The interchangeability of tandem queues with heterogeneous customers and dependent service times

Consider m queueing stations in tandem, with infinite buffers between stations, all initially empty, and an arbitrary arrival process at the first station. The service time of customer j at station i is geometrically distributed with parameter pi, but this is conditioned on the fact that the sum of the m service times for customer j is cj. Service times of distinct customers are independent. We show that for any arrival process to the first station the departure process from the last station is statistically unaltered by interchanging any of the pi's. This remains true for two stations in tandem even if there is only a buffer of finite size between them. The well-known interchangeability of ·/M/1 queues is a special case of this result. Other special cases provide interesting new results.