This paper deals with the performance assessment of a textile industry, namely African Textile Manufactures (ATM) Ltd., in Chalawa, Kano industrial estate, Federal Republic of Nigeria. The entire system ATM has different sections of machines for processing various assignments. The machines in the sections have diverse capabilities. We assumed the following subsystems: spinning, weaving, pretreatment, printing, and steaming. The failure rates of subsystems are constant, and they follow exponential distribution, while the repairs follow two types of distributions, namely general distribution and Gumbel-Hougaard family copula distribution. The system analyzed by employing supplementary variable and the Laplace transform. Various measures of reliability have been computed for differential values of failures and two types of repair. The conclusions and some recommendations are addressed to improving the management of ATM. (original abstract)