Time-dependent solution for the manufacturing line with unreliable machine and batched arrivals

Time-dependent queue-size distribution in a finite-buffer manufacturing line with unreliable machine is investigated. Successive jobs arrive in batches (groups) with sizes being generally distributed random variables, and are being processed individually with exponential service times. Applying the approach based on the memory less property of exponential distribution and the total probability law, a system of integral equations for the transient queue- size distribution conditioned by the initial level of buffer saturation is derived. The solution of the corresponding system written for Laplace transforms is found via linear-algebraic approach..