A First-Principles Approach to Constructing Transfer Functions for Admission Control In Computing Systems

There has been considerable interest in applying linear control theory to admission control in computing systems, which are notoriously nonlinear. This paper develops a first principles approach t,o constructing parameterized transfer function models for an abstraction of admission control, the M/M/l/K queueing system. We linearize this system using the first order model y(k + 1) = ay(k) + h(k), where II is the output (e.g. number in system) and u is buffer size. The pole, a, is estimated as the lag 1 autocorrelation of y at steady state, and b is estimated using dy/du. We use the analytic model to study the effects of workload (i.e., arrival and service rates) and sample times on a and b. We show that, a and b move in opposite directions, an effect that can have significant implications on closed loop poles. In particular, a increases with workload intensity (the combined effect of arrival rates and service times) and decreases with arrival rates (if workload in-tensity is fixed) and sample times. Further, the DC gain for response time and number in system drops to 0 as buffer size increases, and the DC gain of number in sy&em converges to 0.5 as workload intensity
becomes large. These insights may aid in designing robust and/or adaptive controllers for computing systems. Last, we use our models to explain why the integral control of a Lotus Notes email server has an oscillatory response to a change in reference value.