In this paper, we propose mapped twoway water filling (MTWF) scheme to maximize energy efficiency (EE) for hybrid bursty services with quality of services (QoS) requirements in twoway multirelay (TWMR) OFDM networks. The bursty traffic is first analyzed by strictly proved equivalent homogeneous Poisson process, based on which the QoS requirements are converted into sumrate constraints. The formulated nonconvex EE maximization problem, including subcarrier assignment, relay selection (RS) and rate allocation, is NPhard involving combinatorial optimization. To conduct optimal RS on each subcarrier without priori bursty traffic knowledge, we utilize some approximate relationships under high data rate demands to remove its dependence on twoway data rates, and simplify the whole optimization problem as well. After the optimal channel configuration is obtained, which only depends on channel conditions, subcarrier assignment is attained through elitist selection genetic algorithm (ESGA), and rate allocation of each service is fulfilled by deducing twoway water filling principle. A new equivalent optimization objective function is proposed next as the simple evaluating index in ESGA to reduce complexity. Finally, simulations are carried out to verify the superiority and convergence of our scheme, as well as the applicability for different scenarios.
The work was supported by National Nature Science Foundation of China Project (Grant No. 61471058), Hong Kong, Macao and Taiwan Science and Technology Cooperation Projects (Grant Nos. 2014DFT10320, 2016YFE0122900), Beijing Nova Program (Grant No. xx2012037), Shenzhen Science and Technology Project (Grant No. 20150082), and Beijing Training Project for the Leading Talents in S&T (Grant No. Z141101001514026).
Appendix
Proof of the average performance equivalent
With the same data packet departure (that is homogeneous Poisson process), there are two kinds of data packet arrival, which are equivalent about the average performance, i.e., delay in this paper. The two types of data packet arrival can be described as follows.
Homogeneous Poisson arrival: The data packets arrive with a fixed rate $\lambda$ with a period of time $T$.
Heterogeneous Poisson arrival: The data packets arrive with a changed rate $\lambda(t)$ with regard to time $t$ within a period of time $T$, and the following constraint is satisfied, as given by
Since the departure process is identical, the equivalence of the two queue systems can be ensured if the equivalence of the two arrival processes can be proved within time duration $T$. Observing the whole queue system at time $T$, it is seen that the waiting time, depending on the state of queue system, will be same for different queue systems as long as the probability of queue length is identical. It is noted that only same average performance at time $T$ is guaranteed, but not the transient performance or any time before $T$.
As for homogeneous poisson arrival, the probability of arriving $k$ data packets within time $T$ can be expressed as
Similarly, for heterogeneous poisson arrival, this probability of arriving $k$ data packets can be obtained, as given by
The average time duration can be written as $T_1^s+\frac{1}{\Lambda_1^s}$ for service $s$ in downlink from a burst arrival to another burst arrival periodically. The burst data packet arrival is a heterogeneous poisson process, thus the following relationship can be obtained with the arrival rate $\lambda_1^s$ during bursty duration while $0$ during bursty interval, as given by
Since ${\lambda_1^s}^*=\frac{\lambda_1^s~\cdot~T_1^s}{{T_1^s+\frac{1}{\Lambda_1^s}}}$, the new homogeneous poisson arrival with the parameter of ${\lambda_1^s}^*$ can meet the constraint
Therefore, it is completely proved that it is equivalent on average performance for the bursty arrival and the homogeneous poisson arrival with the parameter of ${\lambda_1^s}^*$.
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Figure 1
(Color online) Hybrid bursty services transmission in TWMR OFDM networks.
Figure 2
(Color online) Bursty traffic of single service.
Obtain sumrate constraints from the QoS requirements through equivalent queue analysis for all services; 
According to the integratednoisechannel selection criterion, the optimal relay for each subcarrier is selected and the optimal channel conditions are recorded; 
Calculate the comprehensive twoway channel coefficients for each subcarrier; 
Initialize the subcarrier assignment sequence satisfying the $\{C_s^n\}$ constraints of each service for chromosome $i$; 


Resolve water level according to 
Conduct the rate allocation and obtain the data rate ${r_1}$ and ${r_2}$ according to water level for service $j$; 

Let these negative data rates be 0 and exclude these subcarriers. Turn to Step 


Calculate the fitness according to the equivalent optimization function ( 

Select and record the best assignment scheme; 
Obtain the next generation by crossover and mutation; 
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