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Queuing theory : Notes

The document discusses queuing theory and its applications in telecommunications and networking. It covers various concepts including probability, random variables, Poisson processes, and service rates, highlighting their mathematical properties and implications for system performance. Additionally, it summarizes the importance of these theories in analyzing and optimizing service systems and customer wait times.

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Queuing theory : Notes

  • 1.
    ASSMAteDate Page L INTRODUCTLON:AUELOINGTH EORY S 7oYSmins 0 7x5fo.3 X 20= hoomvh ya ekpect to Wait on 475 3oY. 20mins avesaq 2 Ctoettng tonmeLo 2-S total tme 7xs + 20* 3 5 60 S mins avg.watingtme 3sx 60 20/ 7.23 0.8 5 min 2x30+_8xs .22 mins 60 avg tattHngHme 05 4 030 , AK erLang Caoa) SHudieeand ormalized queing theorybor tale phone- networKS robbic_enggenagCirpor take-oh londing
  • 2.
    alassMAte Data 13 %20t Paga PROBABLITN THEORy eventsprobabilt Sample Space F p F C 2 P Fg2 i s discrete F 2** ih s centinuoLK F 2 AXIOMS OF PROBABILITY pCa):L E0E E,EeF hen pCE) + p (E p CEvE) CoROLLARIES iACB p(A) p(B) Au (en A) Since pC)o A0CE
  • 3.
    elAsSmAte Date 13 82r2D Page 2CEUE p ) p(E) ENE) pCEUE) E pE v (EaE)) pCE)p (E,E EnE2 pE)p(CE,OE) CEVE.)) EOE (EE NE)p)-pE, 0E PCE DE)pCE) (SE) p (E p(E)-plEOE) CoNDITLDNALPROBABILTY CABD pr bability of evet A given B as ocured A BeF Q tS dkcrete Sat auu outcomes qually KalyaB p (AIB) NCANB N)AnS
  • 4.
    classmate Data138.2020 Paga PROBABILITY MASS FONCILONPME C)= HWrOll_a_dice. wtcaj X (Mi,n)>t na- (XESu, 123 pi(5,) L6,5, C,O 36 ContinUOUS_random vaCiahle noX S l i m Ux probabt lity alens'ty fpnethian- CPDE) cmulative distribution_Aunction (CDEE E() Cxs) E Co FxC-s)-o
  • 5.
    edasSmate Date 3,3-22 ( Page ECaE (X) ACBCA eR ()E O1 asSupp.ort opX- auSS distrilbutoy potssion distyibutto) C- 27 O Standavd deulato). O T tS Catlednormaldietrdbvtion
  • 6.
    elassmate Data'17:8:202p Paga - ExPECTATION HI0. Aindout.binomial distribuhon-g0cmetrit distvebution prove that E Cantb): af b VE ax+b y Ccw) a XCw +o RECAP probabilty axioms derivedeD ¢mportant resolks Condiionok probabilitp random v artableS Cdiscret& ContinuoUs) pm Pdg, cdt- Co):1 EL,E e'E pE AUE pCE tp(E,) CALB)p (ADB)ipB)2o PCB)
  • 7.
    elAsSMAte Date 72.2026, Page qau sslandiStributon 2 2 t 2 V2a M0anmean T Standard oleviet'on0here2, 4uo t s ts called Dormal ouktrtbu ExPECTATION OF RVV X hime3. b manuheadl_we ekpect)toss a_Coin N HTH..H T 2"Outorn R x o,L,2,.nt ( ) =_# heads_dn wto Semple Kexperiments lxp erimevt S sSingcan n Hme 14 Knz hmes
  • 8.
    JasSMe vnte (7.22020 ECx) 0.Kt_K2 4 n.Kn+ Samp Ov9 im 2 n.m kne6 im_K Ko K im um K K pX-o)) m p (X=(-) C - n K erpecteion Ex pX-) neX(en) hor DRV Xw) Ex. da pectation - dCRV HO ind out binomial distribution gcomctri distbtHm Provethat E CoX+b) aEx+b = aX+b Nw X(w)+6 (l YC) C a X()+b=y)
  • 9.
    elassMAte Date 18 202 Page bernoullRV Cec toss pCH)p X HX1 1-p Prob ELpA O. Ct-p):p. y S.143 -2e V 5x+3 TY: S0 +33. Ey 2p +3(1-) Eis imear opArator, Spt3 binomtal RV Chp.) Soces ces n thdepeolent bernoili tralk Y *hbernoulle trtal s Succea O_eihertote EY2EX 2P.p
  • 10.
    alnssmate Data 17-32020 Paga 5 poiss.onRV X o2 px) K! Ex2 Ke k k-1) e VARIAN CE much Spread he distribunoo valves Qvenow Prove ECX-EX) E-[Ex1 M St momenk Mean Cu alled nsret mome RV moment generahng nctionsSataAE tS Called asthe Se ond momen E s taued he ih maman_
  • 11.
    lAssmate Date 17:20 Page 6- MEMORYLESSRV expenantial RV )2e O-0 Cc Ey)1 - eA p IB AOB)PCB) P(x>t P(x)to) p(AC) - pCA)_ eato - ( - E - C1-s)p(t t.) Cxt|X4 pX t-t markov prepa
  • 12.
    goma pay eorningelassate phisics engine Dete 17.3.2t Poge X.ime atcwhith_ (S packet arYivel 0 0 t los t 1oSec ooesn_matfe momory e. mar kov serversS2ved savorage cr xpected packet costcmersrevesh d 2rvers Aima Spas the uHe L -l(- avG or ixpeCkEd ) theSustern LtLs
  • 13.
    clsSMAte Date 17.4-2ta Page averoge rateo arrival moAE) Aron t t+At pdhe amval At o (at) o_ptmore thon ene armival V4 IN avrivals in non-overlapping hterval ara cndepent oeach other Stata n U fime t npackek _Qrmved H=prob. that thare om k arnvak ti time. b_ 4 (-2At) A t ) EAE)- )_ AQ--A r K-0 +At -2Ae)
  • 14.
    lassmate Date17:8.2020 Page 9CtHAL t )a) n da,) -2a)im dt , ) Co) C C ) e im Ao da) dt daCt 29,ce)29,4)_ ) (a) k (mepvyle
  • 15.
    elSSMAt Date 2-8.»a Page UE UINTHEORy RECAP memorless andem vaYiahloS expenental (2) poiess.n (At) geometri t h twoSucces.Sive airivals indepenolent At pLexaty one arrival i At AAt + oAt mare than one armival in At soCat) hen the proCassis a PotsSon process oith param2ter 2A._ n_arrarrivals in t tima units) e ( t nl a vindhya KCIS
  • 16.
    claMAt Dote 202202o Paga urd ,12,)t).nKarnivals cn t time units) e + O cn Second queu E K mhtrst qveve E k- in Lirst quevet n Second qveve E th trstqueve K n Second GUve pK packets in queweI& K-K packets en quwe2) K 0 tdepenclt (A (ate Ko K1! Ck-K)I e K poisson C2+2) poisson RV X(A) Ex 2L xpenentinl RvX(p) Ex V
  • 17.
    AsSMAteDate 20 822 Page3 tTt ho packas ànt. ext pa.ckat arrival time -Tst) PT 1- exponeryHiat- A=5 packels Second O. 03 5 5 packok h se.conda) e 1o 0:175 packais in Lsa.cond e 5 51 lo pLCKak in econd ey O183 A/B/c/D/E_ r A/B/ AOrmivalprocess mean 2 servite rate () Servey max no o CUxfaners on packek he Sstem, pepulahien max. nc o Cusfomer toill ouer need fb Serv
  • 18.
    elasSMAte Dota20:8.2020/ Poge 4- M/M M-memcrylessCpotsson)2 M-Senuice processmemoryleg Cu exponential -oneServer cobur copopulattonsize BIRTH-DEATH PRoCESS load bactor P A P(E probabikity onceluctomers ch th system at Ame t n t4At)At) : honew arrial9noneu) olepartre yo were in State n at time t._ OR one ne arrival, yoULwer in Mate. at ime t R M neu olepartora &OUwerr in_State nH at ime
  • 19.
    SSMAte Date 26.82020 Page AL) AAt )+uAtPTt kaking lim_At>o dp, (Atup,t) +Agt) 4uE) or Steady &iate dnct)o olt Calculatinq loP et
  • 20.
  • 21.
    elassMAteDate 20 2:0 Page M/Mtlizathon-P P- 1 .9999 0,93 75 b.9 3 5 p (o 3) -28 35p C2 Col2 tPtP+P+ P+R -6.999 12timeA (6 2 . 5 2 5 S O = S . . 2 2 b12 2 O-0 2
  • 22.
    clasSMAate Date 21: 8.2020 Page UEUINGTHEORY_ RECAP poLsSon process on pois&on PrOcessS 2 potssooProceseA birthcleafh proces Stare n ocostornars/ packeis in Hhe sstem AlBlc/D/E A- arrval pracess (A) mean B Service processs Cmean uD C senverS D- t maxek Cusfonarspacketk (bohtersize E*populafion &ize thct uorl need service AlBl A/BleD Eo MM on Serven afivalproes smemes he service proASA- s mema le paiscan arivo orth_ Cexpoantiad Oth Seviea Yote
  • 23.
    elassnateData 24:826 Page dpC d tCo we willassomedh o AP tALPE (ltu_p loadacto . C-pP 2x Ct-p) +
  • 24.
    aAssMte Data 2482020 Paga 3- LpL(1p)p(tP+¢2 L .9 L 0 - 5 4 P:5 S07 L L 20Mbpt 99 Mbpss SDMbps L Lo t Ls tnivy 2 mn A Lmin 09ost min
  • 25.
    elasSmate Date 2182020 Page . QSLITTLE'SLALD L_A avg. ma spent b each pacdet Qvg.t pockek/Aime Cvg. H packek|austamers K=Vz E EY. Ez X not aluwauE Costemers Qrived roein lo,t1):At avg Rach costomRn sperds w unit of ma total tme spent o ol the E D0Stomors thosa uoha arcivedin to,t]. L Fotal time Spnt by thae Customers : lt t Lt Atw tmtn 2mA g0ee temy mt 5m L Au t lau
  • 26.
    alassmate Data 24 82020/ Paga S- 0 . A: 9 customens/min mins A 9 Ccostomasmn MloCostornars min. m i n : Wa twsoe LL Lsar L:0.9 9 Wser W L- 3A=49 castonersmin ioCustorerimin houo much time coil yo0 penchhe Quetx o averac22 C I min u - Wser min-0:1 0. mn 54Se. - 1 CUstomermin 2o Customan mn_ Conct She xpectedfime to be Sp@nt in the Queue P: 0.9 1-P 6.5min18 05-O.05 _o15min 21se2D
  • 27.
    elasSmate Date28-2020 Page GAl9costormes/min 20 Costdmer/min cohatis L:19 tO -L 20 20 2custmers min = loCUstomare/min O.2 min SO 20 Cust/min to ceminHOO wating ime u attng HmR 2e tstmy A/MLN AA- -4 R P u 'N
  • 28.
    lAsSmate Date 24. 2020 Paga dt ON -P P 1-P -pN GUIZ bloc kine prababilif mon / hu. M// MM/m 0n mooodle,_ sla buS: Dehob:theor mml
  • 29.
    classmate Date 27:202t Page QUEOINTHEORY- RECAP N (N+1) M/M/1/N A/elcin/e N) maxt CUStoMAri hthe APE Pra P NP -p Po t1-P oNblotkingprDpabilhy P (T1-pNH Pe.Chen quee i holl or M/M[I_ P 2 N 2 2
  • 30.
    elassMAte Data27.8.2C20/ Paga M/M//N qprobable.N+J N+1 o9 Px 1o - 0-S o.9999 0 0 5 P 508 P O.0l6 P 67 LP=2 = :ol6 PNE OSO8 L5 5 P P::0002 o . 9 N 4 7 2 PN 3 N 20 N (00b P 27x1o L E n) . +LP +2 P t NRNR P + 2p2 + 3p3 NN NH) + 2 N)e+N
  • 31.
    elassAte Date 27.2 207 Page3 LPL p+ p N N N+ LC-P P -pN) NpNN_pN LC- N oNA (1- Ne Nt pN+ N 2 LpN+I or LaA+2NNN+1 M/M/oo UEING SVSTEM Seruer Ce StomeNS L Lo+ Ls L ACo 0populatio L -(N) Cu tCH
  • 32.
  • 33.
    elASSMAt Date Page.Menoryles seruiCA MM/ QUELNGSYSTEM m servers omemery4leA arnival 2 m mA P m P n>mmiBem-m n- m LJ m-mCA) in MM/m m Cn-m) Pm Cn-m).1n= m1 m m- :PenBLmI C1-)
  • 34.
    elassmate Data 27.8.220/ Paga 6 aHerqveue siartskillingth he SySMem etartsbebeavin Simarto M/M/1twith ServireYate ML M/Mm/m UEUING SYSTEM Memery less memaru leK- m server mM t Customes Qrriva Service pacok (n Sidem MG1 M/D/1 enera Seruice olo terminicHo Service procesSrecess
  • 35.
    dassmate Date 312 2 Page Lmin P 386 3 78 2C-068) L: w minO 2C1-P o,99x + O.olx30 0.43min3 0.olx3o 3 0 063 3 O63 0. 1715 14-115
  • 36.
    classmate Date 3t.82020 Page. SO wHATDID WESEE_LAST_CLASS RECAP M/MLlN 4ueve M/M/0 L P blo.cking probabit N NtI o L N/2 M/M/m s P h L4 1-P m-p) Pcob:oh ueLing9 Pa Pm Pmz L Cn-m)p 2 Cn-m)m"p" p M/Mm m max m packets n4S1em mser vers L memory iesS_arVa memoA(ess Servic
  • 37.
    sSMAte Date 31.8.2a Page 2 (o)pzuDP AAlc-p A L I+ An C ult 2 m ua)=2u )-3uu 1 +
  • 38.
    elasSMate Date 21-8. 2026 3Page M/GM- mmorles arava C- qeneroseuice PCAL. One Serer tsepi time faken fo provicle ervia to the Customev bumo CefCrM,tsr isexpontniallyE distributed) er Vas Ct ECtsar) How hat matche weth L twhen 1-P L +2 2C-P) fSer cexponentiolll, distnibutedl.. BAD PoST OFECE tca 20Sec O.9q prob 30min 0.01 prrb Onecustomer amva por inor (a 1 CCUhat co be the vg. im Spontby the CUStome th he post otcRi On overage hoLo mony_CoStommre Co be hov nthe posS L,w,us EL1se Ex 2xP:-
  • 39.
    classmateata Page 2 0 xO.gq+_g00X 60 37.8See, 37-8 ser Ex2-(E) 0 0 x 0Qq (&O02 0.61 3 7 . 8 2 1367.6 78 0.626 (0.63) 12.9Y CU 2.9 min- A =_LcUstomer/min san 7-8Se C (2.9 mn 0:63 M/D/ M-mamoyule arrrlal t sskxecllcenSant Ddteministic gervce 2C1-)
  • 40.
    lassmate Date 2 202o Page LM/M)L (M/o) O.11 O 06 O2s0 O.225o.2 O1S 2 49S 50 2PC-et 2-P)L 2C1L 2Ct-P) + 2 C-PLo M/M2 LD LqoCil D/D MlCalL HDL MM/I Opproximates MM/
  • 41.
  • 42.
    clAssmate- - Date 3-82e Poge 7- 20Mps 7OMbpSH u D e x e r c s e 2 s0Mbps ISDML- MM/L toobps PeOML tTSDM&ps tooMLpsM// MA 1-h-6 & 2-6 _Strotagy Ohich ane t beter 3 6 2-5-6 9raiag
  • 43.
    clAssMAteDate 3.920 Page ROUTNC maka vilgoata- Vorse 33 Cwa'l_do our begt Varse there is a uuanlda A to meven t DO dg that isvrtualand.dikerent uwalra upto he test t can b So cold_ to tOrn mis D.orld ar. makes us standl up pr uohats rignt- Or hopR throluqh our ile minichorusaaKge_reset ttto the start mini-thorvs Chonus hereLoe are goina-a to Save oul tha we lave giua, all wcv.got minu-cheruswe LotlLMake it through here uan, likea star shininabright en yr ioerd tocda Cmaka evil goaueay Noam Kania acals) eode lyoka well,esetall wnttencodeyeke b, there twhen you eall Yga Amar Fronck kellercdelyeko tronenatera NISe2 o uorld omachines AwORLD wITHOUT DANGER on shadou homan natun eODE LYokO CTne 6 200)that twe need e the wy to od the answen ondcne thinq is onsore chorus
  • 44.
    lGsate Dete 39 2020 RECAP MIMImn MMIm/m_Quauing Sstem MIal M/DII M/M/1 tabular SommaruWs Ltle Ls 70 Meps (too ns 50 w 33-33 routing 3-6 & 2-S-6 -4-6 & 2--62 fouhng ,7o 2 10 07 ot delay fem23 o op s33:3.ns 30 7 3 2.C1-) 3x76 Xo
  • 45.
    nte . w:33:33nsA3 1oxto Ch2 33:33 ns routing2 2-6 1 o 50 150 L 76 12.5 nS 70 C8o) xLo% 2- w _12.5 n 50 Met okela- 0 r wt O U2-Sns olalays loo Control outing algorithm altered toad fajected. hroughpt oeedlload reected load
  • 46.
    aSMAtr ROUTINCn delays algol ensuredelays arminimized ocxèmize throughputs pocr= protocalsreuttna atge2 algorithms gcod CStathio NS dummio hroughput lat vS. hienarchical SSet C-ttinta-donMainvS 0ter-clomaun distribuied vs. Centralized ingle VS. multipath liok stiate vsdustonca Necton nead toknoa Chieranhia bellman-ord |whole netoonk routi Cdjstra RoOTNG ALOORITHMS DLJKSTRA CShotestpath algorithms) ChCV,E V et oh vertic8 0oiKshra Set o elges ob 2 G)bellmarn focc 5 s( 2 Ve Step w add nocde Shoctst pat e
  • 47.
    ASSMAte ute 20 nitia zation O P} ALCORITHM PD mi D PPO Si DD minD, Dt dij 2 + 3 gcto until Dz G D 0 Dniki alazhio D,:o D n : 2 P:i . P: 23 D,2, D2 D,o D Peoo .P 2,31 D 2 D 2 P ,2, 3,s D D 2 D S D-5S GPa,2,s, s41 done
  • 48.
    elassmate Data 3.9 2AL Pnga rUnningtime v) Nlv terati ons N- OCM)n each deration o) o(NlogM)OCIEI+V)loqtvl) di Ksira disadlvaniago )-Ve edges are no1 allowod BELLMAN-FORD ve edger ora allow d ho riesf path k no odekined bvt, no-v w1 Cuclec 0 D i 2 - Mgorithm D min oltD, Ct_D turminate i D : D" min Cest ot reaching usgatmost n edgei min Sd +D"D JF averyStep shortestpoth with edge is alczlatedto each node
  • 49.
    elasSMate Date 3.4.e Page 2 5 Do D:D D = D co Yonning am oCIvllEL) oCN v: N O (mIE1 # terotion equred m N ElN
  • 50.
    alassMAte Data3,9.200 Page 8. PROFOSITION a Ds aegeneated hyhe aloarthmare qual to he ShoctestpaBh ramifu 1_o t ecoes h. healgarithm terminates k alculesNOt_Cotamng nccle Lare having non--Le costk. elgoferminates ttSo dn hNe at tecmination De 's he cof of horteai path romtolor to HwKh prove htL D. D 4 K h._ FLOYD-wARSHALL all-pair sortest path d Or h:0..N h minD P PeK terahion-h: te have the shortest path costromi fo, 'vsin9 0d- 2 h (h=O wlo any Yoole)
  • 51.
    elassmate Date 7.9.2020 Page KoUTING ANDVPN RECAP TOUhg-algorithms hartest path) olkstra-gre0elu bellman-tprel goydwarshall progfaongUntast TOUHng- broadast min(s,1) 3 Kruskal's rmin. Cs) MST prims mu tcast 33,53 s M St M Spans 3,55 Cost C)> tW ts minimiaedeeM min coSt Steiner tree NP (omplete PIM
  • 52.
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