This paper deals with the profit analysis of a single-server two-unit (one priority and the other ordinary) cold standby system with two modes of each unit normal and total failure. A switch is used to operate the standby unit (ordinary unit) and it works successfully at the time of need with known probability p (= 1 − q). The priority unit gets preference both for operation and repair. After working for a random length of time, the ordinary unit needs rest for a random length of time. System failure occurs when both the units fail totally. Identifying the system at suitable regenerative epochs, integral equations are set up for the probabilities of the system being in the ‘up’, ‘down’ or ‘failed’ state; to solve these equations the Laplace transform technique is adopted. Various reliability characteristics are obtained to carry out the cost-benefit analysis, and numerical results pertaining to a particular case are also presented.