3.5.1 Alternative Congestion Feedback
So far, it is proved that distributed congestion control algorithms (e.g. the primal-dual algorithm) might become unstable when buffers are modeled explicitly. To remedy this issue, a slightly different pricing mechanism will be proposed below to ensure the stability under both the primal-dual and dual algorithms. For simplicity, the result will be explained for a constant Φ. Here, we again assume that the buffer dynamics can be modeled as (3.39).
Theorem 5. The dual algorithm (3.53) and the primal-dual algorithm (3.38) are both globally asymptotically stable with the unique equilibrium point (x∗,Φ−1p∗), provided the source price vector q(t) is taken asRTΦp(t) as opposed toRTp(t).
Proof: The proof will be provided here only for the primal-dual algorithm because the proof for the dual algorithm is similar. By defining ˜p(t) as Φp(t) and considering the source priceq(t) asRTp(t), it can be shown that the modified primal-dual algorithm turns out to˜
be
˙
xs(t) =ks Us0(xs(t))−qs(t)
, ∀s∈ S (3.56a)
p˙˜l(t) =hl(yl(t)−cl), ∀l∈ L, (3.56b)
where ˜p1, ...,p˜L denote the entries of ˜p(t). Notice that the above algorithm is a special type of the standard primal-dual algorithm given in (3.6), for which it is known that xs(t) → x∗s and ˜pl(t) → p∗l as t goes to infinity. This result implies that x(t) → x∗ and p(t) = Φp(t)˜ → p∗ as t increases. Consequently, the states x(t) and p(t) of the modified primal-dual algorithm with q(t) = RTΦp(t) converge tox∗ and Φ−1p∗, respectively. This
completes the proof.
As illustrated in Examples 3 and 4, if each source s ∈ S updates its transmission rate xs(t) based on the price qs(t) obtained by just adding up the link prices along the route of source s, the corresponding congestion control algorithm may not be stable. Instead, Theorem 5 suggests taking the source price qs(t) as the sth entry of the vector RTΦp(t).
This can be implemented using the following scheme.
For every s ∈ S, let a zero price value be initially assigned to the flow of source s. As this flow passes through every link, say link l, it reports its price value to the link and then increases its price by θlsbl(t). On the other hand, each link maintains a price for itself (say
˜
pl(t)) by adding up the source prices reported to the link by its incoming flows. Now, the acknowledgment of the flow in the return path accumulates the new link prices in its route and reports it back to the source.
It can be verified that the above strategy corresponds to the new price updating q(s) = RTΦp(t) that is needed in the strategy proposed by Theorem 5.
To illustrate the aforementioned idea, let the new pricing mechanism proposed in Theo- rem 5 be deployed to fix the instability of the primal-dual algorithm for the network studied in Example 3. To this end, consider the initial parameters
x(0) = h
10 10 · · · 10 i
, b(0) =v=h
20 20 · · · 20 i
.
(3.57)
The transmission rates and buffer sizes associated with the modified primal-dual algorithm
0 10 20 30 40 50 60 70 80 0
5 10 15 20 25 30
Time
Transmission Rate
Figure 3.5: This figure illustrates the stability of the primal-dual algorithm using the mod- ified buffer-size pricing mechanism for Example 3.
are plotted in Figures 3.5 and 3.6 to demonstrate that all these signals converge and therefore the resulting congestion control algorithm is stable.
3.5.2 Nonzero Buffer Assumption
Despite the fact that Theorem 1 derives a mathematical model for buffer sizes in a general setting, the main results developed in this work (say in Sections IV-A and IV-B) rely on the assumption that the buffer sizes b1(t), ..., bL(t) never become zero. In what follows, we elaborate on the validity of this assumption and justify why the removal of this assumption does not change the conclusions drawn in this chapter.
First, consider the constant-buffer-partitioning case studied in Subsection IV-A-1. A mild assumption used in this part was that all links of the network would be bottleneck links under the optimal transmission rates of all users. This implies that p∗ is a strictly positive vector and as a result b∗ > 0. Hence, if the initial buffer sizes b1(0), ..., bL(0) as well as the initial transmission rates x1(0), ..., xS(0) are all in the neighborhood of their optimal values, then the assumption of positivity of b1(t), ..., bL(t) at all timest≥0 would be met as long as the algorithm is stable. The same argument also holds for the case with state-dependent buffer partitioning coefficients studied in Subsection IV-A-2.
Recall that Examples 3 and 4 demonstrate that the primal-dual algorithm may not be stable for both constant and state-dependent partitioning coefficients. Since the assumption of strict positivity of all buffer sizes is used in these examples, it could be conjectured that
0 10 20 30 40 50 60 70 80 5
10 15 20 25 30
Time
Buffer size
Figure 3.6: This figure illustrates the stability of the primal-dual algorithm using the mod- ified buffer-size pricing mechanism for Example 3.
this instability phenomenon may not occur in practice due to the underlying assumption not being valid. Nonetheless, it can be argued that the primal-dual algorithm is still unstable for Examples 3 and 4 even if the buffer sizes are allowed to become zero. Indeed, it follows from the definition of stability that if the network in either Example 3 or Example 4 is stable (in the absence of the aforementioned assumption), thenb1(t), ..., bL(t) always stay positive and bounded for the initial values b1(0), ..., bL(0), x1(0), ..., xS(0) sufficiently close to their optimal values. Now that the buffer sizes are always positive in the transient time, the underlying assumption is automatically satisfied, under which the instability of the network was already proved.
As far as the primal-dual algorithm with the new pricing mechanism q(t) = RTΦp(t) given in Theorem 5 is concerned, the relation b∗ = Φ−1p∗ is satisfied. Now, notice that although p∗ is a positive vector, Φ−1p∗ might have some negative entries. This yields the contradictory result that the optimal buffer sizes b∗1, ..., b∗L may not be all nonnegative, which implies that some of the buffers 1, ..., L must become empty during transient time.
To remedy this problem, note that if the pricing mechanism q(t) = RTΦ(p(t)−ν) is used instead, for some positive vector ν, then the steady-state vector b∗ becomes equal to Φ−1p∗ +ν. Hence, a proper choice of ν makes the buffer sizes at the equilibrium point strictly positive and, therefore, it resolves the problem. Note that the interpretation of this technique is simply as follows: the buffer size that each link reports to the corresponding users should be less than the true value so that users transmit data at higher rates to keep
the buffers nonempty. This idea of reporting some virtual values for buffer sizes is also applicable to the primal-dual algorithm with the standard pricing mechanism.