A note on sojourn time analysis of a two-stage queueing system

The paper considers an M/G/1-type, two-stage queueing system, in which the two stages in series are attended by a single server alternatively and exhaustively. A double transform for the stationary joint distribution of the queue length in each stage and the remaining service time is obtained. Using the double transform, Laplace–Stieltjes transforms of the total sojourn time distribution in the system and the sojourn time distributions in each stage are also provided. As a result, it is shown that the queueing system is a rare example among M/G/1-type, infinite-capacity queueing systems which have no distributional form of Little's law as studied by Keilson and Servi. Some comments are given on a previous total-sojourn-time analysis having an error due to correlation of the arrival process and the total sojourn time.