Quasi-reversible multiclass queues with order independent departure rates

A solvable queueing network model for railway networks and its validation and applications for the Netherlands

“…Hence, the network can be modeled by a continuous-time Markov chain ðCðsÞ; s > 0Þ, where CðsÞ denotes the state of the network at time s. The steady-state distribution pðCÞ lim s!1 PrfCðsÞ ¼ Cg of this Markov chain is found observing that all queues in the model are quasi-reversible (see [2,9]). Since the network is open, pðÁÞ is given by the product of the steady-state distributions of the queues in isolation, as if they were fed by independent Poisson streams [9, p. 70].…”

“…Since the network is open, pðÁÞ is given by the product of the steady-state distributions of the queues in isolation, as if they were fed by independent Poisson streams [9, p. 70]. Using the results of [1,2] and Appendix B for the distributions of the states of the queues in isolation, the following theorem is obtained.…”

“…Proof. According to [2], a queue with exponential service times is an order independent queue, if the following three conditions hold.…”

“…n ð4Þ j ¼ n 4 g, and define p j ðnÞ ¼ 0 if n i < 0 for some i. For non-negative n with at least one positive n i , p j ðnÞ satisfy the following recursive relation (see [2]):…”

“…; p À 1, whose routes differ from each other and from the route of class c, and only one of these two trains is crossing the junction. The last, rather artificial, condition permits modeling a junction by the so-called order-independent queue (see [2]), as is shown in Lemma B.1 in Appendix B. It is easily seen that this condition applies very rarely: three trains have to be simultaneously at the junction, and even then it mostly coincides with the first one.…”