1. Stability of computer network for the set delay Jolanta Tańcula

2. 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.

3. TCP-DCR protocol model  TCP-DCR protocol can be described by non-linear differential equations where:  P D – 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

4. 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.  Then, we present this equation on the blok diagram

5. Block diagram of model  On the basis of equations and the network model is obtained, presented on the block diagram Fig 1.

6.  we perform to isolate as the high frequancy (parasitic). Substituting differential values to this scheme and simplifying the scheme, we obtain fig.2

7. fig. 3 simplified diagram

8. 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

9.  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).

10. 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 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.

11.  The characteristic quasi-polynomial has the form where means the dominant unit, we assume the following labelling of the dominant unit Definition of quasi-polynomial

12. 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 parameters, which will serve us to calculate the deviation of these parameters and delay as a result, we obtain

13. 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:

14. 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

15.  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.

16.  Theorem 1 The quasi-polynomial of delayed or neutral type is asymptotically stable only when on the variable complex plane, the graph of function does not go through the beginning of the coordinate system of the variable complex plane. If then means the reference polynomial of degree n and is where a is any real positive number. Stability for agreed values of delay

17. 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, p 0 = 0.05, W 0 = 10, R 0 = 0.22, α = 0.1, P D = 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, its stability is confirmed. For the above parameters, it is

18. Characteristics function of the system has a form for and is shown on the graph

19. fig. 4 characteristics function fig.5 enlarged graph of the function

20. Conclusion The function graph does not cross the beginning of the coordinate system, which means that for the set delay the system is stable.

21. Thank you for your attention