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Homewk 7

This document provides the homework assignment for Professor Whitt's IEOR 6711 Stochastic Models I course. The assignment includes solving 11 problems from Chapter 3 of Stochastic Processes by Sheldon Ross on renewal theory. The problems cover topics like renewal functions, renewal reward processes, and the key renewal theorem. Hints are provided for some of the problems.

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Songya Pan
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71 views2 pages

Homewk 7

This document provides the homework assignment for Professor Whitt's IEOR 6711 Stochastic Models I course. The assignment includes solving 11 problems from Chapter 3 of Stochastic Processes by Sheldon Ross on renewal theory. The problems cover topics like renewal functions, renewal reward processes, and the key renewal theorem. Hints are provided for some of the problems.

Uploaded by

Songya Pan
Copyright
© Attribution Non-Commercial (BY-NC)
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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IEOR 6711: Stochastic Models I Fall 2012, Professor Whitt Homework Assignment 7, Tuesday, October 16 Chapter 3: Renewal Theory

Due on Tuesday, October 23.


Problems from Chapter 3 of Stochastic Processes, second edition, by Sheldon Ross. Problem 3.12 (Hint: Just nd an appropriate function h.) Problem 3.13 Problem 3.14 Problem 3.15 Problem 3.16 Problem 3.17 (answer in back) Problem 3.18 Problem 3.22 (Hint: See Example 3.5A on page 125.) Problem 3.23 (Hint: See Example 3.5A on page 125.) Problem 3.21 (Hint: Use Example 3.5A plus Walds equation.) Problem 3.24 (answer in back) Problem 3.25 (Hints: (a) There are two standard approaches: The rst standard approach is to condition on the time of the rst renewal and uncondition. That produces a renewal equation (an integral equation) of the form
t

g (t) = h(t) +
0

g (t x) dF (x) ,

which we then show has the unique solution


t

g (t) = h(t) +
0

h(t x) dM (x) ,

where M is the renewal function associated with the cdf F , i.e.,


M (t) E [N (t)] =
n=1

P (Sn t) =
n=1

Fn (t) ,

and Fn is the cdf of X1 + + Xn , with Xi being IID with cdf F . For the second step, we can use Laplace transforms. In that step, observe that (s) M
0

esx M (x) dx

is not the same as m (s)


0

esx dM (x)

Indeed,

(s) = m M (s)/s.

The second approach is to condition on the time of the last renewal before time t and directly obtain the solution above, as is done in the proof of Lemma 3.4.3 on page 113. (b) Apply the key renewal theorem. Problem 3.27 (answer in back) Problem 3.28

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