Congestion Feedback Control for Computer Networks with Bandwidth ...

Congestion feedback control for computer networks with bandwidth estimation Sławomir Grzyb

Przemysław Orłowski

Faculty of Electrical Engineering West Pomeranian University of Technology Szczecin, Poland [email protected]

Faculty of Electrical Engineering West Pomeranian University of Technology Szczecin, Poland [email protected]

Abstract— Effective congestion control allows networks to operate in region of low delay and high throughput. These characteristics seem to be the most required from network environment properties. To achieve high transfer rates, variety of methods and algorithms are proposed in the literature. This paper focuses on congestion control using a method combining a particle swarm optimization algorithm and the use of finite impulse response filer for non-stationary, discrete, dynamical model of communication channel. Proposed control strategy tunes the model to alleviate the results of sudden, unexpected network state changes. That helps to avoid undesirable congestion effects like packet dropping, retransmissions, high transfer latency and low network throughput. Keywords— congestion control, discrete-time systems, dynamical model, FIR filters, particle swarm optimization.

I. INTRODUCTION Congestion management in its broadest sense plays the key role in maintaining efficient and reliable networks [1], [2], [3]. This challenge is widely presented in [4], [5], [6]. This unfavorable phenomenon affects different sorts of grids like power networks and data exchange networks. Congestion management and overload avoidance in power networks are discussed in [1]. This aspect concerning communication channels in computer networks is presented in [3], [5]. To minimize the effects of blockage formation, variety methods and algorithms are proposed like sliding-mode algorithm [5], [7], fuzzy logic [8] as well as others [9], [10], [11]. In recent time, remarkable attention paid to piece-wise affine (PWA) algorithm, can be observed [12], [13], [14], [15]. This attention is also directed toward nonlinear systems [16], [17] whose stability matter are discussed e.g. in [18], [19], [21], [22]. Particle swarm optimization algorithm, used in this research is successfully used in power control systems to alleviate results of overload and minimize operational costs [1], [20], [23], [24]. Model of communication channel taken under consideration in this paper is described in details in [25]. The analysis of this model with the use of simplified frequency characteristics are provided in [26]. Other control strategies involving this model are presented in [15]. The aim of research presented in the paper is to control the length of outgoing packet queue in such way, to ensure maximal available throughput utilization as well as minimal egress buffer utilization necessary to meet this requirement. The finite impulse response (FIR) filter is used as trajectory

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planner. It relates estimated throughput to the reference buffer utilization. The trajectory planner enables to control the number of packets accumulated in egress buffer of congested node for the discrete non-stationary, dynamical system. Particle swarm optimization is applied in order to calculate coefficients for the trajectory planner as well as for the controller. In the first section of the paper presents non-stationary, discrete, dynamical model. It is followed by description of general issues of taken congestion modeling . Then the control strategy chosen for presented model is discussed. Subsequent section describes applied particle swarm optimization algorithm. The summary of taken considerations and calculations is illustrated in numerical example. II.

MATHEMATICAL MODEL OF THE NETWORK

The network model taken into consideration in this paper is like in [25]. Distinguished key elements are: a source, congested node, destination and a certain number of intermediate nodes. Graphical illustration is shown in Fig. 1. Available throughput h(k) determines the amount of data that can potentially be sent from the congested node CN. It can be modeled as a certain, unknown, restricted function of time d  k  . 0  d  k   d max

(1)

where: d max - maximum bandwidth at any moment of time The number of packets sent from the congested node CN can't exceed the amount of data stored in its output buffer y(k ) . The inequality satisfying this statement is following: 0  h  k   d  k   d max

(2)

and 0  hk   y k 

(3)

where: h( k ) - the number of packets successfully transmitted from the congested node toward the destination in time k.

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III. Packets flow direction

node n-1

S node 2 node 1

h(k) Congested Node

h(k)

D

u(k) y(k)

Fig. 1. Block diagram of communication channel with delay varying in time.

Limited throughput between source and congested node required to state following assumption: u  k   u max

(4)

A full model in vector-matrix notation can be written in the following way: x  k  1   q1  k    q1  k  0   0 

0

0



q2  k  0  q2  k  q3  k  0  

 0

 qn  1  k 

0 1  0  0 0 0 (5)       x k     u k      hk       0 0 0 0  1 1     

y  k   0 0  0 1 x  k 

(6)

CONGESTION CONTROL STRATEGY

Ability of network environment to avoid blockages strongly depends on buffers utilization of network nodes. Utilization level of these buffers should be adequate to accumulate packets in case of congestion appearance. When a buffer is full up, the node is not able to accept any incoming pieces of data. In this state incoming packets are being dropped. That leads to retransmission attempts by terminals involved in data exchange process. It results in increasing network occupancy and decreasing effective network throughput. But in case of sudden throughput rise, a buffer should contain enough data, not to be completely emptied before new data portion flow in. It can be concluded from these facts, that some relation between varying in time available throughput and expected buffer occupancy yr of CN should be found. This available bandwidth may be impacted by some noise. Heuristic and numerical analysis of the problem of controlling the outgoing packet queue length clearly shows that the reference queue length should be independently controlled. The most important variable that have impact on the reference queue length yr(k) is the available bandwidth d(k). In this paper the linear relationship between the reference queue length yr(k) and the available bandwidth is assumed. Due to the presence of perturbations/noise in the signal d(k) and the delay between input and output, 3rd order FIR filter is applied. It relays the reference buffer queue length to varying egress bandwidth . This relation takes the following form:

where:

yr  k   z1d est  k  1  z2 d est  k  2   z3 d est  k  3 

x j (k ) - the number of packets in node j in time k

The controller in feedback loop has simple proportional gain form:

0 - transmission q j  k  - queuing factor q j  k    1 - congestion q j k   1  q j k 

u  k   z 4  yr  k   y  k  

u (k ) - number of packets requested by congested node from the source In each time step k, node j sends its buffer content xj to the next node j+1 toward the destination. It can occur only when introduced queuing factor qj=0. For simulation purposes it is assumed that no bandwidth restrictions have impact on the throughput between internodes. In case the factor equals 1, all packets are stored in node j buffer. The difference between number of packets flowing into the congested node and packets transmitted towards the destination increases the queue length in congested node when it is positive or decreases when it is negative. The controller is in charge to keep this queue length of the right level. Variable u(k) is the control signal sent backward to the source to adjust the number of packets sent from source to destination through the congested node to current network conditions.

(7)

(8)

Available bandwidth d(k) is a function unknown in time. Moment value can be directly calculated only when there is sufficient number of packets stored in egress buffer to fill this bandwidth. Otherwise d(k) can be estimated by following function: h (k )   d est  k    d max 0.5  h( k  1)  h (k  2)  

y (k )  0 y ( k  1)  0  y ( k  2)  0 (9) otherwise

Fig. 2 presents a block diagram of control system with block representation of key components and variables used in mathematical model. FIR filter is used for purpose to generate optimal reference signal yr and to alleviate the influence of noise in d(k).

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Assume that in space S has n dimensions and consists on N particles. Individual particle has in time its own position Z and speed V. It can be described by:

d(k) h(k)

Plant

Estimator

y(k)

u(k)

_

Controller

Z ti   z1i , z2i , z3i , , zni  and Vt i  v1i , v2i , v3i , , vni 

dest(k) yr(k) +

FIR filter

Fig. 2. Block diagram of system controlling the queue length for the system with feedback and proportional controller.

One of plant's inputs is an available bandwidth d(k) which is not measurable. As is mentioned, it can be estimated by measurable h(k) and in further steps this variable is processed. Particle swarm optimization algorithm is used to obtain a vector: Z   z1 z2 z3 z4  (10) We define a cost function to adjust the control to the selected design requirements. It can be described as follows:

J ( z1 , z2 , z3 , z 4 ) 

N

  y(k )  1000  d (k )  h(k )  

(11)

k 30

This construction of cost function increases its value in case, when the outgoing bandwidth is not fully utilized. That means that it could be sent more packets, then is stored in egress buffer: d(k)-h(k). The weight of this component is 1000, in order to balance its impact with the other components. The second component of this function tries to minimize CN buffer utilization as much as possible: y(k). Due to assumed system order and present delays, integration algorithm starts from step 30th. Before this time step, presented sum depends not on controlling strategy, but on system order. Given that the system is described with model (5) and relations (7) and (8), the optimal controller and filter coefficients can be determined by solving the following optimization problem: min J ( z1 , z2 , z3 , z4 ) IV.

(13) (14)

These parameters in the next time step can be calculated as Vt i1  w  Vt i  c1  1  ( Pt i  Z ti )  c2  2  ( Pt i , g  Z ti ) (15) Z ti1  Z ti  Vt i1 , i  1, , N

(16)

where N - number of particles in the swarm, n - number of elements in a particle t - generation number w - inertia weight of the particle c1,c2 - acceleration constant  1 , 2 - random factors in the range [0,1] Pt i - best particle i position achieved so far Pt i , g - global best position of particle in the population Acceleration constant c1 directs each particle towards local best position and constant c2 pulls the particle towards global best position. The inertia weight is an important factor for the convergence of particle swarm optimization algorithm. A large value of it facilitates global searching of new area. Slight weight factor facilitates exploration of local area. Therefore, it is good practice to choose large weight factor for initial iterations and gradually reduce weight factor in successive iterations. This can be achieved by using w  wmax 

wmax  wmin  iter itermax

(17)

where wmax , wmin - maximum and minimum inertia weight iter , itermax - iteration number, maximum iteration number

(12)

PARTICLE SWARM OPTIMIZATION

Particle swarm optimization is a popular and efficient stochastic searching optimization method [23]. PSO is biologically-inspired solution motivated by social behavior of animals such as bees and birds flocking. In this algorithm potential solutions are called particles and evolve in a multidimensional problem space. Population of particles is called swarm. Individual particle in a swarm flies in the search space towards the optimum solution based on its own knowledge of the environment, individual's previous history and global best position among other particles in the swarm.

Efficiency in finding best position depends also on the velocity of particles in the swarm. If it is very low, particle may not explore sufficiently whereas if it is very high, it may oscillate about optimal solution. Finally it can increase towards infinity. Solution of limitation and controlled velocity decrease is as followed V  Vmin Vmax,t  Vmax  max (18)  iter itermax where Vmax,t - maximum velocity at generation t Vmax ,Vmin - initial and final velocity.

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For simulation purpose, initial particle position has uniform random distribution in the range between and initial population velocity is assumed to be 0.

Solid blue line represents the fundamental bandwidth without noise. 100

V.

90

SIMULATION RESULTS

80

In order to illustrate proposed control system, the following discrete non-stationary linear model is assumed. In this example, the numerical simulation of the communication channel with time-varying delay is made. The analysis subject is the length of the outgoing packet queue in the congested node. To fit model delays to those existing in real computer networks an order of system is assumed to be 30. For the purposes of simulation, it is assumed that the system can be described by the following discrete non-stationary linear model: x  k  1  A  k  x  k   B  k  u  k   F  k  h  k  y  k   C k  x  k 

(19)

70 60 50 40 30 20 10 0 0

0.2

0.4

0.6

0.8

1 1.2 Time [s]

1.4

1.6

1.8

2

Fig. 2. Available outgoing bandwidth for the congested node.

On the basis of presented relation (9), calculated dest(k) is illustrated in Fig. 3

where:

A  k   A ,

100 90

B  k   B  [1,0,0,...,0]T , C  k   C  [0,0,...,0,1],

80 70

(20)

60

F  k   F  [0,0,...,0,1]T

50

 k    floor  rem  ,0,04      

40 30 20 10 0 0

0.2

0.4

0.6

0.8

1 1.2 Time [s]

1.4

1.6

1.8

2

Fig. 3. Calculated dest(k)

(21)

On the basis of relations (7) and (8) of the model (5), using cost function (11), with the use of the algorithm described in section 4, the following coordinates are obtained:

Vectors B(k), C(k), and F(k) are constant in time. Matrix A0 represents a model of the communication channel without congestion in the intermediate nodes. Matrix A1 mirrors the state when blockage occurs in the 7th intermediate node. It is based on matrix A0 with following differences: element A1(7,7)=1 and element A1(8,7)=0. The system state, in which node 21st can’t forward packets is mapped by a matrix A2. It’s built on A0 pattern, taking into consideration that A2(21,21)=1 and A2(22,21)=0. Matrix A3 represents a model of congestion in the 14th node. Like previous values it’s based on A0 with two different elements: A3(14,14)=1 and A3(15,14)=0. For purposes of numerical simulation of model (19) -(21), it has been assumed that umax and dmax are 100 packets per sample and the maximum buffer capacity of congested node is 5000 packets. Sampling period is 10 ms. Initial conditions equal:

Z *   z1*

z2*

z3*

z *4   [14.03 7.90 5.07 4.84]

(22)

Appling the controller (8), FIR (7) with (22) and model (19), numerical simulation is performed. The setpoint value yr(k) is illustrated in Fig. 4. 3000

2500

2000

yr(k)

0 0 0 0 0 0  1 0 0 0 0 0    0 1 0 0 0 0  A0           0 0 0 1 0 0    0 0 0 0 1 1 

1500

1000

x  0   [0, 0, 0,..., 0]T

500

Assumed bandwidth d(k) actually available for the congested node towards destination is shown in Fig. 2. Available outgoing bandwidth with additive noise for the congested node, applied for optimization and simulation is plotted in red.

0 0

0.2

0.4

0.6

Fig. 4. Setpoint value vs. time

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0.8

1 1.2 Time [s]

1.4

1.6

1.8

2

[3]

The queue length of congested node is illustrated in Fig. 5. [4]

4000 3500

[5]

y(k) [packets]

3000 2500

[6]

2000 1500

[7]

1000 500 0

0

0.2

0.4

0.6

0.8

1 1.2 Time [s]

1.4

1.6

1.8

2

[8]

Fig. 5. Congested node’s queue length.

Fig. 6 illustrates controller response. It presents the requested amount of packet that the source receives from the controller deployed in congested node.

[9]

[10]

100

[11]

90 80

u(k) [packets]

70

[12]

60 50 40 30

[13]

20 10 0

[14] 0

0.2

0.4

0.6

0.8

1 1.2 Time [s]

1.4

1.6

1.8

2

Fig. 6. Amount of packets requested by controller form the source. [15]

VI.

CONCLUSION

The paper discusses a network congestion avoidance issue by controlling egress buffer utilization. A method of queue length control in active network nodes in this paper is proposed. The main achievements of the method are: effective bandwidth utilization and low buffer occupancy in order to ensure uninterrupted data flow. Particle swarm optimization combined with finite impulse response filter are used to evaluate controller settings and filter noised signal of available throughput. This method enables improvement of network throughput by more effective bandwidth utilization.

[16]

[17]

[18]

[19]

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[2]

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