Interactions of time scales

The identification of suitable time-scales is of crucial importance to any dynamical problem. Time-scales control rates of competition which lead to balance, and also rates of change.

It is easy to show that human administrators only compete with automatic systems in speed and efficiency at times of the day when they have nothing pressing to do. Indeed, it is always possible to arrange for an automatic system to beat a human, provided it can run in overlapping instantiations. A straightforward comparison of the time-scales involved in automated maintenance, to those of manual human maintenance can be made for any operation which is programmable in an automatic system with available technology.

Alarm systems which merely notify humans of errors and then rely on a human response are intrinsically slower than automatic systems which repair errors, provided the alarms represent errors which can be corrected with current automation.

The response time $t_{\rm auto}$ of a automatic machine system M, falls between two bounds (see figure 1)

$$n\, T_p +T_e(A) ~\geq ~ t_{\rm auto}~ \geq ~ T_e(A)$$ (16)

where T_p is the scheduling period for regular execution of the system (e.g. the cron interval, typically half-hour to an hour), T_e(A) is the execution time of the automatic system (typically seconds). The integer $n \ge 0$ since the number of iterations of the automatic system required to fix a problem might be greater than one. The time required to make a decision about the correct course of action T_d(A) is negligible for the automatic system.

 

Figure 2: Overlapping work rates of human and automatic systems.

For a human being, making a decision based on a predecided policy, the response time $t_{\rm human}$ falls between the limits:

$$\infty \geq t_{\rm human} \geq T_w(H) + T_d(H) + T_e(H).$$ (17)

T_d(H) is again the decision time, or time required to determine the correct policy response (typically seconds to minutes). T_e(H) is the time required for a human to execute the required remedy (typically seconds to minutes). T_w(H) is the time for which the human is busy or unavailable to respond to the request, i.e. the wait-time. The availability of human beings is limited by social and physiological considerations. In a simple way, one can expect this to follow a pattern in which the response time is greatest during the night; simplistically, if one assumes that humans sleep 8 hours,

$$T_w(H) > 4(1 +\sin(t/24)),$$ (18)

where time is measured in hours, whereas

$$T_w(A) \simeq 0.$$ (19)

We can note that human response times are usually much longer than the corresponding machine response times,

$$T_d(A) \ll T_d(H)$$

$$T_e(A) \ll T_e(H)$$ (20)

and that the periodic interval of execution of the automatic system is generally taken to be greater than the execution time of the automatic system

$$T_p \geq T_e,$$ (21)

thus avoiding overlapping executions (though this is not necessarily a problem, see the discussion of adaptive locks[9]) It is always possible to choose the scheduling interval to be arbitrarily close to T_e(A) (i.e. as short as one likes). Then provided,

T_w(H) > T_e(A) (22)

the automatic system can always win over a human. This last inequality requires qualification however, since very long jobs (such as backups or file tree parses) increase exponentially in time with the size of the file tree concerned. This makes a prediction: it tells us that one should always arrange to allow such long jobs to be run last in a sequence of maintenance tasks, and also in overlappable threads. This means that long jobs will not hinder the rapid execution of a maintenance program.

Cfengine[6] allows overlapping runs using its scheme of adaptive locks[9]. Thus, by scheduling long jobs last in a cfengine program, it is virtually always possible for cfengine to beat a human, unless it is prevented from running, or the human is given the chance to respond with a head-start; this seldom happens by chance.