15通信网理论基础-第4章-网内业务分析-排队论基础

3.1.3.1.1.3.1.2.3.2.3.2.1.3.2.2.3.2.3.4.1.−≡∆∆∆∆≡∆•‹‹‹‹•‹‹•‹‹4.1.1.−=≡≡≡≡≡≡=≡∆≡∆∆••••≡∆∆≡∆∆=≡∆∆≡∆∆∆≡∆∆≡∆∆≡−=Kendall123∆X/Y/Z•X•Y•Z≡∆M∆D∆EkkErlang∆GI∆G∆HRR≡∆M/M/1∆D/M/CC∆M/M/1∆G/M/1M/G/1∆G/G/1G/G/m•Er/G/1G/Er/1Er/Ek/1/≡1971∆Kendall∆X/Y/Z/A/B/C••ANN•Bm•CFCFSLCFS∆X/Y/Z/∞/∞/FCFS−=≡≡≡=≡≡≡≡≡=≡Ls≡Lq∆≡Ws≡Wq∆∆Little•WsLs≡∆∆≡η∆∆m∆r∆r/m•∆mr=η∆η∆η≡∆∆≡ρ∆μρ/μ•ρm•ρm∆•ρm=≡∆nn∆•n012•Nn012N•Cn012Cn≡∆∆t•tnPnt≡Pnt∆Pnt•t•n∆Pnt•t•n∆••∆•nntPtP=∞→)(lim•∆••t0Pn0n≥0••t∞••Pnt∞PntPn't04.1.2.≡∆∆∆−=≡Nt[0,t)t0≡Pnt1t2[t1,t2)t2t1nn≥0∆[t1,t2)n∆Pnt1t2PN(t2)N(t1)=nt2t1n≥0=≡Pnt1t21∆2t[t,t+∆t)∆t∆t∆P1ttttOt4.1-2•Ott0t•∆3t[t,t+∆t)∆∆)(),(2tOtttPnn∆=∆+∑∞=4.1-3∆=n≡20Pn0tPnt∆≡23[t,t+∆t))(1),(),(1......),(),(),(1),(213210tOttttPtttPtttPtttPtttPtttPnn∆+∆⋅−=∆+−∆+−=−∆+−∆+−∆+−=∆+∑∞=λ4.1-4≡Pnt∆∆Pnttt+t∆tt+t∆[0,t+∆t)[0,t)[t,t+∆t)∆n),0[t),[ttt∆+),0[tt∆+AnPnt01∆tO(∆t)nPn(t)[1∆tO(∆t)]Bn-1Pn1t1∆tnPn-1(t)∆tCn-2n-30Pn2tPn3tn23nO∆tO∆tO∆tnnnO∆tO∆tO∆t∆Pnt+tOtPnt+tPnt1tPn1ttOt∆ttOtPtPttPttPnnnn∆∆++−=∆−∆+−)()()()()(1λλ∆t0≥=≥+−=−10)0(1)()()(1nPntPtPdttdPnnnnλλ4.1-5∆n0BC=−=1)0()()(000PtPdttdPλ4.1-6∆(4.1-5)(4.1-6)•⋅⋅⋅=⋅=−,2,1,0;0!)()(ntenttPtnnλλ4.1-7•(4.1-6)P0tcetP00c=1P0tet(4.1-5)et)()()(1tPeetPdttdPenttnnt−=+λλλλλ)(])([1tPedtetPdnttn−=λλλττλλτλdePetPtntn∫−=01)()(n1tdeedePetPtttλτλττλλτλτλτλ===∫∫−0001)()(P1ttetn22)()()(20012tdeedePetPtttλτλτλττλλτλτλτλ=⋅==∫∫−λτλ−=ettP!2)()(224.1-7⋅⋅⋅=⋅=−,2,1,0;0!)()(ntenttPtnnλλ≡∆Pnttn∆N(t)N(st)N(s)∆E[N(t)]t4.1-8teetitetiteeititPitNEttiitiitiiii⋅=⋅⋅⋅=−⋅⋅⋅=−=⋅=⋅=−∞=−−∞=−−∞=∞=∑∑∑∑λλλλλλλλλλλτ11100)!1()()!1()(!)()()]([≡∆D[N(t)]t4.1-8’D[N(t)]E[N2(t)]E[N(t)]2{}{}tttitteteitiittPiitNEtNtNEtNtNtNEtNEiittiiiiλλλλλλλλλλ+=+−=+−=+−=+−=+−=∑∑∑∞=−−−∞=∞=2222002)()!2()()(!)()1()()1()]([]1)()[()(]1)()[()]([D[Nt]t2tt2t∆•−=T∆≥=−000)(ttetftTλλ4.1-9∆T≡≥−=−0001)(ttetFtTλ4.1-10≡E[T]=1/≡D[T]=1/2≡σ[T]=1/=≡M∆PTts|Ts=PTt4.1-11∆•T•s•≡∆T∆•),0[t1tttntneteeenttPλλλλλ−−∞=−−=−=⋅=−∑10)1(!)()(1104.1-7•:PT≥t=FT(t)1ett0•4.1-10T=≡∆∆1/≡v∆v∆∆v∆tvetF⋅−−=µ1)(tvetf⋅−=µµ)(4.1-12∆μ∆1/μEv−Erlang=≡v1,v2,,vkk≡kμ≡Tv1v2vk≡T0)!1()()(1⋅−=⋅⋅−−tekktktbtkkkµµµ4.1-13≡Tk≡E[T]1/μ≡D[T]1/kμ24.1-14=1≡k≡kμ≡kk=2≡≡∆k≡tk∆k(k)∆ttkektf′⋅⋅−⋅=′λλ)(∆t=t1+t2++tk∆t∆tf(t)k∆tZtkkiZkkZ−=λλφ)(∆4.1-13tkkkekktktb⋅⋅−−⋅−=λλλ)!1()()(1=≡≡≡k1∆≡k≡k≥30≡k∞D[T]0b(t)δ(t1/μ)∆D≡k∆−RHR=≡∑=−∂=Ritiiieta1)(λλ≡∑==∂Rii11=≡R∆∆12R≡12R≡∆HR4.1.3.M/M/1=≡≡≡=≡M/M/1M/M/1/∞/∞/FCFS≡M/M/1M/M/1/N/∞/FCFS≡M/M/1M/M/1/∞/M/FCFS−M/M/1M/M/1/∞/∞/FCFS=M/M/1≡∆∆∆∆≡∆∆∆≡∆∆≡=≡tnPnt∆tn∆∆μ∆[t,t+∆t)1∆tO∆t1∆tO∆t21μ∆tO∆t1μ∆tO∆t3O∆t∆t∆tnn02t,t∆ttt∆tOtABCDnn1n1n××××nnnntttPtttPtttPtttPnnnn∆⋅∆⋅∆−⋅∆⋅∆⋅∆−⋅∆−∆−⋅−+µλµλµλµλ)()1()()1()()1)(1()(11∆Pn(t∆t)∆tPn(t∆t)Pn(t)(1∆tμ∆t)Pn1(t)μ∆tPn1(t)∆tO(∆t)•ttotPtPtPttPttPnnnnn∆∆++−+⋅=∆−∆++−)()()()()()()(11µλµλ•∆t0Pnt⋅⋅⋅⋅⋅⋅=+−+⋅=+−2,1)()()()()(11ntPtPtPdttdPnnnnµλµλ4.1-15∆•n0ABP0t∆tP0t1∆tP1t1∆tμ∆t•)()()(100tPtPdttdP⋅+⋅−=µλ4.1-16•n∆4.1-154.1-16•∆•PnttPn0•4.1-154.1-16−=+−+−=+−+−)181.4(00)()171.4(01110nPPPPPnnnµλµλµλ•Pnn-1012nn+1•01P010μP1•0P0μP1•n≥14.1-18Pn1μPn+1μPn•4.1-1701PP⋅=µλ•4.1-18n1002)(PPPλµλµλµ−⋅⋅+=⋅022PP⋅=µλ•0PPnn⋅=µλ•ρ/μ1()10=∑∞=nnPPn111000=−⋅=∑∞=ρρPPnn≥−=−=1)1(110nPPnnρρρρ4.1-19n≡4.1-191LS)10(1)32()32()1()(324323210−=⋅⋅⋅+++=⋅⋅⋅+++−⋅⋅⋅+++=−⋅=⋅==∑∑∞=∞=ρρρρρρρρρρρρρρnnnnSnPnnEL2Lqρρρ−=−=−⋅=−=−=∑∑∑∞=∞=∞=1)1()1(2111SnnnnnnqLPPnPnnEL3Ws∆WM/M/1μ0)()(1)()()(≥⋅−=−=−−−−wewfewFwwλµλµλµ4.1-20∆nWnnTiTn1WnT1+T2+T3++Tn+Tn+1T1fwn+1Wnw)1()(0+=∑∞=nwfPwfnnTii1,2,n+1μTs4.1-13Wn!)()1(newnwfwnµµµ−⋅=+wnnwnwnnenweenwwf⋅−−∞=−∞=−⋅−=⋅⋅⋅⋅⋅−=⋅⋅⋅⋅−=∑∑)(00)(!)()1(!)()1()(λµµµλµµρµρµµρρ•,∆WSWSE[w]1/μ4Wqµρρλµρµ111⋅−=−=−=sqWW∆λµρλµλµλρρρλµλρρ−=−=−−⋅=−=−=−=qSqSWWLL1)211.4(112)211.4(1112−−=−⋅=−=−=−=−=λµρλµλρρρλµλµλρρqqSSWLWL∆Littleρρµλµρµλλ−=⋅=+=−−=⋅=+=⋅=⋅=SSqqSSSqqSqqSSLLLLL)221.4(11=≡∆100∆n4.1-8∆100∆v4.1-94.1-84.1-9nfnvfv0100.00.2381280.20.4252290.40.6173160.60.894100.81.06561.01.25611.20≡∆Σnfn/1002.1/∆Σvfv/1000.4/∆1/0.42.5/≡2.1μ2.5χ2∆2.1∆μ2.5≡ρ/μ2.1/2.50.84∆84∆16≡∆LS2.1/(2.5-2.1)5.25∆Lq0.845.254.41∆WS1/(2.5-2.1)2.5∆Wq0.84/(2.5-2.1)2.1−M/M/1M/M/1/N/∞/FCFS=≡N≡≡N1≡N≡≡N1≡N∞=≡n-101nn+1N-1N≡1,)(11101−≤⋅=⋅⋅+=⋅+⋅⋅=⋅−−+NnPPPPPPPNNnnnλµµλλµλµ(4.1-23)≡∆P0P1PN1∆ρ/μ∆≤⋅−−=≠−−=++NnPPnNnNρρρρρρ11011111(4.1-24)≡ρ1≡ρ≠1∆•ρ1••∆N•ρ1∆11,1)1(1)(110≠−+−−===++=∑ρρρρρNNNnnSNnPnEL2)1()1()1(01PLPnnELSNnnq−−=−=−=∑=3•4.1-22Little••nN0•e1PN•μe1P0μ•eμe•1P0e/μ•µλµ1)1()1(0+−=−=NqSSPLPLW4WqWs1/μ∆−=−=−−=−+−−=++µµρρρρ1)1()1(1)1(10011SqSSSqNNSWWPLWPLLNL4.1-25=≡∆∆3/∆15∆•N7•3/•μ4/1••2778.0431431111710=−−=−−=++NPρρ239.1)2778.01(11.2)1(11.243143843143088=−−=−−==−×−−=PLLLSqS3eμ1P0410.27782.89/4WSLS/e=2.11/2.89=0.7343.85•N7%7.3431431431187877≈−−⋅=−−=µλµλµλP•≡3/4/LSLqWSWqP0N72.111.390.730.480.2783.7%32.251.00.750.250−M/M/1M/M/1/∞/m/FCFS=nm≡∆∆∆∆∆mLS∆ee=(mLS)(4.1-26)=≡n-1012nn+1m(m-1)(m-n+1)(m-n)m-1m≡n=0dP0(t)/dtμP1(t)mP0(t)0n≤m-1dPn(t)/dt(m-n+1)Pn1(t)μPn1(t)[(m-n)μ]Pn(t)n=mdPm(t)/dtPm1(t)μPm(t)≡=⋅−≤≤+⋅−=⋅+−+⋅=⋅−−+1110111,])[()1(mmnnnPPmnPnmPnmPPmPλµµλλµλµ∆∆10=∑=miiP/μ1∆≤≤⋅⋅−=−=∑=)1()!(!)!(!1000mnPnmmPimmP