Multi-Server Queues: Priorities and Application

MULTI-SERVER QUEUES: PRIORITIES and APPLICATION of JACKSON'S THEOREM

The Multi-Server Queue

The objective of this work is to describe multi-server queues including the idea of priorities and how Jackson's Theorem applies. The multi-server queue is a queue that is shared by multiple servers. When an item arrives and when at least one server is available the item will be then dispatched immediately to that server. The assumption is made that all servers are identical therefore; it makes no difference which server is chosen for the item.

When all servers are busy the queue begins its formation and upon one server becoming free the item is then dispatched from the queue through use of the dispatch discipline being utilized. Other than the parameter of utilization all other parameters carry over to the multiserver case "with the same interpretation.

The study of queues with multiple servers has been ongoing for approximately fifty years and dates back to the work of Kiefer and Wolfowitz. The work of Kiefer and Wolfowitz resulted in the enhancement of call centers, checkout lines, and high performance computing systems since the multiple server enables performance to be increased and as well is "cost-effective and easily scalable." (NSF & GSIA, 2004)

II. Multiple-Server Queues and Multiple Priority Classes

The work of Harchol-Balter, Osogami, Scheller-Wolf and Wierman (2003) entitled: "Multi-Server Queuing Systems with Multiple Priority Classes" states that a great deal of queuing theory is devoted to the analysis of priority queues, in which jobs or customers are "labeled and served in accordance with a priority scheme: high-priority jobs preempt medium-priority jobs, which in turn preempt low-priority jobs in the queue." (Harchol-Balter, Osogami, Scheller-Wolf and Wierman, 2003) in fact, many applications involve priority queuing.

Sometimes determination of the job priority is derived according to the job's owner and through what is known as a 'Service Level Agreement' (SLA) such as when customers make a choice to pay a larger amount in order to get "high-priority access to some high-demand resource." (Harchol-Balter, Osogami, Scheller-Wolf and Wierman, 2003) However, at other times, the job's priority is created artificially in order to "maximize a company's profit or increase system utilization." (Harchol-Balter, Osogami, Scheller-Wolf and Wierman, 2003)

Conducting analysis of the mean response time and analysis of higher moments of response time for various classes of jobs is important and this is important as the problem is more difficult in the context of a multi-server system. 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… & #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)