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In order to illustrate the gains and operation of the randomized feedback scheme we will consider a single cell downlink with N = 15 channels. The channels are assumed to be i.i.d. across users, frequencies, and time slots and the achievable rates (in bits per time slot) are as in Table 5.1 (the rates are calculated according to the LTE specifications, with Ts= 1ms ).

Rate (bits/slot): 0 25 39 63 101 147 197 248 Probability: 0.03 0.04 0.05 0.05 0.06 0.06 0.09 0.09 Rate (bits/slot): 321 404 458 558 655 759 859 933 Probability: 0.1 0.1 0.09 0.06 0.06 0.05 0.04 0.03

Table 5.1: Achievable rates and probabilities used for the simulations We set the traffic patterns to be i.i.d. Poisson, with the same arrival rate, λ bits per slot for each user. We run simulations lasting 10000 time slots each for different arrival rates and plot the average total queue length at each simulation for SSF, randomized SSF with probing probability as derived in Theorem 5.3.1

5.3. Randomized scheme 400 450 500 550 600 650 700 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5x 10 4

Arrival Rate (bits/timeslot)

Average total queue length (bits)

SSF

"optimized" randomized SSF (p>1.0767) "approximate" randomized SSF (p=0.5161)

Figure 5.1: Average Total Queue Length for Different Mean Arrival Rates for 9 users and β = 0.1

(denoted ”Optimized randomized SSF”) and the approximate probability as set in Section 5.3.3.

At first we simulate the system with β = 0.1 and K = 9 users. In this case full probing is possible. The results are plotted in Fig. 5.1. We can see that the randomized version of the algorithm obtained via optimizing the upper bound is the same as SSF here, while the probability of probing in the approximate algorithm is smaller. Also, the performance of the approximate algorithm is better from the other two.

In Fig. 5.2 we present the results of a scenario with K = 25 users and two different values of β, namely 0.05 and 0.01. Note that in both of these cases full probing is not possible.

Again, the approximate version of the algorithm results in a lower probability to feed back than the version that optimizes the upper bound and performs better. In turn, the latter version performs better than SSF. Also, from Figures 5.1 and 5.2 we can see that the stability region of the system shrinks under all algorithms as β and/or the number of users K grow larger. In the case of SSF this happens because as these parameters grow larger, more time needs to be devoted to channel feedback, leaving fewer time for transmission. However, in the randomized versions the main reason for the rate decrease is that it becomes more possible that the user with maximum weight will not feed back his channel state and subsequently another user will be scheduled instead. As we can see in the figures, the decrease in the stability region is slower in the case of the randomized algorithms. This demonstrates that there is a gain with respect to SSF algorithm and moreover that the relative gains of randomizing the SSF

5.4. Scheme based on stopping 10 60 110 160 210 260 310 340 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5x 10 4

Arrival rate (bits/timeslot)

Average total queue length (bits)

SSF, β=0.1 "optimized" randomized SSF, β=0.1 (p=0.3589) "approximate" randomized SSF, β=0.1 (p=0.1649) SSF, β=0.05 "optimized" randomized SSF , β=0.05 (p=0.8075) "approximate" randomized SSF, β=0.05 (p=0.3711)

Figure 5.2: Average Total Queue Length for Different Mean Arrival Rates for 25 users and different values of β

algorithm are bigger when there are more users and/or channel probing is more costly.

The main reason why the approximate algorithm outperforms the one op- timizing the probability so that the increase guaranteed by Theorem 5.3.1 is that the bound to which the optimized probability corresponds to is not tight. In fact, the theoretical analysis in this paper has been done in terms of region increase guarantee and this has been studied using the lower bounds developed in the previous sections. These bounds are not necessarily tight which means that the real expansion is higher than the lower bound. Recall that in the course of derivation of equation (5.5), the quantity was bounded assuming implicitly that (i) if the user with the maximum weight has not probed a channel then no user is scheduled in the channel and (ii) the user polled by the base station is never the user with the maximum weight. As seen previously the approximate probability p∗_uniis less or equal than the one obtained through full optimization.

5.4

Scheme based on stopping

The randomized scheme discussed in Section 5.3 does provide an increase in the stability region of the system with respect to the SSF algorithm, however its main drawback is that the derivation of the probability with which a user should feed back is based on a loose upper bound on this increase. In addition, this probability depends in general on the channel statistics and queue lenghts every slot (the approximate version seems to yield good performance but it is still an approximation). In this Section we propose an alternative scheme, again

5.4. Scheme based on stopping

based on the SSF rule and with no need of the channel statistics, where the base station decides when to stop the feedback procedure and schedule a user, in an attempt to make the quantity of (5.2) as big as possible.