Majorization and arrangement orderings in open queueing networks

Consider an open Jackson network of queues. Majorization and arrangement orderings are studied to order, respectively, various loading and server-assignment policies. It is shown that under these order relations, stochastic and likelihood ratio orderings can be established for the maximum and the minimum queue lengths and for the total number of jobs in the network. Stochastic majorization and stochastic orderings are also established, respectively, for the queue-length vector and the associated order-statistic vector. Implications of the results on loading and assignment decisions are discussed.