  We consider queueing output processes of some elementary queueing models such
as the M/M/1/K queue and the M/G/1 queue. An important performance measure for
these counting processes is their variance curve, indicating the variance of
the number of served customers over a time interval. Recent work has revealed
some non-trivial properties dealing with the asymptotic rate at which the
variance curve grows. In this paper we add to the results by finding explicit
expressions for the second order approximation of the variance curve, namely
the y-intercept of the linear asymptote.
  For M/M/1/K queues our results are based on the Drazin inverse of the
generator. It turns out that by viewing output processes as MAPs (Markovian
Arrival Processes) and considering the Drazin inverse, one can obtain explicit
expressions for the y-intercept, together with some further insight regarding
the BRAVO effect (Balancing Reduces Asymptotic Variance of Outputs). For M/G/1
queues our results are based on a classic transform of D.J. Daley. In this case
we represent the y-intercept of the variance curve in terms of the first three
moments of the service time distribution.
  A further performance measure that we are able to calculate for both models,
is the asymptotic covariance between the queue length and the number of
arrivals or departures. In addition we shed light on a classic conjecture of
Daley, dealing with characterization of stationary M/M/1 queues within the
class of stationary M/G/1 queues, based on the variance curve.
