- Introduction
- Overall Proposed QoS Mapping Framework in a DiffServ Network
- Video Packet Categorization Based on the Relative Priority Index (RPI)
- QoS Mapping Problem Formulation and Solution
- Experimental Results
- Conclusions

## 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
by paying unit price *p*_{q}. (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 [*q*_{min}, *q*_{max}]
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
by paying unit price *p*_{q}. 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)}:

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

and *q*(*i*) is the DS level to which the *i*th 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 :

where
is the number of packets belonging to category *k* for flow *f*. Note
that .
The QoS mapping decision is denoted by the *K*-dimensional vector ,
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, *p*_{q(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
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
is inversely proportional to the DS level *q*.^{4}
If unit price *p*_{q} is assumed to be proportional to DS
level *q*, the optimal mapping solution
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 ,
say (λ),
minimizing

satisfies the equality constraint

then
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

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
*QD*_{k}
versus cost. The cost *p*_{q(k)} corresponding to
the contact point of the tangent line is the cost of *q*(*k*) that
minimizes *J*_{k}(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
*p*_{q(k)} is proportional to *q*(*k*).)

**Figure 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 *l*_{q(i)} = *L* / *q*(*i*) and unit price function *P*_{l} · *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

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
and applying equality constraint
*p*_{q(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 *J*_{i}(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
and

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): *C*_{1} *U*(i)^{2} for “Akiyo,”
*C*_{2} *U*(*i*) for “Hall,” and
for the “Stefan” and “Foreman” sequences.

**Figure 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 *C*_{1} *U*(*i*)^{2},
*C*_{2} *U*(*i*), and ,
respectively. We can get some QoS mapping guidance from types such as *RLI*_{i},
which represent *C*_{1} *U*(*i*)^{2}, *C*_{2}
*U*(*i*), and
as the ascending order for source category *k* shown in Figure
3.12(a). The loss rate function of each DS level *l*_{q}(*i*)
is assumed to be two typical types, such as
or ,
which are shown in Figure
3.12(b).

**Figure 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 *C*_{2} *U*(*i*). Now,

We can get a closed form in case of
using
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:

If *RLI*_{i} has another pattern, such as *C*_{1}
*U*(*i*)^{2}, ,
then an optimal QoS mapping takes the forms that are proportional to *U*(*i*)
and ,
respectively. It matches the actual numeric solutions of the video test sequences
of “Akiyo,” “Hall,” and “Stefan,” which
are patterns similar to *C*_{1} *U*(*i*)^{2},
*C*_{2} *U*(*i*), and ,
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 *p*_{q} is also a linear function
of *q* (e.g., *p*_{q} = *P*_{l}
· *q* for some *P*_{l}). In such a case, the
expected quality degradation caused by packet *i*,
is a linear function of price *p*_{q}.

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*).
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 .
Figure 3.14 also
illustrates how to optimally map the grouped source categories *k* with
different
to network DS levels *q*.

**Figure 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
is greater than that of .

As a simple illustration, let us suppose that we map both packets *i* and *j* to the same DS level that incurs cost *p*_{0}. Next, we insert a perturbation of the QoS mapping. Now, say that we reduce the cost of packet *i* by amount *p*_{0}–*p*_{1} and increase the cost of packet *j* by *p*_{2}–*p*_{0} while preserving the total cost. This can be done by choosing *p*_{1} and *p*_{2} in such a way that *p*_{0}–*p*_{1} = *p*_{2}–*p*_{0}. (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.