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

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 0, t. It is shown that if the interarrival time distribution has increasing mean residual life with mean μ, then the expected forward recurrence time is increasing in t 0, as is M(t) - t/μ. 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 0.