Computation - Operations Research Models and Methods

Computation

Stochastic Analysis

CTMC

	Continuous-Time Markov Chain

Selecting the Markov Process item under Stochastic Analysis, provides the opportunity to construct a Markov Process Model. We describe below the various analyses possible for the example which is described below. All the analyses available for the Markov Chain's are also available for the Markov Process. Since the Markov Process is a continuous time model, there is some expansion of the analyses. Both are implemented by the same add-in, the Stochastic Analysis add-in (stoch_anal.xla).

The Relink Buttons command on the OR_MM menu is useful when opening a workbook containing a Markov Process model in a different computer than the one in which the model was created. The command deletes the old buttons and replaces them with buttons linked to the add-in in the new computer.

When the Markov Process item is selected the dialog below is presented.

The dialog accepts a name for the model and the number of states. When the Make Random Problem checkbox is checked. A random problem is generated.

Example: Providing ATM Service

To illustrate the elements of the stochastic process model, we use the example of a single Automated Teller Machine (ATM) located in foyer of a bank. The ATM performs banking operations for people arriving for service. The machine is used by only one person at a time, and that person said to be in service. Others arriving when the machine is busy must wait in a single queue, and these people are said to be in the queue. Following the rule of first-come-first-served, a person in the queue will eventually enter service and will ultimately leave the system. The number in the system is the total of the number in service plus the number in the queue. The foyer is limited in size so that it can hold only five people. Since the weather is generally bad in this part of the country, when the foyer is full, arriving people do not enter. We have gathered statistics on ATM usage that show the time between arrivals averages 30 seconds (or 0.5 minutes). The time for service averages 24 seconds (or 0.4 minutes). Although the ATM has sufficient capacity to meet all demand, we frequently observe queues at the machine and occasionally customers are lost.

We want to perform an analysis to determine statistical measures that describe the number of people in the system, the waiting time for customers, the efficiency of the ATM machine, and the number of customers not served because there is no room in the foyer. We intend to use these statistics to guide managers in design questions such as whether another ATM should be installed, or whether the size of the foyer should be expanded.