Analytical Modelling of Weighted Round Robin Service
Strategy
Dapeng Yang1, Sijun Liu2, and Keming Wang1, 3
1
College of Industrial Engineering, Zhejiang University, Hangzhou, 310058, China
2
College of Computer Science, Shenzhen University, Shenzhen, 518060, China
3
School of Mechanical Engineering, Tsinghua University, Beijing, 100083, China
E-mail: sijunliu@163.com
Abstract. This paper presents a mathematical description of Weighted Round Robin service
strategy, on which basis all the perspective QoS tools that serve to manage congestion in
converged packet networks work. On the basis of the presented mathematical model, it is
possible to suitably configure the operational parameters (maximum queue length, distribution
of the available transfer capacity) of these tools according to required values of packet delays.
The implementation of the analytical model is demonstrated on a real network segment using
advanced network data traffic emulator. This paper presents a mathematical description of
Weighted Round Robin service strategy, on which basis all the perspective QoS tools that
serve to manage congestion in converged packet networks work. On the basis of the presented
mathematical model, it is possible to suitably configure the operational parameters (maximum
queue length, distribution of the available transfer capacity) of these tools according to
required values of packet delays. The implementation of the analytical model is demonstrated
on a real network segment using advanced network data traffic emulator.
1. Introduction
In converged IP networks, it is necessary to provide, apart from the conventional data operation, also
the video and voice communication support. Because IP packet architecture was not originally
intended for real-time transfers, from the point of view of interactive communication it is necessary to
solve the problem of quality guarantee of multimedia services—Quality of Service (QoS). Suitably
implemented QoS tools ought to guarantee a certain quality level of processing of time-sensitive
interactive applications at the expense of the quality of processing of other applications that do not
necessarily require real-time transfers. Without properly implemented QoS tools, the applications that
do not require real-time processing might use up the available transfer capacity and in this way make
impossible the transfer of time-sensitive interactive applications. The main problem of IP packet
network is thus the appropriate support of multimedia communication taking place in real time and the
connected suitable management of potentially occurring congestion. The appropriate implementation
of QoS tools mainly requires correct setting of the operation parameters of these tools. In order to
provide high Quality-of-Service (QoS) in today’s high-speed converged networks, WRR mechanism
assigns different priorities to different queues. WRR also ensures fair selection interval among all
2
To whom any correspondence should be addressed.
active queues with minimal delay and jitter [1]. In this scheduling algorithm, a weighting coefficient
for each queue determines how many bytes of data the system delivers from the queue before it moves
on to the next queue. The WRR mechanism cycles through the queues. For each queue, packets are
sent until the number of bytes transmitted exceeds the bandwidth determined by the queue’s weighting
coefficient, or the queue is empty [2, 3]. Then the WRR mechanism moves to the next queue. If a
queue is empty, WRR mechanism will send packets from the next queue that has packets ready to
send. This mechanism guarantees a minimum bandwidth to each queue, and allows the minimum to be
exceeded if one or more of the port’s other queues are idle. However, when all the queues are located,
each is limited to its maximum bandwidth according to its assigned weight—no queue achieves more
than a predetermined portion of transfer capacity when the transmission line is under stress [4]. WRR
includes several significant benefits. This scheduling mechanism can be implemented in hardware, so
it can be applied to high-speed interfaces in both the core and at the edges of the network. WRR
mechanism also ensures that all service classes have access to at least some configured amount of
network capacity to avoid bandwidth starvation. WRR queuing provides coarse control over the
percentage of output port bandwidth allocated to each service class. Classification of traffic by service
class provides more equitable management and more stability for network applications than the use of
priorities or preferences. WRR queuing is based on the belief that resource reduction is a better
mechanism to control congestion than resource denial [1, 2]. Other information about WRR models
and detailed description of the status quo can be found in source [4].
2. Experimental Method
The properties of three core samples used for the experiments are given in Table 1.
Table 1．The properties of core samples
Core number
Length/ cm
Diameter /cm
Permeability/mD
A#
5.24
2.511
0.017
B#
5.659
2.535
0.0287
C#
4.535
2.508
0.0675
These core samples are utilized after washing out any oil in the sample and then drying. Dry
nitrogen is used as an experimental gas source. The mass flow meter is used to measure gas flow.
Confining pressure is controlled and regulated using a hand pump. The experiments are conducted by
changing the pore pressure according to the development mode of the gas reservoir. In accordance
with the well depth of the core samples and the data of the oilfield, the initial formation pressure was
30Mpa, and the overburden pressure was 50Mpa. The confining pressure was kept 1Mpa more than
the pore pressure. And the confining pressure and the pore pressure were added in the meantime until
the pore pressure reach at 30Mpa. Then, the confining pressure was added to 50Mpa. The confining
pressure was kept constant, and the pore pressure was gradually reduced as the step was 2Mpa. The
permeability was measured by steady state method.
3. Result and Analyses
To evaluate formation stress sensitivity, we first normalize the permeability and the effective stress.
Figure 1-3 present the relationship between dimensionless permeability and dimensionless effective
stress.
Dimensionless Permeability
1
0.8
0.6
0.4
-0.9338
y = 0.9929x
R2 = 0.9931
A#,0.0107mD
Power relation
0.2
0
0.1
0.6
1.1
1.6
2.1
Dimensionless Effective Stress
2.6
Figure 1. The relationship between dimensionless
permeability and dimensionless effective stress of A#.
Dimensionless Permeability
1
0.8
0.6
y = 1.084x-1.1061
2
R = 0.9751
0.4
B#,0.0287mD
Power relation
0.2
0
0.1
0.6
1.1
1.6
2.1
Dimensionless Effective Stress
2.6
Figure 2. The relationship between dimensionless
permeability and dimensionless effective stress of B#.
4. Echo signal analysis in the 2D frequency domain
Consider radar emitting a series of chirp pulses
 ( )  p(t  kTr ) exp[ j 2 fc (t  kTr )]
(1)
where p(t )  rect (t Tp ) exp   j t  ， t is the time, k is the number of pulses, Tr is the pulse repetition
2
interval, f c is the carrier frequency, Tp is the pulse width,  is the frequency modulation rate and
  B Tp , B is the bandwidth (the pulse compression ratio D  BTp ).
Now suppose this pulse train illuminates a moving target at range R(t ) , and the echo signal is
recorded in a two-dimensional array s(t ' , tk ) , where t '  t  kTr is known as the “fast time” and tk  kTr
is known as the “slow time”. After downconversion, the received baseband signal can be written as

 4 f c

(2)
 exp   j c R(tk ) 



where A is constant and determined by the RCS of the target, T is the coherent integrated time, the
2 R(tk ) 

 tk
s (t ' , tk )  Ap t ' 
 rect  T
c



range of the target can be expressed as
1
R(tk )  R0  vtk  atk2
2
(3)
where R0 is the initial range, v is the radial velocity of the target, a is the radial acceleration of the
target. The Fourier transform of Equation (2) over the fast time t ' can be written as
 4
1 2 


 exp   j c ( f  f c )  R0  vtk  2 atk  




where f is the fast frequency, P( f ) is the Fourier transform of p (t ' ) , and we know that
t
S ( f , tk )  AP( f )rect  k
T
P( f ) 

f2
f 
rect   exp  j

 

B

1
(4)
(5)
After matched filtering(pulse compression) in fast-time-domain corresponding to Equation (4), we can
obtain
 
2R(tk ) 
AB
 tk 
 4 f c

(6)
s(t ' , tk ) 
sin c  B  t ' 
R(tk ) 
 rect   exp  j

c 
c

T 


 
We can see that the position of the signal peak varies with R(tk ) . That is to say, range cell migration
occurs.
5. A novel method of long term coherent integration detection
5.1. The process of the novel method and the mathematical derivation
The basic idea of the method in this paper is that transform S ( f , tk ) to S ( f , ) by keystone transform
in the fast frequency-slow time domain firstly to correct the range cell migration. Then quadratic phase
compensation function H ( f , ) is multiplied, and we can get the expression S ' ( f , ) in fast frequencyslow time domain. Next, we use the expression S ' ( f , ) to construct the objective


function max FFT IFFT f  S ' ( f , )  , thus, we use this function to search the acceleration in the
compensation function. The acceleration corresponding to the maximum value of the objective
function is the optimal estimator â of the target’s real acceleration a . At the same time, the maximum
value of the objective function equals to the coherent integration value. The flowchart of this method
is shown in Figure 1.
What we can know from (15) is that the range cell migration is already corrected, and the coherent
integration is performed in slow time domain. The peak of the spectrum is the intersection of
t '  2 R0 c and f d   2v  , and the corresponding value is the time delay of the target’s initial range
and the Doppler frequency of the initial velocity, respectively.
Because the acceleration of the quadratic phase compensation function H ( f , ) is unknown, it must be
estimated. We use the variable step length search method in this paper. Firstly, we construct the
objective function:
5.2. The problems of realizing the novel method
5.2.1. The implementation method of the keystone transform. There are many approaches to realize
the keystone transform such as sinc interpolation algorithm, DFT-IFFT algorithm, Chirp-Z-IFFT
algorithm, etc[7,10]. The realization of these methods can be inquired in the references, so we don’t
introduce here. We use the Chirp-Z-IFFT algorithm in this paper because the sinc interpolation
algorithm and the DFT-IFFT algorithm are more complicated.
5.2.2. The keystone transform of undersampled in the slow time domain. Range cell migration
correction using keystone transform can still be done even if the data is undersampled in the slow time
domain. But it must be multiplied by a fold factor. The expression of the fast frequency-slow time
after quadratic phase compensation is
6. Conclusions
DVE technology has the characteristics of regional and all objects in the virtual world need part into
different regions. The operation of the virtual world also needs server support. On the one hand, users
in DVE system are moving. On the other hand, these users can enter and leave the network at any
time. Therefore, this needs to provide a good and effective mechanism to solve the problem of
regional division and server distribution in the system. This paper presents a cluster-based server
divide method to realize part region and server distribution. In order to achieve clustering division, we
use DCA method to calculate each user’s view which combines the MinPts value with Theorem 1. In
addition, theoretical analysis and experimental simulation is carried out to verify the validity and
rationality of the DCA algorithm in this paper.
Acknowledgements
The results discussed in this paper have been obtained by the CBELSA/TAPS and Crystal
Barrel/TAPS collaborations. They are part of the theses works of M. Dieterle, I. Jaegle, Y. Magrhbi, F.
Pheron, D. Werthmuller, and L. Witthauer. This work was supported by National Natural Science
Foundation of China (60171009).
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
Zimmermann F, Haak T and Hill C 2004 Galileo System Simulation Facility, 8th Int. Workshop
on Simulation for European Space Programmes (Noordwijk)
Zimmermann F, Haak T, Steindl E, Vardarajulu S and Kalden O 2005 Generating Galileo Raw
Data – Approach and Application. Data Systems in Aerospace (Edinburgh)
Xie G 2009 Principles of GPS and Receiver Design (Beijing: Publishing House of Electronics
Industry) pp 71-104
Tsui J 2005 Fundamentals of Global Positioning System Receivers: A Software Approach (New
York: Wiley&Sons) pp 205-206
Lewandowski W, Azoubib J and Klepczynski W J 1999 GPS: Primary Tool for Time Transfer.
Proceedings of the IEEE 87 163
WU D 2007 Master thesis. Beijing University of Technology 1 p 1
Allan D W and Weiss M A 1980 Accurate Time and Frequency Transfer during Common-View
of A GPS Satellite. Proc. 34th Ann. Frequency. Control Symp (Monmouth) pp 334-346
Strite S and Morkoc H 1992 J. Vac. Sci. Technol. B 10 1237
Jain S C, Willander M, Narayan J and van Overstraeten R 2000 J. Appl. Phys. 87 965
Nakamura S, Senoh M, Nagahama S, Iwase N, Yamada T, Matsushita T, Kiyoku H and
Sugimoto Y 1996 Japan. J. Appl. Phys. 35 L74
Akasaki I, Sota S, Sakai H, Tanaka T, Koike M and Amano H 1996 Electron. Lett. 32 1105
O’Leary S K, Foutz B E, Shur M S, Bhapkar U V and Eastman L F 1998 J. Appl. Phys. 83 826
Qian Z G, Shen W Z, Ogawa H and Guo Q X 2002 J. Appl. Phys. 92 3683
Kurata M 1982 Numerical Analysis for Semiconductor Devices (Lexington, MA: Heath)
Selberherr S 1984 Analysis and Simulation of Semiconductor Devices (Berlin: Springer)