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… & #8230;.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." 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...

If there is feedback in the network). (Bose, 2002)
Bibliography

(2004) CMU Workshop on Multiserver Scheduling (WORMS) NSF and GSIA Carnegie Mellon University. Pittsburg, PA 18 April -- 19 April 2004. Online available at: http://www.cs.cmu.edu/~harchol/WORMS04/

Harchol-Balter, Mor, Osogami, Takayuki, Wolf-Scheler Alan, and Wierman, Adam (2003) Multi-Server Queuing Systems with Multiple Priority Classes. NSF Grant, Theory and IBM Corporation via Pittsburgh Digital Greenhouse Grant 2003. Online available at: http://www.cs.caltech.edu/~adamw/papers/multi2.pdf

Bose, Sanjay K. (2002) Open and Closed Networks of M/M/m Type Queues (Jackson's Theorem for Open and Closed Networks) Online available at: http://www.iitg.ac.in/skbose/qbook/Slide_Set_14.PDF