3.1.1 Additional measures of interest

Once the stationary probability vector ${\mbox{\boldmath {$\pi$}}}$ is known, we can obtain various performance measures of interest such as the average queue length which we showed in the previous subsection. There are additional measures of interest that one can compute knowing $\mbox{\boldmath {$\pi$}}^{(0)}$, $\mbox{\boldmath {$\pi$}}^{(1)}$, and ${\mathbf{R}}$. In particular, the tail distribution of the number of jobs in the system can be expressed as

$$\mbox{{\bf\sf P}}\left[ Ql > x \right] \; = \; \sum_{k=x+1}... ... {({\bf I} - {\mathbf{R}})}^{-1} {\bf e} , \;\;\;\; x \geq 0 ,$$ (3.5)

with the corresponding expectation given by

$${\rm E}{[Ql]} \, = \, {{\mbox{\boldmath$\pi$}}}^{(1)} \sum_{k=0}^... ...th$\pi$}}}^{(1)} {\mathbf{R}}{({\bf I} - {\mathbf{R}})}^{-2} {\bf e} . \;\;\;\;$$ (3.6)

The expected waiting time of a job in the system can then be calculated using Little's law [51] and Eq.(3.6), which yields

$${\rm E}{[W]} \; = \; {\lambda}^{-1} \left( \, {{\mbox{\bol... ...\mathbf{R}}{({\bf I} -{\mathbf{R}})}^{-2} {\bf e} \, \right) .$$ (3.7)

Let $\eta$ denote the spectral radius of the matrix ${\mathbf{R}}$, which is often called the caudal characteristic [68]. In addition to providing the stability condition for a QBD, $\eta$ is indicative of the tail behavior of the stationary queue length distribution. Let ${\bf u}$ and ${\bf v}$ be the left and right eigenvectors corresponding to $\eta$ normalized by ${\bf u}{\bf e} = 1$ and ${\bf u}{\bf v}= 1$. Under the above assumptions, it is known that [94]

$${\mathbf{R}}^x \; = \; \eta^x \, {\bf v}\cdot {\bf u}\: + \: \mbox{o}(\eta^x) , \;\;\;\; \mbox{as } \; x \rightarrow \infty ,$$

which together with Eq.(3.4) yields

$${{\mbox{\boldmath$\pi$}}}^{(x)} {\bf e } \; = \; {{\mbox{\b... ...o}(\eta^{x-1}) , \;\;\;\; \mbox{as } \; x \rightarrow \infty .$$ (3.8)

It then follows that

$$\mbox{{\bf\sf P}}\left[ Ql > x \right] \; = \; \frac{ {{\mb... ...box{o}(\eta^x) , \;\;\;\; \mbox{as } \; x \rightarrow \infty ,$$ (3.9)

and thus

$$\lim_{x \rightarrow \infty} \, \frac{ \mbox{{\bf\sf P}}\lef... ... \frac{ {{\mbox{\boldmath$\pi$}}}^{(1)} {\bf v}}{ 1 - \eta } ,$$ (3.10)

or equivalently

$$\mbox{{\bf\sf P}}\left[ Ql > x \right] \; \sim \; \frac{ {{... ...a } \, \eta^x , \;\;\;\; \mbox{as } \; x \rightarrow \infty .$$ (3.11)

The caudal characteristic can be obtained without having to first solve for the matrix ${\mathbf{R}}$. We define the matrix ${{\mathbf{A}}}^{\ast}(s) = {\mathbf{A}}_0 + s {\mathbf{A}}_1 + s^2 {\mathbf{A}}_2$, for $0 < s \leq 1$. Since the generator matrix ${\mathbf{A}}$ is irreducible, this matrix ${{\mathbf{A}}}^{\ast}(s)$ is irreducible with nonnegative off-diagonal elements. Let $\chi(s)$ denote the spectral radius of the matrix ${{\mathbf{A}}}^{\ast}(s)$. Then, under the above assumptions, $\eta$ is the unique solution in (0,1) of the equation $\chi(s) = 0$. A more efficient method is developed in [9].