1. Problem 15.10 A communication system is either in the on-state (state 1) or the off-state (state 0). Every millisecond the state of the system may change.

An off-state is changed into an on-state with probability α and an on-state is changedintoanoff-statewithprobabilityβ,where0 <α,β<1.Useinduction to verify that the n-step transition probabilities of the Markov chain describing the state of the system satisfy p(n)

00 = β

α +β +α(1−α−β)n where p(n)

01 = 1− p(n)

α +β and p(n)

11 = α

α +β +β(1−α−β)n 00 and p(n)

10 = 1− p(n)

α +β , 11 . Remark: the reader familiar with linear algebra may verify this result from the eigenvalues and eigenvectors of the matrix of one-step transition probabilities.