- 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
- 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
- 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