y的协方差（Covariance），记为covX,Y或 即covX,Y EXEXYEY若 D(X)0且 D(Y)0，则XY XY

cov(X,YD(X) D(Y为X与Y的相关系数（Correlation©UniversityacademyofNetworkDesign©Universityacademyof

Data 协方差常用下 计算covX,YEXYE(X)E(Y©UniversityacademyofNetworkDesign©Universityacademyof

Data 当XY1时，X与Y依概率1特别当XY1时,Y随X的增大而线性增大,此时称X与Y线性正相关;当XY1时,Y随X的增大而线性地减小，此时称X当XY变小时,X与Y的线性相关程度就变弱如果XY=0,X与Y之间就不存性关系，此时称X与Y不相 ©UniversityacademyofNetworkDesign©Universityacademyof

Data 独立与不相关都是随量之间相互联系程度的一种反映, 量(X,Y)而言，X与Y的独立性 X,Y~N,,2,2,

©UniversityacademyofNetworkDesign©Universityacademyof

Data 协方差的性质cov(aXbY)=ab.cov(XY其中a,b为常数；D(XY)=D(X)+D(Y)2cov(X,©University©UniversityacademyofNetworkDesign

Data ©UniversityacademyofNetworkDesign©Universityacademyof

Data 随，而这些有时可以通过变换来解决，和随量的分布变量的pmf和连续随量的密度函数可以分别由它们的z变换和Lace变换确定。如果两个随量具有相同的变换，则它们具有相同的分布函数，反之亦然。因此，在很多情况下，随量的参数可Lace-©UniversityacademyofNetworkDesign©Universityacademyof

Data ： ： ，则X的变换定义为 ©UniversityacademyofNetworkDesign©Universityacademyof

Data ©UniversityacademyofNetworkDesign©Universityacademyof

Data pmf©UniversityacademyofNetworkDesign©Universityacademyof

Data ©UniversityacademyofNetworkDesign©Universityacademyof

Data 定理令X为离散 GY(z)zGX(z (z)z1

©UniversityacademyofNetworkDesign©Universityacademyof

Data ©Universityacademyof©UniversityacademyofNetworkDesign ©UniversityacademyofNetworkDesign©Universityacademyof

Data ©UniversityacademyofNetworkDesign©Universityacademyof

Data ©UniversityacademyofNetworkDesign©Universityacademyof

Data 非负连续 量 ©UniversityacademyofNetworkDesign©Universityacademyof

Data ©UniversityacademyofNetworkDesign©Universityacademyof

Data ©UniversityacademyofNetworkDesign©Universityacademyof

Data ©UniversityacademyofNetworkDesign©Universityacademyof

Data ©UniversityacademyofNetworkDesign©Universityacademyof

Data ©UniversityacademyofNetworkDesign©Universityacademyof

Data ©UniversityacademyofNetworkDesign©Universityacademyof

Data ©UniversityacademyofNetworkDesign©Universityacademyof

Data ©UniversityacademyofNetworkDesign©Universityacademyof

Data ©UniversityacademyofNetworkDesign©Universityacademyof

Data ©UniversityacademyofNetworkDesign©Universityacademyof

Data ©UniversityacademyofNetworkDesign©Universityacademyof

Data ©UniversityacademyofNetworkDesign©Universityacademyof

Data 量，令LZ(s)LX(s)LY©UniversityacademyofNetworkDesign©Universityacademyof

Data ©UniversityacademyofNetworkDesign©Universityacademyof

Data 设有一个过程X(t) ，若对于每一个固定的时tj(j=1,2,…) ，X(tj)是一个随 量，则X(t)称为随机过©UniversityacademyofNetworkDesign©Universityacademyof

Data 随机过程X(t,e–当te固定时，X(t)是一个确定值–当t固定，e可变时，X(t)是一个 量–当te固定时，X(t)是一个确定的时间函数

X(t)是一个随机过程©UniversityacademyofNetworkDesign©Universityacademyof

Data

©UniversityacademyofNetworkDesign©Universityacademyof

Data .©UniversityacademyofNetworkDesign©Universityacademyof

Data 设X(t),tT是一 tT,X 是一个EXX(t),t即

存在，记为mX

mX(ttTmXt

E[X(t)]

©UniversityacademyofNetworkDesign©Universityacademyof

Data 设X(t),tT是一S.P.,tT,X 如果D[X(t)]存在，记为DX(t),DX(t),tT

X(ttT的方差函数DX tDX(t)EXtm 2称σt

©UniversityacademyofNetworkDesign©Universityacademyof

Data s,tT,X(s),X(t)是两r.v.,如果cov(X(s),X(t))存在，记为CX

为X(ttT的协方差函数©UniversityacademyofNetworkDesign©Universityacademyof

Data 设X(ttT是一S.PtTX(t)X如果E[X 称X(t),tXX(ttT的均方值函数设X(t),tT是一 s,tT,X(s),X(t)是两r.v.,如果E[X(s)X 存在，记为

(st),RX(ststT为X(tt

©UniversityacademyofNetworkDesign©Universityacademyof

Data 随机过程X(ttT的协方差函数、相关函数和CX(st)RX(stmX(s)mX(tst DX(t)

(ttt X(t)RX(t,©UniversityacademyofNetworkDesign©Universityacademyof

Data 设X(t),tT和Y(t),tT是两个 设X(t),Y(t),tT是二维S.P.,stTX(s),Y 如果E[X(s)Y 存在，记为RXY(s,t)则RXY(s,t),s,t

为X(t),Y(ttT的互相关函数©UniversityacademyofNetworkDesign©Universityacademyof

Data 如果cov(X(s),Y(t))存在，记为CXY(s,t)则称CXY(s,t),s,tT 为X(t),Y(t),tT的互协方差函数．关系CXY(s,t)RXY(s,t)mX(s)mY(t),s,tT 设X(t),tT和Y(t),tT是两个S.P.，如CXY(s,t)0或 (s,t)m(s)m(t),s,tT X(ttT和Y(ttT不相关©UniversityacademyofNetworkDesign©Universityacademyof

Data ©UniversityacademyofNetworkDesign©Universityacademyof

Data 。©UniversityacademyofNetworkDesign©Universityacademyof

Data Inanarrowtimeinterval,theprobabilityofanarrivalisproportionaltothetimeinterval.Inanarrowtimeinterval,theprobabilityoftwoormorearrivalsisnegligibleincomparisonwiththeprobabilityofonearrival.Numbersofarrivalsinnonoverlaptimeintervalsaremutuallyindependentofoneanother.©UniversityacademyofNetworkDesign©Universityacademyof

Data ©UniversityacademyofNetworkDesign©Universityacademyof

Data DerivationofthePoissonConsideratimeinterval(0,T].Dividethisintervalintonequalparts.Asnincreasesandtendsto∞,T/n→0andwehaveanarrowsub-intervaltendingto0.Therefore,ineachsuchsub-interval,wehaveonearrivalwithprobabilityλT/nandzeroarrivalswithprobability1−λT/n.Twoormorearrivalsoccurwithzeroprobability.Theseargumentsareaccurateinthelimit,asn→∞.Thenumberkofsub-intervalswitharrivalsinatotalofnsub-intervalsisbinomiallydistributed.©UniversityacademyofNetworkDesign©Universityacademyof

Data DerivationofthePoissonThisisthePoissonpmf.Thispmfgivestheprobabilitiesoffindingvariousnumbersofpossiblearrivalsinagiventimeinterval,ifthearrivalschemesatisfiesthepreviouslymentionedthreeproperties.©UniversityacademyofNetworkDesign©Universityacademyof

Data MeanofthePoissonrandomFromthePoissonmean,theaveragenumberofarrivalsinaunitamountoftimeisλ.VarianceofPoissonrandom©UniversityacademyofNetworkDesign©Universityacademyof

Data ©UniversityacademyofNetworkDesign©Universityacademyof

Data ©UniversityacademyofNetworkDesign©Universityacademyof

Data ©UniversityacademyofNetworkDesign©Universityacademyof

Data 象都可以描述为指数分布，如呼叫的持续时间，两次电分布随量X的分布函数定如下：©UniversityacademyofNetworkDesign©Universityacademyof

Data 对于带有参数的指数分布 ©UniversityacademyofNetworkDesign©Universityacademyof

Data 指数分布的重要性是由于它是唯一的拥有无特性的连续分布随量，即无特性更准确地说常常是指Markov指数分布的另一个重要特性是它和离散的Poisson随量间隔时间之间是独立同分布，且有均值X，则在时间区间内到达的顾客数量的随量具有Poisson分布，而 ©UniversityacademyofNetworkDesign©Universityacademyof

Data ©UniversityacademyofNetworkDesign©Universityacademyof

Data ©UniversityacademyofNetworkDesign©Universityacademyof

Data 指数分布中cX 1，该结果可以作为 seacademyofNetworkDesignseacademyof

Data ©UniversityacademyofNetworkDesign©Universityacademyof

Data ©UniversityacademyofNetworkDesign©Universityacademyof

Data ©UniversityacademyofNetworkDesign©Universityacademyof

Data ©UniversityacademyofNetworkDesign©Universityacademyof

Data N UniversityN ©UniversityacademyofNetworkDesign©Universityacademyof

Data 性的重要参数是Hurst系数H，是以水文学家H.E.©UniversityacademyofNetworkDesign©Universityacademyof

Data ©UniversityacademyofNetworkDesign©Universityacademyof

Data ©UniversityacademyofNetworkDesign©Universityacademyof

Data ©UniversityacademyofNetworkDesign©Universityacademyof

Data ©UniversityacademyofNetworkDesign©Universityacademyof

Data ©UniversityacademyofNetworkDesign©Universityacademyof

Data IfXisPareto,itsOr ©Universityacademyof©Universityacademyof

a ©UniversityacademyofNetworkDesign©Universityacademyof

Data AnalternativecommonformofrepresentationusestheHurstparameterHinsteadofα.HaroldEdwinHurst(1880–1978)wasaBritishhydrologist.TheHurstparameterforaParetorandomvariableis©UniversityacademyofNetworkDesign©Universityacademyof

Data TheParetorandomvariableXcanhaveanyphysicaldimension,suchaslength,mass,time,orbits(approximatingnumberofbitsbyarealnumber).Theparameterαisdimensionless,andβhasthesamedimensionasX.©UniversityacademyofNetworkDesign©Universityacademyof

Data ©Universityacademy©Universityacademyof

Data Therangeα∈(1,2]isofinteresttous.Inthisrange,meanisfinitebutthevarianceisDatatrafficinpresentdayLANsisverybursty,havinganoverallfiniteaverageModelingin rrivaltimesbetweensuccessivedatapacketsbyaParetorandomvariablewithα∈(1,2]isgainingpopular