Stochastic Fluid Model with Jumps: The Bounded Model
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Stochastic Fluid Model with Jumps: The Bounded Model Lanying Hu1 · Yong Ren2 Accepted: 13 November 2020 © Springer Nature India Private Limited 2020
Abstract As one of the important hybrid models, stochastic fluid models (SFMs, in short) have many applications. In this work, we discuss a class of SFMs modulated by Markovian chain with lower and upper jumps, where the fluid level has a positive upper bound and a lower bound zero. The joint limiting distribution of the buffer level and the environment state is expressed by a series of differential equations. An example is give to illustrate the theory obtained. Keywords Stochastic fluid flow model · Limiting distribution · Jump
Introduction A typical stochastic fluid model (SFM, in short) is an input-output system, in which a continuous fluid enters and leaves a storage device—a buffer, depending on an underlying Markov chain, which is referred to as a phase process (or stochastic environment). Generally, the phase process is an irreducible, continuous time Markov chain. One can see Anick, Mitra and Sondhi [5] for the historic overview and Kulkarni [16] for the recent survey. SFMs are motivated by their usage in modeling high-speed communication networks [5,14,21,22]. Nowadays, there are many applications for SFMs in many other areas (see [2,7,17]). One of the most interesting works in SFMs is to derive the steady-state distribution for the buffer content, which is generally expressed by a set of differential equations, whose solution could be solved by means of linear combinations of exponentials (see [5,16,17]). Alternative methods have been used to determine the steady-state distribution, for more details, one can see [3,4,6,8,9,11,12,19,20] and the references therein. In the above mentioned works, the buffer content changes continuously. However, in order to exactly depict the stochastic production-inventory systems, the instantaneous jumps should be considered. Some works have been done in the SFMs with jumps [1,15,18,23]. In
This work is supported by the National Natural Science Foundation of China (11871076).
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Yong Ren [email protected]; [email protected] Lanying Hu [email protected]
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Department of Mathematics, Anhui Normal University, Wuhu 241000, China
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School of Science, Beijing University of Civil Engineering and Architecture, Beijing 100044, China 0123456789().: V,-vol
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Int. J. Appl. Comput. Math
(2021) 7:1
the practical application, many buffers have upper boundaries (see [11,13]). However, the unbounded SFMs can not provide a good approximation for the bounded ones. Thus, it is need to consider bounded SFMs. In this paper, we consider the bounded SFMs with the lower and upper jumps, where the fluid level has a lower boundary and a positive upper boundary. The joint limiting distribution of the buffer level and the environment state is expressed by a series of differential equations. This paper is organized as follows. The model of bounded SFMs with jumps is given in Sect. 2. In Sect. 3, we show the analysis for the
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