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We consider queueing output processes of some elementary queueing models suchas the M/M/1/K queue and the M/G/1 queue. An important performance measure forthese counting processes is their variance curve, indicating the variance ofthe number of served customers over a time interval. Recent work has revealedsome non-trivial properties dealing with the asymptotic rate at which thevariance curve grows. In this paper we add to the results by finding explicitexpressions for the second order approximation of the variance curve, namelythe y-intercept of the linear asymptote.  For M/M/1/K queues our results are based on the Drazin inverse of thegenerator. It turns out that by viewing output processes as MAPs (MarkovianArrival Processes) and considering the Drazin inverse, one can obtain explicitexpressions for the y-intercept, together with some further insight regardingthe BRAVO effect (Balancing Reduces Asymptotic Variance of Outputs). For M/G/1queues our results are based on a classic transform of D.J. Daley. In this casewe represent the y-intercept of the variance curve in terms of the first threemoments 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 ofarrivals or departures. In addition we shed light on a classic conjecture ofDaley, dealing with characterization of stationary M/M/1 queues within theclass of stationary M/G/1 queues, based on the variance curve.

The Second Order Terms of the Variance Curves for Some Queueing Output Processes