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f (Xk ) =

for any given function f Thus the Metropolis Hastings algorithm can be used to nd performance measures of the Markov chain {Xn } such as the long-run average cost per time unit when a cost structure is imposed on the Markov chain The most widely used version of the Metropolis Hastings algorithm is the Gibbs sampler Suppose that (N1 , , Nd ) is a d-dimensional stochastic vector whose probability distribution p(x1 , , xd ) = P {N1 = x1 , , Nd = xd } is known up to a multiplicative constant This situation will be encountered in Section 56 in the context of a closed queueing network In this particular application the univariate conditional distribution P {Nk = xk |Nj = xj for j = 1, , d with j = k} (3417)

.

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is explicitly known for each k = 1, , d In order to apply the Gibbs sampler, it is required that the univariate conditional distributions in (3417) are known The Gibbs sampler generates a sequence of successive states (x1 , , xd ) from a Markov chain whose equilibrium distribution is given by p(x1 , , xd ) Gibbs sampler Step 0 Choose an initial state x = (x1 , , xd ) Step 1 For the current state x choose a coordinate which is equally likely to be any of the coordinates 1, , d If coordinate k is chosen, then generate a random variable Y whose probability distribution is given by P {Y = y} = P {Xk = y|Xj = xj for j = 1, , d with j = k} If Y = y, let the candidate state y = (x1 , , xk 1 , y, xk+1 , , xd ) Step 2 The next state x = (x1 , , xd ) is set equal to y Repeat step 1 with this new state x The Gibbs sampler uses the Metropolis Hastings algorithm with the choice 1 P {Xk = y|Xj = xj for j = 1, , d with j = k} d for the Markov matrix M It is not dif cult to verify that for this choice the acceptance probability x,y is given by mx,y = x,y = min p(y)p(x) , 1 = 1 p(x)p(y)

In this section we give some background material First the state classi cation of Markov chains is discussed Next we prove the results that were used earlier in the analysis of the long-run behaviour of Markov chains 351 State Classi cation The concepts of a transient state and a recurrent state were introduced in Section 32 and the following lemma was proved for the Markov chain {Xn } Lemma 351 A state i is transient only if (n) rent only if pii = n=1

< and a state i is recur-

To analyse the transient states and recurrent states in more detail, we need the concept of accessibility

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