Operations Research Models and Methods / Computation / Stochastic Models

Dedicated Queues Markov Process Model

 

The figure below shows the queueing system under consideration. Customers requiring some service are the small circles and servers are the numbered rectangles. Customers arrive to the system from an infinite input source. For the Markov Process Model that we illustrate here, we assume that the time between arrivals is governed by an exponential distribution with an arrival rate specified by the data.

For this model we assume that each server has a dedicated queue of a fixed length. When a customer arrives and finds one or more servers either idle or with available spaces its queue, the customer moves directly to the server. She chooses the server with the smallest number of customers. If two servers have the same number, the one with the smallest index is selected.

If all dedicated queues are full and space remains in the pooled queue, the customer will join that queue. When the pooled queue is full, the customer does not enter the system and is not served.

Customers in a dedicated queue are served on a first-come-first-served basis. Once in a dedicated queue the customer cannot switch. After completing service, the customer departs. Immediately after a departure, if the pooled queue is not empty, a customer moves to the available server. As required by a Markov Process, we assume service times are exponentially distributed. The service rate may be the same for each server or may be different.

 

We create the Dedicated Queues model by clicking the appropriate button on the Model Dialog. There are three parameters: The Maximum for Each Server, The Number of Servers and The Maximum in the System. The data for the example is shown in the dialog. With a total of 10 allowed in the system, the pooled queue can hold up to 4 customers. Although not illustrated here, the maximum in the system can be specified as less than total allowed at the servers (6 in this case). With this number, there is no pooled queue, and customers are not allowed into the system when the maximum number of customers are already in the system.


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Updated 3/10/01
Operations Research Models and Methods

by Paul A. Jensen and Jon Bard, University of Texas, Copyright by the Authors