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j k+c zj k+c (j k + c)! k z k = e (1 z) k + k=0 k=c c 1 k=0 k zk c e = e (1 z) This gives j =0 k + z c k z k c 1 k=0 j z = e (1 z)c 1 k=0 z c z k k zc e (1 z)|z| 1 The generating function j =0 j zj is the ratio of two functions N (z) and D(z) Both functions can be analytically continued to the whole complex plane The denominator D(z) is indeed a nice function in an explicit form (the function N (z) involves the unknowns 0 , , c 1 ) Denote by x0 the unique solution of the116 equation DISCRETE-TIME MARKOV CHAINS x c e (1 x) = 0 on the interval (1, ) and let = 1/x0 Then it can be veri ed from Theorem C1 in Appendix C that j j as j for some constant > 0 Thus the geometric approach enables us to compute the j by solving a nite and relatively small system of linear equations 343 Metropolis Hastings Algorithm In the context of stochastic networks, we will encounter in 5 Markov chains with a multidimensional state space and having the feature that the equilibrium probabilities are known up to a multiplicative constant However, the number of possible states is enormous so that a direct calculation of the normalization constant is not practically feasible This raises the following question Suppose that 1 , , N are given positive numbers with a nite sum S = N i How do i=1 we construct a Markov chain whose equilibrium probabilities are given by j /S for j = 1, , N For ease of presentation, we restrict ourselves to N < To answer the question, we need the concept of a reversible Markov chain Let {Xn } be a Markov chain with a nite state space I and one-step transition probabilities pij It is assumed that {Xn } has no two disjoint closed sets Then the Markov chain has a unique equilibrium distribution { j } Assume now that a non-null vector (gj ), j I exists such that gj pjk = gk pkj , Then, for some constant c = 0, gj = c j (3412) j, k I (3411)