Encuesta dirigida a los profesores de la Unidad Educativa “Riobamba”

The rate proportional fair method is proposed for non-real-time flows. Non-real-time flows are service flows without a strict time deadline. Because of the flexibility in time deadline, absolute bit rate targets are not needed in the short-term and hence these flows can be supported in non- guaranteed bit rate mode. An example is a file download from the Internet, for which the service quality is perceived by users in the order of few seconds, while the transmit time interval is in the order of few milliseconds. Moreover, there is no strict time deadline to deliver the packets within some seconds. Thus a scheduler can deliver the packets in a flexible time duration under a fairness scheduling rule. Proportional fairness is a rule which looks for a fine balance between the fairness and efficiency in scheduling. The ultimate target of the proportional fair method in cellular wireless is to serve each user on its own relative channel peak, so that both efficiency and fairness are achieved.

In OFDMA systems, proportional fairness can be achieved by utilising both time slots and frequency subcarriers; unlike CDMA systems where only time slots are utilised. A possibility in OFDMA from the perspective of downlink is that fairness can be achieved by utilising an independent MCS value in each frequency resource block; unlike the CDMA systems where a single MCS level is loaded in an entire time slot. Below we describe algorithms that can exploit this advantage.

4.2.1.1 Optimal algorithm

Assume a frequency selective channel, where a specific MCS is supported on each subcarrier. Suppose ukf is the MCS that can be supported for user k in subcarrier f and xtkf denotes the

resource allocation in terms of time slots that is allocated to userkon subcarrierf. The objective of the proportional fair scheduler is to maximise the log-sum-of-rates1 _{subject to resource constraints}

that is denoted as maxT k=K X k=1 f=LM X f=1 log[xt_kfukf] (4.1) X k xkf = 1 ∀f (4.2) 0_≤xkf ≤1 (4.3)

From the definitions of MCS and resource blocks, we know that the same MCS is applied for the

L subcarriers in a resource block. Thus we employ this information for computation.

By denoting a resource block using variablem, there areM such resource blocks of L subcarriers each. Thus ukm is the MCS that is loaded for userk on resource block m. Replacing Pxtkf with

xkm, we get, maxLT k=K X k=1 m=M X m=1 logxkmukm (4.4) X k xkm= 1 ∀m={1..M} (4.5) xkm ≥0. (4.6)

Thus in (4.4)-(4.6), we drop the time indextand denotexkmas the amount of subcarrier allocation to user kout of each resource block m. We have also dropped the contraint xkm ≤1 because this condition becomes implicit w.r.t (4.5) and (4.6).

The objective in (4.4) is convex in terms ofxkm and the constraints are linear and therefore this is a convex optimisation problem.

Below we show the optimality condition of the variables xkm using the Lagrangian multipliers

approach. For this purpose, let the multiplersλm be introduced for the equality constraint in (4.5) and the multiplers βkm for the inequality constraint in (4.6).

Upon taking the derivate of Lagrangian function of (4.4)-(4.6), equating it to zero and using the fact that βkm= 0, for xkm >0, the following condition is written

xkm = 1 λk−βkm − X j6=m xkj ukj ukm , (4.7) where βkm>0,if xkm = 0, (4.8) βkm= 0,if xkm >0. (4.9)

The optimal values of xkm can not be obtained trivially from (4.7). The optimal values of λk first need to be obtained, which however is dependent on all other xim, for i 6= k because of the constraint (4.5). Thus finding the optimal solution via Lagrange multipliers requires a multi dimensional search to update all the multipliers based on the constraints in (4.5) and (4.6). Methods for efficient computation exist, for example the interior-point method in convex optimisation [105]. An efficient implementation of the interior-point techniques with some advancements is available in Matlab-based tool such asfmincon. We use this function to compute the optimal solutions. We remark that optimal solutions would result in sub-resource block scheduling to users (i.e resource block scheduling with fractional values) which is not part of LTE. However optimal solutions serve as a useful upper bound to scheduling performance.

4.2.1.2 Suboptimal algorithm

As can be deduced from the optimality conditions above, the optimal solution for proportional fairness is not easy to obtain in OFDMA. Thus, suboptimal resource allocation approaches for fast and efficient maximisation of the objective in (4.4) are needed. Here we present a new scheme which extends the W-CDMA version to include frequency subcarriers of OFDMA. To develop suboptimal algorithms, we treat the problem as a selection problem of user-resource block pairings and thus

look for binary integer (0 or 1) solutions of the variables xkm. Specifically, we wish to use the MCS information of user-resource block pairing. A formal description of the suboptimal algorithm is presented below, which we term the gradient search scheme. We have presented part of these results in [120].

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

2. Calculate weight factors for users as

wk=

X

m

xkmukm+δ, (4.10)

whereδ is a regularisation factor to set a non zero weight valuewk.

3. Select the resource block and users pairing according to (k∗, m∗) = arg maxkmaxm∈V wkukm, i.e, search for the best combination of user index and resource block. Setxk∗_m∗= 1. Remove

the resource blockm∗ _{from the poolV. Repeat until V is empty.}

The gradient search scheme allocates a resource blockmto userkif the pairing (m, k) corresponds to the maximum bit rate gradient. To perform the selection to a user, a resource block sort is needed for each user in each iteration based on the MCS values. This effectively means that the resource blocks are not allocated in some fixed order.

Simulation results for the following schemes : a) Round robin scheduling b) Suboptimal gradient search and c) Optimal proportional fairness are presented in Figure 4.3