Quantitative Analysis for Management, 13/e

Key Equations

(14-1)

$$\pi \left(i\right)=\left({\pi }_1, {\pi }_2, {\pi }_3,\dots ,, {\pi }_n\right)$$

Vector of state probabilities for period i.

(14-2)

$$P=\left[\begin{array}{ccccc}{P}_{11}& P_{12}& P_{13}& \cdots & P_{1n}\\ P_{21}& P_{22}& P_{23}& \cdots & P_{2n}\\ ⋮& ⋮\\ P_{m1}& P_{mn}\end{array}\right]$$

Matrix of transition probabilities—that is, the probability of going from one state into another.

(14-3)

$$\pi \left(1\right)=\pi \left(0\right)P$$

Formula for calculating the state 1 probabilities given state 0 data.

(14-4)

$$\pi \left(n+1\right)=\pi \left(n\right)P$$

Formula for calculating the state probabilities for the period $n+1$ if we are in period n.

(14-5)

$$\pi \left(n\right)=\pi \left(0\right)P^n$$

Formula for computing the state probabilities for ­period n if we are in period 0.

(14-6)

$$\pi =\pi P$$

Equilibrium state equation used to derive equilibrium probabilities.

(14-7)

$$P=\left[\begin{array}{cc}I& O\\ A& B\end{array}\right]$$

Partition of the matrix of transition for absorbing state analysis.

(14-8)

$$F={\left(I-B\right)}^{-1}$$

Fundamental matrix used in computing ...