Descripción sumaria del procedimiento utilizado por Gestora Canaria (2008)

5. CAnARiAS Un CAMinO PROPiO En EL CAMPO DEL RECiCLAJE Y LA ADAPTACiÓn A LAS nUEVAS DiRECTiVAS

5.3. Descripción sumaria del procedimiento utilizado por Gestora Canaria (2008)

In this section, we discuss several issues in implementing the algorithm described in Section 9.1.3. The non-preemptive delay composition theorem, used in deriving the delay constraint 9.4 assumes prioritized scheduling at each intermediate node. As exact prioritized scheduling is impossible to achieve in a distributed wireless network, we implement approximate prioritized scheduling as follows. Each node maintains independent queues for packets of each priority class and always chooses to transmit a packet of higher priority before transmitting a packet of lower priority. Further, in order to prioritize packet scheduling across neighboring nodes, we allow packets of higher priorities to have a smaller minimum contention window size compared to packets of lower priority. In our simulations, we support eight priority levels. This implemen-

tation is similar to the emerging 802.11e standard [1] for the Enhanced Distributed Coordination Function (EDCF), except that the queues do not act as virtual terminals, and packets from higher priority queues are always picked ahead of packets from lower priority queues. Prioritized scheduling can also be achieved using MAC protocols such as [30]. This problem has been addressed in past literature and is orthogonal to the problem we address. Better solutions to MAC layer prioritization will enhance the performance of our algorithm.

Note that the update equations for the algorithm require that nodes are aware of the dual costs and values for X’s and Y ’s computed by nodes in their interference neighborhood, and not just the communication neighborhood (for example, the capacity constraint ensures that the sum of all flow rates in the interference neighborhood of each link is at most as large as the capacity of the link). This behooves the presence of a protocol at the network layer that can obtain this information. Identifying nodes that lie within interference range of a given node is a challenging problem that has been addressed in prior literature in wireless networks (such as [98]). We empirically measured the interference range to be 440m for each link when the communication range was 200m, and used this value in our simulations.

The NUM formulation assumes concave utility functions for the flows, and the optimization objective is to maximize the sum of utilities of all flows in the network. Utility functions serve as a measure of user satisfaction, and can also be used to control the trade-off between efficiency and fairness. For example, [65] defines a family of utility functions targeted at fairness. As the problem of choosing utility functions has been dealt with in literature, we keep our theoretical framework general and not dependent on any particular utility function. In our simulations, we consider two simple logarithmic utility functions to be used with the NUM formulation. In the first utility function, the utility is assumed to be proportional to priority:

fs(xs) = (M ax P riority − Ws+ 1) log xs, (9.20)

where Ws is the priority of flow s (higher value implies lower priority), and M ax P riority is the maximum

permissible priority value. Remember, however, that priority in our framework is set according to urgency, which in general, may not be proportional to utility. Hence, we also consider a utility function that is orthogonal to priority (i.e., urgency), where all flows have the same importance:

fs(xs) = log xs, (9.21)

As mentioned earlier, the above utility functions are true only when delay constraints are met. If delay constraints are violated, the utility is zero. In practice, however, applications such as video streaming can tolerate some deadline misses. Hence, in the evaluation section, rather than dropping utility to zero abruptly,

we drop it linearly in the miss ratio as follows:

App U tility = fs(xs) ∗ (1 − 10 ∗ DM R), DM R < 0.1

= 0, otherwise (9.22)

where fs(xs) is defined as in Equations (9.20) or (9.21), and DM R is the deadline miss ratio for the flow.

Note that the above application utility function reduces to fs(xs), when no deadline misses occur. As

the NUM-based algorithms operate within the feasible region where no deadline misses occur, the optimal solution also lies within this feasible region. At the optimal solution (and at all points within the feasible region), the value of the above application utility function would be the same as that of fs(xs). Hence, the

operation of the NUM-based algorithm remains the same regardless of how utility is defined outside the feasible region. We therefore use fs(xs) defined as per Equations (9.20) or (9.21) as the utility function

for the NUM-based algorithms, but use the application utility defined in Equation (9.22) as a metric of performance in our evaluations in Section 9.1.5.

Finally, the update equations are based on update parameters α. For the update equation for λ, we used a value of 0.5 (α3 = 0.5). For µ, we used a value of 0.2. For all the other update equations we used a

value of 1.0. While we observed that changing these values affect the rate of convergence, for most values of the update parameters, the flow rates converged within 100 iterations of the algorithm. Convergence of a number of NUM problems has been studied in the past [15, 35]. Theoretically studying the convergence and stability properties of the algorithm will be a useful future work.

9.1.5 Simulation Results

In this section, we present results from simulations on ns2 [67]. Our default system consists of 50 nodes placed uniformly at random. We assume that nodes are stationary. The MAC layer protocol is assumed to be 802.11, with prioritized scheduling as described in Section 9.1.4. We consider a default of 5 elastic flows in the system, whose sources and destinations are chosen uniformly at random. The average hop length of the flows was between 3 and 4. The routing protocol is assumed to be DSDV. Link bandwidth is assumed to be 1 Mbps. One iteration of the rate allocation algorithm executes every 0.5 seconds. The flows are assumed to transmit at a constant bit rate chosen by the rate allocation algorithm. For traffic that is bursty, such as video, application-level buffers can be used to smoothen the burst and transmit at the average (roughly constant) rate. Prior theory such as [21] can be used to bound the delay due to buffering as a function of the original burstiness. While the end-to-end delay of a packet is the sum of this buffering delay and the network delay, in this work, we concern ourselves only with the network delay. The buffering delay can be

estimated as discussed in prior work [21].

In our evaluation, we compare the proposed deadline-aware rate allocation algorithm (which we call ‘NUM with Delay Constraints’), with four other algorithms. The first is a rate allocation algorithm determined by the traditional NUM formulation without the delay constraints [59, 51], that maximizes utility in the presence of capacity constraints alone. This algorithm uses prioritized scheduling at the MAC layer and is referred as ‘NUM without Delay Constraints’. In order to demonstrate that prioritized scheduling broadens the space of acceptable flow rates, we choose the second algorithm to be the same as the first except that scheduling at the MAC layer is FIFO. We call this ‘NUM w/o Delay Constraints w/o Prioritized Scheduling’. For the third algorithm, we simulate prioritized scheduling at the MAC layer, in the absence of any rate control (called ‘No Rate Control’). Here, each source of flow is assumed to transmit at the application- specified maximum packet generation rate. This serves as a baseline to analyze the advantages of applying rate control. Finally, to show that our algorithm can work with any prioritized MAC protocol, we evaluate the deadline-aware rate allocation algorithm with 802.11e as the MAC layer protocol (we call this ‘NUM with Delay Constraints with 802.11e’). Table 9.2 shows the differences between the different algorithms. For the rate allocation algorithms based on the NUM formulations with and without the delay constraints, we evaluated both utility functions specified in Section 9.1.4 under Equations (9.20) and (9.21). The results when all flows have the same utility function (Equation (9.21)) are labeled with the suffix ‘Eq. Util. Flows’. The results when the utilities of flows are directly proportional to their priority (Equation 9.20) do not have this suffix. The above algorithms are compared in terms of the achieved throughput and deadline miss ratio for each priority class.

Algorithm Rate Delay Prior.

Control Constr. Sched.

No rate control No No Yes

NUM w/o delay Yes No Yes

NUM w/o delay

Yes No No

w/o prioritization

NUM with delay Yes Yes Yes

NUM with delay,802.11e Yes Yes 802.11e Table 9.2: The different algorithms being compared

All values presented are averaged over 50 simulation runs each lasting for 100 seconds. For each run, results were collected after a duration long enough to ensure that the rate control algorithm had stabilized. Y error bars indicate 95% confidence intervals.

Audio and video flows are typically governed by a maximum traffic generation rate. We imposed different maximum application traffic generation rates that would act as a ceiling for the transmission rates for each flow. We considered traffic generation values equal to 25, 50, 75, 100, 125, and 200 Kbps. A traffic generation

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 200 125 100 75 50 25 0

Deadline Miss Ratio