# When a breakdown occurs all of the people leave since

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Unformatted text preview: i 1 , xi ) &gt; 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 ) &gt; 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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