I am trying to create an optimization model for a problem that involves two different types of queues. Given Poisson demand (for both), there is a queue with constant service time and another queue with another constant service time that allows the service of the first queue to be performed. Please see the following example for clarification. My question is what queueing formulation would represent such a system? My goal is to find a closed-form formulation to calculate the average system waiting time.
Assume there is a battery swapping station for electric vehicles and owns only a single battery swapping machine (i.e., single server). The demand follows Poisson distribution. The station replaces batteries with an FCFS fashion. The service time for battery swapping is deterministic leading to an M/D/1 queue for the swapping service. However, the battery is swapped only if there is a fully-charged battery available at the station. Empty or less-than-fully-charged batteries are charged at the station with an FCFS fashion and a single server (i.e., charger) as well. Assume, the state of the charge for batteries coming into the station is known (because we know where the vehicles are coming from). Hence, the charging time is also deterministic leading to another M/D/1 queue for the charging service. Is there any closed-form solution to find the average waiting time or queue length for these vehicles in such a system?
My interpretation: The closest queue seems to be a semi-open queue (Jia and Heragu, 2009). Another alternative is (Madan and Saleh, 2001, pp. 35, Case 3). In the semi-open queue, I couldn't find a closed-form formulation. In the alternative one, the problem would be modeled somewhat differently as we introduce a probabilistic approach to whether there is a battery available or not at each swapping visit, which is not really matching our problem one-to-one. Can you please advise?
Also, please feel free to change my question title if you come up with a better one.