Technical note: A graphical approach to the analysis of travel times in an automated storage and retrieval system

Operating time is closely related to the performance of an order-picking system. Knowing the probability distribution of operating time will greatly facilitate system performance analysis and optimization. When using an automated storage and retrieval system (AS/RS), travels of the storage and retrieval machine can be modeled using the Chebyshev metric. However, because of the dependence among the travel legs, analyzing the variance of travel time can be quite challenging. In this paper, a graphical approach based on contour lines is proposed for deriving the probability density function (PDF) of travel time in an AS/RS. With the PDF, the expected value and variance of travel time can be obtained. To demonstrate the use of the proposed approach in evaluating the performance of an AS/RS under stochastic conditions, we derive the PDF of travel time and apply it in a queueing analysis. Based on the queueing performance measures and an overall arrival rate, the minimum number of operation requests waiting or being served occurs when the arrival rates of storage and retrieval requests are approximately equal. Also, when arrival rates of storage and retrieval requests are equal, the number of operation requests waiting or being served is minimized when the rack is square-in-time.