# solution

In our usual speech sounds, there are three distinct states, V for voiced as
in vowels, U for unvoiced as in fricatives such as in the syllables â€œfâ€ and
â€œs,â€ and S for silence that occurs between words and for short times before
plosive sounds such as during the pressure build up to say syllables like â€œpâ€
and â€œt.â€ A good model for the sequence of such states is a continuous time
Markov chain with the average transition times from one state to another
given in milliseconds in the following matrix. The rows are for V , U , and S in
order from top to bottom, and the columns, similarly in order from left to right.

Note that the state of this Markov chain is not a random variable (since a
random variable is required to take numerical values for outcomes). It is a
Markov chain of three non-numerical states. This model is useful to determine
the effective bit rate necessary to code speech for transmission, etc. Answer
the following.
(a) Draw the Markov chain with correct values and dimensions on the arcs.
(b) L et the equilibrium state probabilities be pv , pu, and ps. Wr ite the three
global balance equations for the same.
(c) Write any set of a minimum number of equations that uniquely determine
the equilibrium state probabilities.

(d) Express the same set of equations in the above part in a matrix form.
(e) Find the inverse of the coefficient matrix.
(f) With the help of the inverse, find the equilibrium state probabilities.
(g) Over a long time interval T , what is t he expected number of times the
state moves into V ? Note t hat the answer to this is not pv T . Give s imilar
results for U and S.
(h) W ith the help of a computer program, simulate an equilibrium sequence
of 30 states. Plot a straight line with segments marked by notches. The
distance between successive notches should correspond to the time inter-
vals of the states. Write the state names above each interval. You can use
a com puter program to generate this plot.