## The Annals of Probability

### Bounds, Inequalities, and Monotonicity Properties for Some Specialized Renewal Processes

Mark Brown

#### Abstract

Renewal processes with increasing mean residual life and decreasing failure rate interarrival time distributions are investigated. Various two-sided bounds are obtained for $M(t)$, the expected number of renewals in $\lbrack 0, t\rbrack$. It is shown that if the interarrival time distribution has increasing mean residual life with mean $\mu$, then the expected forward recurrence time is increasing in $t \geqslant 0$, as is $M(t) - t/\mu$. If the interarrival time distribution has decreasing failure rate then $M(t)$ is concave, and the forward and backward recurrence time distributions are stochastically increasing in $t \geqslant 0$.

#### Article information

Source
Ann. Probab. Volume 8, Number 2 (1980), 227-240.

Dates
First available in Project Euclid: 19 April 2007

http://projecteuclid.org/euclid.aop/1176994773

Digital Object Identifier
doi:10.1214/aop/1176994773

Mathematical Reviews number (MathSciNet)
MR566590

Zentralblatt MATH identifier
0429.60084

JSTOR