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Unformatted text preview: 4. CHAPTER SUMMARY 3.4 Chapter Summary This chapter shows the power of the law of large numbers to give simple derivations of useful results. The work horse of the chapter is the result for renewal
reward processes. If the times between renewals and the rewards earned in these
periods (ti , ri ) are an i.i.d. sequence then the limiting rate at which rewards are
earned is Eri /Eti . Taking ri = 1 this reduces to the law of large numbers
(Theorem 3.1) for the renewal process
N (t)/t ! 1/Eti If the ti = si + ui with the (si , ui ) are i.i.d. representing the times in states 1
and 2 then taking ri = si we get see that the limiting fraction of time in state
1 is
Esi /(Esi + Eui )
our result (Theorem 3.4) for alternating renewal processes. Other applications
of renewal reward processes gave us results for the limiting behavior of the age
and residual life in Section 3.3.
A second theme here was the simple minded scheme of computing costs
two di↵erent ways to prove quantities were equal. For the GI/G/1...
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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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