7.3.2   Service and Production Rates

Consider an M/M/1 system in which the customer represents an interrupt request
of a certain type and the server represents the processing required for that request.
In this single-processor model, waiters in the queue represent a time-overloaded
condition. Because of the nature of the arrival and processing times, this condition
could theoretically occur. Suppose, however, that the arrival or the processing
times can vary. Varying the arrival time, which is represented by the parameter λ,
could be accomplished by changing hardware or altering the process causing the
interrupt. Changing the processing time, represented by the parameter μ could
be achieved by optimization. In any case, fixing one of these two parameters,
and selecting the second parameter in such a way as to reduce the probability
that more than one interrupt will be in the system simultaneously, will ensure
that time-overloading cannot occur within a specific confidence interval.

For example, suppose 1/λ, the mean interarrival time between interrupt requests,
is known to be 10 milliseconds. It is desired to find the mean processing
time, 1/μ, necessary to guarantee that the probability of time overloading (more
than one interrupt request in the system) is less than 1%. Use Equation 7.15
as follows:

or

then

Thus, the mean processing time, 1/μ, should be no more than 1 millisecond to
guarantee with 99% confidence that time overloading cannot occur.

As another example, suppose the service time, 1/μ, is known to be 5 milliseconds.
It is desired to find the average arrival time (interrupt rate), 1/λ, to guarantee
that the probability of time-overloading is less than 1%. Using Equation 7.19,
yields

or

Hence, the average interarrival time between two interrupt requests should be at
least 50 milliseconds to guarantee only a 1% risk of time overloading. This result
is different from the guarantee that the rate-monotonic theorem, which states that
if a periodic interrupt occurs at exactly a 10-ms rate then a 1/10 = 20% utilization
will be realized. The result of Equation 7.15 applies if an aperiodic interrupt is
arriving at an average of every 10 milliseconds.

Of course, context switching time and blocking due to semaphore waits are
not incorporated in these analyses. Nevertheless, this approach can be useful in
exploring the feasibility of the system with aperiodic or sporadic interrupts.