Flow time distributions in queues with customer batching and setup times

This paper examines the relationship between batch size and flow time in a single server queue in which individual customers are grouped before processing. A setup time precedes the processing time of each batch. The model has applications in manufacturing systems where flow and job shops follow one another or in assembly type manufacturing. In the first part of the paper, the Laplace-Stieltjes Transforms and the first and second moments of several flow time measures are derived. The second part of the paper presents numerical results which relate the batch size and the mean and variance of the flow time. It is shown computationally that the mean and variance of the flow time are generally minimized at different batch sizes, and hence tradeoffs exist in selecting an optimal batch size. The numerical results also show that the variability of the setup time and processing time increases the mean and variance of the flow time as well as the corresponding optimal batch sizes.