T eti 32 proof multiplying and dividing by n t we

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Unformatted text preview: b burns out, and N (t) = max{n : Tn t} is the number of light bulbs that have been replaced by time t. The picture is the same as the one for the Poisson process, see Figure 2.1. If renewal theory were only about changing light bulbs, it would not be a very useful subject. The reason for our interest in this system is that it captures the essence of a number of di↵erent situations. On example that we have already seen is Example 3.1. Markov chains. Let Xn be a Markov chain and suppose that X0 = x. Let Tn be the nth time that the process returns to x. The strong Markov property implies that tn = Tn Tn 1 are independent, so Tn is a renewal process. Example 3.2. Machine repair. Instead of a light bulb, think of a machine that works for an amount of time si before it fails, requiring an amount of time ui to be repaired. Let ti = si + ui be the length of the ith cycle of breakdown 101 102 CHAPTER 3. RENEWAL PROCESSES and repair. If we assume that the repair leaves the machine in a “like new” condition, then the ti are independent and identically distributed (i.i.d.) and a renewal process results. Example 3.3. Counter processes. The following situation arises, for example, in medical imaging applications. Particles arrive at a counter at...
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