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.
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