Measures and characters

As an empirical science, system administration suffers from many shortcomings. It has all of the problems associated with the social sciences: statistical measures are seldom forthcoming, experimental repeatability is a luxury, and sufficient repetition to obtain statistically meaningful samples is a near impossibility. The conditions under which measurements are made are constantly changing. The situation is somewhat analogous to that of non-equilibrium statistical mechanics in physics, but markedly less controlled.

The characteristics which are of interest to us refer to the actions and results which inter-weave in the dynamical behaviour of the system. These include the quality of actions of the system administrator, in relation to the prescribed policy, a typical characterization of the environment which affects the system. The measurements which are most useful are those based on persistent variables, since these have a stable value. Other fluctuating values can be treated stochastically or averaged out into persistent values.

The following measures will be useful in formulating `pay-off' matrices for administration models, as in the example to follow below. The accuracy with which a policy is implemented by an agent of system management (human or automatic system) can be gauged with the following ratio:

$${\rm Accuracy} = \frac{\rm Number~ of~ policy ~actions}{\rm All~ actions performed}$$ (11)

i.e. the fraction of work which is within prescribed guidelines. In algebraic terms:

$$\alpha = \frac{N_p(t)}{N(t)} = \frac{\displaystyle \sum_{(a\subset P)} N_a}{\displaystyle \sum_{(\forall a)} N_a}$$ (12)

For humans $\alpha \leq 1$. For any bug-free automatic system, $\alpha = 1$. Similarly, one may define the efficiency of a system by its use of resources (memory and CPU share):

$${\rm Efficiency} = {\rm Accuracy} \times \left( 1 - \frac{\rm Resources~used}{\rm Resources~available}\right)$$ (13)

In algebraic terms:

$$\varepsilon = \alpha\left(1 - \frac{\displaystyle \sum_{(a\subset P)} r_a}{\displaystyle \sum_{(\forall a)} r_a}\right)$$ (14)

i.e. the more resources which are consumed in implementing a policy, the less efficient it can be considered to be.

Other measures are more useful for describing the relationship of a computer system to its environment, or the influential forces which steer its dynamical evolution. The response of a computer system to its users is characterized by averages which fluctuate in time. Human society's diurnal work pattern imposes a twenty four hour periodic character on these measurements[7,2] and a also a weekly work pattern, which is dominant during weekdays and slight at weekends (at least in the Western world). The periodic topology implies that the distribution of resource usage takes on the special form of a Planck distribution with a Gaussian component, by analogy with statistical physics at temperature T:

$$D(\lambda) = A\;e^{-\left(\frac{(\lambda-\overline\lambda)^2}{2\sigma^2}\right)} + \frac{B}{(\lambda-\lambda_0)^3(e^{1/(\lambda-\lambda_0) T}-1)}.$$ (15)

$\lambda$ is the deviation of a measurement from its average value over a period. The values of the constants A, B, $\lambda_0$and T may be chosen to fit the behaviour of any variable which is strongly coupled to periodic usage. Their absolute values have no significance, since there is no `standard candle' computer system to compare to, but changes relative to the local norm could be interpreted as anomalies. Non-zero A allows for the presence of additional Gaussian noise in some measurements.