EE549_ps6solns_spring08web

EE549_ps6solns_spring08web - UNIVERSITY OF SOUTHERN...

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Unformatted text preview: UNIVERSITY OF SOUTHERN CALIFORNIA, SPRING 2008 1 EE549: Problem Set #6 Solutions I. A NON-ERGODIC PROCESS a) t Z(t) T Fig. 1. A timeline for part (a). b) Clearly we have for any time t > : ≤ 1 t Z t Z ( τ ) dτ ≤ 1 t Z T e τ dτ where equality holds when t ≥ T . Thus: ≤ 1 t Z t Z ( τ ) dτ ≤ e T- 1 t Where T is a random variable, but is finite with probability 1 , and does not depend on the time index t . Taking limits as t → ∞ yields: ≤ lim t →∞ 1 t Z t Z ( τ ) dτ ≤ Therefore, Z = 0 . c) For any time t , we have: E { Z ( t ) } = E { Z ( t ) | t ≤ T } Pr [ t ≤ T ] + 0 = e t Pr [ T ≥ t ] = e t e- αt Therefore: E { Z ( t ) } = e (1- α ) t For α = 1 , we have E { Z ( t ) } = 1 for all t , and so lim t →∞ E { Z ( t ) } = 1 . Thus, Z av = 1 6 = Z = 0 . d) This does not contradict the ergodicity theorem, because we do not have the boundedness property that Z ( t ) ≤ Z max for a fixed constant Z max . In fact, the Z ( t ) process has no a-priori upper bound, and can be arbitrarily large. e) Because E { Z ( t ) } = e (1- α ) t , we have: • If α > 1 : lim t →∞ E { Z ( t ) } = Z av = 0 • If α = 1 : lim t →∞ E { Z ( t ) } = Z av = 1 • If α < 1 : lim t →∞ E { Z ( t ) } = Z av = ∞ Thus, for this problem, Z av = Z whenever α > 1 . UNIVERSITY OF SOUTHERN CALIFORNIA, SPRING 2008 2 II. THE EVOLUTION EQUATION FOR A GI/B/ 1 QUEUE a) If Bernoulli arrivlals, then we have a = (1- λ ) , a 1 = λ , a k = 0 for k > 1 . Thus, for j = 0 we have: Pr [ L ( t + 1) = 0] = Pr [ L ( t ) = 0](1...
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EE549_ps6solns_spring08web - UNIVERSITY OF SOUTHERN...

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