P 2 (t) = o(t). e λt. 1. keep on adding independent, exponentially distributed (with mean 1 λ ) inter-arrival times, or

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1 POISSON PROCESS PP): State space consists of all non-negative integers, time is continuous. The process is time homogeneous, with independent increments and P 0 t) = λ t+ot) P t) = λ t+ot) P 2 t) = ot). WiththehelpofthegeneratingfunctionPz,t)oftheP 0 t),p t),... sequence, we have derived: P n t)= λn t n e λt n! ThecorrelationbetweenNt)andNt+s)is + s t TheconditionaldistributionofNs),giventhatNt)=nands<t)isBn,p= s t ). Tosimulatetheprocesscreateonerandomrealizationofit),fromtime0to timet,wecaneither. keep on adding independent, exponentially distributed with mean λ ) inter-arrival times, or 2. draw one random integer from a Poisson distribution with the mean of Λ λ T,thengenerateaRISfromU0,T),andtakethearrivaltimeto be the corresponding order statisticsi.e. sorting the sample). Non-homogenous version of PP allows λ to bea specific, given) function of toften stepwise). The probability ofexactly) n arrivals during time interval t,t 2 )isthencomputedby Λ n n! e Λ where t 2 Λ= λt)dt t We can split ahomogeneous) PP into two independent processes by flipping a coin to decide whether the current arrival goes left or right. The correspondingλassplitaccordinglyλ left =p λ,λ right =q λ). Twohomogeneous) PPs competing. Probability that the first will reach thevalueofnbeforethesecondonereachesmis p nm i=0 n +i ) i q i

2 PPin2ormoredimensions. Thenumberof dandelions inanareaofsize AhasthePoissondistributionwithΛ=A λ,whereλistheaveragedensity. M/G/ queue. Xt)[customers being served] and Yt)[customers who have already left] are independent, Poisson-type RVs, with Λ x = λ t p t and Λ y =λ t q t,where p t = t t 0 [ Gu)]du Gu)beingthedistributionfunctionoftheservicetimes. Notethatlim t Λ x = λ µ sev. Compoundcluster) PP. Its MGF is exp{ λt[ Mu)]} M being the MGF of the size distribution. For integer-type size distribution cluster precess), Mu) can be replaced by the corresponding PGF Pz), gettingthepgfofyt).ineithercase E[Yt)] = λtµ size Var[Yt)] = λtσ 2 size+µ 2 size) PPof randomduration. ThePGFofNT)is M[λz )] wherem isthemgfoft.thisimplies E[NT)] = λµ T Var[NT)] = λµ T +λ 2 σ 2 T PURE BIRTH PROCESSPB): Only one axiom changes, namely P n,n+ t)=λ n t+ot) AspecialcaseistheYule processwith We define the corresponding PGF which must meet the following PDEλ λ n n λ P i z,t) P i,n t)z n n=i P i z,t)=λzz )P iz,t) subject to P i z,0)=z i 2

3 The solution: negative binomial), where zpt Pz,t)= zq t p t =e λt PURE DEATH PROCESS decreases rather than increases), one unit atatime,ataratedenotedµ n. Special case: µ n n µ leads to and ) i P i z,t)=µ z)p iz,t) P i z,t)=q t +p t z) i wherep t =e µt. Time till extinction has the following distribution function the expected value and variance PrT t)= e µt ) i + µ ) i µ i 2 ) BIRTH AND DEATH PROCESSEScanbothincreaseattherateof λ n )ordecreaseattherateofµ n )oneunitatatime. Wewillconsiderseveral special cases. Linear Growth: leading to and λ n = n λ µ n = n µ P i z,t)=z )λz µ)p iz,t) p t z P i z,t)= r t + r t ) q t z a composition of Binomial and Geometric distributions), where p t λ µ)e λ µ)t λ µe λ µ)t r t µ e λ µ)t ) λ µe λ µ)t 3 ) i

4 The corresponding expected value and variance: i e λ µ)t i λ+µ λ µ eλ µ)t )e λ µ)t Probability of extinction already) at time t is r i t, probability of ultimate extinctionisr i certainwhenλ µ). Expected time till extinctionmeaningful only when this is certain): wheni=,andgeneratedby ω λ ln µ µ λ ω i+ = + µ λ) ωi µ λ ω i i λ wheni>utilizingω 0 =0). Linear Growth with Immigration: leading to λ n = n λ+a µ n = n µ P i z,t)=z )λz µ)p iz,t)+az )P i z,t) non-homogeneous), having the following i = 0 solution: P 0 z,t)= pt zq t ) a/λ modifiednegativebinomial),withthemeanof aq t andvariance aq t λp t λp 2. t The generalany i) solution is: P i z,t)= pt zq t ) a/λ p t z r t + r t ) q t z Thestationarydistributiononlywhenλ<µ)is M/M/ queue: P i z, )= λ n = a ) a/λ µ λ µ λz µ n = n µ ) i 4

5 leads to Its solution is P i z,t)=µ z)p iz,t)+az )P i z,t) ) aqt P i z,t)=exp µ z ) q t +p t z) i wherenowp t =e µt.thisisaconvolutionsi.e. independentsum)ofpoisson withthemeanof aqt µ )andbinomial. Clearly,theasymptoticdistributionis ) a P i z, )=exp µ z ) N welders: leading to Solution: λ i = λ N n) µ i = µ n P i z,t)= z)µ+λz)p iz,t)+nλz )P i z,t) P i z,t)=q ) t +p ) 2) t z) i q t +p 2) t z) N i a convolution of two binomials), where Clearly p ) t = λ+µe λ+µ)t λ+µ p 2) t = λ e λ+µ)t ) λ+µ P i z, )= ) N µ+λz µ+λ Othermore complicated) models: WegiveuponP i z,t),andtrytogetthecorrespondingstationarydistributiont )only,by: where p 0 = p n = λ 0λ λ 2...λ n µ µ 2 µ 3...µ n p 0 n=0 λ 0 λ λ 2...λ n µ µ 2 µ 3...µ n ) If the sum is infinitediverges), stationary distribution does not exist. 5

6 Examples: M/M/ queue: whereρ λ µ. p n =ρ n ρ) Average number of people in the system: Average length of idle period: λ,busyperiod: two. M/M/c queue: where p n = Γ= c i=0 ρ n ρ ρ,peoplewaiting: ρ 2 µ λ n c n!γ ρ n c!γc n c n c ρ i i! + ρ c c! ρ c ) ρ.suf:ρ.,busycycle: thesumofthe Averagenumberofbusyserversisρ,averagelengthoftheactualqueueis ρ c c!γ ρ c ρ c )2 Theexpectedlengthoffull-utilizationperiodisc µ λ). When State 0 is absorbing λ 0 = 0), stationary mode is impossible, and the main issue is the probability of ultimate absorption, given the process startsinstatemdenoteda m ).Thisiscomputedby n k= µ k λ k d n = Sum m a m = certain when the Sum diverges). When absorption is certain, the expected time till it happens is computed, recursively, by i= n=0 d n ω 0 = 0 λ λ 2...λ i ω = µ µ 2...µ i µ i ω n+ = + µ n λ n )ω n µ n λ n ω n λ n mayturnouttobeinfinite-nottypicalthough). GENERAL TIME-CONTINUOUS MARKOV PROCESSES 6

7 can move, instantaneously, from any state to any other state. For simplicity, we consider only finitely manyusually a handful) of states. The corresponding rates can be organized in a matrix form leaving the main diagonal elements blank). When the main diagonal elements are defined to be the negative sum of the remaining row elements, the resulting matrix A is called the infinitesimal generator of the process. The probability of being in State j at time t, given thatattime 0 the process startsin State i, is given bythe i,j) th elementof Pt),wherePt)isasolutionto Pt)=APt) Kolmogorov s forward equations), namely Pt) = expat) Thisrequirescomputingafunctionofasquarematrix,sayB,whichisdone in general by fb)=c fω )+C 2 fω 2 )+...+C N fω N ) whereω k k=..n)arethematrix seigenvalues,andc k arethecorresponding constituent matricesn is the matrix s size). They can be computed based on N k= N C k = I k= N ω k C k = A k= N ω 2 kc k = A 2 k= ω N k C k = A N When two eigenvalues say ω = ω 2 ) are identical, C fω )+C 2 fω 2 )+... changestoc fω )+C 2 f ω )+...throughoutallthis. BROWNIAN MOTION hasboththestatespace,i.e. Xt),andtimerunonacontinuousscale. Itis based on the assumptions of continuity and of independent increments, implying that Xt) Nx 0 +d t, c t) wheredisthedriftparameter,andcisthediffusioncoefficient. Whend=0, we have: ProbabilitythataisneverreachedbytimeT,or Pr max 0 t T Xt)<a X0)=x 0 7. ) = 2Pr Z> a x ) 0 c T

8 whena>x 0 withtheobviouschangesfora<x 0 ). ProbabilityofXT)>ywithouteverreachingz,or ) Pr XT)>y min Xt)>z X0)=x = 0 t T Pr{Z> y x c T } Pr{Z> y+x 2z c T } wherey>zandx>zwiththeobviouschangesfory<zandx<z). Probabilityofreturningtothestartingvalueatleastonce)duringat 0, t ) interval, or ) Pr max Xt)>x min Xt)<x X0)=x = t 0 <t<t t <t<t 2 π arccos t0 t Whendhasanyvalue,wehaveonlyoneresult: ThedistributionofXt),giventhat...Xt 0 )=x 0,Xt )=x,xt 2 )=x 2, Xt 3 )=x 3,...where... t 0 <t <t<t 2 <t 3 <...)isnormal,withthemean of x t 2 t)+x 2 t t ) t 2 t linear interpolation) and the standard deviation of c t 2 t)t t ) t 2 t 8