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3.4 QoS Mapping Problem Formulation and Solution

In the proposed framework, properly matching the RPI of a video packet with DiffServ classes differentiated by loss rate and by delay QoS of the network service is key to successful video delivery. For this, we first formulate the interaction framework into the QoS mapping problem described in the next section. Then, we derive the solution to this mapping problem to use it as a per-flow QoS mapping guideline.

3.4.1 QoS Mapping Problem Formulation

QoS mapping a relative prioritized packet onto a DS level can be formulated into the following optimization problem for a single flow. Each packet i of the flow, mapped to a certain network DS level q ({0, 1, . . . , Q-1}), gets an average packet loss rate 062fig01.gif by paying unit price pq. (DS level refers to a DiffServ class or traffic aggregate in IETF terminology, that is, a DS level is associated with a particular DSCP.) For the sake of simplicity, fixed RDI per flow is used to limit the allowed range of [qmin, qmax] to which packets can be mapped without violating the delay requirement. Then, the problem only needs to address the RLI-based optimization as follows. Each packet i of flow f assigned to a certain network DS level q will get an average packet loss rate 062fig02.gif by paying unit price pq. Given the acceptable total cost P(f), the effort to achieve the best end-to-end quality for flow f can be formulated by minimizing the generalized quality degradation QD(f):

Equation 3.11

03equ11.gif


 

 

 

where N(f) is the total number of packets in flow f. The QoS mapping is denoted by vector

062equ01.gif

and q(i) is the DS level to which the ith packet is mapped.

If we fix the mapping decision for all packets that belong to category k, Eq. (3.11) is simplified to Eq. (3.12). Note that the QD in Eq. (3.12) is an expected QD, where the loss effect of a packet belonging to category k is represented by the average loss effect 062fig03.gif:

Equation 3.12

03equ12.gif


where 063fig01.gif is the number of packets belonging to category k for flow f. Note that 063fig02.gif. The QoS mapping decision is denoted by the K-dimensional vector 063fig03.gif, where K is the number of the source’s packet categories.

3.4.2 Ideal Case for Per-Flow QoS Mapping

To solve the above optimization problem, consider the following factors. First, in a real situation, we must limit the resource allocation based on the TCA specified in the SLA with the DiffServ network. Thus, the above formulation must be constrained further by the traffic conditioning agreement before practical utilization. Next, the cost function given above may better reflect a real situation if the added complexity and out-of-order arrival handling cost due to dynamically scattered packets over multiple DS levels are included. These factors are assumed negligible at the current stage. Finally, pq(k) depends on DiffServ provisioning, which will be decided by the service provider and negotiated in the SLA.

To use this as a guiding solution, we assume that a DS domain provides a proportional DiffServ and maintains the specified proportionality persistently in a wide range of time scales. The optimal solution in such an idealized situation can be derived as follows. In an idealized, proportional, and differentiated service, the packet loss rate 063fig04.gif can be specified as a particular monotonically decreasing function of the DS level q. (In an proportional DiffServ, the actual value of the loss rates experienced can be a certain factor times the function, and the multiplicative factor varies with time.) We consider the case that the loss rate 063fig05.gif is inversely proportional to the DS level q.4 If unit price pq is assumed to be proportional to DS level q, the optimal mapping solution 063fig06.gif to (3.12) can serve as a guide for the mapping decision to be practiced. We can use the Lagrangian technique [108] for both constrained optimization problems (3.11) and (3.12).

Let us assume for simplicity that the value of q(k) can be taken from a continuum. We can then replace the inequality constraint of Eq. (3.12) with an equality constraint because the optimal solution obviously will have to spend all of the allowable cost P(f). According to the Lagrangian principle, there is a useful sufficiency condition that states that if a value λ can be found such that a value of 064fig01.gif, say 064fig02.gif(λ), minimizing

Equation 3.13

03equ13.gif


satisfies the equality constraint

Equation 3.14

03equ14.gif


then 064fig02.gif solves Eq. (3.12). Thus, we can guess a value of λ that minimizes Eq. (3.13). If the minimum point satisfies the equality constraint (3.14), then the solution is found. Otherwise, we can adjust the value λ and minimize Eq. (3.13) again. Furthermore, Eq. (3.13) is separable, so we can separately minimize

Equation 3.15

03equ15.gif


for each k. This minimizing value of q(k) can be found by drawing a line with slope -λ that is tangent to the convex hull of curve QDk 064fig03.gif versus cost. The cost pq(k) corresponding to the contact point of the tangent line is the cost of q(k) that minimizes Jk(λ) for the given λ. Figure 3.11 illustrates such minimizing values of q(k) for different values of k for a given λ. (Note that we are assuming that cost pq(k) is proportional to q(k).)

03fig11.gifFigure 3.11. Illustration for mapping from RLI to the DS level. (Given the relationship for RLI categorization, the loss rate per DS level, and the pricing strategy, the Lagrangian formulation leads to optimal mapping.)

With the assumption of loss rate function lq(i) = L / q(i) and unit price function Pl · q(i) (as illustrated by two of the curves in Figure 3.12(b)), the Lagrangian formula for Eq. (3.11) for each i becomes

Equation 3.16

03equ16.gif


Then, by searching around the convex hull of the graph of QD versus cost, we can obtain the optimal mapping solution for a particular value of λ. Also, we can obtain a closed-form solution by searching for stationary points where 064fig04.gif and applying equality constraint 064fig05.gif pq(i) = P(f). For a given operating λ, the minimum can be computed independently for each packet. The point on the QD versus cost characteristic that minimizes Ji(λ) is the point at which the line of absolute slope λ is tangent to the convex hull of the QD versus cost curve. The resulting closed-form solution is expressed by 065fig01.gif and 065fig02.gif

Further assuming knowledge of the video source’s RLI distribution, we can finally obtain a solution to Eq. (3.11). For this, we approximate the RPI distribution of sequences shown in Figure 3.10 as one of the three different RLI patterns shown in Figure 3.12(a): C1 U(i)2 for “Akiyo,” C2 U(i) for “Hall,” and 065fig03.gif for the “Stefan” and “Foreman” sequences.

03fig12.gifFigure 3.12. (a) RLI pattern with typical categorized video types; (b)an example of the packet loss rate and unit cost of DS level q.

This closed-form solution usually matches the actual numeric solutions depicted in Figure 3.13 for “Akiyo,” “Hall,” and “Stefan,” respectively. Note that the mapping distribution resembles that of Figure 3.10, which is again represented by patterns C1 U(i)2, C2 U(i), and 066fig03.gif, respectively. We can get some QoS mapping guidance from types such as RLIi, which represent C1 U(i)2, C2 U(i), and 066fig03.gif as the ascending order for source category k shown in Figure 3.12(a). The loss rate function of each DS level lq(i) is assumed to be two typical types, such as 066fig05.gif or 066fig06.gif, which are shown in Figure 3.12(b).

13Figure 3.13. Effective QoS mappings for different RLI source patterns in the loss rate of (a) (b) .

One Lagrangian formula of Eq. (3.16) is for the case of an RLI using C2 U(i). Now,

Equation 3.17

03equ17.gif


 

We can get a closed form in case of 066fig01.gif using 066fig08.gif and the equality constraint. Then, we can obtain the solution q(i), an optimal QoS mapping with the λ determined by total cost constraint, as follows:

Equation 3.18

03equ18.gif

If RLIi has another pattern, such as C1 U(i)2, 066fig09.gif, then an optimal QoS mapping takes the forms that are proportional to U(i) and 066fig10.gif, respectively. It matches the actual numeric solutions of the video test sequences of “Akiyo,” “Hall,” and “Stefan,” which are patterns similar to C1 U(i)2, C2 U(i), and 067fig01.gif, respectively. The numeric solutions under the same total cost constraint (i.e., all k is assigned to q = 4) are shown in Figure 3.13(a) as a categorized source version.

Now, let us consider the case in which the loss rate is a linear function of q and the cost pq is also a linear function of q (e.g., pq = Pl · q for some Pl). In such a case, the expected quality degradation caused by packet i, 067fig02.gif is a linear function of price pq.

Let us now narrow our scope further and consider the case in which source packets have been grouped into a small number of categories K, as assumed by Eq. (3.12). We denote the category of packet i by k(i). 067fig03.gif denotes the average RLI value of category k(i) or k. Figure 3.14 illustrates that the expected quality degradation measures, QDs, caused by packets i and j, are both linear functions of price. The slope of the line for packet i (or j) is affected by 067fig04.gif. Figure 3.14 also illustrates how to optimally map the grouped source categories k with different 067fig05.gif to network DS levels q.

03fig14.gifFigure 3.14. Optimization process with linear loss rate function.

Now, let us see how the optimal QoS mapping is determined in this linear loss rate case. In the example case in Figure 3.14, category k(i)’s QD is more sensitive to the price of q than category k(j)’s because the absolute value of 067fig06.gif is greater than that of 067fig07.gif.

As a simple illustration, let us suppose that we map both packets i and j to the same DS level that incurs cost p0. Next, we insert a perturbation of the QoS mapping. Now, say that we reduce the cost of packet i by amount p0p1 and increase the cost of packet j by p2p0 while preserving the total cost. This can be done by choosing p1 and p2 in such a way that p0p1 = p2p0. (For conceptual illustration, we are currently assuming that q is in a continuum.) After such a perturbation of QoS mapping, the resulting total of the expected degradation is reduced, while the total cost remains the same. Thus, for any pair of packets belonging to different categories k, we can keep adjusting their DS levels (prices) while keeping the total cost unchanged. If this process is continued, the resulting optimal mapping q(i) has monotonicity with respect to k(i). (The higher k(i), the higher q(i).) Furthermore, the majority of packets are mapped to the extreme values of q. In other words, most packets are mapped to the highest DS class or lowest DS class. The numerical solution in Figure 3.13(b) illustrates this.

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