Pessimistic approximations of the reliability for regenerative models (Q2758440)

Let \((X_n)_{n\geq 1}\) be a sequence of independent and identically distributed positive random variables. Furthermore, let \((Y_n^+)_{n\geq 1}\) be another sequence of independent and identically distributed positive random variables. Finally, let \(Y^-\) be a positive random variable, and let \(\vartheta\) be a geometric random variable. Assume that all these random variables are independent. Consider the three random sums: NEWLINE\[NEWLINE W_0=\sum_{n=1}^\vartheta X_n,\quad W_1=\sum_{n=1}^{\vartheta-1}Y_n^++Y^-,\quad W=W_0+W_1. NEWLINE\]NEWLINE The authors study some models in reliability theory where \(W_0\) is often used to approximate \(W\). They obtain, under various conditions, upper bounds on \(P\{W>t\}-P\{W_0>t\}\).