**Homework **

*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**n*:*n*customers in the system,*n >= k*- Draw the state transition diagram of the model.
- Write the steady state balance equations.
- Show that the steady state probabilities of the sleeping states are the same.
- Express the steady state probability for the state
*(2,W)*in terms of*l*and*m*and the steady state probability for state 0