Sequences of Improved Two-Sided Bounds for the Renewal Function and the Solutions of Renewal-Type Equations

Renewal-type and renewal equations usually do not have analytical solutions, and hence bounds for the functions satisfying such equations have a great practical importance. In this paper, sequences of monotone non-decreasing general lower bounds and sequences of monotone non-increasing general upper bounds for a general renewal-type equation converging to the function under interest, are given. Similar sequences of such two-sided bounds are given for the renewal function of an ordinary renewal process which converge to the renewal function and are improvements of the famous corresponding bounds of Marshall (1973). Also, such sequences of bounds converging to the ordinary renewal function, are obtained for several reliability classes of the lifetime distributions of the inter-arrival times. Finally, sequences of such two-sided bounds are given for the ordinary renewal density as well as for the right-tail of the distribution of the forward recurrence time (excess lifetime).