Discrete-time stochastic processes

For the nite state case 632 in matrix notation is dp

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Unformatted text preview: ③ ♥ 2 ② ∏+µ ∏/(∏+µ) µ/(∏+µ) ③ ♥ 3 ∏+µ ... Figure 6.1: The Markov process for an M/M/1 queue. Each node i is labeled with its corresponding rate ∫i to the next transition, and each transition is labeled with the corresponding transition probability in the embedded Markov chain. The embedded Markov chain is a Birth-death Markov chain, and its steady state probabil- 6.1. INTRODUCTION 237 ities can be calculated easily using (5.27). The result is π0 = πn = 1−ρ 2 1 − ρ2 n−1 ρ 2 where ρ = for n ≥ 1. ∏ µ (6.4) Note that if ∏ << µ, then π0 and π1 are each close to 1/2, whereas because of the large holding time in state 0, the process spends most of its time in state 0 waiting for arrivals. We will shortly learn how to treat the expected time in a state as opposed to the expected fraction of transitions to that state, but for now we return to the general study of Markov processes. The evolution of a Markov process can be visualized in several ways. First, assume the process starts in a known state X0 = i at time 0. The next state X1 is determined by the probabilities {Pij ; j ≥ 0} of the embedded Markov chain, and the interval U1 is independently determined by the exponential distribution with rate ∫i . Given that X1 = j , the next state X2 and next interval U2 are independently determined by {Pj k ; k ≥ 0} and ∫j respectively. Subsequent transitions and intervals evolve in the same way. For a second viewpoint, suppose an independent Poisson process of rate ∫i is associated with each state i. When the Markov process enters a given state i, the next transition occurs at the next arrival epoch in the Poisson process for state i. At that epoch, a new state is chosen according to the transition probabilities Pij . Since the choice of next state, given state i, is independent of the interval in state i, this view describes the same process as the first view. For a third visualization, suppose, for each pair of states i and j , that an independent Poisson process of rate ∫i Pij is associated with a possible transition to j conditional on being in i. When the Markov process enters a given state i, both the time of the next transition and the choice of the next state are determined by the set of i to j Poisson processes over all possible next states j . The transition occurs at the epoch of the first arrival, for the given i, to any of the i to j processes, and the next state is the j for which that first arrival occurred. Since such a collection of Poisson processes is equivalent to a single process of rate ∫i followed by an independent selection according to the transition probabilities Pij , this view again describes the same process as the other views. It is convenient in this visualization to define the rate from any state i to any other state j as qij = ∫i Pij . If we sum over j , we see that ∫i and Pij can be expressed in terms of qij for each i, j as X ∫i = qij ; Pij = qij /∫i . (6.5) j This means that the fundamental characterization of the Markov process in terms of the rates ∫i and the embedded chain transition probabilities Pij can be replaced by a characterization in terms of the set of transition rates qij . In many cases, this is a more natural 238 CHAPTER 6. MARKOV PROCESSES WITH COUNTABLE STATE SPACES approach. For the M/M/1 queue, for example, qi,i+1 is simply the arrival rate ∏. Similarly, for i > 0, qi,i−1 = µ is the departure rate when there are customers to be served. Figure 6.2 shows Figure 6.1 incorporating this notational simplification ♥ 0 ② ∏ µ ③ ♥ 1 ② ∏ µ ③ ♥ 2 ② ∏ µ ③ ♥ 3 ... Figure 6.2: The Markov process for an M/M/1 queue. Each transition (i, j ) is labelled with the transition rate qij in the embedded Markov chain. Note that the interarrival density for the Poisson process from i to a given j is qij exp(−qij x). On the other hand, given that the process is in state i, the probability density for the interval until the next arrival, whether conditioned on an arrival1 to j or not, is ∫i exp(−∫i x). 6.1.1 The sampled-time approximation to a Markov process As yet another way to visualize a Markov process, consider approximating the process by viewing it only at times separated by a given increment size δ . The Poisson processes above are then approximated by Bernoulli processes where the transition probability from i to j in the sampled-time chain is defined to be qij δ for all j 6= i. The Markov process is then approximated by a Markov chain and self-transition probabilities from i to i are required for those time increments in which no transition occurs. P Thus, as illustrated in Figure 6.3, we have Pii = 1 − j qij δ = 1 − ∫i δ for each i. Note that this is an approximation to the Markov process in two ways. First, transitions occur only at integer multiples of the increment δ , and second, qij δ is an approximation to Pr {X (δ )=j | X (0)=i}. From (6.3), Pr {X (δ )...
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