Availability Analysis Of A Multi-State Repairable Bubble Gum Production System

This paper present a multi-state repairable bubble gum production system which consists of fifteen sub-units working in series-parallel configuration. By choosing state space and defining operator of system, we transfer model into an abstract Cauchy problem. Then, we prove the operator is densely defined; using the method of ordinary differential equations integration factor to solve non-homogeneous equations, we prove the operator is a resolvent positive operator, and generate a non-negative Co-compression semigroup; Thus, the system has a unique non-negative time dependent solution. And we figure out the adjoint operator of the operator by choosing adjoint state space and defining adjoint operator; using the method of ordinary differential equations integration factor to solve homogeneous equations, we prove0is the simple eigenvalue of the operator, and we figure out that (1,1...,1)is the eigenvector of the eigenvalue; we prove0is the eigenvalue of the operator with geometric multiplicity one; using reductio ad absurdum we prove that algebraic multiplicity of0in X is one; we obtain that there exist at most countable isolated spectral point with limited algebraic multiplicity on the left half plane; and we proved that C0-semigroup is quasi-compact. Based on the above analysis of the semigroup nature, dynamic solution of the system exponentially converge to the steady-state solution of the system. In the case of fixed failure rates and the repair rates, respectively, to obtain the failure rates and the repair rates of the extruder, subsystem B, and the main conveyor into the value changes, the results exhibit the influence of the values of the degree of system availability.