A modified two-moment (MTM) method is proposed for waiting-time analysis of constant-service tandem queueing systems. Input to an intermediate queueing station consists of internal stream(s) provided by the output from previous queue(s) in addition to external Poisson traffic. Internal traffic at a queue is assumed to have initially originated from external Poisson sources and is permitted to exit from the tandem path with fixed independent probability. Analysis of such complex configuration is not tractable and known exact results are restricted to very special cases. It is shown that the MTM method provides a systematic and more accurate tool for waiting-time analysis of such queueing structures as compared to other known approximations.