Two-sided Bounds for some Quantities in the Delayed Renewal Process

In this paper we obtain some “general” two-sided bounds for the delayed renewal function, in the sense that the bounds are valid for any arbitrary distributions of the inter-arrival times. Also, we give a sequence of monotone non-decreasing (non-increasing) lower (upper) general bounds converging to the delayed renewal function. By considering several aging or reliability classes for the distribution of the interarrival times (e.g., \(DFR\), bounded mean residual lifetime, \(NBUE\), \(NWUE\), bounded failure rate, \(DMRL\), \(IMRL\)) we give upper and lower bounds for the delayed renewal function, and moreover by assuming the usual stochastic order between the first and the subsequent interarrival times, we give sequences of monotone non-decreasing (non-increasing) lower (upper) bounds converging to the delayed renewal function. Also, some sequences of bounds for the delayed renewal function in terms of the ordinary renewal function are given. Sequences of monotone non-decreasing (non-increasing) lower (upper) bounds for the delayed renewal density are also given. Finally, we obtain upper and lower bounds for the expected number of renewals over a finite interval, and as a result, we get an improvement of the upper bounds obtained by Lorden (Ann Math Statist 41:520–527, 1970) and Losidis and Politis (2022) for the expected number of renewals over a finite interval under the ordinary renewal process.