1. Queueing with vacation:  Consider an M/M/1 queue with arrival and service rates l  (lambda) and m  (mu) respectively with the following variation of FCFS scheduling discipline. When the system becomes empty, the server goes to sleep and does not wake up until there are k customers in the system. This system can be modeled using a continuous time Markov Chain with the following types of states:
      • 0: system empty
      • (n,S)n customers in the system, with server asleep, 1 <= n <= k-1
      • (n,W): n customers in the system and server awake, 1 <= n <= k-1
      • nn customers in the system, n >= k
    1. Draw the state transition diagram of the model.
    2. Write the steady state balance equations.
    3. Show that the steady state probabilities of the sleeping states are the same.
    4. Express the steady state probability for the state (2,W) in terms of l  and m  and the steady state probability for state 0