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Unformatted text preview: i 1 , xi ) > 0 for 1 i n, we have
n
Y i=1 q (xi 1 , xi ) = n
Y q (xi , xi 1) i=1 (a) Show that if q has a stationary distribution that satisﬁes the detailed balance
condition, then the cycle condition holds. (b) To prove the converse, suppose
that the cycle condition holds. Let a 2 S and set ⇡ (a) = c. For b 6= a in S let
x0 = a, x1 . . . xk = b be a path from a to b with q (xi 1 , xi ) > 0 for 1 i k let
k
Y q (xi 1 , xi )
⇡ (b) =
q (xi , xi 1 )
j =1 Show that ⇡ (b) is well deﬁned, i.e., is independent of the path chosen. Then
conclude that ⇡ satisﬁes the detailed balance condition.
Hitting times and exit distributions
4.25. Consider the salesman from Problem 4.1. She just left Atlanta. (a) What
is the expected time until she returns to Atlanta? (b) Find the answer to (a)
by computing the stationary distribution.
4.26. Consider the two queues in series in Problem 4.8. (a) Use the methods
of Section 4.4 to compute the expected duration of a busy period. (b) calculate
this from the stationary d...
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 Spring '10
 DURRETT
 The Land

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