Discrete-time stochastic processes

First note from p p 618 that if i i i 1 then j pj 1

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Unformatted text preview: r the backward Markov chain, and interpret it as a model for residual life. Exercise 5.12. Consider the sample time approximation to the M/M/1 queue in figure 5 a) Give the steady-state probabilities for this chain (no explanations or calculations required– just the answer). In parts b) to g) do not use reversibility and do not use Burke’s theorem. Let Xn be the state of the system at time nδ and let Dn be a random variable taking on the value 1 if a departure occurs between nδ and (n + 1)δ , and the value 0 if no departure occurs. Assume that the system is in steady-state at time nδ . b) Find Pr {Xn = i, Dn = j } for i ≥ 0, j = 0, 1 c) Find Pr {Dn = 1} d) Find Pr {Xn = i | Dn = 1} for i ≥ 0 e) Find Pr {Xn+1 = i | Dn = 1} and show that Xn+1 is statistically independent of Dn . Hint: Use part d); also show that Pr {Xn+1 = i} = Pr {Xn+1 = i | Dn = 1} for all i ≥ 0 is sufficient to show independence. f ) Find Pr {Xn+1 = i, Dn+1 = j | Dn } and show that the pair of variables (Xn+1 , Dn+1 ) is statistically independent ofDn . 5.10. EXERCISES 233 g) For each k > 1, find Pr {Xn+k = i, Dn+k = j | Dn+k−1 , Dn+k−2 , . . . , Dn } and show that the pair (Xn+k , Dn+k ) is statistically independent of (Dn+k−1 , Dn+k−2 , . . . , Dn ). Hint: use induction on k; as a substep, find Pr {Xn+k = i | Dn+k−1 = 1, Dn+k−2 , . . . , Dn } and show that Xn+k is independent of Dn+k−1 , Dn+k−2 , . . . , Dn . h) What do your results mean relative to Burke’s theorem. Exercise 5.13. Let {Xn , n ≥ 1} denote a irreducible recurrent Markov chain having a countable state state space. Now consider a new stochastic process {Yn , n ≥ 0} that only accepts values of the Markov chain that are between 0 and some integer m. For instance, if m = 3 and X1 = 1, X2 = 3, X3 = 5, X4 = 6, X5 = 2, then Y1 = 1, Y2 = 3, Y3 = 2. a) Is {Yn , n ≥ 0} a Markov chain? Explain briefly. b) Let pj denote the proportion of time that {Xn , n ≥ 1} is in state j . If pj > 0 for all j , what proportion of time is {Yn , n ≥ 0} in each of the states 0, 1, . . . , m? c) Suppose {Xn } is null-recurrent and let pi (m), i = 0, 1, . . . , m denote the long-run proportions for {Yn , n ≥ 0}. Show that pj (m) = pi (m)E [time the X process spends in j between returns to i], j 6= i.} Exercise 5.14. Verify that (5.49) is satisfied by the hypothesized solution to p in (5.53). Also show that the equations involving the idle state f are satisfied. Exercise 5.15. Replace the state m = (m, z1 , . . . , zm ) in Section 5.6 with an expanded state m = (m, z1 , w1 , z2 , w2 , . . . , zm , wm ) where m and {zi ; 1 ≤ i ≤ m} are as before and w1 , w2 , . . . , wm are the original service requirements of the m customers. a) Hypothesizing the same backward round-robin system as hypothesized in Section 5.6, find the backward transition probabilities and give the corresponding equations to (5.475.50) for the expanded state description. b) Solve the resulting equations to show that m ≥ ∏δ ¥m Y πm = π + φ f (wj ). 1 − ∏δ j =1 c) Show that the probability that there are m customers in the system, and that those customers have original service requirements given by w1 , . . . , wm , is √ !m m Y ∏δ Pr {m, w1 , . . . , wm } = πφ (wj − 1)f (wj ). 1 − ∏δ j =1 d) Given that a customer has original service requirement w, find the expected time that customer spends in the system. 234 CHAPTER 5. COUNTABLE-STATE MARKOV CHAINS Exercise 5.16. A taxi alternates between three locations. When it reaches location 1 it is equally likely to go next to either 2 or 3. When it reaches 2 it will next go to 1 with probability 1/3 and to 3 with probability 2/3. From 3 it always goes to 1. The mean time between locations i and j are t12 = 20, t13 = 30, t23 = 30. Assume tij = tj i ). What is the (limiting) probability that the taxi’s most recent stop was at location i, i = 1, 2, 3? What is the (limiting) probability that the taxi is heading for location 2? What fraction of time is the taxi traveling from 2 to 3. Note: Upon arrival at a location the taxi immediately departs. Exercise 5.17. Consider an M/G/1 queueing system with Poisson arrivals of rate ∏ and expected service time E [X ]. Let ρ = ∏E [X ] and assume ρ < 1. Consider a semi-Markov process model of the M/G/1 queueing system in which transitions occur on departures from the queueing system and the state is the number of customers immediately following a departure. a) Suppose a colleague has calculated the steady-state probabilities {pi } of being in state i for each i ≥ 0. For each i ≥ 0, find the steady-state probability pii of state i in the embedded Markov chain. Give your solution as a function of ρ, πi , and p0 . b) Calculate p0 as a function of ρ. c) Find πi as a function of ρ and pi . d) Is pi the same as the steady-state probability that the queueing system contains i customers at a given time? Explain carefully. Exercise 5.18. Consider an M/G/1 queue in which the arrival rate is ∏ and the service time distributin is uniform (0, 2W ) with ∏W < 1. Defin...
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This note was uploaded on 09/27/2010 for the course EE 229 taught by Professor R.srikant during the Spring '09 term at University of Illinois, Urbana Champaign.

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