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Figure 1

The conceptual structure of a Convolutional Neural Network based classifier used to model an Active Queue Management mechanism.

In order to prepare the training set for the proposed neural network model, network simulations were performed, reflecting the queueing behavior of a fractional order controller P I α . The values of the fractional order P I α controller parameters have been presented in the Table 1 . These values were determined based on our previous work [ 26 ]. The results of these articles have shown that the choice of controller parameters significantly affects the queue length control properties. The process of choosing proper AQM/PI controller parameters is non-trivial. It has a significant impact on the packet dropping function (i.e., for an integral order α it can strengthen and accelerate the response of a controller). Properly selected AQM parameters should allow us to obtain adaptation to the changing transmission conditions and desired queue behavior. We discussed the influence of these parameters on queue behavior in papers [ 15 ]. The controller parameters were chosen in such a manner that controller P I α 1 was the weakest controller, and controller P I α 3 was the strongest one, which implies a large number of packet rejections and ease of maintaining the desired queue length.

To obtain training data for an AQM model based on Convolutional Networks, network simulations were performed using the AQM mechanism. For this purpose, the discrete event simulator SimPy (written in Python) was used. This software is available under the MIT License and has been used in our previous works regarding the evaluation of AQMs [ 21 , 26 ].

Our simulation model was a discrete model of a G / M / 1 / N queue. The simulation time was divided into discrete time intervals of length d t . Arrival of a packet was generated (or not) in a given time slot by a traffic source. The source of traffic was self-similar and based on Fractional Gaussian Noise (FGN) process. The advantages of such a source have been described previously in the articles [ 10 , 15 , 25 ].

All experiments considered different degrees of traffic self-similarity expressed using Hurst parameter. In experiments the Hurst parameter changed between H = 0.5 (no correlation) and H = 0.9 (high degree of LRD).

The input intensity coefficient was set to a constant value λ = 0.5 . Thus, the simulation packet source always had a constant intensity. Parameter μ represents the time of packet processing and dispatching (probability of taking a packet from the queue). Different values of this coefficient were used in the experiments. The parameter μ took values between μ = 0.5 (moderately stressed system) to μ = 0.15 (highly stressed system). This choice of simulation parameters allowed us to observe all properties of the AQM mechanism.

In our experiments, we considered different numbers of items from queue occupancy history taken into consideration in the samples used to train Convolutional Networks. For simplicity, we refer to this number of samples as ’CNN History’. This length corresponded to the number of time slots in the simulation model that were used as training data for the network. For example C N N = 200 refers to 200 * d t time intervals taken into consideration. Throughout this time, we observed the behavior of the AQM queue.

Thus, the training data consisted of:

1. Learning features:
   • (a)
     The last n items from the queue’s occupancy history (CNN History).
   • (b)
     History of packet rejections in n last queue states

     where n ∈ [ 20 ; 100 ; 200 ; 300 ; 400 ; 500 ; 1000 ] .

2. Classes:
   • (a)
     11 labels that mapped the probability of packet rejection to the current transmission conditions, according to the principle shown in Table 2 .

     Table 2

     Decision class labels representing ranges of probabilities of packet being dropped.

     Decision Class	Probability Interval [%]
     1	[0;5)
     2	[5;15)
     3	[15;25)
     4	[25;35)
     5	[35;45)
     6	[45;55)
     7	[55;65)
     8	[65;75)
     9	[75;85)
     10	[85;95)
     11	[95;100]

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Therefore, we considered different lengths of queue occupancy history, because from the perspective of the router, which is a low resource device, minimizing the length of the history would be beneficial. In our study, we tried to determine the minimum acceptable length of n last items of the queue’s occupancy history.

For each probability interval, one million one-dimensional learning records were prepared. Therefore, the training set consisted of 11 million records. They contained transmission information such as the length of the queue in each consecutive time slot, the number of dropped packets, and the value of the P I α controller’s packet rejection probability function. We present the process of data preparation in Figure 2 . This amount of data seemed to be sufficient in comparison with the cardinality of data reported in the literature [ 56 ].

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Figure 2

Training data preprocessing process.

Input data prepared in such a manner were used in the process of supervised learning of the neural network models. In order to train the model and minimize the cost function, the optimizer Adaptive Moment Estimation (Adam) was used with the following parameters:

η = 10 − 3 , β 1 = 0.9 , β 2 = 0.999

(2)

where: η is the learning rate, β 1 is the exponential decay rate for the first moment estimates and β 2 is the exponential decay rate for the second moment estimates. The Adam optimizer is expressed by the equation [ 57 ]:

v t = β 1 v t − 1 + ( 1 − β 1 ) g t s t = β 2 s t − 1 + ( 1 − β 2 ) g t

(3)

where v is the first moment, which resembles momentum that records the past normalized gradient, s is the second moment and g denotes the gradient descent.

In both the four convolutional layers and the two dense layers, ReLU was used as the activation function and Sigmoid/Softmax functions were used to determine the activation of the output layer. Categorical cross-entropy was used as a cost function. Figure 1 shows the conceptual structure of a neural network model used for the purpose of active queue management mechanism.

We limited the training process to 10 epochs. This value was sufficient, since the values start to stabilize after only 5–6 epochs, as confirmed by the results in Table 3 , Table 4 and Table 5 . We also compare the accuracy of the model, when Softmax activation function ( Table 3 ) and Sigmoid activation function ( Table 4 ) were used in the output layer. Higher results were obtained for the Sigmoid function.

Table 3

The accuracy measurements for testing the CNN model trained on data regarding three P I α 1 , P I α 2 and P I α 3 controllers, n last items in queue occupancy history taken into consideration (we used Softmax function as an activation function of the last layer).

	Softmax
	n History Length
		20	100	200	300	400	500	1000
CNN by behavior PI α 1	5 epochs	52.27	54.48	54.50	56.67	58.40	58.81	51.59
	6 epochs	52.36	54.88	54.68	56.70	58.42	58.84	51.72
	10 epochs	52.40	55.50	54.83	56.74	58.47	58.90	51.82
CNN by behavior PI α 2	5 epochs	48.37	47.23	40.29	41.99	43.68	44.06	43.94
	6 epochs	48.45	47.61	40.32	42.06	43.74	44.06	44.00
	10 epochs	48.54	48.42	40.31	42.30	43.78	44.24	44.04
CNN by behavior PI α 3	5 epochs	47.07	41.83	33.80	32.30	32.92	33.79	38.45
	6 epochs	47.18	42.09	33.81	32.31	32.93	33.82	38.44
	10 epochs	47.41	42.62	34.18	32.32	32.92	33.83	38.50

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Table 4

The accuracy measurements for test data for a neural network model trained with data representing the behavior of P I α 1 , P I α 2 and P I α 3 controllers, n last items in queue occupancy history taken into consideration (we used Sigmoid function as an activation function of the last layer).

	Sigmoid
	n Last Items in Queue Occupancy History
		20	100	200	300	400	500	1000
CNN by behavior PI α 1	5 epochs	55.98	73.95	80.76	83.67	85.22	86.15	88.46
	6 epochs	56.04	74.06	80.88	83.80	85.39	86.31	88.74
	10 epochs	56.16	74.38	81.19	84.13	85.69	86.64	89.46
CNN by behavior PI α 2	5 epochs	50.34	70.70	75.94	76.92	77.03	77.20	83.71
	6 epochs	50.38	70.83	76.08	77.06	77.18	77.32	84.04
	10 epochs	50.53	71.13	76.40	77.36	77.57	77.79	84.81
CNN by behavior PI α 3	5 epochs	48.77	67.16	67.54	65.65	64.39	64.04	77.46
	6 epochs	48.82	67.29	67.70	65.83	64.57	64.24	77.76
	10 epochs	49.06	67.60	68.03	66.23	64.99	64.68	78.68

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Table 5

The accuracy measurements for test data for a neural network model trained with data representing the behavior of three controllers simultaneously, given n recent queue occupancy history elements.

	Sigmoid
	n Last Items in Queue Occupancy History
		20	100	200	300	400	500	1000
CNN by behavior 3 PI α	5 epochs	49.52	67.03	68.55	68.48	68.40	68.42	70.98
	6 epochs	49.58	67.12	67.70	68.65	68.60	68.61	71.30
	10 epochs	49.74	67.34	68.99	69.03	69.15	69.21	72.10

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In the case of Softmax function ( Table 3 ), the minimum accuracy was 32.3%, and the maximum 58.9%. For the models in which we applied the Sigmoid activation function for the last layer the minimum accuracy was 48.77% (for the network trained on the data from the P I α 3 controller, where the CNN History = 20 ), and the maximum 89.46% (for the network trained on the data from the P I α 3 controller, where the CNN History = 1000 ). Taking all the results into consideration, the best results were obtained for the CNN History ≥ 500 , and the worst for the CNN History < 100 ( Table 4 ).

In the case of the model trained on data representing the behavior of three controllers simultaneously and the use of the Sigmoid activation function of the output layer, the maximum accuracy was 72.1% for the CNN History ≥ 500 (see Table 5 ).

5. Evaluation of the Neural Network-Based AQM

This section presents the behavior of the trained neural network (as assumed in Section 4 and evaluates its effectiveness as an AQM mechanism. This evaluation was performed using previously described simulation mechanisms. During the study, we evaluated the number of packets dropped from the queue and the average queue occupancy. We compared the effectiveness of the neural network-based AQM mechanism with the results of the P I α controller-based AQM mechanism. We used the network traffic with different degrees of self-similarity during the experiments.

To increase the readability of the paper, we present only two extreme cases - the results obtained for a non-self-similar traffic ( H = 0.5 ) and for a traffic with high degree of LRD ( H = 0.9 ).

The intensity of the packet source in the simulation was assumed to be ( l a m b d a = 0.5 ). On the other hand, the packet service time in a system was set to a constant value ( μ = 0.25 ) in order to obtain a heavily loaded system.

In our experiments, we evaluated four separate neural network models. The first three neural networks were trained with the data obtained from controllers P I α 1 , P I α 2 , and P I α 3 . The fourth model was trained with data regarding all of these controllers. In the first phase of the experiment, we considered two neural network models (see Figure 1 ): the first one with Softmax, and the second one with Sigmoid activation function of the last layer.

A comparison of Table 3 and Table 4 shows that although Softmax function is more commonly used in the literature as an activation function of the output layer of the neural network for multiclass classification, Sigmoid function performs better in our case. In the worst case, in which the network obtained accuracy of 32.31%, changing the activation function to Sigmoid resulted in significant accuracy increase (65.65%). Additionally, in the best obtained case accuracy changed from 58.90% to 86.65%. Figure 3 and Figure 4 show average queue lengths for AQM mechanism based on neural network. Detailed results are compared on Table 6 and Table 7 for Sigmoid function and on Table 8 and Table 9 for Softmax function. Both presented networks imitate the behavior of the first controller— P I α (see Table 1 ). Comparing the number of discarded packets and the average queue sizes, we find that they are similar regardless of the chosen network activation function in the last layer. As Hurst increases, the number of dropped packets decreases slightly in the case of Sigmoid function (<1%).

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Figure 3

Distribution of queue length obtained for CNN model with the last layer activation function Sigmoid, trained using data regarding P I α 1 controller and parameters: K P = 0.0001 , K I = 0.0004 , α = − 0.4 , H = 0.5 ( left ), H = 0.9 ( right ).

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Figure 4

Distribution of queue length obtained for CNN model with the last layer activation function Softmax, trained using data regarding P I α 1 controller and parameters: K P = 0.0001 , K I = 0.0004 , α = − 0.4 , H = 0.5 ( left ), H = 0.9 ( right ).

Table 6

Detailed results of queue occupancy obtained for CNN model with the last layer activation function Sigmoid, trained using data regarding P I α 1 controller and parameters: K P = 0.0001 , K I = 0.0004 , α = − 0.4 and H = 0.5 .

AQM	Packet Dropped	Average Queue Length
P I α 1	249,878	168.98
CNN History = 20	251,198	172.97
CNN History = 100	248,936	176.45
CNN History = 200	250,063	175.69
CNN History = 300	249,510	166.87
CNN History = 400	250,104	166.23
CNN History = 500	250,800	174.31
CNN History = 1000	249,561	173.33

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Table 7

Detailed results of queue occupancy obtained for CNN model with the last layer activation function Sigmoid, trained using data regarding P I α 1 controller and parameters: K P = 0.0001 , K I = 0.0004 , α = − 0.4 and H = 0.9 .

AQM	Packet Dropped	Average Queue Length
P I α 1	263,387	139.16
CNN History = 20	261,678	145.66
CNN History = 100	262,304	145.81
CNN History = 200	262,518	147.22
CNN History = 300	262,935	139.28
CNN History = 400	263,872	140.89
CNN History = 500	263,440	142.22
CNN History = 1000	263,654	143.14

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Table 8

Detailed results of queue occupancy obtained for CNN model with the last layer activation function Softmax, trained using data regarding P I α 1 controller and parameters: K P = 0.0001 , K I = 0.0004 , α = − 0.4 and H = 0.5 .

AQM	Packet Dropped	Average Queue Length
P I α 1	249,878	168.98
CNN History = 100	250,038	182.21
CNN History = 300	250,455	164.06
CNN History = 500	250,017	174.73

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Table 9

Detailed results of queue occupancy obtained for CNN model with the last layer activation function Softmax, trained using data regarding P I α 1 controller and parameters: K P = 0.0001 , K I = 0.0004 , α = − 0.4 and H = 0.9 .

AQM	Packet Dropped	Average Queue Length
P I α 1	263,387	139.16
CNN History = 100	261,271	148.89
CNN History = 300	263,609	134.64
CNN History = 500	264,569	145.64

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Taking into consideration higher accuracy obtained using Sigmoid function, we chose this function to be used in further experiments.

Figure 3 compares the behavior of two AQM mechanisms: P I α 1 controller and the CNN-based AQM trained on the data reflecting the behavior of this controller.

For the CNN model, different lengths of the last n elements of the queue occupancy history (input to the neural network) were considered. Regardless of the value of n , the resulting queue length distributions are similar to the queue length distribution of the P I α 1 controller. For Poisson traffic (non-self-similar traffic, H = 0.5 ), the average queue length oscillates between 166 and 176 packets (see Table 6 ). For highly self-similar traffic (parameter H = 0.9 ), the average queue length was between 139 and 147 (see Table 7 ). In this case, all the Convolutional Neural Network models (with different numbers of CNN History) obtained larger values of the average queue length, with fewer packets dropped, than the P I α 1 mechanism.

Figure 5 presents the results for stronger AQM mechanism P I α 2 . The detailed results of dropped packets numbers and queue lengths are presented in Table 10 and Table 11 . Because of the fact that the P I α 2 controller was stronger than the one presented above, the obtained average queue lengths were smaller.

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Figure 5

Distribution of queue length obtained for CNN model with the last layer activation function Sigmoid, trained using data regarding P I α 2 controller and parameters: K P = 0.0001 , K I = 0.0004 , α = − 0.5 , H = 0.5 ( left ), H = 0.9 ( right ).

Table 10

Detailed results of queue occupancy obtained for CNN model with the last layer activation function Sigmoid, trained using data regarding P I α 2 controller parameters: K P = 0.0001 , K I = 0.0004 , α = − 0.5 and H = 0.5 .

AQM	Packet Dropped	Average Queue Length
P I α 2	250,314	134.72
CNN History = 20	249,610	135.27
CNN History = 100	250,657	140.37
CNN History = 200	249,633	137.95
CNN History = 300	249,752	142.29
CNN History = 400	248,852	134.86
CNN History = 500	249,960	138.85
CNN History = 1000	249,744	129.60

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Table 11

Detailed results of queue occupancy obtained for CNN model with the last layer activation function Sigmoid, trained using data regarding P I α 2 controller parameters: K P = 0.0001 , K I = 0.0004 , α = − 0.5 and H = 0.9 .

AQM	Packet Dropped	Average Queue Length
P I α 2	264,819	109.37
CNN History = 20	263,014	117.57
CNN History = 100	264,135	112.98
CNN History = 200	264,217	115.69
CNN History = 300	264,668	116.24
CNN History = 400	265,538	110.05
CNN History = 500	264,533	112.47
CNN History = 1000	265,839	105.25

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Figure 6 compares the last pair of controllers: controller P I α 3 , with the corresponding models based on Convolutional Neural Networks. The results prove that this controller is the strongest one. The AQM mechanism increased considerably the number of dropped packets and decreased the obtained queue lengths. In the case of traffic without LRD (see Table 12 , for parameter H = 0.5 ) the average queue occupancy oscillateds between 116 and 139 packets, and in the case of traffic characterized by a high degree of LRD (see Table 13 , for parameter H = 0.9 ) between 94 and 121 packets.

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Figure 6

Distribution of queue length obtained for CNN model with the last layer activation function Sigmoid, trained using data regarding P I α 3 controller and parameters: K P = 0.0001 , K I = 0.0004 , α = − 0.6 , H = 0.5 ( left ), H = 0.9 ( right ).

Table 12

Detailed results of queue occupancy obtained for CNN model with the last layer activation function Sigmoid, trained using data regarding P I α 3 controller and parameters: K P = 0.0001 , K I = 0.0004 , α = − 0.6 and H = 0.5 .

AQM	Packet Dropped	Average Queue Length
P I α 3	250,840	117.53
CNN History = 20	251,892	139.26
CNN History = 100	250,362	136.35
CNN History = 200	248,878	117.67
CNN History = 300	250,533	116.40
CNN History = 400	250,011	118.85
CNN History = 500	250,166	118.67
CNN History = 1000	249,801	118.20

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Table 13

Detailed results of queue occupancy obtained for CNN model with the last layer activation function Sigmoid, trained using data regarding P I α 3 controller and parameters: K P = 0.0001 , K I = 0.0004 , α = − 0.6 and H = 0.9 .

AQM	Packet Dropped	Average Queue Length
P I α 3	265,707	95.13
CNN History = 20	265,737	121.19
CNN History = 100	263,952	110.79
CNN History = 200	265,383	97.28
CNN History = 300	266,295	94.09
CNN History = 400	266,366	95.79
CNN History = 500	265,592	97.31
CNN History = 1000	266,184	96.90

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It should be noted that for all three CNN-based AQM models, a more efficient AQM model was obtained compared to the controllers that were used to create the test data. Even for the model that obtained the smallest accuracy during the learning process (48.77%, see Table 4 ), based on non-integer controller data of order P I α 3 , for CNN History < 100 , the obtained average queue length was larger than for the base mechanism P I α 3 . This situation occurred both for traffic without LRD (see Table 12 ) and for traffic characterized by a high degree of LRD (see Table 13 ).

In the next simulation step, we evaluated the AQM-CNN mechanism whose learning data were generated from the behavior of all three P I α controllers. Figure 7 shows the queue distribution, and Figure 8 shows the changes in queue occupancy over time. Details of the number of packets dropped and the resulting average queue occupancy are presented in Table 14 for the traffic without LRD and in Table 15 , for traffic with a high degree of LRD.

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Figure 7

Distribution of queue length obtained for CNN controller trained using data regarding three P I α controllers, H = 0.5 ( left ), H = 0.9 ( right ).

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Figure 8

Queue occupancy obtained for CNN controller trained using data regarding three P I α controllers, with CNN History = 500 , H = 0.5 ( left ), H = 0.9 ( right ).

Table 14

Detailed results of queue occupancy results obtained for CNN model trained using data regarding three P I α controllers and H = 0.5 .

AQM	Packet Dropped	Average Queue Length
CNN 3 History = 20	249,727	128.75
CNN 3 History = 100	249,988	174.64
CNN 3 History = 200	250,583	164.87
CNN 3 History = 300	249,907	169.98
CNN 3 History = 400	249,593	173.41
CNN 3 History = 500	250,157	138.08
CNN 3 History = 1000	249,334	170.46

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Table 15

Detailed results of queue occupancy results obtained for CNN model trained using data regarding three P I α controllers and H = 0.9 .

AQM	Packet Dropped	Average Queue Length
CNN 3 History = 20	262,841	120.94
CNN 3 History = 100	263,859	152.15
CNN 3 History = 200	262,298	137.21
CNN 3 History = 300	263,205	131.49
CNN 3 History = 400	263,818	129.81
CNN 3 History = 500	263,872	127.37
CNN 3 History = 1000	263,554	138.32

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The results show that when the number of last n elements of queue occupancy history taken as a CNN input is too small (CNN History < 100 ), then, independent of the degree of self-similarity of the traffic, the number of dropped packets, and the average queue length, approximates the results obtained using the sets of controllers P I α 2 and P I α 3 (see Table 10 , Table 11 , Table 12 and Table 13 ).

On the other hand, when the considered number of last n queue occupancy history elements is larger (CNN History ≥ 100 ), the obtained average queue length increases by 46 packets for traffic without LRD ( Table 14 , for H = 0.5 ), or by 32 packets, for traffic characterized by a high degree of LRD (see Table 15 , for H = 0.9 ). This means that the resulting queue distribution matches the one of the original and the most efficient controller P I α 1 ( Figure 3 ).

This feature indicates that for the AQM model based on Convolutional Networks, as the number of story elements used increases, the ability of the mechanism to adapt to current Internet transmission conditions also improves.

6. Conclusions

The paper presents a new Active Queue Management mechanism based on Convolutional Neural Networks and supervised learning.

To train the Convolutional Networks used in the experiments, data obtained through simulation have been used. The training data of the CNN model reflect the behavior of the AQM mechanism, based on a fractional order controller P I α .

In our experiments, we took into account the effect of the degree of traffic self-similarity and long-term dependence on the performance of the proposed mechanism.

We also considered the effect of the number of last n elements of the queue occupancy history, used as input of the neural network, on the efficiency of the proposed mechanism. The best results were obtained for CNN History = 500. The minimum length of CNN History for which results are still acceptable is 100.

In the experiments, neural networks with different number of convolutional layers and different optimizers and cost functions were considered to build the AQM model. After comparing the results obtained with different activation functions, the results have shown that the most efficient model used Sigmoid activation function in the output layer, therefore we chose this function for further experiments. The decisions made in this work were also influenced by our previous work regarding traffic classification in terms of the degree of self-similarity [ 25 ].

The most efficient AQM obtained in our study was based on the Convolutional Neural Network model, trained using the data reflecting the behavior of all three P I α controllers jointly.

The results confirmed that the model based on Convolutional Neural Networks can effectively reproduce the results of the classical AQM algorithm and effectively manage the data transmission. Such a model maintains the assumed average number of packets in the queue and reduces the total number of dropped packets, independent of the degree of traffic self-similarity.

It seems that the proposed mechanism exhibits some advantages over previously proposed mechanisms encountered in the literature. Our previous study [ 26 ] demonstrated that the reinforcement learning methods are well suited for maintaining the assumed queue size. However, in computer networks, the process of controlling packet traffic is more complex. The objective is to maximize the transmission efficiency. This efficiency is characterized by: throughput, delay, and possible retransmissions. Efficiency of AQM mechanisms is influenced by self-similarity of network traffic. The higher the Hurst parameter value is, the greater problems with correct packet management occur. The proposed solution addresses this problem much more effectively. The biggest disadvantage of this solution is greater computational and memory complexity of solutions based on Convolutional Neural Networks. This complexity may affect the difficulty of implementing this solution in real routers.

In our previous study [ 58 ], we used a Linux-based computer as a router. In that study, we used a special router implementation based on a special forwarding mechanism (based on the iptables mechanism), which delivered all packets to the user program implementing AQM. This solution greatly simplifies the research model. Unfortunately, the tests have shown that forwarding packets from kernel space to userspace requires a significant amount of time and is not optimal. In the target solutions the whole implementation should be realized in the kernel of the system. The implementation may be a great challenge on routers with low hardware resources. In such solutions instead of multiplication operations bit shifting is used, which causes calculation errors. For CNN calculations these errors may be too high. However, it seems that the computational power of routers will increase in the future. We want to devote a separate article to the problems of implementing AQMs in real routers.

Author Contributions

Conceptualization, J.S. and A.D.; methodology, J.S.; software, J.S.; investigation, A.D., J.S. and S.M.; validation, J.D., D.M.; project administration, J.S.; formal analysis, J.D. and K.F.; data curation, J.S. and D.M.; funding acquisition, J.S.; writing—original draft preparation, J.S., J.D., A.D. and K.F.; writing—review and editing, J.S., J.D.; visualization, J.S., D.M. and S.M.; supervision, A.D. All authors have read and agreed to the published version of the manuscript.

Funding

Publication supported by Own Schoolarship Fund of the Silesian University of Technology in year 2019/2020, grant number: 22/FSW18/0003-03/2019.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

Footnotes

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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