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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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 Spring '10
 DURRETT
 The Land

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