For studying and modeling the time to failure of a system or component, many reliability practitioners used the hazard rate and its monotone behaviors. But, nowadays, there are two problems, first, the modern components have high reliability and second their distributions are usually have non-monotone hazard rate, such as, the truncated normal, the Burr XII, and the inverse Gaussian distributions. So, modeling these data based on the hazard rate models seems to be too stringent. Zimmer et al. (1998) and Wang et al. (2003, 2008) introduced and studied a new time to failure model in continuous distributions based on log-odds rate (LOR) which is comparable to the model based on the hazard rate. There are many components and devices in industry, that have discrete distributions with non-monotone hazard rate, so, in this paper, we introduce the discrete log-odds rate which is different from its analogue in continuous case. Also, an alternative discrete reversed hazard rate which we called it the {\\\\it second reversed rate of failure} in discrete times is also defined here. It is shown that the failure time distributions can be characterized by the discrete LOR. Moreover, we show that the discrete logistic and log logistics distributions have property of a constant discrete LOR with respect to $t$ and $\\\\ln t$ respectively. Furthermore, properties of some distributions with monotone discrete LOR such as, the discrete Burr XII, the discrete Weibull, and the discrete truncated normal are obtained.

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