The process capability (PC) indices are considered numerical measures for process performance. Hence, they are widely adopted in manufacturing industry in Japan. This paper introduces a new generalized PC index, denoted as C_{pyk}, for discrete processes, to determine the generalized
C_{pyk} index for discrete distributions-an area not previously explored in the literature. Some estimation methods are employed to estimate the discrete generalized PC index, C_{pyk}, for the discrete Lindley distribution. The performance of these methods is also explored using simulation results. Additionally, non-parametric bootstrap methods, such as standard, percentile, student, and bias-corrected percentile, are used for interval estimation that provide more information than point estimates. In the simulation study, the coverage probabilities and average widths are examined for the bootstrap methods. The simulation results demonstrate that the maximum likelihood and bias-corrected percentile bootstrap methods outperform other estimation techniques. Two real-world applications are used to validate the theoretical findings.