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Random walks with additive increase and multiplicative decrease are widely used in the modern networking environments for distributed control of the communication parties activity. Thus Additive Increase Multiplicative Decrease algorithm implements the random walk to provide flow control at the Internet transport layer. Different variations of the algorithm are used by more than ten TCP protocol implementations. More sophisticated variations of the algorithm are proposed to provide distributed performance control for highly congested publish/subscribe IoT environments. As for data communication networks the algorithms are used by data sources and in IoT environments they are implemented by the clients subscribed for semantic brokers service notifications. Wide scope of the applications and strict demands to their performance define the topicality of modeling and analysis studies of the important properties of the random walks mentioned above.

In many cases their key performance metrics could be described by step-wise random process with semi-markovian or renewal properties. The state space of the process is the set of non-negative integers and multiplication is followed by the floor operation due to the nature of the communication protocols. Nevertheless in most researches the step-wise process is substituted by piece-wise processes with polynomial (as usual linear) growth periods with markovian or renewal properties as well and the floor operation is neglected. Therefore their space of state is formed by non-negative real numbers. The substitute allows application of the powerful analytical methods and hence yields simpler models and stronger results. Meanwhile there are few works those research a connection between the step-wise and corresponding piece-wise process. In the paper we study connection between parameters of such processes and and their asymptotic behavior.