Jackson Open Queuing Network with Finite Capacity Using Stopping Protocol

Question

I am studying about Jackson Open Network that consist of M/M/S/K queues. Based on what i had read, using stopping protocol (that all arrival are blocked and all queues stop doing service except the congested queue, until the congested resolve) product form solution for this network of queue still holds. In my understanding, if the product form solution still hold it implies that the queues in the network behave like independent M/M/S/K queues. If that case happened, does the total average people in the network is just the summation of average people in each queue composer? and how about the total average waiting time in the network, can i using Little's Law for this case?

Answer 1

Based on my verification in simple case like tandem queues that consist of two M/M/1/K queues. I observed that using stopping protocol the balance equation is satisfied by product form solution. From that we can see the network as independent queue one to another. It is means that average number of costumers in the network is the sum of average number of costumer in each M/M/S/K queues. But, for calculating the average waiting time in network we must consider about effective arrival rate to the network. Considering the stopping protocol will block all arrival. for the arrival that comes from the other queue will be hold in the previous queue while the arrival from outside the network will get rejected. By seeing those facts we can conclude that effective arrival rate to the network is equal to the arrival rate times $(1-(P_{N1}+P_{N2}))$, where $P_{Ni}$ is the probability rejecting costumer arrival from outside network to queue-i. This case later can be expanded to Jackson network.