Introduction to Renewal Theory

Moshe Haviv
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Moshe Haviv: The Hebrew University

Chapter Chapter 2 in Queues, 2013, pp 21-36 from Springer

Abstract: Abstract Let {X i } i = 1 ∞ be a series of independent and identically distributed nonnegative random variables. Assume they are continuous. In particular, there exists some density function f X (x), x≥0, such that $$F_{X}(x) \equiv \mathrm{P}(X_{i} \leq x) =\int _{ t=0}^{x}f_{X}(t)\,dt$$ , i ≥ 1. Imagine X i representing the life span of a lightbulb. Specifically, there are infinitely many lightbulbs in stock. At time t = 0, the first among them is placed. It burns out after a (random) time of X 1. Then it is replaced by a fresh lightbulb that itself is replaced after an additional (random) time of X 2, etc. Note that whenever a new lightbulb is placed all statistically starts afresh. Let $$S_{n} = \Sigma _{i=1}^{n}X_{i}$$ , n ≥ 1, and set S 0 = 0. Of course, $$S_{n+1} = S_{n} + X_{n+1}$$ , n ≥0.

Keywords: Lightbulb; Erlang Random Variables; Total Life Span; Limiting Joint Distribution; Inspection Paradox (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1007/978-1-4614-6765-6_2

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