# An Improved Algorithm to Enhance the Performance of FAST TCP Congestion Control for Personalized Healthcare Systems

Authors: Jiyang Ye; Bowen Huang; Xiaolong Chen
Journal: Wireless Communications and Mobile Computing
 (2021)

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## Abstract

The development of personalized medical systems should be supported by a fast and stable network system. The FAST TCP network system is the appropriate support system for this purpose. However, when the FAST TCP is deployed, the static mapping selection method for protocol parameters is unable to guarantee the small queuing delay and fast convergence of the network simultaneously. By conducting theoretical analysis and simulation experiments, the relationships among FAST TCP protocol slow start condition, control law gain parameters, and FAST TCP system convergence rate were examined. To ensure the stability of the FAST TCP system and to select the smallest protocol parameters, an improved method to effectively accelerate the convergence velocity of the FAST TCP system is proposed in this study. In this method, the number of packets for staying in the buffer for FAST TCP connections was taken as the criterion of the slow start, and the gain parameter of the control law was dynamically adjusted according to the local information of each FAST TCP connection. Using this improved method, the FAST TCP system can achieve a stable and small queuing delay, whilst the FAST TCP system could converge quickly to the equilibrium point simultaneously.


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## Body

## 1. Introduction

With the continuous advancement of the digital wave, more and more hospitals are actively rolling out personalized healthcare systems with the goal of achieving rapid development in the field of intelligent medical care. However, personalized healthcare system requires unilateral transmission of the lossless compression radiographic images and pathology slides, as well as networks with low latency and high reliability [1]. At the same time, remote control business scenarios such as remote ultrasound, remote surgery, and remote ward rounds also present the characteristics of unilateral transmission, large bandwidth, and low delay. Although operators are constantly increasing the bandwidth, the speed of the medical network did not change with the increase of bandwidth in the same proportion [1, 2]. In the application of a personalized healthcare system, the network environment is still not very satisfactory. In order to improve the network speed in the medical environment, increasing the investment of hardware is not the best choice. Therefore, it is necessary to design a new algorithm to improve the network utilization rate. FAST TCP [2, 3], a new type of transmission control protocol based on the delay calculation for the next generation of high-performance networks with high bandwidth, long delay, and large capacity, can support the requirements of personalized medical systems. FAST TCP uses unilateral acceleration technology, keeping compatible with traditional TCP flow control mechanism, whilst not making any changes to the TCP/IP header field. Therefore, FAST TCP can communicate with all the traditional TCP protocols, and in the application of personalized medical system, it only needs to be deployed on the server side to bring significant acceleration effect. The FAST TCP protocol features a congestion controller at each connection source end and the application of the estimated queue delay as the congestion feedback signal. Based on the design idea of the FAST TCP network equilibrium, its performance indexes, such as stability, fast response, throughput, and fairness, have been proven to be better than those of TCP RENO, HSTCP, and STCP [3, 4]. However, choosing the appropriate protocol parameters to ensure that the FAST TCP protocol shows consistent good performance remains an open problem [2].Studies have shown that protocol parameterα, control law gain parameter γ, and slow start threshold m of FAST TCP must be initialised when establishing a FAST TCP connection [2, 5]. The protocol parameter α represents the number of packets expected to remain in the link buffer when FAST TCP reaches equilibrium. Thus, under the precondition of a stable network system, a smaller protocol parameter α leads to a shorter queuing delay of the link [2, 6, 7]. However, past studies [8–10] have reported that setting a smaller protocol parameter α would lead to slower convergence speed of the system to the equilibrium point and reduced utilization of the network. Therefore, the existing method of static mapping selection for FAST parameters could not guarantee that the system would obtain a smaller queue delay and a faster convergence rate at the same time [11, 12]. A study [13] used measurement distributed facilities in the network to monitor the backbone links and then applied fuzzy control technology to guide the source end of each FAST TCP connection to select the appropriate protocol parameter α according to the link performance, so as to stabilise queue delay. However, the algorithm did not consider the improvement of the convergence speed of the system.Protocol parameterα, control law gain parameter γ, and slow start threshold m of FAST TCP are found to be the key factors affecting the system [14]. If the settings are not appropriate, they could seriously affect system performance [15]. Given that the protocol parameter α refers to the number of packets expected to remain in the link buffer when FAST TCP reaches equilibrium, therefore, under the precondition of system stability, the smaller the protocol parameter α, the smaller the queue delay of the link [16]. However, past studies [2, 3] have shown that choosing a smaller protocol parameter could lead to a slower convergence speed of the system to the equilibrium point and the reduced utilization of the network. Literature [13] used measurement facilities distributed in the network to monitor the backbone link state and then applied fuzzy control technology to guide the source end of each FAST TCP connection to select appropriate protocol parameters according to the link state to stabilise queue delay; however, their proposed system did not improve the convergence speed of FAST TCP.To overcome these shortcomings, the relationships among FAST TCP slow start criteria, control law gain parameters, and FAST TCP system convergence speed were discussed through theoretical analysis and NS-2 simulation. To ensure the stability of the FAST TCP system and select the smallest possible protocol parameters, an improved method to enhance the convergence speed of the system is presented. The key finding of this method considered the number of packets for staying in the buffer for FAST TCP connections as the criterion of a slow start and adjusted the control law gain parameters according to the local information of each connection. The proposed method also ensured that the system was stable and had a small queue delay, thereby improving the speed at which the FAST TCP system converges to the balance point.The main contribution of this paper is to overcome the above defects; in the case of ensuring FAST TCP system stability, the FAST TCP flow chooses the protocol parameters as small as possible, adaptive dynamic adjustment slow start threshold and gain parameters according to the local information source end. This improved method can ensure that the system is stable and has small queuing delay at the same time, improving the speed of convergence to equilibrium system.The rest of the paper is organised into sections. Section1 introduces the basic concepts and algorithms of the FAST TCP network model. Section 2 analyses the relationship between the slow start threshold and system convergence speed through simulation experiments. This section also presents and analyses the reasons, proposes the number of packets for staying in the buffer for FAST TCP connections as the criterion to determine the slow start threshold, and determines the improved algorithm of slow start threshold based on the parameters of each connection protocol. In Section 3, the relationship between gain parameters and system convergence rate is discussed theoretically. Under the condition of system stability, the operation state of the network system was evaluated according to the local information of each connected source end. When the system was far from the equilibrium point, larger dynamic gain parameters were selected to bring the system as close to the equilibrium point as possible in the least amount of time. When the system was near the equilibrium point, smaller gain parameters were selected to reduce the instability of the system. When the system tended to balance, choosing a smaller gain parameter reduced the system instability. Section 4 discusses the simulation results under the dynamic environment, and the final section presents the conclusion.


## 2. Related Research Works

### 2.1. Basic Symbols

For the convenience of reading, the relevant basic symbols are introduced at first.As shown in Figure1, we assumed that the dumbbell network system contained N FAST TCP connections with the same link attributes and that they shared the unique bottleneck link l. We also defined the connections set I=1,2,⋯,N and assumed that the buffer at the link end was large enough. We adopted the tail-dropping queue management algorithm. For the FAST TCP system, the following performance indicators were defined:Figure 1
Dumbbell network topology.δ: overshoot refers to the difference between the maximum transmission flow obtained by each connection and the steady-state value and the ratio of this difference to the steady-state valuest: convergence time refers to the time when the new connection converges to the equilibrium pointFor any giveni connection, we defined i∈I and set the following parameters:Wit: size of sending window on the source side (packets)di: propagation delay(s) (we assumed that all connections obtained accurate propagation delay and that there was no fairness problem [7])qit: queuing delay(s)Dit: go and back delay, Di=di+qi (s)xit: transmission traffic (packets/s), where xit=Wit/Ditzit=xit·qit: the number of packets actually left in the link buffer [1, 6] (packets)αi: protocol parameters (packets), at equilibrium, the number of packets expected to stay in the link buffermi: slow start threshold(s)γi: gain parameters of control laweit=Wit/di: normalised window size, sum of packets left in buffer and packets left per unit time [1]T: window update cycle(s)For linkl, we set the following:C: the bandwidth is packets/sblt: queue length is packetsqlt: queuing delay, ql=bl/cl (s)


### 2.2. Research Works

The stability of the FAST TCP system is related to FAST TCP parameters and network parameters [2, 5]. The FAST TCP parameters include protocol parameter α representing the number of packets of data expected to remain in the link side buffer, control law gain parameter γ, and window update period T. The network parameters include propagation delay d and link bandwidth c. When the two sets of parameters do not cooperate, the stability of the system would be affected. Therefore, establishing the relationship between the two sets of parameters to guide the choice of protocol parameters is an effective solution to the disclosure problem [7–9].Literature [2] proposed a guidance scheme for selecting protocol parameters based on link bandwidth by using a static mapping table. On the basis of experimental simulation, another study [8] pointed out that in a single-link network, the value should be greater than 0.0075 times of the link bandwidth. However, the above two schemes do not consider another network parameter d, which is an empirical value strategy, making it difficult to ensure the stability of the system under different network environments.Assuming that window update cycleT is an RTT, Tan et al. [9] established a continuous congestion control model for a single-link single-connection FAST TCP network. The following conditions for the local asymptotic stability of the FAST TCP system are given by using the Routh–Hurwitz stability criterion [10].Given network parametersc and d and protocol parameter γ, the protocol parameter α was selected to meet the following expression α>fγ·c·d, where fγ=γ2+2γ−4+γ16γ+γ2/2γ2+γ+2, thus indicating that the FAST TCP network congestion control models are asymptotically stable.The stability condition connected the two sets of parameters and enabled the decoupling of the protocol parameters from other related parameters, thus guiding the selection of such parameters. However, the FAST TCP model used for stability analysis only considered that the window update periodT is a special case of RTT and ignored the exponential smoothing filter used to estimate the congestion feedback signal at the source end. There are also some defects, such as conservative guidance scope and limited application scope. Thus, according to the stability conclusion found in the literature [11], the following guidance scheme is presented.Under the constraint that parameterT is an RTT, γ∈0,0.93, c>0, and parameter d is bounded, we can guide the selection of protocol parameters α to ensure system stability within the scope of 1,∞.The guidance scheme relaxed the selection range of the protocol parameters but restricted others [14, 15]. When other parameters do not meet the constraint conditions, the guidance scheme would be ineffective; thus, it can be said that the application range of the guidance scheme is limited.According to the stability conclusion presented in past studies [8, 16], the following guiding scheme could be given: we can select α>0 or α>cd under constrained γd/T∈0,1 conditions. Similarly, this scheme is limited in terms of guiding the application scope.


## 2.1. Basic Symbols

For the convenience of reading, the relevant basic symbols are introduced at first.As shown in Figure1, we assumed that the dumbbell network system contained N FAST TCP connections with the same link attributes and that they shared the unique bottleneck link l. We also defined the connections set I=1,2,⋯,N and assumed that the buffer at the link end was large enough. We adopted the tail-dropping queue management algorithm. For the FAST TCP system, the following performance indicators were defined:Figure 1
Dumbbell network topology.δ: overshoot refers to the difference between the maximum transmission flow obtained by each connection and the steady-state value and the ratio of this difference to the steady-state valuest: convergence time refers to the time when the new connection converges to the equilibrium pointFor any giveni connection, we defined i∈I and set the following parameters:Wit: size of sending window on the source side (packets)di: propagation delay(s) (we assumed that all connections obtained accurate propagation delay and that there was no fairness problem [7])qit: queuing delay(s)Dit: go and back delay, Di=di+qi (s)xit: transmission traffic (packets/s), where xit=Wit/Ditzit=xit·qit: the number of packets actually left in the link buffer [1, 6] (packets)αi: protocol parameters (packets), at equilibrium, the number of packets expected to stay in the link buffermi: slow start threshold(s)γi: gain parameters of control laweit=Wit/di: normalised window size, sum of packets left in buffer and packets left per unit time [1]T: window update cycle(s)For linkl, we set the following:C: the bandwidth is packets/sblt: queue length is packetsqlt: queuing delay, ql=bl/cl (s)


## 2.2. Research Works

The stability of the FAST TCP system is related to FAST TCP parameters and network parameters [2, 5]. The FAST TCP parameters include protocol parameter α representing the number of packets of data expected to remain in the link side buffer, control law gain parameter γ, and window update period T. The network parameters include propagation delay d and link bandwidth c. When the two sets of parameters do not cooperate, the stability of the system would be affected. Therefore, establishing the relationship between the two sets of parameters to guide the choice of protocol parameters is an effective solution to the disclosure problem [7–9].Literature [2] proposed a guidance scheme for selecting protocol parameters based on link bandwidth by using a static mapping table. On the basis of experimental simulation, another study [8] pointed out that in a single-link network, the value should be greater than 0.0075 times of the link bandwidth. However, the above two schemes do not consider another network parameter d, which is an empirical value strategy, making it difficult to ensure the stability of the system under different network environments.Assuming that window update cycleT is an RTT, Tan et al. [9] established a continuous congestion control model for a single-link single-connection FAST TCP network. The following conditions for the local asymptotic stability of the FAST TCP system are given by using the Routh–Hurwitz stability criterion [10].Given network parametersc and d and protocol parameter γ, the protocol parameter α was selected to meet the following expression α>fγ·c·d, where fγ=γ2+2γ−4+γ16γ+γ2/2γ2+γ+2, thus indicating that the FAST TCP network congestion control models are asymptotically stable.The stability condition connected the two sets of parameters and enabled the decoupling of the protocol parameters from other related parameters, thus guiding the selection of such parameters. However, the FAST TCP model used for stability analysis only considered that the window update periodT is a special case of RTT and ignored the exponential smoothing filter used to estimate the congestion feedback signal at the source end. There are also some defects, such as conservative guidance scope and limited application scope. Thus, according to the stability conclusion found in the literature [11], the following guidance scheme is presented.Under the constraint that parameterT is an RTT, γ∈0,0.93, c>0, and parameter d is bounded, we can guide the selection of protocol parameters α to ensure system stability within the scope of 1,∞.The guidance scheme relaxed the selection range of the protocol parameters but restricted others [14, 15]. When other parameters do not meet the constraint conditions, the guidance scheme would be ineffective; thus, it can be said that the application range of the guidance scheme is limited.According to the stability conclusion presented in past studies [8, 16], the following guiding scheme could be given: we can select α>0 or α>cd under constrained γd/T∈0,1 conditions. Similarly, this scheme is limited in terms of guiding the application scope.


## 3. FAST TCP Congestion Control Algorithms

In accordance with past studies [1, 2, 5] and analyses of the FAST TCP pseudocode, protocol parameter αi was selected according to link bandwidth and slow start threshold mi=0.00075 (s) and control law gain parameter γi=0.5 when the FAST TCP connection was established. Then, the sending window was updated according to Algorithm 1 [1].Algorithm 1
Each acknowledgement frame is received {If connection establishment time is less thanmi,wik′+1=wik′+1//Execute slow start window update algorithmElse ifzik′≠αiEvery window update cycleT///T is the window update cyclewik+1=1−γi·wik+γidi/di+qik·wik+αi//Execute FAST TCP window update algorithm}The description of Algorithm1 showed that the convergence speed of the FAST TCP protocol was related to the parameters αi, mi, and γi [1], among which αi can be used to determine the balance point location of the FAST TCP system. If the αi size was smaller, the convergence rate was slower. Meanwhile, the parameter mi can be used to determine when the source changes from a slow start algorithm to a FAST window update algorithm. Parameter γi can be used to determine the proportion of window changes each time the FAST TCP algorithm is executed. Hence, the selection of these two parameters can affect the convergence rate. Therefore, under the condition of ensuring system stability, a smaller parameter αi (smaller queue delay) was selected to promote the existing static selection method of parameters miandγi and to adjust them dynamically using local information available at each connection source. In this way, we can improve the FAST TCP protocol convergence speed and maintain a smaller queue delay.


## 4. Analysis and Improvement of Slow Start Threshold for FAST TCP Network Convergence Speed

### 4.1. Slow Start Queue Delay Ratio

As can be seen from Algorithm1, the size of the slow start threshold was set to decide the time at which the slow start algorithm switched to the FAST TCP window in order to update the algorithm.Firstly, we defined the meaning of each variable as shown below:T1=mi is the execution time from the start of establishing the connection to the end of the slow start.T2=N·αi/C is the queue delay for executing the FAST TCP window update algorithm when the system was balanced.Queuing delay ratio was obtained from past studies [1, 5, 6]:
(1)T1T2=miN·αi/C=miC1N·αi.Formula (1) represents the queue latency ratio of executing the slow start algorithm and executing the FAST TCP algorithm for a FAST TCP connection from the start of the establishment to the achievement of the equilibrium state.We can see that the value obtained through Formula (1) changes when the relevant parameters N, αi, mi, and C change, thus indicating that the relative time of slow start algorithm execution also changes.


### 4.2. Simulations to Verify the Relationship between Queue Delay Ratio of Slow Start and Convergence Speed of FAST TCP System

Experiment 1.
The number of connectionN was set to 3, the simulation time was set to 10 (s), and the size of each packet was set to 1000 bytes. Next, we set the link transmission bandwidth C to 96 Mb/s (C=12000 packets/s). FAST TCP sets the static mapping parameter as follows: αi=200, mi=0.00075 (s), and γi=0.5. The transmission delay at the FAST TCP source side was set to di=0.1s. We tracked the data sending rate of FAST TCP connections. The simulation results are shown in Figure 2 and T1/T2=0.015 were obtained by using Formula (1). As shown in Figure 2, the convergence time of all connections (fast 1, fast 2, and fast 3, representing the flow rates of the three connections of the system, respectively) is 2 (s), and the overshoot is 25. Thus, the system had a faster convergence speed and a smaller overshoot.Experiment 2.
The link transmission bandwidthC was set to 960 Mb/s (C=120000 packets/s). The other parameters were the same as in Experiment 1. The simulation results are shown in Figure 3.Figure 2
Each connection transmits traffic, connection start time, and overshoot whenC=120000 (packets/s), αi=200.Figure 3
Each connection transmits traffic, connection start time, and overshoot whenC=120000 (packets/s), αi=200.The value ofT1/T2=0.15 obtained from Formula (1) indicated that each connection accounted for a large proportion in the slow start phase compared with Experiment 1. Therefore, in Figure 3, we can see that the convergence time of the system is st>10 (s), and the overshoot is δ>200%.Using Formula (1), we determined that when the bandwidth increased, the (1) ratio can be kept constant by increasing the protocol parameter αi.Experiment 3.
Here, we increased the protocol parameterαi=2000, but the other parameters were still the same as in Experiment 2. The simulation results are shown in Figure 4. The value of T1/T2=0.015 obtained from Formula (1) was the same as that in Experiment 1. Therefore, in Figure 3, it can be seen that the convergence time of the system is st=2 (s) and the overshoot δ=25%, which are the same as in Experiment 1. The performance of the proposed method is better in Experiment 1 compared to Experiment 2. However, it is known from the literature [1] that the queuing delay of link in Experiment 2 is 0.005 s in a balanced state, whilst the bottleneck queuing delay in Experiment 3 increased to 0.05 s when the value of αiincreased to 2000.Experiment 4.
Table1 presents the simulation values of system overshoot and convergence time when the network is guaranteed to be in a stable state. The relevant parameters of the network and FAST TCP protocol are different from those in Table 1.Figure 4
Each connection transmits traffic, connection start time, and overshoot, whenC=120000 (packets/s), αi=2000.Table 1
Relationships among slow start threshold, system overshoot, and convergence rate.
SNαimiC (packet/s)Nd (s)T1/T2δst (s)12000.000751200020.10.022512.4%2.221000.00751200040.10.22561.5%2.232000.00151200020.10.04514.5%2.141000.0031200040.10.098%1.754000.0041200010.10.1227.5%1.962000.0061200020.10.1843.4%2.072000.000751200040.10.0112541.6%1.982000.00751200040.10.122571.5%2.294000.00151200020.10.023023.5%1.95104000.0031200020.10.04727.6%1.85112000.0051200040.10.08121.3%1.8122000.0061200040.10.09920%1.75132000.0071200040.10.10566%2.1142500.000751200040.150.00921.5%3.05152500.00751200040.150.0920.5%2.75165000.0051200020.150.0633.3%3.3175000.0091200020.150.10829.1%2.8The experimental results of Experiments1–4 indicate the following:
(1)
When network parameters changed, the queue delay ratio of the slow startup queue of (1) changed, which in turn affected the performance of the FAST TCP protocol (overshoot and convergence time)(2)
Increasing the protocol parameterαi under the same conditions improved the convergence speed of the system; however, doing so also increased the link queuing delay(3)
When different slow start thresholds were chosen, the networks exhibited different levels of performance. Comparing the data of groups 1, 7, and 14 with those in 4, 12, and 15 in Table1, we can see that the default value of the slow start threshold currently used was not optimal. From the data of groups 4, 12, and 15, the network showed smaller overshoot and faster convergence speed when the corresponding slow start threshold was set at T1/T2∈0.09,0.1


### 4.3. FAST TCP Window Update Algorithm for Improving Slow Start Conditions

On the basis of the above experiments and on Formula (1), we used the current queuing delay parameter as the criterion of the slow start threshold. When the network-related parameters changed, the value T1/T2 also changed along with the performance of the FAST TCP protocol. Therefore, we defined the parameter number li=αi/G (G=10) of packets for each connection left in the buffer, which was used as the slow start threshold for each connection. In this way, even if the bandwidth changes along with the number of connections and other parameters, the value will be kept within the range of (0.09, 0.1), thus ensuring good network performance. The specific improved algorithm is presented below.Algorithm 2
li=αi/10.Each acknowledgement frame is received {If (it was the first time a connection is made andzik′<li)//zik′: the number of data packets actually left in the link buffer at the time of connection i at k·T [8]wik′+1=wik′+1//execute slow start window update algorithmElse if (zik≠αi)Every slotT {wik+1=1−γi·wik+γidi/di+qik·wik+αi//Execute FAST TCP window update algorithm}The validity of Algorithm2 is verified below.Experiment 5.
In this experiment, the parameter settings were the same as those in Experiment2, but the source side used the improved Algorithm 2. After tracking the data sending rate of the FAST TCP streams, the simulation results indicated that the convergence time of the FAST TCP system was st=6 (s), as shown in Figure 5. The overshoot δ was less than 6%. Compared to Experiment 2, Algorithm 2 significantly reduced the overshoot and improved the convergence speed of the system without increasing the protocol parameter αi (i.e., queuing delay).Figure 5
Each connection transmits traffic, connection start time, and overshoot based on improved Algorithm2 when c=120000 (packets/s), αi=200.


## 4.1. Slow Start Queue Delay Ratio

As can be seen from Algorithm1, the size of the slow start threshold was set to decide the time at which the slow start algorithm switched to the FAST TCP window in order to update the algorithm.Firstly, we defined the meaning of each variable as shown below:T1=mi is the execution time from the start of establishing the connection to the end of the slow start.T2=N·αi/C is the queue delay for executing the FAST TCP window update algorithm when the system was balanced.Queuing delay ratio was obtained from past studies [1, 5, 6]:
(1)T1T2=miN·αi/C=miC1N·αi.Formula (1) represents the queue latency ratio of executing the slow start algorithm and executing the FAST TCP algorithm for a FAST TCP connection from the start of the establishment to the achievement of the equilibrium state.We can see that the value obtained through Formula (1) changes when the relevant parameters N, αi, mi, and C change, thus indicating that the relative time of slow start algorithm execution also changes.


## 4.2. Simulations to Verify the Relationship between Queue Delay Ratio of Slow Start and Convergence Speed of FAST TCP System

Experiment 1.
The number of connectionN was set to 3, the simulation time was set to 10 (s), and the size of each packet was set to 1000 bytes. Next, we set the link transmission bandwidth C to 96 Mb/s (C=12000 packets/s). FAST TCP sets the static mapping parameter as follows: αi=200, mi=0.00075 (s), and γi=0.5. The transmission delay at the FAST TCP source side was set to di=0.1s. We tracked the data sending rate of FAST TCP connections. The simulation results are shown in Figure 2 and T1/T2=0.015 were obtained by using Formula (1). As shown in Figure 2, the convergence time of all connections (fast 1, fast 2, and fast 3, representing the flow rates of the three connections of the system, respectively) is 2 (s), and the overshoot is 25. Thus, the system had a faster convergence speed and a smaller overshoot.Experiment 2.
The link transmission bandwidthC was set to 960 Mb/s (C=120000 packets/s). The other parameters were the same as in Experiment 1. The simulation results are shown in Figure 3.Figure 2
Each connection transmits traffic, connection start time, and overshoot whenC=120000 (packets/s), αi=200.Figure 3
Each connection transmits traffic, connection start time, and overshoot whenC=120000 (packets/s), αi=200.The value ofT1/T2=0.15 obtained from Formula (1) indicated that each connection accounted for a large proportion in the slow start phase compared with Experiment 1. Therefore, in Figure 3, we can see that the convergence time of the system is st>10 (s), and the overshoot is δ>200%.Using Formula (1), we determined that when the bandwidth increased, the (1) ratio can be kept constant by increasing the protocol parameter αi.Experiment 3.
Here, we increased the protocol parameterαi=2000, but the other parameters were still the same as in Experiment 2. The simulation results are shown in Figure 4. The value of T1/T2=0.015 obtained from Formula (1) was the same as that in Experiment 1. Therefore, in Figure 3, it can be seen that the convergence time of the system is st=2 (s) and the overshoot δ=25%, which are the same as in Experiment 1. The performance of the proposed method is better in Experiment 1 compared to Experiment 2. However, it is known from the literature [1] that the queuing delay of link in Experiment 2 is 0.005 s in a balanced state, whilst the bottleneck queuing delay in Experiment 3 increased to 0.05 s when the value of αiincreased to 2000.Experiment 4.
Table1 presents the simulation values of system overshoot and convergence time when the network is guaranteed to be in a stable state. The relevant parameters of the network and FAST TCP protocol are different from those in Table 1.Figure 4
Each connection transmits traffic, connection start time, and overshoot, whenC=120000 (packets/s), αi=2000.Table 1
Relationships among slow start threshold, system overshoot, and convergence rate.
SNαimiC (packet/s)Nd (s)T1/T2δst (s)12000.000751200020.10.022512.4%2.221000.00751200040.10.22561.5%2.232000.00151200020.10.04514.5%2.141000.0031200040.10.098%1.754000.0041200010.10.1227.5%1.962000.0061200020.10.1843.4%2.072000.000751200040.10.0112541.6%1.982000.00751200040.10.122571.5%2.294000.00151200020.10.023023.5%1.95104000.0031200020.10.04727.6%1.85112000.0051200040.10.08121.3%1.8122000.0061200040.10.09920%1.75132000.0071200040.10.10566%2.1142500.000751200040.150.00921.5%3.05152500.00751200040.150.0920.5%2.75165000.0051200020.150.0633.3%3.3175000.0091200020.150.10829.1%2.8The experimental results of Experiments1–4 indicate the following:
(1)
When network parameters changed, the queue delay ratio of the slow startup queue of (1) changed, which in turn affected the performance of the FAST TCP protocol (overshoot and convergence time)(2)
Increasing the protocol parameterαi under the same conditions improved the convergence speed of the system; however, doing so also increased the link queuing delay(3)
When different slow start thresholds were chosen, the networks exhibited different levels of performance. Comparing the data of groups 1, 7, and 14 with those in 4, 12, and 15 in Table1, we can see that the default value of the slow start threshold currently used was not optimal. From the data of groups 4, 12, and 15, the network showed smaller overshoot and faster convergence speed when the corresponding slow start threshold was set at T1/T2∈0.09,0.1


## 4.3. FAST TCP Window Update Algorithm for Improving Slow Start Conditions

On the basis of the above experiments and on Formula (1), we used the current queuing delay parameter as the criterion of the slow start threshold. When the network-related parameters changed, the value T1/T2 also changed along with the performance of the FAST TCP protocol. Therefore, we defined the parameter number li=αi/G (G=10) of packets for each connection left in the buffer, which was used as the slow start threshold for each connection. In this way, even if the bandwidth changes along with the number of connections and other parameters, the value will be kept within the range of (0.09, 0.1), thus ensuring good network performance. The specific improved algorithm is presented below.Algorithm 2
li=αi/10.Each acknowledgement frame is received {If (it was the first time a connection is made andzik′<li)//zik′: the number of data packets actually left in the link buffer at the time of connection i at k·T [8]wik′+1=wik′+1//execute slow start window update algorithmElse if (zik≠αi)Every slotT {wik+1=1−γi·wik+γidi/di+qik·wik+αi//Execute FAST TCP window update algorithm}The validity of Algorithm2 is verified below.Experiment 5.
In this experiment, the parameter settings were the same as those in Experiment2, but the source side used the improved Algorithm 2. After tracking the data sending rate of the FAST TCP streams, the simulation results indicated that the convergence time of the FAST TCP system was st=6 (s), as shown in Figure 5. The overshoot δ was less than 6%. Compared to Experiment 2, Algorithm 2 significantly reduced the overshoot and improved the convergence speed of the system without increasing the protocol parameter αi (i.e., queuing delay).Figure 5
Each connection transmits traffic, connection start time, and overshoot based on improved Algorithm2 when c=120000 (packets/s), αi=200.


## 5. Analysis and Improvement of Control Law Gain Parameters on the Protocol Convergence Rate

### 5.1. Theoretical Analysis

Previous studies [2, 3] have introduced a more accurate FAST protocol model, including self-synchronization and integral link, and the Nyquist stability criterion was applied to show that the system was asymptotically stable for bounded round-trip delay in a single bottleneck link when the value range of control rate gain parameter γ was 0,0.93.Firstly, we analysed the influence of gain parameterγi of control law on the convergence rate of FAST TCP system when γi∈0,0.93.We divide both sides of Equation (2) by propagation delay di, thus arriving at
(2)Sit+T=1−γi·Sit+γi1di+qit·Sit+αidi,which we organised as follows:
(3)Sit+T=1−γi·qtdi+qt·Sit+γi·α⌢i,whichα⌢i=αidi.Theorem 1.
We also assumed that theγi values of all FAST TCP connections were equal and satisfied γi∈0,0.93, that St=∑iSit, and that the sizes of the FAST TCP system at exit slow start and equilibrium point were S0 and S∗, respectively. Thus, for any small positive number ε, the convergence time st satisfies the following:
(4)st≥lnε/S0−S∗ln1−γi,so∣Sst−S∗<ε,whichS∗=c+∑iα⌢i..Proof.
From (3), we arrive at the following:
(5)Sit+T=1−γi·Sit+γidit·Sitdi+qt+α⌢i.
According to a past study [1], ∑idit·Sit/di+qt=C, and this can be obtained by summing up the source terminals of Equation (5) as follows:
(6)St+T=1−γi·St+γi·c+α⌢,where α⌢=∑iα⌢i.☐Then, we can transform Equation (6) into
(7)St+T−c+α⌢=1−γi·St−c+α⌢.Equation (7) means an equal ratio sequence wherein the first term is S0−c+α⌢, and the common ratio is 1−γi for any st=k·T. Thus, we have
(8)Sst=c+α⌢+Y0−c+α⌢·1−γist.From Equation (8), we know that Sst converges to equilibrium point S∗, and S∗ satisfies
(9)S∗=c+α.⌢Therefore, Equation (8) can be expressed as follows:
(10)Sst=S∗+S0−S∗·1−γist.From Equation (10), we know that for any small positive number ε, when the convergence time is
(11)st≥lnε/S0−S∗ln1−γi,we have Sst−S∗≤ε.The numerator of Equation (11) is negative, which means that the larger the gain parameter γi of the control law, the shorter the time for the system to converge to the equilibrium point.Each FAST TCP connection can calculate the numberzit=wit/Ritqit of the link buffers actually left by the connection according to the local information wit,qit,di,αi obtained by the source. Then, we calculate the absolute position ηi=zit−αi and relative position ηi/αi of the actual number of link buffers, after which the expected equilibrium point and the relative position are normalised to diffi=minηi/αi,1. Under the same conditions, the larger the absolute position ηi relative to the normalised position diffi, the larger the convergence time st. As shown in Equation (11), the convergence time st decreases with the increase of the gain parameter γi of the control law. Therefore, the following exponential function formula can be used to ensure the stability of the system:
(12)γi=0.5+0.5·eκ·diffi−1eκ−1κ<−1.The gain parameterγi was selected according to the standardised relative position. When the system is far away from the equilibrium point (diffi value is larger), the larger gain parameter γi is selected so that the system could quickly converge to the equilibrium point. When the system is near the equilibrium point (diffi value is smaller), in order to reduce the overshoot and jitter, the smaller gain parameter γi could be selected to ensure that the system has less jitter near the equilibrium point.


### 5.2. FAST TCP Window Update Algorithm for Improving the Gain Control Law Gain Parameters

Algorithm3 presents a FAST TCP window update algorithm, which can improve the slow start threshold and control law gain parameters simultaneously.Algorithm 3: The algorithm is described as follows.
FAST TCP window update algorithm with an improved slow start condition and gain parameters:li=αi/10.Each acknowledgement frame is received {If (it was the first time a connection is made andzik′<li)wik′+1=wik′+1//execute slow start window update algorithmElse if (zik≠αi)Every slotT {{Flag=1;ηi=ABSxit·qit−αi;diffi=minηi/αi,1γi=0.5+0.5·eκ·diffi−1/eκ−1κ<−1wik+1=1−γiwik+γidi/di+qikwik+αi}}


### 5.3. Simulation Verification.

The effectiveness of Algorithm3 is verified below.Experiment 6.
The parameter setting was the same as in Experiment2, but the source side adopted improved Algorithm 3. The parameter of Algorithm 3 was κ=−10. The simulation results are shown in Figure 6. As can be seen in the figure, the convergence time of the system was st≤4 (s), and the overshoot was δ≤5%. Compared with Experiments 2, 3, and 5, the improved Algorithm 3 reduces the overshoot of the system and improves the convergence speed of the system without increasing the protocol parameter αi (i.e., queuing delay).Figure 6
WhenC=120000 (packets/s) and αi=2000, based on the improved Algorithm 3 of each connection transmission flow, connection start time and overshoot.


## 5.1. Theoretical Analysis

Previous studies [2, 3] have introduced a more accurate FAST protocol model, including self-synchronization and integral link, and the Nyquist stability criterion was applied to show that the system was asymptotically stable for bounded round-trip delay in a single bottleneck link when the value range of control rate gain parameter γ was 0,0.93.Firstly, we analysed the influence of gain parameterγi of control law on the convergence rate of FAST TCP system when γi∈0,0.93.We divide both sides of Equation (2) by propagation delay di, thus arriving at
(2)Sit+T=1−γi·Sit+γi1di+qit·Sit+αidi,which we organised as follows:
(3)Sit+T=1−γi·qtdi+qt·Sit+γi·α⌢i,whichα⌢i=αidi.Theorem 1.
We also assumed that theγi values of all FAST TCP connections were equal and satisfied γi∈0,0.93, that St=∑iSit, and that the sizes of the FAST TCP system at exit slow start and equilibrium point were S0 and S∗, respectively. Thus, for any small positive number ε, the convergence time st satisfies the following:
(4)st≥lnε/S0−S∗ln1−γi,so∣Sst−S∗<ε,whichS∗=c+∑iα⌢i..Proof.
From (3), we arrive at the following:
(5)Sit+T=1−γi·Sit+γidit·Sitdi+qt+α⌢i.
According to a past study [1], ∑idit·Sit/di+qt=C, and this can be obtained by summing up the source terminals of Equation (5) as follows:
(6)St+T=1−γi·St+γi·c+α⌢,where α⌢=∑iα⌢i.☐Then, we can transform Equation (6) into
(7)St+T−c+α⌢=1−γi·St−c+α⌢.Equation (7) means an equal ratio sequence wherein the first term is S0−c+α⌢, and the common ratio is 1−γi for any st=k·T. Thus, we have
(8)Sst=c+α⌢+Y0−c+α⌢·1−γist.From Equation (8), we know that Sst converges to equilibrium point S∗, and S∗ satisfies
(9)S∗=c+α.⌢Therefore, Equation (8) can be expressed as follows:
(10)Sst=S∗+S0−S∗·1−γist.From Equation (10), we know that for any small positive number ε, when the convergence time is
(11)st≥lnε/S0−S∗ln1−γi,we have Sst−S∗≤ε.The numerator of Equation (11) is negative, which means that the larger the gain parameter γi of the control law, the shorter the time for the system to converge to the equilibrium point.Each FAST TCP connection can calculate the numberzit=wit/Ritqit of the link buffers actually left by the connection according to the local information wit,qit,di,αi obtained by the source. Then, we calculate the absolute position ηi=zit−αi and relative position ηi/αi of the actual number of link buffers, after which the expected equilibrium point and the relative position are normalised to diffi=minηi/αi,1. Under the same conditions, the larger the absolute position ηi relative to the normalised position diffi, the larger the convergence time st. As shown in Equation (11), the convergence time st decreases with the increase of the gain parameter γi of the control law. Therefore, the following exponential function formula can be used to ensure the stability of the system:
(12)γi=0.5+0.5·eκ·diffi−1eκ−1κ<−1.The gain parameterγi was selected according to the standardised relative position. When the system is far away from the equilibrium point (diffi value is larger), the larger gain parameter γi is selected so that the system could quickly converge to the equilibrium point. When the system is near the equilibrium point (diffi value is smaller), in order to reduce the overshoot and jitter, the smaller gain parameter γi could be selected to ensure that the system has less jitter near the equilibrium point.


## 5.2. FAST TCP Window Update Algorithm for Improving the Gain Control Law Gain Parameters

Algorithm3 presents a FAST TCP window update algorithm, which can improve the slow start threshold and control law gain parameters simultaneously.Algorithm 3: The algorithm is described as follows.
FAST TCP window update algorithm with an improved slow start condition and gain parameters:li=αi/10.Each acknowledgement frame is received {If (it was the first time a connection is made andzik′<li)wik′+1=wik′+1//execute slow start window update algorithmElse if (zik≠αi)Every slotT {{Flag=1;ηi=ABSxit·qit−αi;diffi=minηi/αi,1γi=0.5+0.5·eκ·diffi−1/eκ−1κ<−1wik+1=1−γiwik+γidi/di+qikwik+αi}}


## 5.3. Simulation Verification.

The effectiveness of Algorithm3 is verified below.Experiment 6.
The parameter setting was the same as in Experiment2, but the source side adopted improved Algorithm 3. The parameter of Algorithm 3 was κ=−10. The simulation results are shown in Figure 6. As can be seen in the figure, the convergence time of the system was st≤4 (s), and the overshoot was δ≤5%. Compared with Experiments 2, 3, and 5, the improved Algorithm 3 reduces the overshoot of the system and improves the convergence speed of the system without increasing the protocol parameter αi (i.e., queuing delay).Figure 6
WhenC=120000 (packets/s) and αi=2000, based on the improved Algorithm 3 of each connection transmission flow, connection start time and overshoot.


## 6. Dynamic Environment Simulation

When FAST TCP connections in the bottleneck link arrive and depart dynamically, the improved slow start threshold and control law gain parameter FAST TCP algorithm could also reduce the overshoot of the system and improve the convergence speed of the system without increasing the protocol parametersαi (i.e., queuing delay).Experiment 7.
In this experiment, the link bandwidthC was 960 Mb/s, αi=200, di=0.1 (s), the fast 1 and fast 2 connections were established at the beginning, the fast 3 connection was established at 100 s, and the fast 2 connection was released at 200 s. If there was no improved algorithm, we take mi=0.00075 (s), γi=0.5. The simulation results in Figure 7 revealed the convergence time of FAST TCP system st>20 (s) and overshoot δ>200% at the start of the system. When the new FAST TCP flows joined and departed, the convergence time of the FAST TCP system was st>20 (s). Under the same conditions cited above, the source side adopted the improved Algorithm 3 without setting parameters mi,γi (assuming that each fast connection obtained accurate propagation delay and could allocate bandwidth fairly). The simulation results in Figure 8 revealed that the convergence time of the FAST TCP system was st=4 (s) and the overshoot was δ=5%.Figure 7
The convergene time and overshoot with the original FAST TCP algorithm.Figure 8
The convergence time and overshoot with the improved FAST TCP algorithm.


## 7. Conclusion

In this paper, the number of remaining link buffers was taken as the judgment condition of a slow start. This ensured that the distribution of queuing delay ratio was within the expected range and would not change with the change of bandwidth, the number of connections, and other parameters, thus improving the convergence speed of the system. At the same time, an improved method of dynamically adjusting the gain parameters of the controller was proposed. According to the local information (sending window, protocol parameters, RTT, and propagation delay), which can be obtained by each connection source, this method judged the network operation state and adjusted the gain parameters of control law according to the network operation state within the range of relevant protocol parameters to ensure the stability of FAST TCP system.The simulation results showed that the two improved methods overcame the defects of static selection of protocol parameters in the original FAST TCP algorithm and ensured a stable and small queuing delay in the system. At the same time, the speed of convergence to the equilibrium point was improved, and high speed, large connection, and low delay were achieved for personalized healthcare systems.In the future, each connection cannot communicate directly, so each connection cannot synchronously adjust the protocol parameters, and the convergence speed of the protocol parameter adjustment is slow. The next step is to discuss the synchronization of connection protocol parameters so as to provide fairness in bandwidth allocation.



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*Source: 9947883-2021-06-21.xml*