stability of computer network for the set delay dr jolanta tańcula
TRANSCRIPT
Stability of computer network for the set delay
dr Jolanta Tańcula
TCP-DCR protcol model
• In wireless network, it is difficult to state whether reduction in the efficiency of TCP is caused by errors in transmission whether connection overload. In order to optimize the network efficiency, various modifications of TCP standard have been proposed.
• One variant of TCP is TCP-DCR (Delay Control Rate) delayed response to errors. This protocol improves fault tolerance, which is obtained by adding a small time τ. If the package is not recovered by the retransmission before time τ, TCP starts algorithms against overload.
TCP-DCR protocol model
• TCP-DCR protocol can be described with non-linear differential equations
1))(()(
)(
2
1' D
DD PtRtptRtR
tRtWtW
rtttR
P
tR
PtW
tW
PrtttRPtR
tNC
tWPrtttRPtR
tNC
tq
DD
DD
1)(
)(;0max
1'
Linearization of the model
• We used approximated of the system dynamics by linearisation of the non-linear model around the determined operating point. Taking the window size W (congestion window size, indicating how many packets may be sent without waiting for acknowledgment of the receipt) and the queue length q as constant and packet marking/dropping probability p as the data, the assumed operating point is defined by W’(t)=0 and q’(t)=0 is determined by equation:
000 ,, pqW
• After linearization of equations around the operating point, we achieve a linear model of the system given in the form of the following equations
012
10
0
20
00
DDD Pp
R
tW
rttR
P
R
P
01 00
0
DD PrttRPR
tNWC
where individual differentials are defined
RR
RR
RR
fq
q
fp
p
fW
W
fW
W
fW 11111'
fW
W
fq
22'
rttR
PpW
W
f D
0
001
2
W
f
W
f
R
11
rttR
W
p
f
R
0
201 1
2
2
020
1
)(
//1
rttR
CP
R
CP
q
f DD
20
0201
)(2
/)(
rttR
CPpW
q
f D
R
0
2
R
N
W
f
20
02
CR
NW
q
f
Block diagram of model
• On the basis of equations and the network model is obtained, presented on the block diagram
fig 1.
• we perform to isolate as the high frequancy (parasitic). Substituting differential values to this scheme and simplifying the scheme, we obtain
fig.2
s
fig. 3 simplified diagram
Determination of transfer function of the model
• Transfer function of the open system, in accordance with the block diagram from fig.3 is determined by the formula
0
0
0
0
00
20
20
12
2)( Rs
D
e
CRNW
sRPpW
s
RNW
sP
• The above transfer function represents an inertial element of the second grade with permanent delay and will be used when defining the characteristic quasi-polynomial of a mathematical model. Transmittance P(s) will be used in the analysis of stability. Fig.2 also shows AQM block (Active Queue Management), which represents the traffic control in the network based on RED algorithm (Random Early Detection).
Definition and conditions of stability
• A computer network stability is when the trajectory for any initial conditions tends to zero. If we assume that the router packets are queuing is to take such action to queue decreased to zero and the traffic on the network run smoothly.
• Dynamic stability of linear systems with delays is completely determined by the decomposition of a complex variable plane zeros of its characteristic quasi-polynomial.
Definition of quasi-polynomial
• The notion and the test of stability apply to dynamic systems. The computer network is a special dynamic system and we could the stability test of this system. Let us state a linear dynamic system with one constant delay in the form
n
k
kk htxa
0
)( 0)(
where , are real coefficients and is constant delay. Since in the system there is a delay, therefore, the system stability depends on location of the elements of the characteristic quasi-polynomial.If we assume that the characteristic quasi-polynomial has the form:
where means the dominant unit, we assume the following labelling of the dominant unit
)()/()()( txdtdtx kkk ka0h
shn
kk
k egshsG
0
),(
shn eg
she
• Quasi-polynomial of delayed or neutral type is called asymptotically stable, if there is a positive number ε, for which the following condition is met
0, hsw sRe
Transfer function of the system for the set model
• In order to determine the quasi-polynomial of the dynamic system in Fig. 3 we assume the following symbols for parameters:
• To transfer function P(s) we substitute uncertain parameters , which will serve us to calculate the deviation of these parameters and delay as a result, we obtain
20
20
1 2R
NWd
0
0002
12
CR
PpCWNWd D
20
020
3
12
CR
PpNWd D
321 ,, ddd
0Rh
The main algorithm for Internet routers is RED algorithm. After determining the formula of its transfer function C (s) :
and based on the block diagram from Fig. 3, we create transfer function of the whole system described with the equation:
0
322
1 Rsedsds
dsP
Ks
KLsC
0132
232
2
RseKLddsdsKs
dsdsKLsG
Quasi-polynomial of the system
• The quasi-polynomial of the system has the form:
The characteristic quasi-polynomial will be used to test the stability of the fixed parameters and test method stability for the set delay
0,,,, 100RsedswdswdRsw
0132
20 ,, RseKLddsdsKsdRsw
0)()()(,, 13322
23
0RseKLdKdsdKdsdKsdRsw
• Dynamic systems with a delay have an infinite the number of roots and to test their stability for agreed values of delays, graphic (frequency) are used, e.g. Mikhailov criterion based on the following theorems.
Stability for agreed values of delay
• Theorem 1 • The dominant unit is asymptotically stable
only when graph , prepared in the function of parameter , does not go around the beginning of the variable complex plane or does it go through it. Necessary and sufficient condition
0Rse
,0
jRjRj od/,,~00
• Theorem 2The quasi-polynomial of delayed or neutral type is asymptotically stable only when on the variable complex plane, the graph of function
prepared for does not go through the beginning of the coordinate system of the variable complex plane, and means the reference polynomial of degree n and is
where a is any real positive number.
jRjRj od/,,~00
,0
jod
nod ass
Test of stability for the set delay
• Stability will be tested with frequency method for the set delay, that is , in the space of parameters .
• Assuming N = 60, C = 1000, p0 = 0.05, W0 = 10,
R0 = 0.22, α = 0.1, PD = 0.25 and substituting to equations (1), (2), we obtained appropriate values of parameters:
• Based on the form of the quasi-polynomial, the dominant unit is determined and then using th.1 and th.2, its stability is confirmed. For the above parameters, it is
22.0)( 0 R
321 ,, ddd
750001 d 5.302 d 5.823d
1)( 0 Rse
• A necessary condition of stability is asymptotic stability of the dominant unit. Since it has a constant value and in the complex characteristics, in respect of parameter , it is a point, lying at the beginning of the coordinate system, it is of delayed type, which means that it is asymptotically stable.
• In order to test the necessary and sufficient condition of stability of the quasi-polynomial of delayed type, it is required to select an appropriate reference polynomial
For the aforementioned quasi-polynomial, when a =1, it has the form
Substituting to quasi-polynomial
and dividing by the reference polynomial, we obtain the function in the form:
3)()( asswod
3)1()( sswod
0132
20 ,, RseKLddsdsKsdRsw
We set characteristics (w wavy line) for and
3
22.01332
23
3
0 )1(,,~
j
eKLdKdjdKqdKjqRjw
j
),(~ 0 Rjw
;0 22.0)( 0 R
fig. 4 characteristics
function
fig.5 enlarged graph of the function
),(~ 0 Rjw
),(~ 0 Rjw
Conclusion
The function graph does not cross the beginning of the coordinate system, which means that for the set delay the system is stable.
22.00 R
REFERENCES• Busłowicz M.: Odporna stabilność układów dynamicznych liniowych stacjonarnych
z późnieniami. Wydawnictwa Politechniki Białostockiej, Białystok 2002.• Busłowicz M.: Stabilność układów liniowych stacjonarnych o niepewnych parametrach. Dział
Wydawnictw i Poligrafii, Białystok 1997.• Tanenbaum A.: Sieci komputerowe, Wydawnictwo Helion, Gliwice 2004.• Hollot C.V., Misra V., Towsley D., Wei Bo Gong: A Control Theoretical Analysis of Red, Infocom
2001, Vol. 3 p.1510-1519.• Hollot C.V., Misra V., Towsley D., Wei Bo Gong: Analysis and Design of Controllers for AQM
Routers Supporting TCP Flows, IEEE System and Control Methods for Communication Networks 2002, Vol. 47, No. 6
• Bhandarkar S., Sadry N., Reddy A.L.N., Vaidya N.:TCP-DCR: A novel protocol tolerating wireless channel errors, IEEE Transaction on Mobile Computing 2005, Vol. 4 No. 5
• Czachórski T., Grochla K., Pekergin F.: Stability and Dynamics of TCP-NCR(DCR) protocol in presence of UDP flows, Proceedings of Third international EURO-NGI network of excellance conference on Wireless systems and mobility in next generation internet 2006, Springer-Verlag Berlin, Heidelberg p. 241-254.
• Tańcula J., Klamka J., Analiza D-stabilności protokołu TCP-DCR, Studia Informatica Wyd. Silesian University of Technology, Volume 33, Number 3A(107), p.7-18, Gliwice 2012
• Tańcula J., Klamka J., Examination of robust D-stability of TCP-DCR, Theoretical and Applied Informatics, Vol.24-No.4/2012, p.327-344, Gliwice 2012
Thank you for your attention