Queuing system with working breakdowns has a wide variety of applications such as in computer network, manufacturing industries and transportation systems. It is obvious that server unreliability makes negative impact on performance of any queuing system, and undoubtedly, a reliable server with smooth services has a positive impact on the economic activities of such systems. Therefore, in order to maintain a desired level of service in any queuing system, performance modeling is a first step to providing a solution. Using a set of differential equations, we modeled a single server Markovian queuing system with working breakdowns and recovery policies. The time-dependent solution was obtained using a numerical scheme which was implemented in Matlab. Further, the performance measures were analyzed to understand the sensitivity of the parameters on the model solution. The experience I had during the post-bachelor program was indeed awesome as I was exposed to learn many new things which have added to my academic skills. My special thanks goes to Dr. Michael Winckler for his thorough supervision, and to the HGS MathComp, for providing the Post-bachelor learning platform.