Queueing analysis of the replenishment of multiple in-the-aisle pick positions

A case-picking operation with Multiple In-The-Aisle Pick Positions (MIAPP) is modeled as an M/G/1/N queueing system. Cases are picked manually from pallets located on the bottom level of storage racks. An aisle-captive Narrow-Aisle Lift Truck (NALT) travels rectilinearly to replenish the floor level of the rack by retrieving a pallet load from an upper level of the rack. From a queueing perspective, the NALT is the server and the order-picking positions in need of replenishment are customers. In this article, the replenishment requests from order-picking positions are assumed to occur at a Poisson rate (i.e., homogeneous customers). The corresponding probability density functions of service times are derived, and their Laplace–Stieltjes transforms are obtained, leading to steady-state performance measures of the system. In many situations, the replenishment requests from individual pick positions may not follow a homogeneous Poisson process, and the order-picking operation consists of heterogeneous customers. However, a simulation study indicates that an M/G/1/N queueing model yields accurate performance measures in such situations. Interestingly, when the number of pick positions is large enough to justify an MIAPP-NALT operation, the time between consecutive replenishment requests within a storage/retrieval aisle is approximately exponentially distributed. A numerical example is provided to illustrate the use of the developed model and to show the practical values of the analytical results in the performance analysis of such storage/retrieval systems.