If I have two tandem queues where customer arrive at the 1st queue at poisson process of rate lamba and wait until there turn for service, server 1 then completes his service in Exp(mu1) amount of time (independent). The customer then proceeds to the second where he waits his turn and then is served by server 2 with a service time Exp(mu2) independent of everything. The customer then leaves the network.

If the tandem queue lengths are in equilibrium when the customer arrives why is it that his waiting times in queue 1 and 2 (this is queuing time plus service time) are independent while his queuing times in each queue are not independent.

I know that the equilibrium distribution of the queue lengths of queue 1 and queue 2 are independent at each fixed time t but this seems to mean queuing times should also be independent.

I'm puzzled?? Any help would be great!