Simulation Environment

As shown in Figure 6, the network topology of our simulation consists N remote hosts and one central controller. The system service rate is fixed at

Rs (R_s =300 packets per second). Let w_i denote the quality of service, λ_i denote be the predicted packet arrival rate and b_i represents that the size of uplink-buffer in host i (In our simulation, we suppose that all b_i in the system have the same size). We also denote the number of available time slots in a frame as Λ , that is, the available link bandwidth provided by the uplink channel. The simulation lasts for 20000 seconds. Also we assume that packet arrival process follows a Poisson process.

central controller

WFQ Packet Scheduler Bandwidth

Alloactor by the frame-based

algorithm

Λ

uplink buffer

node 1 uplink buffer

node 2

uplink buffer

node N

Bnadwidth Requirment

Bnadwidth Requirment

Bnadwidth Requirment

The information of scheduled slot The information of allocated bandiwdth

Figure 6: The topology of our simulation.

As shown in Figure 6, the overall operations in the system are divided into four steps:

1. The central controller gains the information of bandwidth requirement of each host.

2. According to the information of bandwidth requirement for all hosts, the traffic controller uses a frame-based bandwidth allocation algorithm to determine how much bandwidth each host can be allocated.

3. According to the allocated bandwidth, the central controller uses the WFQ scheduling algorithm to schedule which packet to be transmitted in each time slot.

4. The central controller delivers the packet scheduling information, produced by WFQ, to the remote hosts. After receiving this information, the remote host can delivers packet according to it.

In the first and fourth step, a propagation delay exists. Those delays are combined to be one round-trip delay and this delay uses a frame in unit. Therefore, the scheduling information calculated in the current frame is not directly used in the next frame, but used in the nextτ frames, whereτ means the round-trip propagation delay.

4.2 Simulation Results

Experiment 0:

In the first simulation, we want to investigate the probability of using congested algorithms, P_c. P_c is defined as the probability of

∑

= ( + )>Λ

1

k j N

j k

xj λ because we use this equation to determine whether the system is congested state or not.

There are 3 hosts in this simulation (N =3), and the weighted values of host 1, host 2 and host 3 are 4, 2, and 1, respectively (w₁:w₂ :w₃ =4:2:1). The other

parameters of simulation are: b₁ =b₂ =b₃ =10000, Λ =60, τ =1 ,λ₁ =λ₂ =λ₃ =65~100. FRA is used to

reduce the waste of bandwidth in non-congested state.

0 10 20 30 40 50 60 70 80 90 100

65 70 75 80 85 90 95 100

load (%)

probability

linear polynomial (n=2) max-min min-max

Figure 7: The probability of using congested algorithms

As shown in Figure 7, when the load less than 0.65, the probability, P_c, for all congested algorithms are almost 0. Because when the load is light, the packet in the system will be quickly delivered and thus few backlog in the queuing. Thus, the probability of using congested algorithm is very small.

As the system load increases, the probability of congestion occurring increases, resulting in the increase of using the congested algorithm. Thus the influence of congested algorithm increase as the system load raise.

For all congested algorithm, the more extra-allocated bandwidth the algorithm is, the more the un-used bandwidth among extra-allocated bandwidth has. By the example 2, for the host with large backlog (host A), the extra-allocated bandwidth using proportional polynomial algorithm is usually more than the extra-allocate bandwidth using proportional linear algorithm. The waste of bandwidth using the proportional polynomial algorithm is usually more than the waste of bandwidth using the proportional linear algorithm.

The proportional max-min algorithm can avoid the extra allocation to exact requirement. Thus, the link bandwidth is used more effective, comparing to proportional linear algorithm and proportional polynomial algorithm. Therefore, the probability of using the proportional max-min algorithm increases slower than those of using the proportional linear algorithm and the proportional polynomial algorithm.

Since the proportional min-max algorithm tries it best to make the maximal )

(x_i +λ_i ′ as small as possible, among the ratios of four proportional algorithms, as the system load increase, the ratio achieved by proportional min-max algorithm increases slower than other algorithm.

Experiment 1: (Fairness)

In this simulation, let the network has two hosts (N =2), where w₁ = w₂ =1, and the other parameters are b₁ =b₂ =10000,Λ =60,τ =1, the system load is from 65% to 100%. FRA is allocated to reduce the waste of bandwidth in the non-congested state.

In order to evaluate the effect the of delay proportion achieved by our proposed algorithm, the proportion of the average delay adopted by host i to the average delay achieved by host j is denoted as R_ij, that is,

j i

ij d

R = d .

0.500 0.700 0.900 1.100 1.300 1.500

65 70 75 80 85 90 95 100

load(%)

proportion

linear polynomial (n=2) max-min min-max

Figure 8 : The Proportion achieved by various congested algorithms .

Since the weighted value of the host 1 is equal to the weighted value of the host 2 (w_A =1,w_B =1), the targeted proportion is 1, that is, the proportion achieved by our proposed algorithm would be close to 1 as the system load increases. As shown in the Figure 8, the proportion R₁₂, obtained by using proportional linear algorithm, proportional polynomial algorithm (n=2), proportional max-min algorithm, and

proportional min-max algorithm, are stable at 1.

In this simulation, the time needed for the proportional linear algorithm, proportional polynomial algorithm, proportional min-max algorithm, proportional max-min algorithm to achieve fair delay is 9000, 2000, 3000, and∞ seconds.

Because of behavior of tardy approach by using proportional max-min algorithm in the case of fair weights, this algorithm is not discussed in the late simulation.

However, in reality, the simulation result shows that proportional max-min algorithm achieve delay proportion (fairness in the case) in heavy load extreme-slowly. It is mainly due to the proportional max-min algorithm allocated the link bandwidth as fair as possible. For the samew_i, if ∀i λi + >xi ^ΛN, then each host i will get ^Λ_N when adopting the max-min algorithm. Thus the host with larger backlog will not be allocated much more bandwidth to decrease the number of backlog, especially when the load is heavy.

Therefore, in this simulation, when using proportional max-min algorithm to provide the proportional delay differentiation in high-loaded situation, once host 2 has larger amount of backlog, it is expected that host 2 would be allocated more bandwidth to decrease the amount of backlog, however, it is true but the amount of the bandwidth for decreasing the backlog of host 2 is rarely small. So, for host 2, its number of backlog still large and that causes its average delay is greater than host 1.

Experiment 2: (Proportion)

Furthermore, we want to investigate whether our proposed algorithms can achieve to the targeted proportion. There are 3 hosts in this simulation (N =3), and the weighted values of them are 4, 2, and 1(w₁:w₂ :w₃ =4:2:1). The parameters of simulation are the same as the first simulation (experiment 0). FRA is adopted to reduce the waste of bandwidth in the non-congested state.

0.00 0.20 0.40 0.60 0.80 1.00

65 70 75 80 85 90 95 100

load (%)

proportion

linear host 1: host 2 linear host 1: host 3 polynomial (n=2) host 1: host 2 polynomial (n=2) host 1 : host 3 min-max host 1: host 2 min-max host 1: host 3

Figure 9: Proportion achieved by various congested algorithms

Because of poor performance by using proportional max-min algorithm in the case of fair weights, this algorithm is not discussed in the late simulation.

Since λ₁ =λ₂ =λ₃, the bandwidths allocated by the FRA for host 1, host2, and host3 are equal, the targeted proportion using FRA is 1. The proportions achieved by those four congested algorithms all begin from 1 because the probability of using congested algorithm at the load 0.65 are almost 0 (from the previous simulation results).

According to example 1 and example 2, it is easy to realize that the un-achievement of delay proportion using proportional linear algorithm and proportional polynomial algorithm. Thus the Figure 9 verifies that the achieved proportions )(R_12,R₁₃ using proportional linear algorithm and proportional polynomial algorithm do not approach to the targeted proportions (0.5, 0.25).

From theorem 3, it indicates that the achievement of delay proportion using proportional min-max algorithm. In this simulation, as the system load increase, the achieved proportions (R_12, R₁₃) using proportional min-max approach to the targeted proportions (0.5, 0.25). The proportional min-max algorithm is the only

method, which can achieve the delay proportion.

From the first simulation result, as the system load increase, the probability of using proportional min-max algorithm increases slower than those of other algorithms. Thus, in the Figure 9, the achieved proportions (R_12,R₁₃) using proportional min-max algorithm are smoothly approach to the targeted proportions (0.5, 0.25).

As shown in Figure 7, the probability of using the proportional linear algorithm suddenly goes from 2% to 90% when the load raises from 75% to 80%. Thus, the achieved proportion R using proportional linear algorithm decreases to 0.2 ₁₂ suddenly when the load raises from 75% to 85 %. The achieve proportion R has ₁₃ the same phenomenon. For the same reason, the performance of using the proportional polynomial algorithm has the similar curve.

Experiment 3: (Robustness)

We will discuss that the scalability and robustness of the proportional min-max allocation algorithm. The system load is fixed at 99%. Supposing there are N hosts ranging from 3 to 10, where w₁:w₂:...:w_N−1:w_N =2^N−1:2^N−2:...:2¹ :1, and the other parameters remain the same.

In order to evaluate the achieved delay proportion in the case where there is a large number of hosts, a performance metric, named variance of proportion, is proposed.

Definition: Variance of Proportion (VP)

Given that the system has N hosts, each host i has a weighted value w_i. Let the achieved average delay of host i bedi, then the VP is defended as

1 ) 1

( ²

1

1

−

−

=

∑

⁻

=

N w w

d d

VP

N

i

N i

N i

.

The smaller VP an algorithm has, the better delay proportion it achieves.

0.0E+00 2.0E-03 4.0E-03 6.0E-03 8.0E-03 1.0E-02 1.2E-02 1.4E-02 1.6E-02

3 4 5 6 7 8 9 10

# of class

variance of proportion

linear polynomial(n=2) min-max

Figure 10: Variance of proportion achieved by congested allocations in case of different number of class

Figure 10 represents the VP when different number of classes for three congested algorithms. From this figure, the VPs of congested algorithms decrease as the number of classes increase. Because the bandwidth provided by uplink channel in a fame size is fixed in this simulation, the more classes the system has, the less bandwidth of each host in a frame is allocated. The less bandwidth of each host will lead to the behavior of bandwidth allocation be more exquisite. In a fixed frame size, the less number of classes in the system, the more bandwidth will be allocated to hosts, the more the un-used bandwidth among allocated bandwidth has. Thus, in this environment, the less number of classes in the system, the larger variance the system will be.

The effect of algorithm in non-congested state:

When congestion does not occur, in order to avoid the waste of bandwidth, we provide some congested algorithm. Now we will study the efficiency of these algorithms and investigate whether these algorithms will affect the delay proportion.

Experiment 4: (Fairness)

Suppose that there are two hosts (N=2), wherew w₁: ₂ =1:1. The proportional min-max algorithm is used as default congested algorithm. Other parameters remain the same with the experiment 1.

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

65 70 75 80 85 90 95 100

load

Proportion

FRA WRA RPA RNA

Figure 11: Proportion achieved by various non-congested allocation algorithm Since the w₁ =w₂ and λ₁ =λ₂, according the bandwidth allocation of FRA, the amount of allocated bandwidth for host 1 is equal to the amount of allocated bandwidth for host 2. Thus, the achieved proportion R₁₂ by using FRA is 0.5. And also according to bandwidth allocation of WRA, RPA, and RNA, the targeted proportion R₁₂ by using WRA, RPA, and RNA, are all the same to be 0.5. Thus, from Figure 11, it shows the proportion, R₁₂ , for different methods during non-congestion. In this case, the proportion steadily maintains to be close 1 and approaches to 1 as the system load increases. For the increasing load, the proportion affected by the proportional min-max algorithm will become obvious; therefore, the proportional should be expected to end at 1.

Experiment 5: (Proportion)

Although the behavior of four non-congested algorithms are almost same in the case of same weights. Do the non-congested algorithms affect the achieved delay proportion? In this experiment, three hosts exist (N=3), where w w₁: ₂:w₃ =4 : 2 :1. The other parameters are still the same as the experiment 2.

0.00 0.20 0.40 0.60 0.80 1.00 1.20

65 70 75 80 85 90 95 100

load(%)

proportion

FRA host1:host2 FRA host1:host3 WRA host1:host2 WRA host1:host3 RPA host1:host2 RPA host1:host3 RNA host1:host2 RNA host1:host3

Figure 12: Proportion achieved by various avoiding waste allocation algorithm Table 1: The probability of using congested algorithm for various non-congested

algorithms

FRA WRA RPA RNA 65 0.00% 8.10% 0.00% 0.01%

70 0.04% 27.19% 0.11% 0.09%

75 0.96% 45.13% 1.37% 1.28%

80 10.11% 61.59% 10.91% 10.60%

85 44.19% 77.70% 43.96% 43.73%

90 86.79% 92.76% 86.59% 86.58%

95 99.69% 99.71% 99.68% 99.68%

99 100.00% 100.00% 100.00% 100.00%

Figure 12 shows the proportion achieved by different methods during non-congestion. Since the influence affected by proportional min-max algorithm on delay proportion increase as the system load increase, in this experiment, no matter what method we use in non-congested state, when the system load is increasing the achieved proportions, R12 and R13,will be close to 0.5 and 0.25, respectively.

Table 1 shows that probability of using min-max algorithm for various non-congested algorithms at different load. When the system load is more than 95%,

the probability is almost 99%, it leads to that the system always uses the proportional min-max algorithm. Thus, from Figure 12 when the system is more 95%, the proportions, R12, achieved by different non-congestion algorithms will be the same. The proportions, R13, achieved for different non-congested algorithms also have the same property.

Since the bandwidth allocations made by FRA, WRA, RPA, and RNA are

)

τ , respectively. For the host

with lager backlog, the bandwidth allocation made by WRA has more extra-allocated bandwidth than other methods. The more extra-allocated bandwidth the host with larger backlog has, the more un-used allocated bandwidth among the extra-allocated bandwidth that the host has. The un-used allocated bandwidth among the extra-allocate bandwidth will cause that the other hosts will not be allocated the bandwidth at proper time and then the backlog of other hosts increase and the congestion will occur. Thus, in the Table 1, the probability of using WRA is obvious different from the probability of using FRA, RPA, and RNA. And the wastes of un-used bandwidth among the extra-allocated bandwidth for using FRA, RPA, and RNA is less then the un-used bandwidth among the extra-allocated bandwidth for using WRA.

The achieved proportion is mainly affected by how large the probability of occurring congestion is. In this simulation, the larger the probability is, the greater influence the proportional min-max algorithm affects. From the table 1, the ratios at the same level of load by different non-congested algorithm are different, so the achieved proportion in Figure13 at the same level of load are different. Also, we can denote the ratio to be as P_c when load at ρ, and suppose the targeted proportion in the case that the system always keeps in the congested state is R_c, the targeted proportion without any congestion is R_nc, and the targeted proportion when load at ρ is R_ρ, then it satisfies: R_ρ = R_c×P_c +R_nc ×(1−P_c). From the table 1, we can easily to perceive that at the same level of load, the method we use in the non-congested state will affectP_c.

Since the proportion of weighted values among the host 1, host 2, and host3 isw w₁: ₂:w₃ =4 : 2 :1, and by the theorem 6, the targeted proportion R₁₂ using FRA is larger than he targeted proportion R₁₃ using FRA when the system load is rather light and there is no congestion. Similarly, when using WRA, RPA, or RNA in the non-congested state, the targeted proportion R₁₂ is larger than he targeted proportion R₁₃. As shown in Figure 12, it indicates that there exists this kind of phenomenon.

Experiment 6: (Frame size)

In this experiment, the load is fixed at 99%. We will investigate how the delay proportion is affected by the frame size. In this experiment, it is supposed that there are three hosts (host 1, host 2,and host 3) in the system, where w₁:w₂ :w₃ =4:2:1 and the frame sizes are 3, 30, 300, and 3000. The other parameters are still the same as experiment 2.

0 0.1 0.2 0.3 0.4 0.5 0.6

1 10 100 1000

frame size (3 unit)

VP

linear polynomial(n=2) min-max

Figure 13: The VP of the congested algorithms for various frame sizes.

Since we use the predictive method to evaluate the bandwidth requirement for the remote hosts at next frame and allocate bandwidth to the remote hosts. If frame size is rather large, the amount of predicted bandwidth requirement at a time is rather large.

Thus, the probability of incorrect prediction would be larger. The large frame size the system has, the loosely an incorrect the traffic controlled by the central controller behaves. From Figure 13, it shows that all the VPs increase as frame size increase.

The Figure 13 shows that the proportion min-max algorithm could perform well on maintaining the proportional delay (At the same frame size, comparing to the proportional linear algorithm and proportional polynomial algorithm), it once again tells us that when the frame size become rather large in some level, the linear and polynomial algorithm can not control the traffic very well due to the larger frame size causing the rougher control.

Experiment 7: (Propagation delay)

In the previous simulation, the propagate delay in the uplink transmission is assumed to be 1 frame unit. In this experiment, the propagate delay are 1, 10, 100,and, 1000 units. Let the load be fixed at 99% and the frame size be fixed at 40.

The FRA is adopted as the non-congested algorithm, and the remaining parameters are the same as the previous experiments.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

1 10 100 1000

papagation delay

VP

linear polynomial(n=2) min-max

Figure 14: The VP of the frame based bandwidth allocation for various propagation delays

Since the propagation delay, the central controller cannot directly perceive the bandwidth requirement of remote hosts and cannot directly make a reaction for the variability of bandwidth requirement and the burst of traffic. Also due to the propagation delay, the scheduled information made by the central controller will be out of date for the remote host, the remote host cannot send the packet at proper time, and there will be much un-expected backlog beyond the bandwidth allocated for the

proportion. The VPs for using proportional linear algorithm, proportional polynomial algorithm and proportional min-max algorithm will become lager as the propagation. Among the congested algorithms, the VP of proportional min-max is the smallest, it shows that the proportional min-max algorithm achieve is better than the others when the frame is large.

Experiment 8: (Packet loss)

In the previous simulations, the sufficient buffer size is provided and thus the system does not have any packet loss. However, the buffer size is limited and the packet loss is also an important issue in the uplink transmission. In this simulation, the buffer size is set as 50, 70, 90, 110, 130, and 150 packets, and propagate delay is fixed at one frame unit.

Packet loss Vs Buffersize

0 0.02 0.04 0.06 0.08 0.1

50 70 90 110 130 150

buffersize

loss rate

linear polynomial(n=2) min-max

Figure 15: The packet loss of the frame based bandwidth allocation for various buffer sizes during non-congested under proportional allocation

Figure 15 shows the results of the loss ratio achieved by various congested algorithms under various buffer sizes .The proportional min-max algorithm could have the lower packet loss rate than others in the case of small buffer size. Because the proportional min-max algorithm tries to maintain the length of weighted backlog (x_i+λ_i)′ as fair as possible, it makes the host with the largest weighted backlog

Figure 15 shows the results of the loss ratio achieved by various congested algorithms under various buffer sizes .The proportional min-max algorithm could have the lower packet loss rate than others in the case of small buffer size. Because the proportional min-max algorithm tries to maintain the length of weighted backlog (x_i+λ_i)′ as fair as possible, it makes the host with the largest weighted backlog

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