Data Exchange and Differential Equations

Data Exchange and Differential Equations


In one of the projects I was working on, a data exchange mechanism was implemented between remote components of the system, which worked according to the following scenario: source component A on its side prepares data to be transmitted; Recipient component B periodically opens a communication session and takes all the data that A has accumulated at the time of connection. Data arriving already during a communication session is postponed until the next connection.

At some point, I realized that the data transfer in such a scheme is described using an ordinary differential equation. Description of the model and the conclusions that were obtained with its help, under the cut.

Denote by $ x (t) $ - the amount of data in some conventional units accumulated for exchange on the component A side at the time point $ t $ . Let the pause between the end of the exchange session and the beginning of the next be $ a_0 & gt; 0 $ units of time, and for transferring one unit of data,  $ a_1 & gt; 0 $ units of time. Then on the transfer $ x (t) $ data units required  $ a_0 + a_1x (t) $ units of time. Data transfer rate is

$ \ frac {x (t)} {a_0 + a_1x (t)}. \ quad (1) $


If the data accumulation rate on side A is indicated by $ f (t) $ , then  $ x (t) $ is the solution of a differential equation:

$ \ frac {dx} {dt} = - \ frac {x} {a_0 + a_1x} + f (t). \ quad (2)  $


Since the unlimited growth of the volume of still undistributed data is a highly undesirable situation, the important task is to obtain conditions for the boundedness of solutions of this equation.

For simplicity, we will assume the function $ f (t) $ continuous. Let

$ f (t) = \ phi_0 + \ phi (t), $


where

$ \ left | \ int_0 ^ t \ phi (s) ds \ right | \ leq K _ {\ phi} & lt; + \ infty $


for all $ t \ geq 0 $ , and  $ \ phi_0 & gt; 0 $ is a constant playing the role of an average value.

Consider a few examples. Let $ f (t) $ periodic and its schedule is:

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< br/> In this case, $ \ phi_0 = 1/3 $ ,  $ \ phi (t) = f (t) - \ phi_0 $ .
Numerically integrating equation (1) for several parameter values ​​ $ a_0, a_1 $ and initial values ​​of  $ x (0) $ , we get the following decision graphics:
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The examples show: when $ 1/a_1 & gt; \ phi_0 $ , solutions are also limited for different values ​​of  $ x (0) $ the system tends to some steady state. The shorter the pauses between sessions $ a_0 $ , the faster this convergence. With $ 1/a_1 & lt; \ phi_0 $ such convergence is not observed, and solutions grow over time. Reducing the duration of pauses slows the growth rate, but the tendency to unlimited increase $ x (t) $ is still saved.

In general, it can be shown that if $ 1/a_1 & gt; \ phi_0 $ , then the solutions to equation (1) are limited, and if  $ 1/a_1 & lt; \ phi_0 $ - unlimited solutions will be obtained. That is, the boundedness of solutions is determined only by the ratio of the rates of accumulation and extraction of data. Duration of pauses between exchanges $ a_0 $ , the only parameter that can be easily controlled, does not fundamentally affect the behavior of the system. Although, as can be seen from relation (1) and examples, with its increase, the exchange rate decreases.

As a result, the analysis of the model leads to the following conclusions. If the exchange rate is insufficient, and the volume of data to send is constantly increasing on the source side, then trying to rectify the situation by reducing the pauses between sessions does not make sense.Help here can only increase system performance.

On the other hand, in the case when the exchange service constantly loads computers to the detriment of other tasks, it would be advisable to increase the pauses within reasonable limits: this will only affect the relevance of the data without risking overflow of the source with unsent data.
Detailed calculations for the conditions of bounded solutions and some other questions concerning the considered model are published in the materials of the school-seminar "Mathematical modeling, numerical methods and program complexes" named after E.V. Resurrection. You can view and download the article using this link .

Source text: Data Exchange and Differential Equations