In this paper the classic tandem queueing problem is examined. An exact procedure is developed for the analysis of a production line consisting of single machines linked in series with no intermediate buffers between them. Arrivals occur only at the first queue, which is assumed infinite, according to a Poisson distribution. All processing times and interarrival times are assumed to be exponentially distributed. Departures from the system may only occur from the last machine. The feedback case is also included. The exact algorithm gives the marginal probability distribution of the number of units in each machine, the mean queue length and the critical input rate, i.e., the throughput of the line. The basic step in the algorithm is the evaluation of the unique positive solution of a quadratic matrix equation using an iterative scheme given by Evans, which has proved faster than that one used by Latouché and Neuts. Results are given for production lines consisting of two, three, four, five six, seven and eight stations. For the last two cases, viz, K = 7 and K =8, results were obtained from a Floating Point System (FPS).