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Unformatted text preview: INUOUS TIME MARKOV CHAINS Example 4.18. M/M/1 queue has jump rates q (i, i + 1) = for i 0 and q (i, i 1) = µ for n 1. The embedded chain has r(0, 1) = 1 and for i 1 r(i, i + 1) = +µ r(i, i µ +µ 1) = From this we see that the embedded chain is a random walk, so the probabilities Pi (VN < V0 ) are the same as those computed in (1.17) and (1.22). Using this with results in Section 1.10 we see that the chain is positive recurrent null recurrent transient if if if E0 T 0 < 1 E0 T 0 = 1 P0 (T0 < 1) < 1 <µ =µ >µ Example 4.19. Branching process has jump rates q (i, i + 1) = i and q (i, i 1) = µi. 0 is an absorbing state but for i 1 the i’s cancel and we have r(i, i + 1) = +µ Thus absorption at 0 is certain if probability of avoiding extinction is r(i, i µ +µ 1) = µ but if P1 (T0 = 1) = 1 > µ then by (1.23) the µ For another derivation let ⇢ = P1 (T0 < 1). By considering what happens when the chain leaves 0 we have ⇢= µ ·1+ +µ +µ · ⇢2 since starting from state 2 extinction occurs if and only if each individual’s family line dies out. Rearranging gives 0 = ⇢2 ( + µ)⇢ + µ = ( ⇢ )(⇢ µ/ ) T...
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