# Travel time in carousel systems

##
Nelly Litvak

### University of Twente, The Netherlands

### Résumé:

A carousel is an automated storage and retrieval system that is widely
used in modern warehouses. The system consists of a circular disk with a
large number of shelves or drawers along its circumference. The disk can
rotate either direction past a picker who has a list of items to be
collected from n different drawers. We model this process by assuming
that the picker occupies a fixed position along a circle and that the
drawers to be visited are distributed independently and uniformly along
the circle. The picker might chose various routes for collecting the
items. We first analyse the most natural Nearest Item heuristic where
the next item to be picked is always the nearest one. Next, we study the
shortest possible route. Along the way, we encounter rather curious
properties of exponential random variables and a probability
distribution with a most unusual behavior linked to Jacobi's theta
functions. We also present some results for the case when the items'
locations are not uniform and discuss the optimal allocation problem.