Wireless Sensor Network (WSN) paradigm is an integral component of ubiquitous computing and Machine-to-Machine communications. Since WSNs are widely used in homeland security, military applications, next generation power lines, critical infrastructure monitoring and smart spaces, they are naturally attractive to the adversaries and vulnerable to natural conditions because of their harsh topologies. Although, there are some solutions against Denial of Service (DoS) attacks conducted against single or multiple sensor nodes in WSNs, WSNs are, at best, weakly defended against more sophisticated attack types. Therefore, the period that the sensor network will stand out against such attacks has a crucial importance to calculate intervention or backup times for WSNs. In this study, we propose a linear programming (LP) approach for modeling the impact of single node attacks on network lifetime in WSNs. We explored the parameter space through the numerical evaluations of the LP model to quantify the impact of elimination of the most critical node on WSN lifetime.