6.262.PS9.sol

6.262.PS9.sol - 6.262 Discrete Stochastic Processes Spring...

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6.262 Discrete Stochastic Processes, Spring 2010 Problem Set 9 — Solutions due: Friday, April 23, 2010 Problem 1 (Exercise 5.10) a) M/M/ 1 : From (5.40), we have π i = ρ i (1 - ρ ) for i 0 where ρ = λ/μ and ρ < 1 (positive recurrent). M/M/m : First note that in the Markov chain, with k customers in service, P { departure in ( t,t + δ ] } = kμδ + o ( δ ) . So, while the forward jump probability is still λδ , the backward jump probability is (min { k,m } μδ ). Recalling that a recurrent birth-death chain is positive recurrent if and only if it satisﬁes Equation (5.38), deﬁning ρ = λ/ ( μm ) we have π i π i - 1 = λ for i < m and π i π i - 1 = ρ for i m, which simpliﬁes to π i = ( ± λ μ ² i π 0 i ! , i < m ρ i π 0 m m m ! , i m . Since i π i = 1, we have π 0 = 1 + m - 1 X i =1 ( λ/μ ) i i ! + X i = m ρ i m m m ! ! - 1 = 1 + m - 1 X i =1 ( λ/μ ) i i ! + ( ) m m !(1 - ρ ) ! - 1 Finally, the other π i can be obtained from the above relationships. M/M/ : Following an analogous reasoning to the M/M/m case without a cutoﬀ at m , we obtain π i = ³ λ μ ´ i π 0 i ! for all i 0 . Using the Taylor series expansion of e λ/μ = i =0 ( λ/μ ) i i ! , we see that π 0 = e - λ/μ . Thus, π i = ³ λ μ ´ i e - λ/μ i ! , 0 . 1

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b) M/M/ 1 : Recalling our results for birth-death chains (Section 5.2), for the chain to be tran- sient we need λ/μ > 1, for null recurrent λ/μ = 1, and for positive recurrent λ/μ < 1. M/M/m : Finitely many states do not aﬀect the transient/recurrent nature of an irreducible chain (why?). It suﬃces therefore to look at the chain corresponding to states m,m + 1 ,... . For the chain to be transient, we need λ/mμ > 1, for null recurrent, λ/mμ = 1, for positive recurrent λ/mμ < 1. M/M/ : For the chain to be transient, we need λ > 0 and μ = 0 (i.e. customers arrive but they do not depart). We cannot have null recurrence for μ > 0, as for any value of λ letting n = b λ/μ c + 1 yields that mμ > λ for all m n . The chain corresponding to states n,n + 1 ,... is therefore positive recurrent, and, taking into account the ﬁnite-state recurrent chain corresponding to states 1 , 2 ,...,n , the result extends to the rest of the chain. Thus, for
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6.262.PS9.sol - 6.262 Discrete Stochastic Processes Spring...

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