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depends on the initial state i The reason is that in this Markov chain example there are two disjoint closed sets of states De nition 331 A non-empty set C of states is said to be closed if / pij = 0 for i C and j C, that is, the process cannot leave the set C once the process is in the set C For a nite-state Markov chain having no two disjoint closed sets it is proved in Theorem 357 that fij = 1 for all i I when j is a recurrent state For such (k) a Markov chain it then follows from (332) that limn (1/n) n pij does k=1 not depend on the initial state i when j is recurrent This statement is also true for a transient state j , since then the limit is always equal to 0 for all i I by Lemma 323 For the case of an in nite-state Markov chain, however, the situation is more complex That is why we make the following assumption Assumption 331 The Markov chain {Xn } has some state r such that fir = 1 for all i I and rr < In other words, the Markov chain has a regeneration state r that is ultimately reached from each initial state with probability 1 and the number of steps needed to return from state r to itself has a nite expectation The assumption is satis ed in most practical applications For a nite-state Markov chain the Assumption 331 is automatically satis ed when the Markov chain has no two disjoint closed sets; see Theorem 357 The state r from Assumption 331 is a positive recurrent state Assumption 331 implies that the set of recurrent states is not empty and that there is a single closed set of recurrent states Moreover, by Lemma 358 we have for any recurrent state j that fij = 1 for all i I and jj < Summarizing, under Assumption 331 we have both for a nite-state and an in nite-state Markov chain (k) that limn (1/n) n pij does not depend on the initial state i for all j I k=1 In the next subsection it will be seen that the Cesaro limits give the equilibrium distribution of the Markov chain 332 The Equilibrium Equations We rst give an important de nition for a Markov chain {Xn } with state space I and one-step transition probabilities pij , i, j I De nition 332 A probability distribution { j , j I } is said to be an equilibrium distribution for the Markov chain {Xn } if j =.

k pkj ,

(335)

An explanation of the term equilibrium distribution is as follows Suppose that the initial state of the process {Xn } is chosen according to P {X0 = j } = j , Then, for each n = 1, 2, , P {Xn = j } = j , j I j I

Event report GPRS (ERG) Furnish charging information GPRS (FCI-GPRS) Initial DP GPRS (IDP-GPRS) Release GPRS (REL-GPRS)

In other words, starting the process according to the equilibrium distribution leads to a process that operates in an equilibrium mode The proof is simple and is based on induction Suppose that P {Xm = j } = j , j I for some m 0 Then P {Xm+1 = j } =

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