I am trying to model the following system:
There is a two server queue that operates as follows:
i) Two different queues are formed in front of the two servers.
ii)The arriving customer joins the shorter of the two queues (the customer in service is counted as part of that queue).
iii)If the two queues are equal, he joins either one with probability $0.5.$
iv) Queue jumping is not allowed.
v) The arrival process is PP($\lambda$) and the service times are iid exp(\mu) at either server
Let $X_i (t)$ be the number of customers in the $i^{th}$ queue, including any in service with server $i, i = 1, 2.$
I am trying to model $\{(X_1 (t), X_2 (t)), ~t \ge 0\}$ as a CTMC. However, this is different from the case of usual CTMC's which contains only $1$ variable which changes with $t$. So I am unable to figure out the structure of CTMC, and the possible trigger events to formulate the generator matrix. Can some one help me with this?