We consider an asymmetric inclusion process, which can also be viewed as a model of n queues in series. Each queue has a gate behind it, which can be seen as a server. When a gate opens, all customers in the corresponding queue instantaneously move to the next queue and form a cluster with the customers there. When the nth gate opens, all customers in the nth site leave the system. For the case where the gate openings are determined by a Markov renewal process, and for a quite general arrival process of customers at the various queues during intervals between successive gate openings, we obtain the following results: (i) steady-state distribution of the total number of customers in the first k queues, k= 1 , ⋯ , n; (ii) steady-state joint queue length distributions for the two-queue case. In addition to the case that the numbers of arrivals in successive gate opening intervals are independent, we also obtain explicit results for a two-queue model with renewal arrivals.
Bibliographical noteFunding Information:
Onno Boxma: research done in the framework of the IAP BESTCOM project, funded by the Belgian government and by the Gravity program NETWORKS of the Dutch government.
Offer Kella: supported in part by Grant 1462/13 from the Israel Science Foundation and the Vigevani Chair in Statistics.
© 2016, The Author(s).
- Asymmetric inclusion process
- Queue length distribution
- Synchronized service
- Tandem network