conventional TCP variants are shown. They perform significantly worse than any Congestion Pricing based TCP variant.
5.4.4 Comparison of SBRM With Other Approaches in a Parking Lot Topology
Several different combinations of source, link, and path price transport algorithms were eval-uated and compared to SBRM in [Ham01]. There, in a parking lot topology, the resulting throughput of the sources, the utilization of the links and the average backlogs were recorded for each combination for several simulation runs. Again, a scenario with a small number of flows (“few flows” scenario) and a scaled up scenario with increased capacities and a larger number of flows (“many flows” scenario) were used to evaluate scalability of the algorithms with regard to the number of flows (cf. Subsection 4.4.1). A summary of the results is shown in Figure 5.4.4. The displayed simulation type numbers are explained in Table 5.4.2.
The figures show significant variation of the resulting rate allocation in the “many flows”
scenario for all single bit marking strategies (simulation types 1, 8 ,15, 22, and 23, cf. Figures 5.4.4a and b) with the exception of SBRM (simulation types 24, 25, and 26). SBRM, on the other hand, yields a rate allocation that is comparable to the strategies with full pricing infor-mation (simulation types 3, 4, 5, 7, 10, 11, 12, 14, 17, 18, 19, and 21). The utilization of the link is also well above 90% for SBRM at comparably low average backlogs (cf. Figures 5.4.4c and d). In summary, SBRM has proven to be the best Congestion Pricing implementation while using only a single bit for path price transport.
5.4.5 Conclusions
Single Bit Resource Marking (SBRM) is a simplified combination of the Direct Window Update source algorithm (4.2.3) and REM’s price computation rule 2 (5.2.4) with some modifications to both algorithms. SBRM can establish the desired weighted proportionally fair rate allocation.
Also, the dynamic behavior of the congestion window is as desired, leading to almost constantly low queue sizes and average bottleneck link utilization around 95%. SBRM is therefore a practical implementation of Congestion Pricing theory using only a single bit that performs comparably to CP-TCP/EPF. SBRM by far outperforms the conventional TCP algorithms as well (cf. Subsection 4.3.1). For these reasons, SBRM is the most promising approach to gain superior performance in a TCP/IP based network by using Congestion Pricing theory.
5.5 Steady-state Analysis of SBRM
In this section a model will be presented for the analytical calculation of the steady-state of SBRM in a single bottleneck link scenario. This model can be used to calculate average rates, average backlog and average marking probability.
A single source solves the optimization problem stated by (3.2.8)–(3.2.9). Thus, in the steady-state,U(xn)−pxwill be at its maximum. This leads to the first order condition:
U0(xn) =p∗n. (5.5.1)
Chapter 5: Single Bit Marking Strategies
8.5 Time Slot 20 − 40 s: Throughput, Scenario 1, Source S3
Simulation Type
[Mbps]
(a) Throughput of source 3 (8 flows)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
2830 Time Slot 40 − 60 s: Throughput, Scenario 2, Source S3
Simulation Type
[Mbps]
(b) Throughput of source 3 (400 flows)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
Time Slot 40 − 60 s: Utilization, Scenario 2, Router R3
Simulation Type
Utilization [%]
(c) Utilization of link 3 (400 flows)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
3000 Time Slot 40 − 60 s: Avg. Queue Size, Scenario 2, Router R3
Simulation Type
packets
(d) Average backlog of link 3 (400 flows)
Figure 5.4.4: Comparison of the resulting rate allocations of different combinations of source and link algorithms [Ham01]
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5.5 Steady-state Analysis of SBRM
Table 5.4.2: Legend to the simulation type numbers in Figure 5.4.4
Simulation Source Algorithm Link Algorithm Rate Estimation Slow Start
1 Vegas PC1 1 Bit RTT yes
2 Vegas PC1 4 Bit RTT yes
3 Vegas PC1 EPF RTT no
4 Vegas PC1 EPF RTT yes
5 Vegas PC1 EPF min. RTT no
6 Direct Window Update PC1 1 Bit RTT yes
7 Direct Window Update PC1 EPF RTT no
8 Vegas PC2 1 Bit RTT yes
9 Vegas PC2 4 Bit RTT yes
10 Vegas PC2 EPF RTT no
11 Vegas PC2 EPF RTT yes
12 Vegas PC2 EPF min. RTT no
13 Direct Window Update PC2 1 Bit RTT yes
14 Direct Window Update PC2 EPF RTT no
15 Vegas PC3 1 bit RTT yes
16 Vegas PC3 4 bit RTT yes
17 Vegas PC3 EPF RTT no
18 Vegas PC3 EPF RTT yes
19 Vegas PC 3 EPF min. RTT no
20 Direct Window Update PC3 1 Bit RTT yes
21 Direct Window Update PC 3 EPF RTT no
22 Direct Window Update VQM 1 Bit RTT yes
23 Direct Window Update VQM n Bit RTT yes
24 Direct Window Update SBRM + PC1 RTT yes
25 Direct Window Update SBRM + PC2 RTT yes
26 Direct Window Update SBRM + PC3 RTT yes
27 TCP Reno RED n/a yes
Chapter 5: Single Bit Marking Strategies
Since the utility function is strictly increasing, p∗n>0. Therefore∑l∈L(n)Cl(yl)>0and thus b∗l >0, wherel is a bottleneck link. Becauseb∗l >0, the aggregate rate in the steady-state at link l cannot be less than the links capacitycl.Since it also cannot be larger than the link’s capacity, y∗l =cl. Therefore:
y∗l =cl=
∑
n∈N(l)
x∗n. (5.5.2)
In case of logarithmic utility functions (3.2.21), U0(xn) =wn
xn. (5.5.3)
Inserting (5.5.1) and (5.5.3), (5.5.2) becomes:
cl=
∑
n∈N(l)
wn
p∗n. (5.5.4)
Now a single bottleneck link is assumed. From theweighted proportionalfairness criterion (3.2.23), it is known that the available rate at the bottleneck link will be shared as follows:
x∗n= wn
∑Ni=1wi·c. (5.5.5)
Thus, using (5.5.4) and (5.5.5) the average marking probability can be determined as:
m∗=p∗=wn
x∗n = ∑Ni=1wi
c . (5.5.6)
It follows that∑Nn=1wn≤cl, which was already discussed in Section 5.1.
The marking probability leads to the average queue length by using the inverse of (5.4.1):
b∗ = b0−1
γln(1−p∗) (5.5.7)
This reveals a problem: As the number of sources increases without adjustment of each source’s willingness to pay wn, the steady-state queue length will also increase. This prob-lematic coupling of congestion and performance measure was already discussed in Subsection 2.4.1. But since the marking probability increases exponentially, linear increase in the number of sources will only lead to a logarithmic increase in queuing delay.
Using (2.3.1), the steady-state congestion window can finally be calculated as follows:
cwndn∗=RT Tm∗·x∗n. (5.5.8) Table 5.5.1 shows the steady-states in a single bottleneck link scenario for both TCP Reno with RED queues2 and SBRM. The TCP Reno+RED model was taken from [Low00]. Again, the table shows that the resulting average rate depends on the round-trip time (RTT) only for TCP Reno.
2Assumption is made that the steady-state queue size is between minimum and maximum thresholds.
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5.6 Compatibility with Conventional TCP