El Reciclaje

The utility proportional fair method 2 is used for real-time flows in the non guaranteed bit rate mode. Packets belonging to the real-time service flows need to be scheduled within a time deadline. An example is a multi media streaming video, for which a user’s experience is affected by delay

2_{we call this utility proportional fair because it targets proportional fairness of a utility metric. Note that pro-}

0 5 10 15 20 25 30 35 0 0.2 0.4 0.6 0.8 1

Log Sum of Rates

Empirical CDF

36 users, 48 resource blocks, 15 dB average SNR Round Robin

Gradient Search − sub optimal Optimal PF

Figure 4.3: Performance comparison of resource allocation schemes in terms of the proportional fair objective.

in packet arrivals, which are called jitters. Thus for these kind of applications, a short term bit rate target, denoted as Ωk is applicable in scheduling to userk. It may not always be possible to meet the requirement of each user when there are too many users. Thus in the case of congestion in the network, a scheduler has to evaluate the benefit of a rate allocation to a user. The benefit can be modeled using a utility function wherein, above a minimum data rate, denoted as Ωmin, the usefulness of the data rate increases steadily. Below this minimum data rate, the service quality becomes unacceptable. Importantly, there is not a significant change in the perceived service quality for data rates from zero until Ωmin. This minimum bit rate is thus used as a reference by the scheduler to measure the quality of service to a user. Similarly, above the target bit rate Ωk, there is no significant change in the perceived service quality. To model this approach, we make use of sigmoidal logistic function. The sigmoidal function is described by three regions : a) minimum bit rate Ωmin , b) inflection point ˜D, where the function changes convexity and c) the saturation bit rate, denoted as Ωk, which is also the target bit rate for the scheduler.

One can approximate linearity in the region between Ωmin and Ωk, in which case the relationship to the inflection point ˜Dbecomes

˜

D_≈ Ωk+ Ωmin

2 . (4.11)

The sigmoidal utility 3 is defined using the value of ˜Das

Ψk = [1 + exp [s[ ˜D−(

X

m

xkmukm)]]]−1, (4.12)

where we recall thatP

mxkmukm is the data rate that is served to a userk.

The parametersin (4.12) characterises the slope of the function and can be set to achieve a certain value of Ψk at Pmxkmukm = Ωmin. Let Ψk = 0.1 forPmxkmukm = Ωmin. Based on this value, the parameters can be found to be _˜ ln 9

D−Ωmin.

Now we wish to perform OFDMA resource allocation using the sigmoidal utility function as a tool. The maximisation problem for K users is written as

max k=K X k=1 logΨk (4.13) X k xkm = 1 ∀m (4.14) 0_≤xkm≤1. (4.15)

It is noteworthy that we use an objective of log-sum-of-sigmoidals using the function (4.12) instead of sum of sigmoidals. The goal is to exploit the fact that the sigmoidal function is a log-convex func- tion. Therefore, any maximal value of the objective in (4.13) is also the global optimal. To obtain the maximal values ofxkm, we may again use an interior point method in convex optimisation. For a suboptimal scheme, we use a gradient approach as before in Section 4.2.1.2 based on the gradients of utilities Ψk.

1. Initiate resource block set V consisting of all the resource blocks. Initiate xkm = 0∀k = 1..K,_∀m = 1...M.

2. Calculate weight factors for users in resource blockm as

wk= Ψk+δ, (4.16)

where Ψk ≥ 0 ∀k is obtained using (4.12) for given values of Ωmin, Ωk and s. δ is a regularisation term to set a minimum value ofwk.

3. Select the resource block and user pairing according to

(k∗, m∗) = arg max

k maxm∈V wk(Ψk(xkm = 1)−Ψk(xkm = 0)), (4.17)

i.e, search for the best gradient of log-sigmoidal using user index and resource block. Set

xk∗_m∗ = 1. Remove the resource block m∗ from the pool V. Repeat until V is empty.

Simulation results are presented in Figure 4.4

−1200 −100 −80 −60 −40 −20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Log Sum of Sigmoidals

Empirical CDF

36 users, 48 resource blocks, 15 dB average SNR Round Robin

Suboptimal gradient search Optimal solution

Figure 4.4: Performance comparison of resource allocation schemes in terms of log-sum-of-sigmoidal utility function. Target rate Ωk=2.25 Mbps. Ωmin=Ω₂k Mbps

These results of utility PF have shown the case when two data rates Ωmin and Ωk characterise the quality of service. An intuitive understanding of utility PF is that, the benefit of allocating a

rate less than Ωmin is low and Ωk acts as an upper limit. The limiting case of Ωmin= Ωk, will be addressed in the following sections as a separate problem, in what is termed as guaranteed bit rate.