Discrete-time stochastic processes

Exercise 518 consider an mg1 queue in which the

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Unformatted text preview: {W > j δ }, and f (j ) = F (j ) − F (j + 1). In this case δ i=1 F (i) lies between E [W ] − δ and E [W ]. As δ → 0, ρ = ∏E [W ], and distribution of time in the system becomes identical to that of the M/M/1 system. 5.7 Semi-Markov processes Semi-Markov process are generalizations of Markov chains in which the time intervals between transitions are random. To be specific, let X (t) be the state of the process at time t and let {0, 1, 2, . . . } denote the set of possible states (which can be finite or countably infinite). Let the random variables S1 < S2 < S3 < . . . denote the successive epochs at which state transitions occur. Let Xn be the new state entered at time Sn (i.e., Xn = X (Sn ), and X (t) = Xn for Sn ≤ t < Sn + 1). Let S0 = 0 and let X0 denote the starting state at time 0 (i.e., X0 = X (0) = X (S0 ). As part of the definition of a semi-Markov process, the sequence {Xn ; n ≥ 0} is required to be a Markov chain, and the transition probabilities of that chain are denoted {Pij , i ≥ 0, j ≥ 0}. This Markov chain is called the embedded Markov chain of the semi-Markov process. Thus, for n ≥ 1, Pr {Xn = j | Xn−1 = i} = Pr {X (Sn ) = j | X (Sn−1 ) = i} = Pij . (5.58) Conditional on X (Sn−1 ), the state entered at Sn is independent of X (t) for all t < Sn−1 . As the other part of the definition of a semi-Markov process, the intervals Un = Sn − Sn−1 between successive transitions for n ≥ 1 are random variables that depend only on the states X (Sn−1 ) and X (Sn ). More precisely, given Xn−1 and Xn , the interval Un is independent of the set of Um for m < n and independent of X (t) for all t < Sn−1 . The conditional distribution function for the intervals Un is denoted by Gij (u), i.e., Pr {Un ≤ u | Xn−1 = i, Xn = j } = Gij (u). (5.59) The conditional mean of Un , conditional on Xn−1 = i, Xn = j , is denoted (i, j ), i.e., Z U (i, j ) = E [Un | Xn−1 = i, Xn = j ] = [1 − Gij (u)]du. (5.60) u≥0 We can visualize a semi-Markov process evolving as follows: given an initial state, X0 = i at time 0, a new state X1 = j is selected according to the embedded chain with probability Pij . Then U1 = S1 is selected using the distribution Gij (u). Next a new state X2 = k is chosen according to the probability Pj k ; then, given X1 = j and X2 = k, the interval U2 is selected with distribution function Gj k (u). Successive state transitions and transition times are chosen in the same way. Because of this evolution from X0 = i, we see that U1 = S1 224 CHAPTER 5. COUNTABLE-STATE MARKOV CHAINS is a random variable, so S1 is finite with probability 1. Also U2 is a random variable, so that S2 = S1 + U2 is a random variable and thus is finite with probability 1. By induction, Sn is a random variable and thus is finite with probability 1 for all n ≥ 1. This proves the following simple lemma. Lemma 5.5. Let M (t) be the number of transitions in a semi-Markov process in the interval (0, t], (i.e., SM (t) ≤ t < SM (t+1) ) for some given initial state X0 . Then limt→1 M (t) = 1 with probability 1. Figure 5.8 shows an example of a semi-Markov process in which the transition times are deterministic but depend on the transitions. The important point that this example brings out is that the embedded Markov chain has steady-state probabilities that are each 1/2. On the other hand, the semi-Markov process spends most of its time making long transitions from state 0 to state 1, and during these transitions the process is in state 0. This means that one of our first ob jectives must be to understand what steady-state probabilities mean for a semi-Markov process. 1/2; 1 r 0→0 r ✿♥ ✘0② 1/2; 10 1/2; 1 ③♥ 1 ② 1/2; 1 r 0→1 1→0 r 0→0 r Figure 5.8: Example of a semi-Markov process with deterministic transition epochs. The label on each arc (i, j ) in the graph gives Pij followed by U (i, j ). The solid dots on the sample function below the graph show the state transition epochs and show the new states entered. Note that the state at Sn is the new state entered, i.e., Xn , and the state remains Xn in the interval [Sn ,n+1 ). In what follows, we assume that the embedded Markov chain is irreducible and positiverecurrent. Define U (i) as the expected time in state i before a transition, i.e., X X U (i) = E [Un | Xn−1 = i] = Pij E [Un | Xn−1 = i, Xn = j ] = Pij U (i, j ). (5.61) j j The steady-state probabilities {πi } for the embedded chain tell us the fraction of transitions that enter any given state i. Since U (i) is the expected holding time in i per transition into i, we would guess that the fraction of time spent in state i should be proportional to πi U (i). Normalizing, we would guess that the time-average probability of being in state i should be X pi = πi U (i)/ πj U (j ). (5.62) j By now, it should be no surprise that renewal theory is the appropriate tool to make this precise. We continue to assume an irreducible positive-recu...
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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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