Example: Light
Bulb Replacement
A bowling alley has a sign made
entirely of light bulbs. There are 1000 bulbs in the sign.
The manager is concerned about the maintenance of the sign.
He wonders how many of the bulbs must be replaced in a monthly
service call and how much should he budget for maintenance
of the sign. |
Historical data and probability analysis
determine the probabilities of replacement based on the
age of a bulb. This information is shown by the matrix below.
Each row of the matrix describes what might occur for a
bulb of a particular age. For example, the row labeled New
indicates that if a new bulb is inserted in a socket during
one of the monthly maintenance inspections, the probability
that it will fail during the month and be replaced with
a new bulb at the next inspection is 0.5. The probability
that it will not fail and survive to an age of one month
is also 0.5. This poor quality seems to be rather extreme,
but we exaggerate for illustration. The remaining rows indicate
similar data for other ages. In every case the bulb is replaced
with a new one or ages by one month. For the example, we
assume the bulb is always replaced after it is four months
old.
The names New, 1-mo, 2-mo., etc. are the states of the system.
At a monthly inspection the bulb must be in one of the states.
The matrix is called the transition matrix because it shows
the probability of transition from each state to every other
state. We call this a Markov process when the transition
probabilities depend only on the state of the system. The
figure appearing at the beginning of this article is called
the State Transition Diagram and represents the same information
as the transition matrix.
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