Abstract：In this paper, we study the power efficiency and delay performance of the IEEE 802.3az Energy Efficient Ethernet (EEE) protocol. A new approach is proposed to analyze the M/G/1 queue with the vacation time that is governed by the arrival process and the parameter  and N of the BTR strategy. Our key idea is to establish the connection between the vacation time and the arrival process to account for their dependency. We first derive the distribution of the number of arrivals during a vacation time based on an event tree of the BTR strategy, from which we obtain the mean vacation time and the power efficiency. Next, from the condition on the number of arrivals at the end of a vacation period, we derive a generalized P-K formula of the mean delay for EEE systems, and prove that the classical P-K formula of the vacation model is only a special case when the vacation time is independent of the arrival process. Our analysis demonstrates that the  policy and N policy of the BTR strategy are compensating each other. The  policy ensures the frame delay is bounded when the traffic load is light, while the N policy ensures the queue length at the end of vacation is bounded when the traffic load is heavy. These results, in turn, provide the rules to select appropriate  and N. Our analytical results are confirmed by simulations.

Abstract：Base station (BS) sleeping operation is one of the effective ways to save energy consumption of cellular networks, but it may lead to longer delay to the customers. The fundamental question then arises: How much energy can be traded off by a tolerable delay? In this paper, we characterize the fundamental tradeoffs between total energy consumption and overall delay in a BS with sleep mode operations by queueing models. Here, the BS total energy consumption includes not only the transmitting power but also basic power (for baseband processing, power amplifier, etc.) and switch-over power of the BS working mode, and the overall delay includes not only transmission delay but also queueing delay. Specifically, the BS is modeled as an M/G/1 vacation queue with setup and close-down times, where the BS enters sleep mode if no customers arrive during the close-down (hysteretic) time after the queue becomes empty. When asleep, the BS stays in sleep mode until the queue builds up to N customers during the sleep period (N-Policy) . Several closed-form formulas are derived to demonstrate the tradeoffs between the energy consumption and the mean delay for different wake-up policies by changing the close-down time, setup time, and the parameter N. It is shown that the relationship between the energy consumption and the mean delay is linear in terms of mean close-down time, but non-linear in terms of N. The explicit relationship between total power consumption and average delay with varying service rate is also analyzed theoretically, indicating that sacrificing delay cannot always be traded off for energy saving. In other words, larger N may lead to lower energy consumption, but there exists an optimal N* that minimizes the mean delay and energy consumption at the same time. We also investigate the maximum delay (delay bound) for certain percentage of service and find that the delay bound is nearly linear in mean delay in the cases tested. Therefore, similar tradeoffs exist between energy consumption and the delay bound. In summary, the closed-form energy-delay tradeoffs cast light on designing BS sleeping and wake-up control policies that aim to save energy while maintaining acceptable quality of service.

Abstract：The performance analysis of peer-to-peer (P2P) networks calls for a new kind of queueing model, in which jobs and service stations arrive randomly. Except in some simple special cases, in general, the queueing model with varying service rate is mathematically intractable. Motivated by the P-K formula for M/G/1 queue, we developed a limiting analysis approach based on the connection between the fluctuation of service rate and the mean queue length. Considering the two extreme service rates, we proved the conjecture on the lower bound and upper bound of mean queue length previously postulated. Furthermore, an approximate P-K formula to estimate the mean queue length is derived from the convex combination of these two bounds and the conditional mean queue length under the overload condition. We confirmed the accuracy of our approximation by extensive simulation studies with different system parameters. We also verified that all limiting cases of the system behavior are consistent with the predictions of our formula.