Multi-Machine Repairable System with One Unreliable Server and Variable Repair Rate

Shengli Lv

doi:10.3390/math9111299

Abstract

This paper analyzed the multi-machine repairable system with one unreliable server and one repairman. The machines may break at any time. One server oversees servicing the machine breakdown. The server may fail at any time with different failure rates in idle time and busy time. One repairman is responsible for repairing the server failure; the repair rate is variable to adapt to whether the machines are all functioning normally or not. All the time distributions are exponential. Using the quasi-birth-death(QBD) process theory, the steady-state availability of the machines, the steady-state availability of the server, and other steady-state indices of the system are given. The transient-state indices of the system, including the reliability of the machines and the reliability of the server, are obtained by solving the transient-state probabilistic differential equations. The Laplace–Stieltjes transform method is used to ascertain the mean time to the first breakdown of the system and the mean time to the first failure of the server. The case analysis and numerical illustration are presented to visualize the effects of the system parameters on various performance indices.

Highlights

The machine repairing system can be applied to many real systems, such as computer networks, telecommunications, manufacturing systems, aircraft maintenance, and others [1]
Reliability function and mean time to system failure (MTTF) were derived from Laplace–Stieltjes transform equations
Yen et al [18] studied reliability and sensitivity analysis of a retrial machine repair problem with working breakdowns operating under the F-policy

Summary

Introduction

The machine repairing system can be applied to many real systems, such as computer networks, telecommunications, manufacturing systems, aircraft maintenance, and others [1]. Yen et al [18] studied reliability and sensitivity analysis of a retrial machine repair problem with working breakdowns operating under the F-policy. They assumed that the server was subject to working breakdowns only when there was at least one failed machine in the system. If the print job has non-preemptive priority to the copy job, the printer will do the coming print work first when there is copy job waiting This means that the failed server has different repair rates which depend on the states of the machines. Laplace–Stieltjes transform technique is used to derive the reliability indices of the machines and the server in a case analysis; the numerical results are presented.

Model Description

Steady-State Indices

Server Reliability

Case Analysis

Conclusions