# References request: two-queue, one-server model with pre-emptive queue priority and finite buffers

Sorry of the title is a mouthful. I'm developing a queue model with the following characteristics:

1. Two queues: One contains an infinite number of people (Queue A) while the other (Queue B) is initially empty and has a finite size of $L$ people.
2. Alternatively, Queue A can be thought of as having a finite size with no arrival rate, but each person served from Queue A goes right back to the end of Queue A (i.e., recycling of persons).
3. Queue B has exponentially distributed interarrival times of rate $\lambda$.
4. The server has exponentially distributed service times of rate $\mu$
5. Service Discipline: Persons in Queue B have preemptive priority over persons in Queue A.

Question/request: Looking for literature that has assessed this type of situation. I have seen literature on a single queue with different classes of persons (with pre-emptive or non-preemptive priority) and on modeling two finite queues, but nothing like this. Hoping someone more familiar with the literature can provide some pointers.

• What type of performance characteristics are you interested in? Customers in queue $B$ do not see any of the customers in queue $A$ due to the preemptive priority. So, for the type-$B$ customers it is just a simple $M/M/1/L$ queue. – Ritz Jul 4 '16 at 8:57
• @Ritz I am interested in the number of each type of customer served, on average, for every 100 completed services. – user237392 Jul 6 '16 at 4:13
• Type-$B$ customers do not see any type-$A$ customers and therefore occupy the server for a fraction of $\rho := \lambda / \mu < 1$ of time. Both types of customers require the same amount of service and the average number of type-$B$ customers served for every $K$ completed services is $\rho K$. – Ritz Jul 6 '16 at 6:57
• @Ritz doesn't the finite-length of queue B complicate matters as does the their preemptive priority? – user237392 Jul 6 '16 at 15:13
• You are right, I overlooked the fact that the the queue for type-$B$ customers has a finite length $L$. The preemptive priority matters a lot: type-$B$ customers do not see any type-$A$ customers. So, the type-$B$ customers experience an $M/M/1/L$ queue. Label the stationary probabilities of having $n$ type-$B$ customers in the system as $p(n)$. Then, in stationarity, the server is working on type-$B$ customers a fraction $1 - p(0)$ of the time. Using the same argument as in the previous comment: the average number of type-$B$ customers served for every $K$ completed services is $(1 - p(0))K$. – Ritz Jul 7 '16 at 8:23