first time question here. I'm having a rough time trying to represent the following CTMC. Any help would be gladly appreciated.
We consider a server with a infinite buffer connected to a network. User send files varying sizes to the server. The treatment is done through a FCFS order.
However at anytime errors or cancelations may happen, resulting in the deletion from the buffer list of the file currently being processed and/or one or more files in the waiting list. These deletions are initiated by some system tasks. For this case the a system task occuring will delete the file being printed.
We make the following assumptions :
- Files arrive to the server according to a Poisson process of rate $\lambda$ (expressed in files/hour)
- Files have an exponential size with a mean of $m$ pages
- The treatment duration of a file is proportional to its size and the server is able to process $p$ pages per minute.
- System tasks of type i are generated by the system according to a Poisson process of rate $\alpha$ (expressed in tasks per hour)
- The queue has an unlimited size.
1) Are the assumptions listed above enough for the stochastic process $ n\{t(0)\}_{t\ge0}$ to be a CTMC ? If no give the missing assumptions.
2) Draw the CTMC.
I am no makov chains guru, but so far I can't do it mostly because on my model the rate $\alpha$ and $p$ go from the same nodes to the same nodes.
Idea 1
1) Supposing $p$ is a Poisson process, therefore the stochastic process can be a CTMC, we can add it to $\alpha$.
Idea 2
1) Supposing $p$ is an exponential distribution, we can now model the CTMC. We also add an extra state to differenciate the system task happening (therefore a deletion) and a file being treated normally.
I apologize for the above not being crystal clear, but I couldn't achieve any better. X,D symbolizes the number of remaining tasks in the system after a deletion. It's a tweak I did to avoid merging the rates.