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The root wk having the largest modulus must be real and positive Why Denoting this root by , the asymptotic expansion (345) then follows Example 313 (continued) The GI /M/1 queue The Markov chain {Xn } describing the number of customers present just prior to arrival epochs satis es Condition B with r = 0 and s = 1, as directly follows from the one-step transition probabilities pij given in (312) The constants k are given by k =
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It is directly veri ed that 1 > 0 and 1 k= k k = 1 / < 0 Thus we can directly conclude from (3410) that the equilibrium probabilities j are of the form j for all j 0 for constants > 0 and 0 < < 1 The characteristic equation (349) coincides with the equation (3315) Next we give an application in which Condition A is used to establish the asymptotic expansion (345) Example 341 A discrete-time queueing model Messages arrive at a communication system according to a Poisson process with rate The messages are temporarily stored in a buffer which is assumed to have in nite capacity There are c transmission channels At xed clock times t = 0, 1, messages are taken out of the buffer and are synchronously transmitted Each channel can only transmit one message at a time The transmission time of a message is one time slot Transmission of messages can only start at the clock times t = 0, 1, It is assumed that < c, that is, the arrival rate of messages is less than the transmission capacity
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To analyse this queueing model, de ne the random variable Xn by Xn = the number of messages in the buffer (excluding any message in transmission) just prior to clock time t = n Then {Xn , n = 0, 1, } is a discrete-time Markov chain with the in nite state space I = {0, 1, } The one-step transition probabilities are given by pij = e pij = e j , j! 0 i < c and j = 0, 1, i c and j = i c, i c + 1,
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By the assumption < c the Markov chain can be shown to satisfy Assumption 331 Hence the equilibrium probabilities j , j = 0, 1, exist and are the unique solution to the equilibrium equations j = e
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in conjunction with the normalizing equation j =0 j = 1 Multiplying both sides j and summing over j , we nd of the equilibrium equation for j by z j =0 j j j c 1 j c+j k=c j =k c k=0
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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 the
116 equation
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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)