On the optimal tradeoff of average service cost rate, average utility rate, and average delay for the state dependent M/M/1 queue

The optimal tradeoff between average service cost rate, average utility rate, and averagedelay is addressed for a state dependent M/M/1 queueing model, with controllable queue lengthdependent service rates and arrival rates. For a model with a constant arrival rate λ for all queuelengths, we obtain an asymptotic characterization of the minimum average delay, when theaverage service cost rate is a small positive quantity, V , more than the minimum average servicecost rate required for queue stability. We show that depending on the value of the arrival rate λ,the assumed service cost rate function, and the possible values of the service rates, the minimumaverage delay either: a) increases only to a finite value, b) increases without bound as log(1V),c) increases without bound as 1V , or d) increases without bound as 1√V, when V ↓ 0. We thenextend our analysis to a complementary problem, where the tradeoff of average utility rate andaverage delay is analysed for a M/M/1 queueing model, with controllable queue length dependentarrival rates, but a constant service rate μ for all queue lengths. We obtain an asymptoticcharacterization of the average delay, when the average utility rate is a small positive V lessthan the maximum utility rate which can be obtained with the queue being stable. However,we find that in this case the minimum average delay always increases without bound, either aslog(1V), 1V , or 1√Vdepending on μ, the assumed utility rate function, and the possible valuesof the arrival rates. We then consider a M/M/1 queueing model, with controllable queue lengthdependent service rates and arrival rates, for which we obtain an asymptotic characterization ofthe minimum average delay under constraints on both the average service cost rate as well as theaverage utility rate. The results that we obtain are useful in obtaining intuition as well guidancefor the derivation of similar asymptotic lower bounds, such as the Berry-Gallager asymptoticlower bound, for discrete time queueing models.