On a make-to-stock production/mountain modeln with hysteretic control

We consider a make-to-stock production-inventory model with one machine that produces stock in a buffer. The machine is subject to breakdowns. During up periods, the machine fills the buffer at a level-dependent rate α(x)>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha (x)>0$$\end{document}. During down periods, the production rate is zero, and the demand rate is either β(x)>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta (x)>0$$\end{document} or γ(x)>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma (x)>0$$\end{document} when the inventory level is x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x$$\end{document}; which of the two demand rates applies depends on a hysteretic control policy. Such a policy is used to avoid undesirable rapid switching. In the context of our paper hysteresis is introduced in the form of bang-bang control. Namely, there are two switchover levels such that when the buffer content reaches the higher level we change the drift downward and when the buffer content reaches the lower level we change the drift upward. We determine the conditions under which the steady-state distribution of the inventory level exists, and we derive that distribution. Other performance measures under consideration are the number of switches from β(·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta (\cdot )$$\end{document} to γ(·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma (\cdot )$$\end{document} per busy period, the busy period distribution, and the overshoot above a particular hysteretic level.