Operations Research Models and Methods / Computation / Stochastic Models

Finite Queue Markov Chain 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 some input source. If some server is not busy, the customer immediately begins to be served. Otherwise, the customer must wait in a queue until a server is available. Some time is required for service, after which the customer departs.


The input source, also called the calling population, is the collection of customers providing inputs to the queueing system. We assume for this model that the calling population is infinite, that is, the rate of arrivals into the system is unaffected by the number already there.

A queue discipline defines the rules by which customers are selected for service. A common discipline, assumed here, is first-come-first-served. Service is provided by one or more servers (or channels) operating in parallel. The servers may or not be identical.

The queueing system is the combination of the queue and the service channels. For this model, we assume that the total number of customers that can be present in the system is limited to some maximum number. That is, the size of the queue is finite and limited by the maximum number in the system less the number of servers. When customers arrive and find the queue to be full, the customer does not enter the system and does not receive service.

We create the model by selecting Finite Queue on the Model Dialog. The two parameters of this system are the maximum number in the system and the number of servers. We have chosen small numbers for these parameters to simplify the presentation of the example. We have selected the Different Servers option. The model constructed is somewhat more complicated than if the servers were the same. By checking the Random Problem box, the server probabilities are randomly generated.


The Model Worksheet

Updated 3/7/01
Operations Research Models and Methods

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