4.13 Queueing network In the queueing network of Figure P4.13, all service times are negative exponential and the two arrival processes shown are independent and Poisson.

        There is space for one waiting user at the indicated point in front of queueing system 3. Whenever that queueing space is occupied, this "blocks" the departure of users from system 1 and from system 2. No space for waiting exists in front of either system 1 or system 2. Prospective users of either system are turned away if that system is either busy or blocked at the time of user arrival.
        Write the steady-state balance equations for this queueing network. You will find it useful to define carefully the states of the network and to draw a state-transition diagram for it. Assume that if both systems 1 and 2 are "blocked" at any time ' system 1 users receive priority whenever the single waiting space in front of system 3 becomes free.