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 average delay is addressed for a state dependent M/M/1 queueing model, with controllable queue length dependent service rates and arrival rates. For a model with a constant arrival rate $\lambda$ for all queue lengths, we obtain an asymptotic characterization of the minimum average delay, when the average service cost rate is a small positive quantity, $V$, more than the minimum average service cost rate required for queue stability. We show that depending on the value of the arrival rate $\lambda$, the assumed service cost rate function, and the possible values of the service rates, the minimum average delay either: a) increases only to a finite value, b) increases without bound as $\log\frac{1}{V}$, c) increases without bound as $\frac{1}{V}$, or d) increases without bound as $\frac{1}{\sqrt{V}}$, when $V \downarrow 0$. We then extend our analysis to (i) a complementary problem, where the tradeoff of average utility rate and average delay is analysed for a M/M/1 queueing model, with controllable queue length dependent arrival rates, but a constant service rate $\mu$ for all queue lengths, and (ii) a M/M/1 queueing model, with controllable queue length dependent service rates and arrival rates, for which we obtain an asymptotic characterization of the minimum average delay under constraints on both the average service cost rate as well as the average utility rate. The results that we obtain are useful in obtaining intuition as well guidance for the derivation of similar asymptotic lower bounds, such as the Berry-Gallager asymptotic lower bound, for discrete time queueing models.