I am stuck with the following problem about modelling a queue. I have an approach to the solution but unfortunately my teacher said it is incorrect. The problem is that he didn't give me any more feedback and I don't know how to do it.
This is the problem:
The messages arrive at a multiplexer with a Poisson rate λ to be transmitted through a capacity communication line with capacity = 'C' octets per second. The lengths of the messages, specified in octets, are exponential with a parameter μ. When the number of messages waiting are 3 or more, the transmission capacity increases to 'Ce' when adding a telephone access line. Time required to prepare the telephone access line to increase capacity is exponential with rate τ. If the connection is completed when there are less than three messages waiting or if the number of messages waiting falls below two at any time while using the additional capacity, the extra line is immediately disconnected.
This was my approach: State Diagram (sorry, I can't embed images yet).
I can't seem to figure where are we wrong and how to do a correct model.
I think you can 'split' the diagram after the second state. The you can have two streams that are parallel: one with the telephone line and one without. So you would have state 4 with telephone and state 4 without telephone (and similar for all states indicating more than 3 messages waiting). The rate between two states with the same amount of messages in the systems is $\tau$. The other rates are similar to the ones in your figure.