Models
Example Problem
 Example Problem ATM Machine
 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.

One way to described the process associated with this situation is the state-transition network shown in Fig. 1. The state is the number of customers in the foyer. A change in state occurs when we have an arrival (a) or a departure (d). When the foyer opens in the morning, the system is empty -- in state 0. As customers arrive the state increases. Since there is only one machine, customers must wait in a queue when the state is greater than one. The state index increases and decreases as customers arrive and depart. When the foyer is full, the state reaches 5 and we assume that no other arrivals occur until there is a departure.

Figure 1. State-transition network for ATM example

The dynamic features of the model are provided by its activities. In this example we identify service and arrival activities. Service begins when a customer first appears in front of the ATM and ends when all the requested banking operations are complete. The culmination of the service activity is the event d in the figure. The duration of the activity that leads to this event is the time for service. The arrival activity is the process that generates customers for the ATM. The activity begins immediately after an arrival occurs and ends with the event of the next arrival. The event is an arrival (a in the figure) and the duration of this activity is the time between arrivals.

Operations Research Models and Methods
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by Paul A. Jensen