Renewal theory

A branch of probability theory describing a large range of problems connected with the rejection and renewal of the elements of some system. The principal concepts in renewal theory are those of a renewal process and renewal equation. A renewal process may be described using the classical scheme of sums of independent random variables in the following manner. Let $ \xi _ {1} , \xi _ {2} , . . . $ be a sequence of independent, non-negative, identically-distributed random variables having distribution function $ F( x) $. Let $ \zeta _ {0} = 0 $, $ \zeta _ {n} = \xi _ {1} + \dots + \xi _ {n} $, $ n \geq 1 $. The renewal process $ N _ {t} $ is defined as

$$ \tag{1 } N _ {t} = \max \{ {n } : {\zeta _ {n} \leq t } \} . $$

If the $ \xi _ {i} $ are interpreted as the lifetimes of some successively replaceable element, the random variable $ N _ {i} $ equals the number of replacements (or renewals) of these elements during time $ t $. The renewal function $ H( t) = {\mathsf E} N _ {t} $ plays an important part in studying $ N _ {t} $. This function satisfies the renewal equation

$$ \tag{2 } H ( t) = \ F ( t) + \int\limits _ { 0 } ^ { t } H ( t - u) dF ( u). $$

For $ F( t) = 1 - e ^ {- \rho t } $, $ t \geq 0 $, there arises an important special case of the renewal process — the Poisson process, in which

$$ {\mathsf P} \{ N ( t) = k \} = \frac{( \rho t) ^ {k} }{k! } e ^ {- \rho t } ,\ \ k = 0, 1 \dots $$

and $ H( t) = \rho t $.

The renewal process $ N _ {t} $ and the renewal equation (2) are very important in the study of various theoretical and applied problems in queueing theory, reliability theory, storage theory, the theory of branching processes (cf. Branching process), etc. A large number of results in renewal theory is connected with the study of the asymptotic properties of the renewal function $ H( t) $ as $ t \rightarrow \infty $. An elementary renewal theorem states that

$$ \tag{3 } \lim\limits _ {t \rightarrow \infty } \ \frac{H ( t) }{t } = { \frac{1}{m} } , $$

where $ m = {\mathsf E} \xi _ {i} $. It was shown by D. Blackwell in 1948 (cf. [1]), that if the distribution $ \xi _ {i} $ is not concentrated on some arithmetical lattice of the type $ \{ 0, d, 2d , . . . \} $, $ d > 0 $, then for any $ h > 0 $,

$$ \tag{4 } \lim\limits _ {t \rightarrow \infty } \ [ H ( t + h) - H ( t)] = { \frac{h}{m} } . $$

Numerous results are available which generalize and precisize equations (3) and (4) in several respects. Results of the kind presented in (3) and (4) are used to study the asymptotic behaviour of the solution $ X( t) $ of the renewal-type equation

$$ X ( t) = \ K ( t) + \int\limits _ { 0 } ^ { t } X ( t - u) dF ( u), $$

in which the free term $ K( t) $ is some function other than $ F( t) $ which meets certain conditions.

The relation

$$ \tag{5 } {\mathsf P} \{ N _ {t} \geq n \} = {\mathsf P} \{ \zeta _ {n} \leq t \} $$

follows from definition (1). Since limit theorems for sums $ \zeta _ {n} $ of independent terms have been thoroughly studied, relation (5) makes it possible to obtain limit theorems for the number of renewals $ N _ {t} $.

There exists a large number of generalizations of the scheme just described. One such generalization, connected with semi-Markov processes (cf. Semi-Markov process), yields the so-called Markov renewal process, in which the system has some number of states, and the lifetimes of the individual elements are random elements which depend on the states of the system before and after the moment of renewal.

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