While addressing some difficulties in the modeling of unsteady-state residence-time distribution (RTD), the care of perfectly mixed systems was considered. By an extension of the boundary condition domain, the traditional unfeasible concept of fresh and original populations was discarded to achieve a solution valid for all the fluid elements in a single vessel. This result enabled the development of the unsteady-state RTD for N vessels in series in the form of a joint probability density function. The relationship between the joint RTD and the combined RTD was also established. Numerical simulations were performed under various time varying flow patterns. The proposed solution can be applied to RTD-based models of chemical reactors operating under unsteady-state conditions. Potential applications in polymer reaction engineering are briefly outlined.