Journal of Mathematical Sciences, Vol. 138, No. 1, 2006 ASYMPTOTIC RESULTS FOR A RUN AND CUMULATIVE MIXED SHOCK MODEL F. Mallor, 1 E. Omey, 2 and J. Santos 1 UDC 519.2 1. Introduction Several system reliability models subject to shocks can be found in the literature. Generally, the models studied assume that the shocks arrive at random times, causing damages of random magnitude, and that the system fails in a specific way. For instance, in the cumulative shock model, studied in [8], the system fails as soon as the sum of the random damages exceeds some prefixed level. The same authors also study in [7] a shock model where the failure was due to an extreme shock. Gut, in [2], combines the previous two models to obtain several asymptotic results where the system breaks down as soon as one of the preceding types of failure occurs. In [5], the run shock model is introduced: the system fails when a “large” run of extreme shocks occurs. We point out that this model turned out to be an extension of the extreme shock model of Shanthikumar and Sumita (see [7]). The aim of this work is, following [2], to study the mixed-shock-model result of the combination of the new run shock model and the cumulative shock model. The paper is organized as follows. In Sec. 2, the involved shock models are presented and, finally, the asymptotic behavior for the mixed one is studied in Sec. 3. 2. Classic Shock Models In the classic shock models, the general setup is a family {(X n ,Y n )} ∞ n=1 of independent two-dimensional vectors identically distributed as the nonnegative vector (X, Y ), where X n represents the magnitude of the nth shock and Y n represents the time between the (n − 1)st and the nth shock. The quantities of interest are the numbers of shocks until failure N , the damage accumulated by the system S N = ∑ N i=1 X i , and the time up to failure T N = ∑ N i=1 Y i . In the following subsections, we will recall the different ways of failure. 2.1. Cumulative shock model. Let ν (z) denote the first passage time associated with the sequence of indepen- dent identically distributed (i.i.d.) random variables {X i } ∞ i=1 , that is, ν (z) = min  n: S n = n  i=1 X i >z  . Note that ν (z) a.s. −→ +∞ as z → +∞. Well-known results for the asymptotic behavior of ν (z) and the stopped sums S ν(z) and T ν(z) , assuming that μ X = E[X 1 ] > 0 and that μ Y = E[Y 1 ] exists and is finite, are as follows: ν (z) z a.s. −→ 1 μ X as z → +∞, (1) S ν(z) z a.s. −→ 1 as z →∞, T ν(z) z a.s. −→ μ Y μ X as z → +∞. 2.2. Run shock model. This model, introduced in [5] and extended in [6], assumes that the first run of a prefixed length k of critical shocks causes the system failure, where a critical shock is defined by a region R q ⊆ R, with q =1 − p = P{X ∈ R q } > 0. That is, the system lifetime T τ (q,k) is given by the stopping time τ (q,k): τ (q,k) = min{n: X n ,X n-1 ,...,X n-k+1 ∈ R q }, T τ (q,k) = τ (q,k)  n=1 Y n . Proceedings of the Seminar on Stability Problems for Stochastic Models, Pamplona, Spain, 2003, Part III. 5410 1072-3374/06/1381-5410 c  2006 Springer Science+Business Media, Inc.