3.7.1 Ramaswami's formula

From Eqs.(3.22) and (3.24) and the aid of matrix ${\mathbf{G}}$, we derive Ramaswami's recursive formula [75], which is numerically stable because it entails only additions and multiplications^3.6. Ramaswami's formula defines the following recursive relation among stationary probability vectors $\mbox{\boldmath {$\pi$}}^{(i)}$ for $i \geq 0$:

$$\mbox{\boldmath {$\pi$}}^{(i)} = - \left( \mbox{\boldmath {... ...{(k)} \S^{(i-k)}\right) {\S^{(0)}}^{-1} ~~~ \forall i \geq 1,$$ (3.26)

where, letting ${\mathbf{F}}^{(0)} \equiv \L$, matrices $\widehat{{\mathbf{S}}}^{(i)}$ and $\S^{(i)}$ are defined as follows:

$$\widehat{{\mathbf{S}}}^{(i)} = \sum_{l=i}^{\infty} \widehat{... ...=i}^{\infty} {\mathbf{F}}^{(l)} {\mathbf{G}}^{l-i}, ~i \geq 0.$$ (3.27)

Observe that the above auxiliary sums represent the last column in the infinitesimal generator $\overline{{\mathbf{Q}}}$ defined in Eq.(3.20). We can express them in terms of matrices ${\mathbf{X}}_i$ defined in Eq.(3.24) as follows:

$$\widehat{{\mathbf{S}}}^{(i)} = \widehat{{\mathbf{F}}}^{(i)} ... ... 1~~~\S^{(i)} = {\mathbf{F}}^{(i)} + {\mathbf{X}}_i,~i \geq 0.$$

Given the above definition of $\mbox{\boldmath {$\pi$}}^{(i)}$ for $i \geq 1$ and the normalization condition, a unique vector $\mbox{\boldmath {$\pi$}}^{(0)}$ can be obtained by solving the following system of $m$ linear equations, i.e., the cardinality of set ${\cal{S}}^{(0)}$:

$$\mbox{\boldmath {$\pi$}}^{(0)} \left[ \left( \widehat{{\ma... ...ight)^{-1} {\mathbf{1}}^T \right] = [{\mathbf{0}}~\vert~1],$$ (3.28)

where the symbol ``$^\diamond$'' indicates that we discard one (any) column of the corresponding matrix, since we added a column representing the normalization condition. Once $\mbox{\boldmath {$\pi$}}^{(0)}$ is known, we can then iteratively compute $\mbox{\boldmath {$\pi$}}^{(i)}$ for $i \geq 1$, stopping when the accumulated probability mass is close to one. After this point, measures of interest can be computed. Since the relation between $\mbox{\boldmath {$\pi$}}^{(i)}$ for $i \geq 1$ is not straightforward, computation of measures of interest require generation of the entire stationary probability vector.