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Unformatted text preview: he root we want is µ/ < 1.
As the last two examples show, if we work with the embedded chain then
we can use the approach of Section 1.8 to compute exit distributions. We
can also work directly with the Qmatrix. Let VA = min{t : Xt 2 A} and
h(i) = Pi (X (TA ) = a). Then h(a) = 1, h(b) = 0 for b 2 A {a}, and for i 62 A
h(i) = X q (i, j )
j 6=i Multiplying each side by i Q(i, i) we have
X
Q(i, i)h(i) =
Q(i, j )h(j )
i = j 6=i which simpliﬁes to X
j Q(i, j )h(j ) = 0 for i 62 A. (4.19) Turning now to hitting times, we work the ﬁrst two examples using the
embedded chain: 4.4. EXIT DISTRIBUTIONS AND HITTING TIMES 135 Example 4.20. M/M/1 queue. This is particularly simple because the the
time in each state i > 0 is expoenential with rate + µ so the new result follows
from the one in discrete time given in (1.28)
E1 T 0 = 1
+µ
1
·
=
+µ µ
µ (4.20) Example 4.21. Barbershop chain. (continuation of Example 4.14) The
transition rates are
q (i, i 1) = 3 for i = 1, 2, 3 q (i, i + 1) = 2 for i = 0, 1, 2 so the embedded chain is
0
1
2
3 0
0
3/...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).
 Spring '10
 DURRETT
 The Land

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