### 4.1 High energy-efficiency OMS based on SC for mixed traffics scheme

Let the transmitter's power be *P*, and λ(0 ≤ λ ≤ 1) be the power allocation coefficient for SC that indicates how much power is allocated to a BMS, thus the transmit power allocated to a BMS is λ*P* and the transmit power allocated to an EMS is (1 - λ)*P*. Take the *i*-th (*i* = 1, 2,...) time transmission for example, the user selection ratio is *p*_{
b
}(*i*) and the transmit rate is *R*_{
b
}(*i*). If the user's received data rate is less than *R*_{
b
}(*i*), this transmission of this time is considered be be interrupted or outage. From (3), the outage probability of the basic layer is (noted that in the following the subscript *i* representing the *i* th transmission is neglected for simplicity):

{P}_{b}^{\mathsf{\text{out}}}=F\left({\text{log}}_{2}\left(1+\frac{\lambda {\rho}_{0}{\left|h\right|}^{2}}{\left(1-\lambda \right){\rho}_{0}{\left|h\right|}^{2}+1}\right)\le {R}_{b}\right)=F\left({\left|h\right|}^{2}\le \frac{\left({2}^{{R}_{b}}-1\right)}{\left[1-\left(1-\lambda \right){2}^{{R}_{b}}\right]{\rho}_{0}}\right)=1-{p}_{b}.

(5)

Substituting (3) into (5), we obtain the transmission rate for the basic layer:

{R}_{b}={\text{log}}_{2}\left(1-\frac{\lambda {\rho}_{0}\frac{\text{ln}{p}_{b}}{{r}_{c}^{n}}}{1-\left(1-\lambda \right){\rho}_{0}\frac{\text{ln}{p}_{b}}{{r}_{c}^{n}}}\right).

(6)

Thus, according to (4) we have the total transmission delay as follows:

D=\sum _{i=1}^{m}\frac{S}{B\phantom{\rule{2.77695pt}{0ex}}{\text{log}}_{2}\left(1-\frac{\lambda {\rho}_{0}\text{ln}{p}_{b}\left(i\right)/{r}_{c}^{n}}{1-\left(1-\lambda \right){\rho}_{0}\text{ln}{p}_{b}\left(i\right)/{r}_{c}^{n}}\right)}.

(7)

When the outage probability for the service transmission is {P}_{b}^{\mathsf{\text{out}}} and when the total transmission times *m* meets {\prod}_{i=1}^{m}\left(1-{p}_{b}\left(i\right)\right)\le {P}_{b}^{\mathsf{\text{out}}}, thus the service completes its transmission and its outage performance can be guaranteed.

Through Lagrangian algorithm, we can get an optimal *m*^{opt} from (7), which can meet the demand of minimizing transmission delay. Here for the OMS based on SC, the user selection ratio for each transmission is assume to be the same for simplicity. Under this condition, then the user selection ratio can be expressed as {p}_{b}^{\mathsf{\text{opt}}}=1-{\left({P}_{b}^{\mathsf{\text{out}}}\right)}^{\frac{1}{{m}^{\mathsf{\text{opt}}}}}. From (5), we can easily obtain the optimal user selection ratio {p}_{b}^{\mathsf{\text{opt}}} which minimizing transmission delay.

As for the proposed scheme, the traffics of BMS and EMS are different, so that the scheme does not totally coherent with the characteristics of SC, which means that even though the EMS can be decoded only by the successful decoding of BMS the user selection ratio for EMS does not need to be less than that of BMS. Exactly on the opposite, in order to let the EMS be transmitted successfully for the same transmission times as the BMS, the user selection ratio of EMS should be no less than BMS. In this way the user selection ratio of the enhanced layer *p*_{
e
}can be expressed as:

{p}_{e}=1-F\left({\left|h\right|}^{2}\le \frac{\left({2}^{{R}_{e}}-1\right)}{\left(1-\lambda \right){\rho}_{0}}\right)\ge {p}_{b}^{\mathsf{\text{opt}}}.

(8)

We define system throughput as the data amount transmitted by BS in a certain time. In our proposed mixed traffics transmission scheme, at each transmission time there are both BMS and EMS services, which makes the transmission rate very high. The system throughput can be expressed as:

{C}_{\text{opt}}(\lambda )=m\left\{{\mathrm{log}}_{2}\left(1+\frac{\lambda {\rho}_{0}{\left|{h}_{b}\right|}^{2}}{(1-\lambda ){\rho}_{0}{\left|{h}_{b}\right|}^{2}+1}\right)\times {p}_{b}^{\text{opt}}+{\mathrm{log}}_{2}\left(1+(1-\lambda ){\rho}_{0}{\left|{h}_{e}\right|}^{2}\right)\times {p}_{e})\right\}.

(9)

where the channel gain for the BMS user is {\left|{h}_{b}\right|}^{2}=\frac{\left({2}^{{R}_{b}}-1\right)}{\left[1-\left(1-\lambda \right){2}^{{R}_{b}}\right]{\rho}_{0}}.

Therefore, our problem is to find a scheme that maximizes the average system throughput. In the process of minimizing transmission delay, the user selection ratio of BMS and the corresponding user channel gain has been determined. Through Lagrangian algorithm, the optimal user selection ratio for an EMS can be obtained from (8) and (9) as:

{p}_{e}^{\mathsf{\text{opt}}}=-\frac{\left(1+\left(1-\lambda \right){\rho}_{0}{\left|{h}_{e}\right|}^{2}\right)\text{log}\left(1+\left(1-\lambda \right){\rho}_{0}{\left|{h}_{e}\right|}^{2}\right)\text{ln}2}{\left(1-\lambda \right){\rho}_{0}}\cdot \frac{1}{\partial {\left|{h}_{e}\right|}^{2}/\partial {p}_{e}^{\mathsf{\text{opt}}}}.

(10)

It should be noted that (10) is still not a closed-form equation. And the user selection ratio {p}_{e}^{\mathsf{\text{opt}}} must be chosen to meet the requirement of {p}_{e}^{\mathsf{\text{opt}}}\ge {p}_{b}^{\mathsf{\text{opt}}}. Combing (3), (8), and (10) the optimal channel gain threshold {\left|{h}_{e}^{\mathsf{\text{opt}}}\right|}^{2} for choosing EMS is calculated as follows:

{\left|{h}_{e}^{\mathsf{\text{opt}}}\right|}^{2}={\left\{\frac{2}{{p}_{e}^{\mathsf{\text{opt}}}n{r}_{c}^{2}}\gamma \left(2/n,{r}_{c}^{n}{\left|{h}_{e}^{\mathsf{\text{opt}}}\right|}^{2}\right)\right\}}^{n/2}.

(11)

Similar to (10), (11) is also not a closed-form equation, however, we can obtain {\left|{h}_{e}^{\mathsf{\text{opt}}}\right|}^{2} using iterations.

Substituting (11) into (10), we can get the optimal user selection ratio for EMS as {p}_{e}^{\mathsf{\text{opt}}}, and then we can have the optimal system throughput in (9). Unless the {p}_{e}^{\mathsf{\text{opt}}} channel gain of a certain kind of non-hot service is higher than the threshold in (11), this kind of service can be superimposed on hot service and be transmitted.

We define energy efficiency as the energy consumption per unit system throughput. In the proposed scheme, according to the definition, we have the energy efficiency expression in a time *T*:

E{E}_{\mathsf{\text{opt}}}\mathsf{\text{=}}\frac{P\cdot T}{{C}_{\mathsf{\text{opt}}}\left(\lambda \right)}=\frac{P\cdot T}{{\text{log}}_{2}\left(1+\frac{\lambda {\rho}_{0}{\left|{h}_{b}\right|}^{2}}{\left(1-\lambda \right){\rho}_{0}{\left|{h}_{b}\right|}^{2}+1}\right)\times {p}_{b}^{\mathsf{\text{opt}}}+{\text{log}}_{2}\left(1+\left(1-\lambda \right){\rho}_{0}{\left|{h}_{e}^{\mathsf{\text{opt}}}\right|}^{2}\right)\times {p}_{e}^{\mathsf{\text{opt}}}}.

(12)

### 4.2 Scheme A

In scheme A of changing unicast to multicast, multicast information and unicast information are transmitted separately and they use full transmission power, respectively. Let the worst channel Gaussian noise power in broadcast users is *N*_{1}, and the Gaussian noise power of unicast user be *N*_{2}, according to delay expression, we have the transmit delay:

{D}_{\mathsf{\text{schemeA}}}=\frac{S}{B\phantom{\rule{2.77695pt}{0ex}}{\text{log}}_{2}\left(1+GP{\left|{h}_{b}\right|}^{2}/{N}_{1}\right)}+\frac{S}{B\phantom{\rule{2.77695pt}{0ex}}{\text{log}}_{2}\left(1+GP{\left|{h}_{e}\right|}^{2}/{N}_{2}\right)}.

(13)

Comparing to the other two schemes, the transmission time of multicast and unicast is half because of its TDM transmission model. Thus for scheme A the system throughput is given as:

{C}_{\mathsf{\text{schemeA}}}\left(\lambda \right)=m/2\left\{{\text{log}}_{2}\left(1+GP{\left|{h}_{b}\right|}^{2}/{N}_{1}\right)+{\text{log}}_{2}\left(1+GP{\left|{h}_{e}\right|}^{2}/{N}_{2}\right)\right\}.

(14)

According to the definition of energy efficiency and combining (13) and (14), the energy efficiency of this scheme can be expressed as:

E{E}_{\mathsf{\text{schemeA}}}=\frac{PT}{{C}_{\mathsf{\text{schemeA}}}\left(\lambda \right)}=\frac{2PT}{{\text{log}}_{2}\left(1+GP{\left|{h}_{b}\right|}^{2}/{N}_{1}\right)+{\text{log}}_{2}\left(1+GP{\left|{h}_{e}\right|}^{2}/{N}_{2}\right)}.

(15)

### 4.3 Scheme B

Scheme B is the hot over non-hot traffic SC transmission scheme, let the worst channel Gaussian noise power in basic layer multicast business be *N*_{1}, we have the transmission delay of a certain service as (16), for the delay is determined by basic layer in this scheme.

{D}_{\mathsf{\text{schemeB}}}=\frac{S}{B\phantom{\rule{2.77695pt}{0ex}}{\text{log}}_{2}\left(1+\frac{\lambda GP{\left|{h}_{b}\right|}^{2}}{\left(1-\lambda \right)GP{\left|{h}_{b}\right|}^{2}}+{N}_{1}\right)}.

(16)

In this scheme, there are BMS and EMS in each transmission, and its transmission rate depends on the worst user channel condition of basic layer. Under a certain power allocation coefficient, the user with the worst channel condition cannot receive EMS completely which leads to outage. Considering the same transmission time as the other two schemes, the throughput in this scheme can be expresses as:

{C}_{\mathsf{\text{schemeB}}}\left(\lambda \right)=m\left\{{\text{log}}_{2}\left(1+\frac{\lambda GP{\left|{h}_{b}\right|}^{2}}{\left(1-\lambda \right)GP{\left|{h}_{b}\right|}^{2}+{N}_{1}}\right)+{\text{log}}_{2}\left(1+\left(1-\lambda \right){\rho}_{0}{\left|{h}_{es}\right|}^{2}\right)\times \left(1-{P}_{es}^{\mathsf{\text{out}}}\right)\right\}.

(17)

where {P}_{es}^{\mathsf{\text{out}}} is the outage probability of EMS in this scheme, and |*h*_{
es
}|^{2} denotes the corresponding enhanced layer user channel gain. As for the basic layer, its transmission rate is determined by the worst channel condition, so the outage probability is 0; while the outage probability of enhanced layer is depending on the worst channel condition of enhance layer users.

According to the definition of energy efficiency and combining (16) and (17), the energy efficiency of this scheme can be expressed as:

E{E}_{\mathsf{\text{schemeB}}}=\frac{PT}{{C}_{\mathsf{\text{schemeB}}}\left(\lambda \right)}=\frac{PT}{{\text{log}}_{2}\left(1+\frac{\lambda GP{\left|{h}_{b}\right|}^{2}}{\left(1-\lambda \right)GP{\left|{h}_{b}\right|}^{2}+{N}_{1}}\right)+{\text{log}}_{2}\left(1+\left(1-\lambda \right){\rho}_{0}{\left|{h}_{es}\right|}^{2}\right)\times \left(1-{P}_{es}^{\mathsf{\text{out}}}\right)}.

(18)