Multi-Server Queues: Priorities and Application Essay

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Adding to the complexity is the fact that the multiserver systems are "prevalent in many applications where prioritization is used…" (Harchol-Balter, Osogami, Scheller-Wolf and Wierman, 2003) the reason for the difficulty in conducting analysis of the priority queuing in multi-server settings is that the jobs "of different priorities may be in service at the same time.

III. Jackson's Theorem

The conduction of analysis of the priority queuing in multi-server settings is addressed when jobs of different priorities are in service simultaneously in what is known as 'Jackson's Theorem' which states that "…provided the arrival rate at each queue is such that equilibrium exists, the probability of the overall systems state (n1… ….nk) for K. queues will be given by the product-form expression as follows:

The work of Bose (2002) states that the Jackson Theorem when used in multiple customer classes or multi-servers requires the assumption that the "service time distribution at a node will be the same for all classes even though they may differ from one another.

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" This results in the service times being "state dependent" according to Bose (2002) and that the. "…external arrival rates and routing probabilities will vary from on e class of customers to another." (Bose, 2002)

Bose 2002 states that the implications of Jackson's Theorem are as follows:

(1) Once flow balance has been solved the individual queues may be considered in isolation;

(2) the queues behave as if they are independent of each other (even though they really are not independent of each other) and the joint state distribution may be obtained as the continued product of the individual state distributions (product-form solution); and (3) the flows entering the individual queues behave as if they are Poisson, even though they may not really be Poisson in nature (i.e. If there is feedback in the network). (Bose, 2002).....

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