stochastic - CDA6530: Performance Models of Computers and...

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CDA6530: Performance Models of Computers and Networks Chapter 3: Review of Practical Stochastic Processes
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2 Definition Stochastic process X = {X(t), t T} is a collection of random variables (rvs); one rv for each X(t) for each t T Index set T --- set of possible values of t t only means time T: countable discrete-time process T: real number continuous-time process State space --- set of possible values of X(t)
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3 Counting Process A stochastic process that represents no. of events that occurred by time t; a continuous- time, discrete-state process {N(t), t>0} if N(0)=0 N(t) 0 N(t) increasing (non-decreasing) in t N(t)-N(s) is the Number of events happen in time interval [s, t]
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4 Counting Process Counting process has independent increments if no. events in disjoint intervals are independent P(N 1 =n 1 , N 2 =n 2 ) = P(N 1 =n 1 )P(N 2 =n 2 ) if N 1 and N 2 are disjoint intervals counting process has stationary increments if no. of events in [t 1 +s; t 2 +s] has the same distribution as no. of events in [t 1 ; t 2 ]; s > 0
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5 Bernoulli Process N t : no. of successes by time t=0,1,…is a counting process with independent and stationary increments p: prob. of success Note: t is discrete When n t, N t B(t, p) E[N t ]=tp, Var[N t ]=tp(1-p) P ( N t = n )= Ã t n ! p n (1 p ) t n
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6 Bernoulli Process X: time between success Geometric distribution P(X=n) = (1-p) n-1 p
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7 Little o notation Definition: f(h) is o(h) if f(h)=h 2 is o(h) f(h)=h is not f(h)=h r , r>1 is o(h) sin(h) is not If f(h) and g(h) are o(h), then f(h)+g(h)=o(h) Note: h is continuous lim h 0 f ( h ) h =0
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8 Example: Exponential R.V. Exponential r.v. X with parameter λ has PDF P(X<h) = 1-e - λ h , h>0 Why?
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9 Poisson Process Counting process {N(t), t 0} with rate λ t is continuous N(0)=0 Independent and stationary increments P(N(h)=1) = λ h +o(h) P(N(h) 2) = o(h) Thus, P(N(h)=0) = ? P(N(h)=0) = 1 - λ h +o(h) Notation: P n (t) = P(N(t)=n)
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10 Drift Equations
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11 For n=0, P 0 (t+ t) = P 0 (t)(1- λ∆ t)+o( t) Thus, dP 0 (t)/dt = - λ P 0 (t) Thus, P 0 (t) = e - λ t Thus, inter-arrival time is exponential distr.
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stochastic - CDA6530: Performance Models of Computers and...

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