Preservation of Unimodality under Shock Models Mini N. Balu, S.V. Sabnis Department of Mathematics, Indian Institute of Technology, Bombay, India Received June 1996; accepted June 15, 1999 Abstract: A number of results pertaining to preservation of aging properties (IFR, IFRA etc.) under various shock models are available in the literature. Our aim in this paper is to examine in the same spirit, the preservation of unimodality under various shock models. For example, it is proved that in a non-homogeneous Poisson shock model if {p k } k≥0 , the sequence of probabilities with which the device fails on the kth shock, is unimodal then under some suitable conditions on the mean value function Λ(t), the corresponding survival function is also unimodal. The other shock models under which the preservation of unimodality is considered in this paper are pure birth shock model and a more general shock model in which shocks occur according to a general counting process. c  1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 952–957, 1999 Keywords: unimodality; shock model; Poisson process; nonhomogeneous Poisson process; pure birth process; variation diminishing property; total positivity 1. INTRODUCTION Suppose that a device is subjected to shocks which occur according to a counting process {N (t),t ≥ 0}. Let { ¯ P k } k≥0 be a sequence in which the kth term ¯ P k denotes the probability of surviving k shocks. Assume that 1= ¯ P 0 ≥ ¯ P 1 ≥ ¯ P 2 ···. Then the survival probability ¯ H(t) of a device is given by ¯ H(t)= ∞  k=0 P [N (t)= k] ¯ P k . (1) If F is the common distribution of independent interarrival times of the process {N (t),t ≥ 0}, then ¯ H(t) can also be written as ¯ H(t)= ∞  k=0 ¯ F (k+1) (t)p k+1 , (2) where ¯ F (k) denotes the k-fold convolution of F and p k = ¯ P k-1 - ¯ P k ,k ≥ 1. First, Esary et al. [6] showed that the discrete aging properties like IFR, IFRA, NBU, and NBUE are preserved in the continuous form by the survival function ¯ H(t) when {N (t),t ≥ 0} is Correspondence to: S.V. Sabnis c  1999 John Wiley & Sons, Inc. CCC 0894-069X/99/080952-06