OR2-Ch4-排队论2—随机服务系统

[e-mail]/M/n4-1M/M/14-2M/M/C4-34-4Exit20031010949Page2of284.1M/M/1™FIFO™M/M/1///FIFO¾¾:¾¾¾¾1.1pρ=µλλλ=iL,2,1,0=µ=µii1110pρρ−=ρ∑∞=iinnPPρρ−=ρ−=)1(;10ρρρρρρρ−=−=−===∑∑∑∞−∞∞1)1()1()(1100nnnsnnPnnELλµλ−=sL20031010949Page3of284.1M/M/1——ρρ=+−=−=∑∞11101PPn2.ssnnnqLLPnPPnLρ=λ−µλρ=ρ−ρ=ρ−=−=−=∑∑∑∞∞∞=1)1(21113.λ=λ−µ==ssLWEW1)(P.3254.λ=ρ=λ−µρ=µ−=qssqLµ/1=−qsWW20031010949Page4of284.1M/M/1——ssWLλ=qqWLλ=µ/1+=qsWWµλ/+=qsLLk111)1()(+∞+=∞+=ρ=ρρ−==∑∑kknnknnPknPtsetTP)()(λ−µ−=tqqetTPtW)1(1}{)(ρµρ−−−=≤=tTq≤tTs≤tssetTPtW)(1}{)(λµ−−−=≤=541.2.3.220031010949Page5of284.1M/M/1——10n+1nn-12M/M/158.0,25.1,1=ρ=µ=λ1.4125.11=−=λ−µλ=sL2.=162.3125.18.0=−=λ−µρ=qW3.2P2/5=12/5=72.052)(=ρλ−µ−e∞//1//NMMNN1NNN-1N-220031010949Page6of284.1M/M/1——λ=µµ+λ=λ+µλ=µ−−+11101)(NNnnnPPPPPPP1,−≤Nn1(0=∑NnP110+==µ=λNPPnnNnPρρρ111+−−=1011+−−=NPρρNn≤≠ρ1p.3171N1NN0is11N1iPL++=ρ−ρ−−ρ−ρ==∑)()()(0sN1iiqP1LP1iL−−=−=∑=esssLPLWλµ=−=)1(0µ1−=sqWW20031010949Page7of284.1M/M/1——PN(1PN))1(NP−λ012n-1nn+1(m-n+1)(m1)m(mn)m-2m-1m2)1()1(0PPNe−=−=µλλmmMM//1//msLm−)(seLm−=λλ20031010949Page8of284.1M/M/1——0Pλm0Pmλ101Pµ111])[()1(011110=λ=µ−≤≤µ+λ−=λ+−+µµ=λ∑=−−+miimmnnnPPPmnPnmPnmPPPm∑=−=miiimmP00)()!(!1µλ0)()!(!PnmmPnnµλ−=11−≤≤mn20031010949Page9of284.1M/M/1——)1(0PmLs−−=λµµ−=λ−−µ=−−=λ−µ+λ−=11)1(1)1)((000sqssqWWPmWPLPmL)1(0PLms−λµ=−20031010949Page10of28Let’stakebreak.Wouldyoulikecoffeeortea?Helpyourselfplease.M/M/CExit20031010949Page11of284.2M/M/C012n-1nn+12n(n+1)cn≤n-1nn+12cccnf4.2.1.M/M/C(M/M/n/∞/∞/FIFO)™¾cc(nc)n(nc)()µλρc=1ρµ+λ=λ+µ≤≤µ+λ=λ+µ+λ=µ−+−+)(,)()1(,)()1(111101cnPcPPccnPnPPnPPnnnnnn∑∞==11iiP1≤ρ20031010949Page12of284.2M/M/C——λλρρρρµλρµλµλµλµλµλssqqcncnqqqsccnnncnnnckckLWLWPccPcnLcLLLcPPcnPcnPcccnPnPckP==−=−=+=+=−==≥≤=•+=∑∑∑∞+=∞=−−−=,)1(!)()()1(!)()()(,)(!1)(,)(1])(!1)(1[0120001100™20031010949Page13of284.2M/M/C——123331=λ4.0=µ352c523c.,.,=µλ=ρ=µλ=()()()()045.0]35.211!35.2!25.2!15.2!05.2[132100=−⋅+++=−P1.20031010949Page14of284.2M/M/C——()515.3045.0)35.21(!335.25.223=×−⋅=qL01.6=+=µλqsLL515.3==λqqLW01.65.2515.3=+=sW()70004501352P133nP2303..)(!.)(!)(.=⋅ρ−=⋅ρ−µλ=3.1MM//3/1=λ4.0=µ83.0=ρ16.4=qL17.00=P5314.031=−=sL15=sW5.12=qW0.8320031010949Page15of284.2M/M/C——4.2.2FIFONcMM/////∞•λµ–λj=λ,j=0,1,...,N–1;λN=0µj=jµ,jc;j=c,jc,j=1,...,N•λj,µj()())]1()(1[)1(!)(!0!11)(!!1200000ρ−ρ−−ρ−ρ−ρρ=≤≤ρ⋅≤≤⋅ρ=≠ρρ−ρ−ρρ⋅+ρ=−−=∑cNcNcqncnnckNcckcNccPLNncPcccnPncPcckcPµλ=ρc20031010949Page16of28)1(NqqPLW−=λµ1+=qsWW)1(NqsPcLL−+=ρnccN=100]!)([−=∑=ckkkcPρ0!)(PncPnnρ=cn≤≤0N=cPcµ/100===sqqWWL()())1(!010!11ccncnccnnsPcnccnPLn−===∑∑∑−=−ρρρρP.33620031010949Page17of284.2M/M/C——4.2.3FIFOmcMM/////∞µλρcm=cmcn≤nc−mnc≤≤cn−1010])!(1!)!(!1[!1−=+=∑∑µλ−+µλ−=ckmckkckckmcckmkmP≤≤+−≤≤−=−mncPccnmmcnPnnmmPncnnn1!)!(!0!)!(!00µλµλ20031010949Page18of284.2M/M/C——∑=mnsnPL1()∑+=−=mcnnqPcnL1()seLm−=λλessLWλ=eqqLWλ=µλ+=eqsLLP.337820031010949Page19of284.2M/M/CTakeabreakM/G/1Exit20031010949Page20of284.31//GM4.3.1M/G/11.;2.34FIFO56TE[T]Var(T)7][/1TEλµλ+=+=qseqsLLLLssWLλ=µ1][+=+=qqsWTEWWqqWLλ=eλ20031010949Page21of284.3.3M/E/1Pollaczek-Khintchine(P-K)][TEλρ=)1(2][22ρλρρ−++=TVarLs][10TEPλ−=ρ−=sqLLλssLW=λqqLW=4.3.2M/D/10µ1][=TE)1(21222µλ−µλ+µλ+µλ=sLµλ−=10P)1(2ρρµλ−=−=sqLLλssLW=λ=qqLWP.3391020031010949Page22of284.3.3M/E/1••kkk12…kkk())!1(),,(1−=−−ketkktfktkkµµµK,3,2,100=ktµ[]µ=kTEi1[]221µ=kTVari[]µ=1TE[]21µ=kTVar•kkkkVVV+++L21kVVVT+++=L2120031010949Page23of284.3.3M/E/1kkk=1k[]0=TVarρ−ρ++ρ=1212kkLsρ−ρ+=1212kkLqλ=qqLWλ=ssLWµλpP.3401120031010949Page24of284.4M/G/c/c/cLqWSPnLs20031010949Page25of284.4M/G/c/c/c)1(csPL−µλ=∑=µλµλ=ciinninP0!/)/(!/)/(Pccc16810%20031010949Page26of284.4M/G/c/c/M/G/c/c/c=3M/G/3/3/3c=n=32105.03333.63333.16/)8/16(2/)8/16(1/)8/16(1/)8/16(6/)8/16(!/)/(!3/)/(321033033==+++=µλµλ=∑=iiiP21.05%21.05%10%4c=4M/G/4/4/420031010949Page27of284.4M/G/c/c/0952.076667.024/)8/16(6/)8/16(2/)8/16(1/)8/16(1/)8/16(24/)8/16(!/)/(!4/)/(4321044044==++++=µλµλ=∑=iiiP9.52%81.1)0952.01(816)1(4=−=−µλ=PLs20031010949Page28of284.4M/G/c/c/HavearestExit