Reversibility and dependence in a U-shaped production line

We deal with a U-shaped production line with multiple machines and a multifunction worker. The worker visits machines in a cyclic fashion and operates an item on each machine. The operation consists of detaching the processed item from the machine, sending it to the next machine, putting the new item on the machine and switching it on. Walking times of the worker and processing and operation times are i.i.d. random variables for each machine. We show that the sum of the first n cycle times is stochastically the same as that in the reversed system where machines are arranged in the reversed order. We give examples where the distributions of waiting and cycle times depend on the machines, called starting ones, where the worker begins operation at time 0. A sufficient condition is derived under which the limiting expected waiting and cycle times do not depend on the starting machines.