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This paper develops a first principles approach to constructing parameterized transfer function models for an abstraction of admission control, the M/M/1/K queueing system. We linearize this system using the first order model y(k+1)=ay(k)+bu(k), where y is the output (e.g., number in system) and u is the buffer size. The pole a is estimated as the lag 1 autocorrelation of y at steady state, and b is estimated using dy/du. With these analytic models for a and b, we study the effects of workload (i.e., arrival and service rates) and sample times. We show that a and b move in opposite directions at large utilizations, an effect that can have significant. implications on closed loop poles. 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 system converges to 0.5 as workload intensity becomes large. These insights may aid in designing robust and/or adaptive controllers for computing systems. Finally, our models provide insight into why the integral control of a Lotus Notes e-mail server has an oscillatory response to a change in reference value.