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.