Figure 1: Probability that both children remain sleeping as a function of time, given that they
were initially asleep. This chart was made by evaluating the matrix exponential at various points
premultiplied by the scalar time, demonstrating the usefulness of this method.
that in our example that both children
are asleep 30 percent of the time, Mary
only is sleeping 18 percent of the time,
Neil only is sleeping 14 percent of the
time, and both children are crying 36
percent of the time. Note that the children do not have equal sleeping behaviors. This is because Mary has a little
lambda.
We’ve (somewhat sloppily) found
the limiting distribution, but we may
do a great deal more. Suppose that
both children are currently asleep.
We wish to compute the probability
that they will still be asleep in one
hour. This is easy; we simply compute
P(1) = eG and pre-multiply the result by
A NA L Y T I C S
the initial condition vector (1,0,0,0),
which strips off the top row, and we
see that there is an 81 percent chance
that both children will still be sleeping
in an hour.
Harrison Schramm (harrison.schramm@gmail.
com) is an operations research professional in
the Washington, D.C., area. He is a member of
INFORMS and a Certified Analytics Professional
(CAP).
NOTES
1. Real children, of course, exist in
many states as they grow older.
2. “G Matrix” is another math-rap
name ripe for the picking!
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