Optimal and Fair Resource Allocation for Multiuser Wireless Multimedia Transmissions

According to the common sense in the field of video signal processing, the PSNR threshold can be set to different values, such as 40 dB, 35 dB, or 32 dB, representing perfect, good and acceptable video quality, respectively. The PSNR threshold can also be set dynamically according to the total resources available, the number of users, and so forth. As an illustration, we choose QoS threshold as PSNR = 35 dB corresponding to good video quality, that is, dB in (4). Denote as representing power required by users to achieve PSNR of 35 dB. Using co-opetition strategy, if ( means calculating the sum of all members in , i.e., .), the lower and upper bounds of achievable PSNR are set at dB and , respectively, and and  dB otherwise. Correspondingly, when we have , lower and upper bounds of rates are and , respectively, and and otherwise. In this paper, it is easy to calculate corresponding to PSNR threshold, for both (21) and (22) are invertible and monotonic increasing functions. Thus, given PSNR threshold, or not can be easily determined, and consequently, both and are known.

Given each user's utility definition in (21) and (22), system utility writes

where refers to as . We assume that capacity approaching channel codes is employed at PHY layer. Then our co-opetition strategy writes

where . Note that (24) has the same form as (8). The first constraint on the sum of the power (24) corresponds to in (8).

Using LOD, maximization of (24) can be decomposed into

where , and

where is defined as the upper bound of transmit power of user corresponding to .

The optimum variable of (25), , can be obtained by simply making the partial derivative of and let it equal to 0,

Then we have

where

As mentioned in Section 3.3, (26) can be solved at PHY layer by the weighted sum rate maximization with the constraints of total and individual power. Note that in (22) is concave and increasing with respect to , thus the item to be maximized in (26) is also concave increasing. The domain of (26) is formed by two linear inequalities, each of which forms a convex domain together with . Thus the domain of (26) is also convex, and (26) is accessible to conventional convex optimization techniques, such as feasible direction method and projected gradient method. In this paper the feasible increasing direction method is employed (see the Appendix for details).

So far, given fixed , two subproblems, (25) and (26), have been solved. We denote the optimal values of them with and , respectively. In the following, the optimum , denoted by , will be determined such that the sum of and is minimized, that is,

Note that, the dual function might not be differentiable or, in other words, (29) is not accessible to classical computational method, such as steepest descent method. In this paper we employ the sub-gradient method, which applies to both differentiable and nondifferentiable dual functions. Much like the feasible increasing direction method, sub-gradient method also searches the optimal and iteratively. The main iteration writes

where is the step-size which can be set as constant, and denotes the sub-gradient at . Note that, at rightly forms a sub-gradient, so the sub-gradient can be obtained almost without any cost.

In this experiment we focus on individual PSNRs in the case of two users. At APP layer, user 1 transmits Foreman sequence of CIF resolution at 30 Hz, and user 2 transmits Mobile sequence of CIF resolution at 30 Hz. At PHY layer, we set the bandwidth to = 250 kHz, and let the receiver noise power to be and for user 1 and user 2, respectively. Total transmit power varies from 50 to 800. Figure 2 shows the individual PSNRs achieved by these two schemes. If NBS_SP is employed, user 1 can achieve higher PSNR that user 2 or, in other words, it is very hard for user 2 to achieve satisfying video quality (PSNR 35). In the case of , user 1 can always be satisfied. Note in this case, user 1's video satisfaction degree increases very slowly as the PSNR increases, but significantly for user 2. Taking this observation into account, co-opetition imposes individual constraint on each user (see (4). For example, with , which can not satisfy two users simultaneously, co-opetition decreases user 1's PSNR to 35 dB, and consequently, user 2's PSNR achieves an improvement about 1 dB. If have , user 2's PSNR is improved such that user 2 is just satisfied. Note, in these two cases, co-opetiton keeps user 1 satisfied, while user 2 either be satisfied or achieve much QoS improvement. It is worth to mention that, under a given total transmit power constraint, NBS_SP can achieve higher total PSNR of two users than that in co-opetition. This is because the NBS_SP maximizes the sum of PSNRs without taking the individual PSNR constraints into account. The co-opetition works in quite a different way. It maximizes the sum of PSNRs under the constraints of individual PSNR. Therefore, the co-opetition is not only optimal (As stated in Section 1, in this paper the optimal means sum utility maximization under certain constraints, differing from unconstrained optimization.) , but also fairer than NBS_SP. This argument is further verified with other experiments

We study a more complicated scenario with nine users, each transmitting a sequence randomly selected from Table 1. They also experience different receiver noises randomly generated from 0 to 25. Figure 3 shows the number of satisfied users and the minimum PSNRs achieved by NBS_SP and co-opetition. We observe that, the co-opetition always outperforms the NBS_SP. For example, in the case of , co-opetition can make all users satisfied, but only 6 users satisfied by NBS_SP. With respect to the minimum PSNR, which is an important criteria evaluating system in the worst case, improvement of around 6 dB can be achieved when . Note that, NBS_SP can only make minimum PSNRs from about 25 dB to 29 dB, corresponding to poor video quality, while above 32 dB for co-opetition leading to acceptable video quality. Recall that, the co-opetition implies a judicious mixture of competition and cooperation. Through competition, the best system efficiency can be achieved. However, pure competition, for example, NBS_SP, might make very high PSNRs for some users, for example, users transmitting simple video content or having good channel quality, but low PSNRs for the others. This disadvantage is eliminated by co-copetition through introducing cooperation among users. Again, this experiment indicates that co-opetition provides a good tradeoff between system efficiency and fairness.

In previous experiments, the threshold PSNR is fixed to be 35 dB. In order to consider more fairness in resource allocation, adaptive threshold can be employed. As an illustration, we present a simple method to set the threshold PSNR. More optimal and fair scheme for determining the threshold PSNR will be investigated in our future work. We employ PSNR = 32 dB, 34 dB and 36 dB to represent acceptable, good and very good quality, respectively. Denote resources required by the three levels with , then threshold PSNR, , can be determined as follows

where is denote as total resources available.

Same system setup as that in previous experiment is used. We observe from Figure 4(a) that, co-opetition employing adaptive still outperforms the NBS_SP. Moreover, adaptive is more concerned with fairness than that using fixed threshold. For example, in the case of low resource, for example, , = 32 dB is selected. Consequently, an improvement of about 3 dB and 2 dB can be achieved for the minimum PSNRs compared to NBS_SP and co-opetition using fixed threshold (see Figure 3(b)), respectively. Note, these improvements are significantly important for users having low PSNRs. Although these improvements come from further decreasing the maximum achievable PSNR, it can provide fairer resource allocation. For instance, in Figure 4(a), it is very easy for all users to achieve similar quality level using co-opetition. Moreover, can also be set to a very high level, for example, 36 dB in the case of . An important advantage of this is that all users can be guaranteed high video quality, but cannot by fixed PSNR threshold and NBS_SP.

Our co-opetition is also optimal. As stated in Section 1, optimal means sum utility maximization (SUM) under individual constraints. The optimality is verified by experimental analysis in the case of two users. Results of two examples of them are shown in Figure 5(a) and Figure 5(b). System setup is the same as that in Figure 2. The optimal average PSNRs are achieved by exhaustive search. Recall that the LOD method consists of inner and outer iterations. In each inner iteration, the power allocation is initiated corresponding to for Figure 5(a) and for Figure 5(b). In the outer iteration, the values of and are initialized randomly. Figures 5(a) and 5(b) show the results of outer iterations.

From these two figures, we can see that our strategy is optimal under individual constraints. In Figure 2, cannot satisfy two users simultaneously. Therefore the PSNR of user 1 is pegged at the threshold PSNR = 35 dB. The optimal average PSNR can be achieved after 14 iterations. In Figure 5(b), can make satisfying PSNR for both the two users. We observe that, user 2's PSNR has only little fluctuation, and converges to the threshold. At the optimal power allocation, both the two users' PSNRs are above or equal to the threshold. All these coincide with the results in Figure 2.

To summarize, threshold PSNR plays importantly in adaptive/nonadaptive co-opetition strategies. It provides radio resource allocation (RRA) with more flexible tradeoff between system efficiency and fairness among users.