Poisson process Markov process

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1 E2200 Quuing hory and lraffic 2nd lcur oion proc Markov proc Vikoria Fodor KTH Laboraory for Communicaion nwork, School of Elcrical Enginring 1

2 Cour oulin Sochaic proc bhind quuing hory L2-L3 oion proc Markov Chain coninuou im Coninuou im Markov Chain and quuing Markovian quuing L4-L7 Non-Markovian quuing L8-L10 Quuing nwork L11 E2200 Quuing hory and lraffic 2

3 Oulin for oday Rcall: quuing Rcall: ochaic proc oion proc o dcrib arrival and rvic propri of oion proc Markov proc o dcrib quuing coninuou-im Markov-chain Graph and marix rprnaion E2200 Quuing hory and lraffic 3

4 Rcall from prviou lcur Quuing hory: prformanc valuaion of rourc haring Spcifically, for lraffic Dfiniion of quuing rformanc riangl: rvic dmand, rvr capaciy and prformanc Srvic dmand i random in im hory of ochaic proc Arrival Blocking Srvic E2200 Quuing hory and lraffic 4

5 Sochaic proc Sochaic proc A ym ha volv chang i a - in im in a random way Random variabl indxd by a im paramr Sa pac: h of poibl valu of r.v. X or Xn Th ochaic proc i: aionary, if all n h ordr aiic ar unchangd by a hif in im: rgodic, if h nmbl aiic i qual o h aiic ovr a ingl ralizaion conqunc: if a proc rgodic, hn h aiic of h proc can b drmind from a ingl infinily long ralizaion and vic vra Sa probabiliy diribuion in im for on ralizaion Sa probabiliy diribuion for an nmbl of ralizaion E2200 Quuing hory and lraffic 5

6 Oulin for oday Rcall: quuing, Quick ovrviw: ochaic proc oion proc o dcrib arrival and rvic propri of oion proc Markov proc o dcrib quuing coninuou-im Markov-chain Graph and marix rprnaion Tranin and aionary a of h proc E2200 Quuing hory and lraffic 6

7 7 oion proc Ky diribuion in h cour oion diribuion Dicr probabiliy diribuion robabiliy of a givn numbr of vn Exponnial diribuion Coninuou probabiliy diribuion E2200 Quuing hory and lraffic [ ] X E k p k X k k,! [ ] [ ] [ ] , 2, 1 1, X V X E X E x X x F x p x f x x Wikipdia Exponnial oion

8 oion proc oion proc: o modl arrival and rvic in a quuing ym Dfiniion: Sochaic proc dicr a, coninuou im X : numbr of vn arrival in inrval 0-] couning proc X i oion diribud wih paramr X k p k k! k, E[ X ] i calld a h inniy of h oion proc no, limiing a probabilii p k lim p k do no xi p k : oion diribuion 0 k vn E2200 Quuing hory and lraffic 8

9 Df: Th numbr of arrival in priod 0,] ha oion diribuion wih paramr, ha i: Thorm: For a oion proc, h im bwn arrival inrarrival im i xponnially diribud wih paramr : Rcall xponnial diribuion: roof: oion proc X k k a la on arrival unil 1 no arrival unil 1 p k k! f, F 1, E[ ] 1 p k : oion diribuion Exp 0 k vn numbr of arrival oion diribuion inrarrival im xponnial E2200 Quuing hory and lraffic 9

10 Th mmoryl propry Df: a diribuion i mmoryl if: > + > > Exampl: h lngh of h phon call Aum h probabiliy diribuion of holding im i mmoryl Your phon call la 30 minu in avrag You hav bn on h phon for 10 minu alrady Wha hould w xpc? For how long will you kp alking? > + 10 > 10 > I do no mar whn you hav ard h call, if you hav no finihd y, you will kp alking for anohr 30 minu in avrag. E2200 Quuing hory and lraffic 10

11 11 E2200 Quuing hory and lraffic Df: a diribuion i mmoryl if: Exponnial diribuion: Th Exponnial diribuion i mmoryl h only coninuou mmoryl diribuion: Exponnial diribuion and mmoryl propry F F f >, 1, > > + >, > > + > > > + > > + > +

12 oion proc and xponnial diribuion oion arrival proc impli xponnial inrarrival im Exponnial diribuion i mmoryl numbr of arrival oion diribuion inrarrival im xponnial For oion arrival proc: h im unil h nx arrival do no dpnd on h im pn afr h prviou arrival oion arrival W ar o follow h ym from hi poin of im Exp E2200 Quuing hory and lraffic 12

13 Group work Waiing for h bu: Bu arrival can b modld a ochaic proc Th man im bwn bu arrival i 10 minu. Each day you arriv o h bu op a a random poin of im. How long do you hav o wai in avrag? Conidr h am problm, givn ha a Bu arriv wih fixd im inrval of 10 minu. b Bu arriv according o a oion proc. S Th hichhikr paradox in Viramo, oion proc. E2200 Quuing hory and lraffic 13

14 ropri of h oion proc S alo problm 2 1. Th um of oion proc i a oion proc Th inniy i qual o h um of h innii of h ummd muliplxd, aggrgad proc 2. A random pli of a oion proc rul in oion ubproc Th inniy of ubproc i i p i, whr p i i h probabiliy ha an vn bcom par of ubproc i 3. oion arrival im avrag ASTA w prov lar Sampling a ochaic proc according o oion arrival giv h a probabiliy diribuion of h proc vn if h arrival chang h a Alo known a RO Random Obrvr ropry 4. Suprpoiion of arbirary rnwal proc nd o a oion proc alm horm w do no prov Rnwal proc: indpndn, idnically diribud iid inr-arrival im E2200 Quuing hory and lraffic 14

15 Oulin for oday Rcall: quuing, ochaic proc oion proc o dcrib arrival and rvic propri of oion proc Markov proc o dcrib quuing Coninuou-im Markov-chain Graph and marix rprnaion Tranin and aionary a of h proc E2200 Quuing hory and lraffic 15

16 Sochaic proc p i Xi Markov proc Th proc i a Markov proc if h fuur of h proc dpnd on h currn a only no on h pa - Markov propry X n+1 j X n i, X n-1 l,, X 0 m X n+1 j X n i Homognou Markov proc: h probabiliy of a chang i unchangd by im hif, dpnd only on h im inrval X n+1 j X n i p ij n+1 - n Markov chain: if h a pac i dicr A homognou Markov chain can b rprnd by a graph: Sa: nod Sa chang: dg 0 1 M E2200 Quuing hory and lraffic 16

17 Coninuou-im Markov chain homognou ca Coninuou im, dicr pac ochaic proc, wih Markov propry, ha i: X X n+ 1 n+ 1 j j X X n n i, X i, n1 0 l, X < 1 < < n 0 m < n+ 1 a Sa raniion can happn in any poin of im Exampl: numbr of pack waiing a h oupu buffr of a rour numbr of cuomr waiing in a bank Th im pn in a a ha o hav mmoryl diribuion xponnial o nur Markov propry: h probabiliy of moving from a i o a j omim bwn n and n+1 do no dpnd on h im h proc alrady pn in a i bfor n. E2200 Quuing hory and lraffic 17

18 Coninuou-im Markov chain homognou ca L u om xampl, ha may b modlld by Coninuou Tim Markov Chain Sochaic proc: dicr a pac, coninuou im I u my phon, for 5 minu in avrag, hn I do no u i for 30 minu in avrag, hn I u i again. Th copi of h cour bindr ar old on by on ack arriv o an oupu buffr, and ar rvd on by on Dfin h a Li h condiion o hav a Markovian modl Dfin h poibl raniion among h a E2200 Quuing hory and lraffic 18

19 Coninuou-im Markov chain homognou ca Sa chang probabiliy: X n+1 j X n i p ij n+1 - n Characriz h Markov chain wih h a raniion ra inad: q ij q ii X + Δ j X i lim, i j - ra inniy of a chang Δ 0 Δ q ij - dfind o ay calculaion lar on j i Traniion ra marix Q: Q q qm 00 0 q 01 q M M 1 q q 0M M 1 M q MM q q Q E2200 Quuing hory and lraffic 19

20 Summary oion proc: numbr of vn in a im inrval ha oion diribuion im inrval bwn vn ha xponnial diribuion Th xponnial diribuion i mmoryl Markov proc: ochaic proc fuur dpnd on h prn a only, h Markov propry Coninuou-im Markov-chain CTMC a raniion inniy marix Nx lcur CTMC ranin and aionary oluion global and local balanc quaion birh-dah proc and rvii oion proc Markov chain and quuing dicr im Markov chain E2200 Quuing hory and lraffic 20

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