Ageing properties of certain dependent geometric sums

Abstract

We study some distribution properties of a random sum of i.i.d. non-negative random variables, where the number of terms is geometrically distributed and not independent of the summands. The results are applied to the system failure time of a one-unit system with a single spare and repair facility. In such a system when the operating unit fails it is immediately replaced by the spare and sent to the repair facility. The system continues operating until the first time when the failed unit has not yet been repaired by the failure of the operating unit. Certain ageing properties such as NBU, NWU, NBUE, NWUE, HNBUE, HNWUE, L+ and L– are shown to be inheritable from the working time of the operating unit to the system lifetime.

On max-sum equivalence and convolution closure of heavy-tailed distributions and their applications

“…This compound geometric convolution is also an interesting distribution arising in many applied probability models such as reliability, queueing, risk theory; see, for example, Brown (1990), Cai and Garrido (2002), Gertsbakh (1984), Kováts and Móri (1992), Willmot and Lin (1996), and references therein. A special form of (3.1) often appearing in ruin theory and queueing is where F itself is a convolution of the distribution G with a distribution H supported on [0, ∞), i.e.…”

Monotonicity and aging properties of random sums

“…Brown (1990) showed that a geometric sum S N ¼ X 1 þ Á Á Á þ X N always has the NWU aging property whatever the distribution of X 1 is, namely if N is a geometric random variable, then PrfS N 4x þ y j S N 4xgX PrfS N 4yg; xX0; yX0 always holds whatever the distribution of X 1 is. Other distributional properties of a geometric sum can be found in Bon andKalashnikov (2001), Gut (2003), Kova´ts and Mo´ri (1992), Li (2004), Lin and Stoyanov (2002), Willmot (2002), Willmot andCai (2001, 2004), and references therein.…”