6. Stochastic processes (2)

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1 Contents Markov processes Brth-death processes Lect6.ppt S Introducton to Teletraffc Theory Sprng 5 Markov process Consder a contnuous-tme and dscrete-state stochastc process X(t) wth state space S {,,,N} or S {,,...} Defnton: The process X(t) s a Markov process f P{ X ( tn+ ) xn+ X ( t) x, K, X ( tn) xn} P { X ( tn+ ) xn+ X ( tn) xn} Process X(t) wth ndependent ncrements s always a Markov process: X ( tn) X ( tn) + ( X ( tn) X ( tn)) Consequence: Posson process A(t) s a Markov process: accordng to Defnton 3, the ncrements of a Posson process are ndependent for all n, t < < t n+ and x,, x n + Ths s called the Markov property Gven the current state, the future of the process does not depend on ts past (that s, how the process has evolved to the current state) As regards the future of the process, the current state contans all the requred nformaton 3 4

2 Tme-homogenety State transton rates Defnton: Markov process X(t) s tme-homogeneous f P { X ( t + ) y X ( t) x} P{ X ( ) y X () x} for all t, and x, y S In other words, probabltes P{X(t + ) y X(t) x} are ndependent of t Consder a tme-homogeneous Markov process X(t) The state transton rates q, where, S, are defned as follows: q : lm P{ X ( h) X () } h h The ntal dstrbuton P{X() }, S, and the state transton rates q together determne the state probabltes P{X(t) }, S, by the Kolmogorov (backwards/forwards) equatons Note that on ths course we wll consder only tme-homogeneous Markov processes 5 6 Exponental holdng tmes State transton probabltes Assume that a Markov process s n state Durng a short tme nterval (t, t+h], the condtonal probablty that there s a transton from state to state s q h + o(h) (ndependently of the other tme ntervals) Let q denote the total transton rate out of state, that s: q : q Then, durng a short tme nterval (t, t+h], the condtonal probablty that there s a transton from state to any other state s q h + o(h) (ndependently of the other tme ntervals) Ths s clearly a memoryless property Thus, the holdng tme n (any) state s exponentally dstrbuted wth ntensty q 7 Let T denote the holdng tme n state and T denote the (potental) holdng tme n state that ends to a transton to state T Exp( q ), Exp( q T can be seen as the mnmum of ndependent and exponentally dstrbuted holdng tmes T (see lecture 5, slde 44) T mnt T Let then p denote the condtonal probablty that, when n state, there s a transton from state to state (the state transton probabltes); q p P{ T T} q ) 8

3 State transton dagram Irreducblty A tme-homogeneous Markov process can be represented by a state transton dagram, whch s a drected graph where nodes correspond to states and one-way lnks correspond to potental state transtons lnk from state to state q > : Markov process wth three states, S {,,} Q q q q q Defnton: There s a path from state to state ( ) f there s a drected path from state to state n the state transton dagram. In ths case, startng from state, the process vsts state wth postve probablty (sometmes n the future) Defnton: States and communcate ( ) f and. Defnton: Markov process s rreducble f all states S communcate wth each other : The Markov process presented n the prevous slde s rreducble 9 Global balance equatons and equlbrum dstrbutons Consder an rreducble Markov process X(t), wth state transton rates q Defnton: Let π (π π, S) be a dstrbuton defned on the state space S, that s: S π It s the equlbrum dstrbuton of the process f the followng global balance equatons (GBE) are satsfed for each S: π q π q (GBE) It s possble that no equlbrum dstrbuton exsts, but f the state space s fnte, a unque equlbrum dstrbuton does exst By choosng the equlbrum dstrbuton (f t exsts) as the ntal dstrbuton, the Markov process X(t) becomes statonary (wth statonary dstrbuton π) Q π + π + π π π π π + π π ( + ) π + π π π, +, (GBE)

4 Local balance equatons Contents Consder stll an rreducble Markov process X(t).wth state transton rates q Proposton: Let π (π π, S) be a dstrbuton defned on the state space S, that s: S π Markov processes Brth-death processes If the followng local balance equatons are satsfed for each, S: π q π then π s the equlbrum dstrbuton of the process. Proof: (GBE) follows from by summng over all In ths case the Markov process X(t) s called reversble (lookng stochastcally the same n ether drecton of tme) q 3 4 Brth-death process Irreducblty Consder a contnuous-tme and dscrete-state Markov process X(t) wth state space S {,,,N} or S {,,...} Defnton: The process X(t) s a brth-death process (BD) f state transtons are possble only between neghbourng states, that s: In ths case, we denote > q : q, : q, + In partcular, we defne and N (f N < ) Proposton: A brth-death process s rreducble f and only f > for all S\{N} and > for all S\{} State transton dagram of an nfnte-state rreducble BD process: State transton dagram of a fnte-state rreducble BD process: N N- N N 3 N N 5 6

5 Equlbrum dstrbuton () Equlbrum dstrbuton () Consder an rreducble brth-death process X(t) We am s to derve the equlbrum dstrbuton π (π S) (f t exsts) Local balance equatons : π π + + Thus we get the followng recursve formula: + π π π π + Normalzng condton : π π S S Thus, the equlbrum dstrbuton exsts f and only f < S Fnte state space: The sum above s always fnte, and the equlbrum dstrbuton s N, π + π π Infnte state space: If the sum above s fnte, the equlbrum dstrbuton s, π + π π 7 8 Pure brth process Q π π + π + ρπ π π ρ ( ρ : / ) π + π + π π ( + ρ + ρ ) π ρ + ρ + ρ 9 Defnton: A brth-death process s a pure brth process f for all S State transton dagram of an nfnte-state pure brth process: State transton dagram of a fnte-state pure brth process: : Posson process s a pure brth process (wth constant brth rate for all S {,, }) Note: Pure brth process s never rreducble (nor statonary)! N N N- N

6. Stochastic processes (2)

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