|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.