Question 7.E.3.11: A particular electronic device consists of a collection of s......
A particular electronic device consists of a collection of switching circuits that can be either in an ON state or an OFF state. These electronic switches are allowed to change state at regular time intervals called clock cycles. Suppose that at the end of each clock cycle, 30% of the switches currently in the OFF state change to ON, while 90% of those in the ON state revert to the OFF state.
(a) Show that the device approaches an equilibrium in the sense that the proportion of switches in each state eventually becomes constant, and determine these equilibrium proportions.
(b) Independent of the initial proportions, about how many clock cycles does it take for the device to become essentially stable?
(a) Draw a transition diagram similar to that in Figure 7.3.1 with North and South replaced by ON and OFF, respectively. Let x_k be the fraction of switches in the ON state and let y_k be the fraction of switches in the OFF state after k clock cycles have elapsed. According to the given information,
x_k = x_{k−1}(.1) + y_{k−1}(.3)
y_k = x_{k−1}(.9) + y_{k−1}(.7)
so that p^T_{k+1} = p^T_k T, where p^T_k= (x_k y_k) and T = \begin{pmatrix}.1 &.9\\ .3 &.7\end{pmatrix}. Compute σ (T) = \{1, −1/5\}, and use the methods of Example 7.3.4 to determine the steady-state (or limiting) distribution as
p^T_∞ = \underset{k→∞}{lim} p^T_k = \underset{k→∞}{lim} p^T_0 T^k = p^T_0 \underset{k→∞}{lim} T^k = (x_0 y_0) = \begin{pmatrix}1/4 &3/4\\ 1/4 &3/4 \end{pmatrix}
= \left(\frac{x_0 + y_0}{4} + \frac{3(x_0 + y_0)}{4}\right) = (1/4 3/4).
Alternately, (7.3.15) can be used with x_1 = \begin{pmatrix}1\\1\end{pmatrix} and y_1 = (1 3) to obtain
p^T_∞ = \underset{k→∞}{lim} p^T_0 T^k = p^T_0 \underset{k→∞}{lim} T^k = p^T_0 G_1 = \frac{(p^T_0 x_1)y^T_1}{y^T_1 x_1} = \frac{y^T_1}{y^T_1 x_1}. (7.3.15)
p^T_∞ = p^T_0 \underset{k→∞}{lim} T^k = p^T_0 \underset{k→∞}{lim} G_1 = \frac{(p^T_0 x_1)y^T_1}{y^T_1 x_1} = \frac{y^T_1}{y^T_1 x_1} = (1/4 3/4).(b) Computing a few powers of T reveals that
T^2 = \begin{pmatrix}.280 &.720\\.240 &.760\end{pmatrix}, T^3 = \begin{pmatrix}.244 &.756\\ .252& .748\end{pmatrix},
T^4 = \begin{pmatrix}.251 &.749\\.250 &.750\end{pmatrix}, T^5 = \begin{pmatrix}.250 &.750\\ .250 &.750\end{pmatrix},
so, for practical purposes, the device can be considered to be in equilibrium after about 5 clock cycles, regardless of the initial configuration.
