### Stability of computer network for the set delay

```Stability of computer network for
the set delay
Jolanta Tańcula
TCP-DCR protocol model


In wireless network, it is difficult to state whether
reduction in the efficiency of TCP is caused by errors in a
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 by non-linear differential
equations
W ' t  
1  PD
R t 

PD 
R t   rtt

W t  W t  R ( t )   
 p ( t  R ( t )   )  PD 1   

2
R t  R ( t )   
N t 


C

W t 








R t 1  PD  R t  rtt PD

q ' t   


N (t )
 max  0 ;  C 
W t 
R ( t ) 1  PD    R t   rtt  PD 




where:
 PD – congestion-independent loss probability in wireless part of the
network,
 rtt – time, after which the wireless protocol is able to recover from
an error with probability α,
 τ – additional time to demonstrate confirmation of transmission
error due to imperfect of transmission media over wireless
networks
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
W , q , length
p 
receipt) and the queue
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.
Then, we present this equation on the blok diagram
0

0
0
Block diagram of model

On the basis of equations and the network model is
obtained, presented on the block diagram
Fig 1.

s
we perform to isolate
as the high frequancy
(parasitic). Substituting differential values to this scheme
and simplifying the scheme, we obtain
fig.2
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
2
NW 0
2
P (s) 
2 R0

2W 0  p 0  PD 1     
NW 0 
s 
 s 





R
CR
0
0 


e
 s  R 0 


The above transfer function represents an inertial
element of the second grade with permanent delay and
will be used when defining the characteristic quasipolynomial 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 is stable 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 quasipolynomial.
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.
Definition of quasi-polynomial

The characteristic quasi-polynomial
has the form
n
G (s, h) 
where

k

s gk e
 sh

k 0
g k e
 sh

means the dominant unit, we assume the following labelling of
the dominant unit e  sh
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:
2
d1 
NW 0
2R
2
0
d2 
NW 0  2W 0 C  p 0  PD 1  
CR 0

2 NW 0  p 0  PD 1   
2
d3 
2
CR 0
d1 , d 2 , d 3

To transfer function P(s) we substitute parameters
, hwhich
 R  will serve us to calculate the deviation of these
parameters and delay
as a result, we obtain
0
P s  
d1
s  d2s  d3
2
e
 s  R 0 

The main algorithm for Internet routers is RED algorithm.
After determining the formula of its transfer function C
(s) :
KL
C s  
sK
and based on the block diagram from Fig. 3, we create
transfer function of the whole system described with the
2

KL
s
 d2s  d3 
equation:
G s  
 s  K s 2

 d 2 s  d 3  KLd 1 e
 s  R 0 

Quasi-polynomial of the system

The quasi-polynomial of the system has the form:
w  s , R 0   , d   w 0  s , d   w 1  s , d e
 s  R 0 
w  s , R 0   , d    s  K s  d 2 s  d 3   KLd 1 e
2
w  s , R 0   , d   s  ( K  d 2 ) s  ( Kd
3
2
2

 s  R 0 

 d 3 ) s  ( Kd 3  KLd 1 ) e
 s  R 0 
The characteristic quasi-polynomial will be used to test
the stability of the fixed parameters and test method
stability for the set delay


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 theorem.
Stability for agreed values of delay

Theorem 1
The quasi-polynomial of delayed or neutral type is
asymptotically stable only when on the variable complex
plane, the~ graph of function
  j  , R 0       j  , R 0    /  od  j  
s  j
does not go through
the beginning of the coordinate
system
then
  s   of
s  the
a  variable complex plane. If
means the reference
polynomial of degree n and is
where a is any real positive number.
n
od
Test of stability for the set delay

Stability will be tested with frequency method for the set
(delay,
R   ) that
 0 . 22 is
d , d , d 
, in the space of parameters
.
Assuming N = 60, C = 1000, p0 = 0.05, W0 = 10,
.5
d substituting
82 . 5
d  75000
R0 = 0.22,
α = 0.1, PdD =300.25
and
to equations
(1), (2), we obtained appropriate values of parameters:
0

1
1

2
2
3
3
Based on the form of the
the dominant
s ( R quasi-polynomial,
)
1
 e
unit is determined and then using th.1, its stability is
confirmed. For the above parameters, it is
0
Characteristics function of the system has a form
~  j , R   , q  
w
0
 j    K  d 3    Kq 2  d 3  j   Kd 3  KLd 1 e
3
for   0 ;  
(R0
is shown on the graph
2
( j   1)
  )and
 0 . 22
3
 0 . 22 j 
fig. 4 characteristics
~ ( j , R   )
w
0
function
~ ( j , R   )
w
0
fig.5 enlarged graph
of the function
Conclusion
The function graph does not cross the beginning of the
  that
0 . 22 for the set delay
coordinate system, which Rmeans
0
the system is stable.
Thank you for your
attention
```