Before I ask my question these are the symbols I used:
$w$ = Customers waiting in queue
$\mu$ = Service rate
$T_s = 1/\mu$ = Average service time
$\lambda$ = Average arrival time
$\rho$ = Server utilization
How come $\rho=\lambda T_s$ for a M/M/1/K System (which is the same for as M/M/1)
This is my intuitive reasoning, please tell me where I am going wrong.
Picture a M/M/1/3 queue system.
- Assume at one point the queue is full. i.e w=3. At this point, a task (n) is born. This will cause this new task to be rejected.
- Now assume a few moments later, the server has processed all the 3 tasks (w=0) and a new task hasn't arrived yet. So the server will be idle until a new task (n+1) arrives
Now in the scenario above, if it was a M/M/1 queue, the task (n) wouldn't have been rejected and so the server wouldn't be idle as shown in the second bullet point above. In other words, a M/M/1 system would serve task (n) and (n+1) where as a M/M/1/3 system would serve only task (n+1)
So clearly, in a M/M/1/K system, the server has more chances to have idle periods, thus lowering its utilization when compared to a M/M/1 system.
So again, $\rho=\lambda T_s$ for a M/M/1/K system doesn't make sense to me. What am I missing?