EP2200 Queuing theory and teletraffic systems. 3rd lecture Markov chains Birth-death process - Poisson process. Viktoria Fodor KTH EES

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1 EP Queung heory and eleraffc sysems 3rd lecure Marov chans Brh-deah rocess - Posson rocess Vora Fodor KTH EES

2 Oulne for oday Marov rocesses Connuous-me Marov-chans Grah and marx reresenaon Transen and seady sae soluons Balance euaons local and global Pure Brh rocess Posson rocess as secal case Brh-deah rocess as secal case EP Queung heory and eleraffc sysems

3 Sochasc rocess =PX= Marov rocesses The rocess s a Marov rocess f he fuure of he rocess deends on he curren sae only - Marov roery PX n+ = X n =, X n- =l,, X =m = PX n+ = X n = Homogeneous Marov rocess: he robably of sae change s unchanged by me shf, deends only on he me nerval PX n+ = X n = = n+ - n Marov chan: f he sae sace s dscree A homogeneous Marov chan can be reresened by a grah: Saes: nodes Sae changes: edges M EP Queung heory and eleraffc sysems 3

4 4 EP Queung heory and eleraffc sysems Connuous-me Marov chans homogeneous case Connuous me, dscree sace sochasc rocess, wh Marov roery, ha s: Sae ranson can haen n any on of me Examle: number of aces wang a he ouu buffer of a rouer number of cusomers wang n a ban The me sen n a sae has o be exonenal o ensure Marov roery: he robably of movng from sae o sae someme beween n and n+ does no deend on he me he rocess already sen n sae before n.,,, n n n n n n n X X P m X l X X X P saes

5 Connuous-me Marov chans homogeneous case Sae change robably deends on he me nerval: PX n+ = X n = = n+ - n Characerze he Marov chan wh he sae ranson raes nsead: PX Δ X lm, - rae nensy of sae change Δ Δ - defned o easy calculaon laer on Transon rae marx Q: Q M M M M M M MM =4 =6 4 Q EP Queung heory and eleraffc sysems 5

6 Oulne for oday Marov rocesses Connuous-me Marov-chans Grah and marx reresenaon Transen and seady sae soluons Balance euaons local and global Pure Brh rocess Posson rocess as secal case Brh-deah rocess as secal case EP Queung heory and eleraffc sysems 6

7 7 e d d d d X X P X X P Q Q, lm, lm EP Queung heory and eleraffc sysems Transen soluon The ransen - me deenden sae robably dsrbuon ={,,,...} robably of beng n sae a me, gven. Transen soluon leaves he sae arrves o he sae

8 Examle ransen soluon =4 =6 4 Q d Q d e Q Transen sae Saonary / seady sae A: =, = B: =, = EP Queung heory and eleraffc sysems 8

9 Transen soluon - commen Marx exonenal Marx exonenals are defned as: e X = And are dffcul o calculae. =! X Therefore, for he small case on he -sae MC I solved he orgnal se of dfferenal euaons nsead: d d ' ' Q To solve lnear dfferenal euaons s sll ough. See relevan course boos Forunaely, mah rograms can do for you I used Mahemaca EP Queung heory and eleraffc sysems 9

10 Saonary soluon seady sae Def: saonary sae robably dsrbuon saonary soluon lm exss s ndeenden from The saonary soluon has o sasfy: d Q, d Noe: he ran of Q MM s M-! =4 =6,.6, 4 6 Q M 4 6.4,, M M M M M MM EP Queung heory and eleraffc sysems

11 Saonary soluon seady sae Imoran heorems whou he roof Saonary soluon exss, f The Marov chan s rreducble here s a ah beween any wo saes and Q, has osve soluon Euvalenly, saonary soluon exss, f The Marov chan s rreducble For all saes: he mean me o reurn o he sae s fne Fne sae, rreducble Marov chans always have saonary soluon. Marov chans wh saonary soluon are also ergodc: gves he oron of me a sngle realzaon sends n sae, and he robably ha one ou of many realzaons are n sae a arbrary on of me EP Queung heory and eleraffc sysems

12 Oulne for oday Marov rocesses Connuous-me Marov-chans Grah and marx reresenaon Transen and seady sae soluons Balance euaons local and global Pure Brh rocess Posson rocess as secal case Brh-deah rocess as secal case EP Queung heory and eleraffc sysems

13 Balance euaons How can we fnd he saonary soluon? Q= Q Sae Sae : : flow n flow ou P P P 3 Global balance condons In eulbrum for he saonary soluon he ranson rae ou of a sae or a grou of saes - mus eual he ranson rae no he sae or saes flow n = flow ou defnes a global balance euaon P 4 3 Q EP Queung heory and eleraffc sysems 3

14 Grou wor Global balance euaon for sae and : Q Sae : Sae : P 4 43 P P 3 3 Is here a global balance euaon for he crcle around saes and? EP Queung heory and eleraffc sysems 4

15 Balance euaons Local balance condons n eulbrum he local balance means ha he oal flow from one ar of he chan mus be eual o he flow bac from he oher ar for all ossble cus defnes a local balance euaon The local balance euaon s he same as a global balance euaon around a se of saes! 4 P 4 43 P P 3 3 EP Queung heory and eleraffc sysems 5

16 Balance euaons Se of lnear euaons nsead of a marx euaon Q 3 flow n Global balance : flow n = flow ou around a sae or around many saes Local balance euaon: flow n = flow ou across a cu M saes M- ndeenden euaons = flow ou 4 P 4 43 P P P 4 43 P P 3 3 EP Queung heory and eleraffc sysems 6

17 Oulne for oday Marov rocesses Connuous-me Marov-chans Grah and marx reresenaon Transen and seady sae soluons Balance euaons local and global Pure Brh rocess Posson rocess as secal case Brh-deah rocess as secal case EP Queung heory and eleraffc sysems 7

18 Pure brh rocess Connuous me Marov-chan, nfne sae sace Transons occur only beween neghborng saes Sae ndeenden brh nensy:, - = = - + Q No saonary soluon Transen soluon: =Psysem n sae a me number of evens brhs n an nerval EP Queung heory and eleraffc sysems 8

19 9 EP Queung heory and eleraffc sysems Pure brh rocess Transen soluon number of evens brhs n an nerval,] Pure brh rocess gves Posson rocess! me beween sae ransons s Exλ = = e e e e for! ' ' ' ',, ' Q Q

20 Euvalen defnons of Posson rocess. Pure brh rocess wh nensy. The number of evens n erod,] has Posson dsrbuon wh arameer 3. The me beween evens s exonenally dsrbued wh arameer P X e revous slde ure brh rocess chec n he bnder number of evens Posson dsrbuon me beween evens exonenal revous lecure EP Queung heory and eleraffc sysems

21 Pure deah rocess Connuous me Marov-chan, nfne sae sace Transons occur only beween neghborng saes Sae ndeenden deah nensy:, - + No saonary soluon Pure deah rocess gves Posson rocess unl reachng sae Tme beween sae ransons s Exµ EP Queung heory and eleraffc sysems

22 Oulne for oday Marov rocesses Connuous-me Marov-chans Grah and marx reresenaon Transen and seady sae soluons Balance euaons local and global Pure Brh rocess Posson rocess as secal case Brh-deah rocess as secal case EP Queung heory and eleraffc sysems

23 Brh-deah rocess Connuous me Marov-chan Transons occur only beween neghborng saes + brh wh nensy λ - deah wh nensy μ for > modells oulaon Q Sae holdng me lengh of me sen n a sae Unl ranson o saes - or + Mnmum of he mes o he frs brh or frs deahs mnmum of wo Exonenally dsrbued random varables: Ex + EP Queung heory and eleraffc sysems 3

24 4 EP Queung heory and eleraffc sysems B-D rocess - saonary soluon Cu Cu + Local balance euaons, le for general Marov-chans Sably: osve soluon for snce he MC s rreducble,, : : Cu Cu Grou wor: saonary soluon for sae ndeenden ranson raes:,.

25 Marov-chans and ueung sysems Why do we le Posson and B-D rocesses? How are hey relaed o ueung sysems? If arrvals n a ueung sysem can be modeled as Posson rocess also as a ure brh rocess If servces n a ueung sysems can be modeled wh exonenal servce mes also as a ure deah rocess Then he ueung sysem can be modeled as a brh-deah rocess λ λ μ μ EP Queung heory and eleraffc sysems 5

26 Summary Connuous me Marov-chans Marovan roery: nex sae deends on he resen sae only Sae lfeme: exonenal Sae ranson nensy marx Q Saonary soluon: Q=, or balance euaons Posson rocess ure brh rocess number of evens has Posson dsrbuon, E[X]= nerarrval mes are exonenal E=/ Brh-deah rocess: ranson beween neghborng saes B-D rocess may model ueung sysems! EP Queung heory and eleraffc sysems 6

27 Summary Connuous-me Marov chans Balance euaons global, local Pure brh rocess and Posson rocess Brh-deah rocess EP Queung heory and eleraffc sysems 7