Sequences of non-decreasing (non-increasing) lower (upper) bounds for the renewal-type equation as well as for the renewal function which are improvements of the famous corresponding bounds of Marshal [(1973). Linear bounds on the renewal function. SIAM Journal on Applied Mathematics 24(2): 245–250] are given. Also, sequences such bounds converging to the ordinary renewal function are obtained for several reliability classes of the lifetime distributions of the inter-arrival times, which are refinements of all of the existing known corresponding bounds. For the first time, a lower bound for the renewal function with DMRL lifetimes is given. Finally, sequences of such improved bounds are given for the ordinary renewal density as well as for the right-tail of the distribution of the forward recurrence time.

The process of deterioration of repairable systems with each repair is modeled using converging geometric-type processes. It is proved that the expectation of the number of repairs in each interval of time is infinite. A new regularization procedure is suggested and the corresponding optimization problem is discussed.

Nonexponential asymptotics for solutions of two specific defective renewal equations are obtained. These include the special cases of asymptotics for a compound geometric distribution and the convolution of a compound geometric distribution with a distribution function. As applications of these results, we study the asymptotic behavior of the demographic birth rate of females, the perpetual put option in mathematics of finance, and the renewal function for terminating renewal processes.

Vinogradov (1973) used the Laplace transform to characterize the IFR class of life distributions and later Block and Savits (1980) extended the characterization to the main reliability classes. Here we use the same transform again to characterize the continuous time renewal equation and some properties of its solution.

A set-valued analog of the elementary renewal theorem for Minkowski sums of random closed sets is considered. The corresponding renewal function is defined as where are Minkowski (element-wise) sums of i.i.d. random compact convex sets. In this paper we determine the limit of H(tK)/t as t tends to infinity. For K containing the origin as an interior point, where hK(u) is the support function of K and is the set of all unit vectors u with EhA(u) > 0. Other set-valued generalizations of the renewal function are also suggested.

Age and block replacement policies are commonly used in order to reduce the number of in-service failures. The focus in this paper is on the block replacement policies, about which relatively less is known than age replacement policies. Several new results which connect the properties of block replacement policies with the properties of the corresponding renewal function and the excess lifetimes are obtained. Some applications and the relationships between these new results and some known results are included.

We use a simple coupling to prove the classical result that the renewal function U of a zero-delayed renewal process satisfies U(t) – λ . t→λ2μ2/2 as t→∞ if the life-length distribution is of non-lattice type and has finite first and second moments μ and μ2 respectively; λ is the renewal intensity, and is equal to 1/μ.

In a discrete-time renewal process {Nk, k = 0, 1, ·· ·}, let Zk and Ak be the forward recurrence time and the renewal age, respectively, at time k. In this paper, we prove that if the inter-renewal time distribution is discrete DFR (decreasing failure rate) then both {Ak, k = 0, 1, ·· ·} and {Zk, k = 0, 1, ·· ·} are monotonically non-decreasing in k in hazard rate ordering. Since the results can be transferred to the continuous-time case, and since the hazard rate ordering is stronger than the ordinary stochastic ordering, our results strengthen the corresponding results of Brown (1980). A sufficient condition for {Nk+m – Nk, k = 0, 1, ·· ·} to be non-increasing in k in hazard rate ordering as well as some sufficient conditions for the opposite monotonicity results are given. Finally, Brown's conjecture that DFR is necessary for concavity of the renewal function in the continuous-time case is discussed.

By using a shot noise process, general results on system size in continuous time are given both in transient state and in steady state with discussion on some interesting results concerning special cases. System size before arrivals is also discussed.

A characterization of the exponential distribution in the class of all distributions which are HNBUE or HNWUE is proved. An upper (a lower) bound is obtained on the renewal function of a renewal process when the underlying distribution is HNBUE (HNWUE).

A system is subject to shocks. Each shock weakens the system and makes it more expensive to run. It is desirable to determine a replacement time for the system. Boland and Proschan [4] consider periodic replacement of the system and give sufficient conditions for the existence of an optimal finite period, assuming that the shock process is a non-homogeneous Poisson process and the cost structure does not depend on time. Block et al. [3] establish similar results assuming that cost structure is time dependent, still requiring that the shock process is a non-homogeneous Poisson process. We show via a sample path argument that the results of [3] and [4] hold for any counting process whose jump size is of one unit magnitude.

In this study, an approximation to the solution of the renewal integral equation is constructed. Performance of the new method is evaluated and it is shown that the approximation provides an upper bound for the renewal function when the hazard function is a non-increasing function of time.

Block replacement policy, under which items are replaced at constant intervals of time and at failure, is wasteful because sometimes almost new items are also replaced. In order to eliminate this kind of waste, the following replacement policy is suggested.

(1) Items are exchanged for new items at time kT (k ··· 1, 2, …).

(2) If items fail in [(k – 1) T, kT – v), they are replaced by new items, and if in [kT – v, kT), they are replaced by used items.

The results of this ‘extended block replacement policy with used items' are studied for the Erlang lifetime distribution.