HW3_sol - vi?!" a) e}The'ecqueuee below sh...

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Unformatted text preview: vi?!" a) e}The'ecqueuee below sh stl'restuffed bileundnflined firefly ‘lit'y: Ullfllllll flfllllllfllfllfllllllflllll flflllllfllfl 1:) Here the flageare munderlined and the moved [destuffedju aex's: flLLLLLfllilllxllflfl111111011111x11111x11 flflflllllllfllfllllllx 'I'hehimsimws flu: diedalasu'lng Ul5fllx1x3_. musthavea zero sraffcdafterfllifims appearin “1115mm... This stuffedpartem wiJlbe indisfingfishehlc fi'DmlhE. original firing D1 pig-n unless smflingis also used after GP in the firing filmfiz-n- Thus sufiugnumbeusedinthiscm.1hegenuflmgumemisthen by Assume That smiling is necessary nfin 1115 an ill suing: of flrfurm DFEIHIJE}... Then Ellfih a stuffied. sequence is Dljflhlxlxz-.. It fallen: as befall: ItllI. smfing is then mam after 015 in the sequenee [Iljflhlxjxgm This snfl‘mgis always necessary after-[Iii <1.334.: 111: Stuffed string isshown be'.ew vim d1e stuffedbits underlinefiandaflagaddedazthe en _ 1lDllfllflflfllDflIDfllllfllflfllfll Thedeeuflgmleismduend:[desmeImemmEtuithybitsmningnlmebegiming A gimflhitiefllmdeletedfimfiresuingifmepreeedeflneedemdedtfimarefllfl.The flagisdeteetedwl'ma l ispreeededbydieflzeedeeocbdhfisfllflnndflmmmreeendy decoded bilwesnotdelexed. Mabeveisagaumflnflefmdflecfingmywpeefflag mmmflwfllanjunfllfl]; fan-this special mitissm'fieiem tolookfurthe subsningfllfll iufllemcdvedmfilfiflmrmfmlhssinqfificaflunisfllflifalhserfim Dem-11mmjnmcflagithaemoeewbyeimplyanepeliimofmefimtflaghh (3.31., utheloggEm] vj. SimejisflleinmgerpanufiuggE{K},wesecflm11mun lie betweenfland l. ExpreaejngA=E{K}2'-i+j+ lhmrmsef1rdeIK}.weget A =21r +leflEllC1—T+1 A-loggElK] =27- 'f+l This funetian afjriseasilyeeen to bee-Dave): (teqit has a posilive seecmd derimtive}. I: hasthevaluulatf=flandatf=lmleessthmlferfld-fdl. Twisembljshcsflm A SluggEHC] + 1 Findhigthnnfixfilmnnolel- TH bydif‘fcrenfiefimi, flmmjnimumacem at T=-luezfln 2} Thevnlue efZT-"H ] a: point is [In 2]'1+luggfl_1 2]+1=l.914...,m A: lungfl'E] + ur. 2}-l+1ug1{1n1;+1 (5-35 Sufind bin-m always fl”: andalwnya foflflrwflac am 4515. Th: inifia-l I] in this patluu tumdhcafitmmnmsnflnddmmW itselfbcaamfi‘ndbil. Asinmtamflyau: of aubacncion 2.5.2, In: igucrclh: cm film-c dfisinifial D is: amffcdbil since it 15. alums: negligiblfi compachwim the ether :35: {3.1503 wd] dcaignad flag damn: wuuld not allow asmfindbitashfuatbimfnflag}. Ifaaufiudbilmecudndbfi'fllsmmadmjls canvamdhynuianinmal.mmitiatakmunflagifflmnmhitiaflmflismkznaaan abartiffllnmtbitisl. Thmmcnm'ilauafl'uflfilwumaflagmapmxwih probabflfiylflflflmempmmdmmhuuffladymdflagsduewmhsmfifid Malaria-5'. Humfilaammdmappmmmnmmmmmmnuy K—fi planesinwdausmamwhmaamffedhitoauldbcinsamdfnflcrwimflfiinflmdam; musammuefimdmswiatlaxmcexpccmnnumbnattalsalpnmmdflagaduetu mars in surfed bitsis [Ii-6323'. fimmfightpanmnfeighhiummmanmminmufflwcightbiawwldm I21: pfll‘tm'n inlu a flag. Two of flan-.5: panama, [31’ and I’D. cannot appear in amffed data. numbermufmnpannms. fllfimandflflfimcanappminsmficdmtmmwmajn aamfiedhtfii.n1hcflfnl‘nwi11g13j.Thcflrstnfflmsacaaeamnaapondsmmacaaain whichanmorinnmflcdbitcauscsafhfimappcar.mflmhawnheadymmlymdifi1 The second ammpmais m a data suing 0015. Thus fine aubau'ings nt‘data for which a inflemminalhnbitcanmscaflafimnppcuuelistedbflm:thewiricminwifich the enter mast appear is shown undcrijnud: G'DCICID HI—I-I—I-I—II: I l I 1 l Hmll-I-I—l-I—l- n—FHH Gilli“: l I I! I. l Hal-any given bitpuu'tiunj in IheKbirdata suingfi SI—Tfiljmpmbatflfiy thaw-n: of than: pawns 513113 an Illin i5 2‘1 + 4.2'3 = 3.1-7. Thus the pmhalfility of 9 False flag being {ht-ECdede ennrnna data hi; staifingm bi1j nfthcdalnis. 3p1-?.'I11ia is also I'M:de Hum-Dim flags, and sunning EHFEfti'IE hi1: rsf1ha. Hm arm-ng119. an;po number is [KJflpl'T Appmximathg by [uplang Ki? by K, and adding his to Lie Exp-mad numbnrnf faL-n: flag: due In arc-rain miffed bitsglu: overall prubabflityof a his: flagin a fimu:ofler.gfll K is {IHIIIKFL HIE-'3r is Mayan}ij by K, mgnimtflutme firs-rpcanern above can also lppnu-amfingntj—K E, rhea mum-mu pmbabfljtj' of a fab: flag is approximami mm: ciwclyhy [lflEHK-fifijp. 2.37 a) Note that a given packet n can never be without the possibility of packets gaming out of order on the line. To see thi cxamplc below. n-l tr n+1? Hmflmmmismpm musmgmlqummuwmuifim‘m simmwmmtmwflamnfuflhubuqm umflbhmm'nInfiLuIm packetmflwmwnifingfaranukunpm H.11usimplmmlnir1hammfimmnd pmbablydlnmfi ' mrm hrlcjisfudmmaniutmminficrmdin; eachpickelfur ' mackurnakfle. finalize-the, run-ulnarme mfaapetiudafflmhbchmmumdfinglflmhsmdnlhh andcrwleawthemmfimudmmfimmmmm.mmmum mmmfifinnsmwhmafimmhtnmhmfimi Thefinnll'mrn-1rnufl MWMWW Thane-undisdntmmmisfimuf ctn—l mkmthnfinrwmdmna]. fiatflhfiflmflfipflkfln-ZMMM mum channel Themfmhewmdmfidmishnmnfisfimfipm Irriwbefiar'emalnfn-lmawngn-lmbemhuhmfwnfl. Thermfurfiwdflrd _ mfimiswufiddnmkfurn-Zbehgnfiumfuflmnkfmn. wgnbcmeflm: uchhn-lwaslmmmfimwemmmemdmuicfimamleflsmme :‘nlluwingnfic. Inwdnrmmmitpacketnufimgmorm nt‘flmfuilnwin: mdifiufimuflhflfifififid: i}1‘.311+T iii-Thu numberofachflofn I quMJSfl-Ia mmbflofmmufn-luptul iiiJMIm micafn-l kmflTmmfiwfil nmmlmmmiflimnfml- Ennddifion,fiumr¢mi¢dminn=urmmafihe Iona-adu-umflifimmfllkobe mum-timuisflufimuwflchn-lmhsl mm: mam-2T - iiflhenumbernfaclfi ufn-Eequaisd'l: numberofn‘mufisfimsufn-Z filmmmfln-lismlemdsflmflwnemtolmmsmissimufn—L b} An algmiflm must {Ital with I11: possibiiqr at" a fame that is last fix" never fives]. Luimtmcmsfuflymmt nfimnfimhhfit Emllgmihnsumadsin dfiscm,fl1mitmunfaflifa rtgudndaslosnlamaufiveswimnnpacmfl EH: 3am: manna: numb-c: mndulc- 2 is upccmd. sum until aftcr n—l is acted; rhisis true: even 5. consider J 2.39 a) TC .—. (K+V)(j—l) + (K+V)fwxl 13) EITC} = (K+V)U-l+(Mfl()+ll2] Differentiating with respect to K and setting the result equal to O. we get c) For 1:1. 1: can be seen directly from Eq. (2.42) that TC is by choosing KMAX greater than the largest possible value of M (thus making all messages one packet long). The approxmaauon Eq. (2.43) is very poor in this case. but the solution Kmm in Eq. (2.44) ts snll valid, as seen above. For fixed length packets, the amount of fill required for very large K is prohibitive, so the approximation used in part b) above is reasonable and the resulting finite value for K is certainly reasonable. 3:3 Wenepteeemtheeynemasehminmefimme.mwfieflanmeeamemm downjgoesinmnpahfienpoirpumniswflabkuthefimmtdmherfiuwfimin queueforarepairpelwnmbecomefi'ee. Note that ifnr-ltlfissymmieidenfiealndte ofEumpleEJ. Letlbedtednnogltpuoftbeeysnemandlethefltewmgefimeahfiten maehinewaiie fwampfimmbmmfimhpphhgflfl'sflmnfltemme mutt-teetth MR+Q+PJ = N (1} from which Mme} s N (2) Sime the number of machines waiting repair can be It most [hi-m}, the average waiting timelQ isaimnstthe average time terepairmvml matinee. which ismvmlPflIhes. fitom Eq. {I} we obtain 1m+m-m]P+P}EN {3} Applying Little's theorem to the repairpfl'scms. we obtaht 11:15:11 {4} The relations (2H4) give the following bounds for the throughput 7. _y_____ . R+ minfNP/mR+P} STSR+ (N-m+I)P 3+6 [:3 Thprahihifityflmflupusmwfllbeflnlaflmkawis lfdbecmltmnxpmwnfia] disuihaumismmylfis.mmmhawmmflmdm¢diwibuflmln minimmdminmflnmmcnmmmrmmnMgmdmeofmhaf dwadmrdmcusmmcrsmedhslfwsumdisuibufimasmamfimfimufm: eusmnm. (b) humafifimcinthntankis ltumavmgcmmsmdmfimflplusflwerpec dmforflwfimwmmermfmish micm'l'hghtmfimzi: lflsincethedapm proucssismfisdcaflyflmflcalmmfla singlesmufanilirywimltfimfilmwm rm.lvlmupracisel}rw¢have P[nocu5mmud:pansinfi1¢mxumins} -Pllstdu¢snmdepminnmtmins *P{2ndducsmrdepm1innexnmhu} *Pifirfldmnmdcpaninnmtnflm} *Pldihduesnoldcpaflinmttnfinsi = [firm-r5“. Wm: Hm: firs: dapanm'c occurs within Ilmncxtt minsl = l - :‘4‘. midflHmpeCtGdfimcmfi-umtdepamla 1M.Suth¢mwisiflmhm5. {‘3} min-5W Will nmdlmge hemme- d“ situation ‘ whm mm _ ‘ . axch mmmmbaflmmundmhcmndlfinm fmiajanddnmndafis firfl mm: ' ‘ ' ' mmflflmfipm mflnpackegsufatmmtmnhhcsuumsniflwaitm “1 mm“ It: mummfimhuu-Efifimlmwmm lame' mm” _ sWsTfl.W¢hvefl-mtmhflfofthepmhumsyumflme' I wmgmmqmmWJ'hamfm fl+w 5mm: 5 Tim: = 1a 1- am m Tm: tram} mfifififlfl‘m ' mm Van-1m: qr Wailing Time ={MHWRMLQJWHP - mu. Somwmm mmm' . behveeuflandfiqfi. “mmflmdmmvmm utwmungumm 3.1{II {b} Lat N1, N: h: m: numb:- nt' arrivals in zwn disjoint inmals oflmgths TI and 11, Then PiNHN: = n} = mm =1; N2=n4€l =m=yPINl=leWz=wkl = mu [GL1] mwfllwwfln-m = u-Mfl -- fllmmqfluglmflflkfln-k}!l = full + fllflltj + lt:)".’n!] my: idemin Efindaw-W‘Ean—kfl] -{a+ hm: cub: slim by indurnimJ {a} Tin: nmntu‘ of mini: nl‘flac :cmbineflpmn: 1"; disjoint immls is ciamiyindtpmdcmsumnmdmshnwmatflmnumbemfmfivfls in an interval :2 Poisson :fisu'ihutazl, i e. P{A.it-I 1:} I | Ak[1+TJ-A1[fl—...—Ai{t}=n] ==.(i.|+__.+1k‘.r:[(11+n_+1kh]nmj For simplicity 1ctk=2: a salami-proof applies for I: :1 1 'I'bcn H31“ +1] +Aifi+fl '31“ 3" $11”: I =n] Ififlpli‘ul’” Tl - All! l= mfizit erranmqfl] _ Pfirlx’ifit +1}' Alli: III- m]P[.."L.1{t +1} - Align _ mm] and Iln: calculation confirms; as- in part (b). Man P11 mfivalfmmhlpu-iurmti] occmdi =P[Ifldvflfiumfl1.flfi'flmhflm[lmud} = ELIE. {:1} [m t he :11: fimnfm-ival. We haw. icsl I nn'ivaircmred} =P[H:5. l mfivalumduPIIaniw-almcdl =PH mivaluccmcdin {111:}. Darrin]; mauled in [5. tflHPil anivalmmn‘ed = {M3 ' Ififi'u' ' “if”: ' '11}; {M13 ' trim-HI: ' Tn} “(5 - til-Kl:- t1} ' TIL'Is film-Ara than. IJII: miran limo: t is 'IJII'lfDJTfilj' in [th t3]. 3.11 {a} Len"hethan-Era:menrdvfl.md1n-1I+1--5.Wchnwfankfl PM. :4} - PIA-15+ 5} - Mal-01w“ lbrfiamdiafimumurmivalsmanMIcfiulflr P1155” -I -E':'" which 1:13.11]. Tommthlltl.tp. Harchdepeniennnmtflmtusilgmmdcpmdmmufflmnmhm c-fan'iwlsindisjmflinmnls} F‘I‘lghil't’llt] -P{fl Walsh (1:. 11:3]I11-t} IPWIn'ivals infl, t+s]] - e~h=P[t-J:I-s} MW 12 “'11 1| ET- ML To vuin (3.12]. we um flu! Putt + E) — Mt) = fl} - e45 so {3.L21wfl] ht Shawn il' Jim” {¢45— 1 + 1.51:3- n Inland. using L'Huspiwl': ruk m; haw 11mm [EH- 1 + 3.5);‘5 It 1ime {‘1c'15+ 1} = I} To vmfy {3.13}wc mu: mm P[m+a}-m1=n-mfiafl m {3.13) will be shown if limM Mr“ n 15% - D This is fillli'i'flllrm m li:::ur..,,4_-r {RM - l} = {I which is dandy m. Tu vmify {3.14} we nut that PEMI+3}-A[t}&2} =1 . Ham-Ej- Mn=m —Pth+fi)-am- l] - 1 v [1 - 15+ ammufiwififiwtfil {b} MNI.N1b¢flJ:nmbflofmivahinmdisjointhmalsoflmgths‘rludfi. Murma- “1-wmflh N2. n-k} a inborn-ll = 1:]l=|*u'~ln a: n-k} # mums. mnmmwmfln-km - a-Mfl *‘QJIFHIQ'IIJ‘EQIgJfi’nHkfln-kfil . g-fifll 4 Wilt. + 11;}l‘fnfl fluid-amin- Makbtu-knqklmm] mm + mum! cmbcshm harm.) {c} Thgnumhu'flfatfivflsoffinmmbimdmindisjoim inmllsis chmiyhmwmmmnedmsmmmmufufivminm inun'alisPoissmdisuthteiic. PIA1[1+t}+. ..+Ak{1+t]vA1{fl-. .. -hl[t]=n] afiMh-Hfifllfl. 1-,. .1-WIPHII Fm‘shnpliaitylflkflusimflarpmofappfiflfmkhlm Halt” 13+ A1fl+ 1}— nm) - am = n] nflnql’mfi: +1] - Ala} -m,A2(t+1:}- A1111] In-m] = Enfiol'flfit +1?! - Alft} =m]F{-&3lft + 1:} - haul-“'11:! mmmaflafimmnfinmnsinpmfblfilsu Pll mivalfiummlriflmn l Occlll‘fld} =P[l arrivalfiummfifrmn A111?“ mum] l flzfi'vmmm] I 1131 (dflmethuhrfivalwchaw P{t<sl lm-rivaloccumd} =F{t«:s. ] udvaJD-ccmtdwfll minimum-ad} -Pl1 arrival nocurbd in[tl,s}.flan'ivalsmwmdin[s,t1]}JP{l arrival = {1(5 - “fur”: 413:“! '9’): {M11 - nit-“‘3 ' "1} = [s - amt; - II} This shows Lhat flu: arrival rim: 1' is unfit-5111]},I distributui in It... 1-2]. ...
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