Order Independent Queues

Abstract

We present a class of queues which are quasi-reversible and therefore preserve product form distribution when connected in multinode networks. The essential feature leading to the quasi-reversibility of these queues is the fact that the total departure rate in any queue state is independent of the order of the customers in the queue. We call such queues Order Independent (OI) queues. A distinguishing feature of the OI class is that, among others, it includes the FCFS, processor sharing, infinite server and MSCCC queues but not the LCFS queue. We next examine OI queues where arrivals to the queue are lost when the number of customers in the queue equals an upper bound. We prove that such queues satisfy partial balance and we obtain the stationary distribution for the OI loss queue by normalising the stationary probabilities of the corresponding OI queue without losses.OI loss queues can be used to model systems with simultaneous resource possession with the option of queueing blocked customers. The OI loss queue thus extends previous loss models where customers are rejected when processing resources are not available. The OI loss class is next extended to include networks of queues which can be used to model systems with complex loss mechanisms. We finally present several applications of OI loss queues and OI loss networks.