A Lindley-type equation arising from a carousel problem

Abstract: In this paper we consider a system with two carousels operated by one picker. The items to be picked are randomly located on the carousels and the pick times follow a phasetype distribution. The picker alternates between the two carousels, picking one item at a time. Important performance characteristics are the waiting time of the picker and the throughput of the two carousels. The waiting time of the picker satisfies an equation very similar to Lindley’s equation for the waiting time in the P H/U/1 queue. Although the latter equation has no simple solution, it appears that the one for the waiting time of the picker can be solved explicitly. Furthermore, it is well known that the mean waiting time in the P H/U/1 queue depends on to the complete inter-arrival time distribution, but numerical results show that, for the carousel system, the mean waiting time and throughput are rather insensitive to the pick-time distribution

Analytic properties of two-carousel systems

Abstract We present analytic results for warehouse systems involving pairs of carousels. Specifically, for various picking strategies, we show that the sojourn time of the picker satisfies an integral equation that is a contraction mapping. As a result, numerical approximations for performance measures such as the throughput of the system are extremely accurate and converge fast (e.g. within 5 iterations) to their real values. We present simulation results validating our results and examining more complicated strategies for pairs of carousels

Capacity Analysis of Sequential Zone Picking Systems

This paper develops a capacity model for sequential zone picking systems. These systems are popular internal transport and order-picking systems because of their scalability, flexibility, high-throughput ability, and fit for use for a wide range of products and order profiles. The major disadvantage of such systems is congestion and blocking under heavy use, leading to long order throughput times. To reduce blocking and congestion, most systems use the block-and-recirculate protocol to dynamically manage workload. In this paper, the various elements of the system, such as conveyor lanes and pick zones, are modeled as a multiclass block-and-recirculate queueing network with capacity constraints on subnetworks. Because of this blocking protocol, the stationary distribution of the queueing network is highly intractable. We propose an approximation method based on jumpover blocking. Multiclass jump-over queueing networks admit a product-form stationary distribution and can be efficiently evaluated by mean value analysis and Norton’s theorem. This method can be applied during the design phase of sequential zone picking systems to determine the number of segments, number and length of zones, buffer capacities, and storage allocation of products to zones to meet performance targets. For a wide range of parameters, the results show that the relative error in the system throughput is typically less than 1% compared with simulation

Non-Equilibrium Statistical Physics of Currents in Queuing Networks

We consider a stable open queuing network as a steady non-equilibrium system of interacting particles. The network is completely specified by its underlying graphical structure, type of interaction at each node, and the Markovian transition rates between nodes. For such systems, we ask the question ``What is the most likely way for large currents to accumulate over time in a network ?'', where time is large compared to the system correlation time scale. We identify two interesting regimes. In the first regime, in which the accumulation of currents over time exceeds the expected value by a small to moderate amount (moderate large deviation), we find that the large-deviation distribution of currents is universal (independent of the interaction details), and there is no long-time and averaged over time accumulation of particles (condensation) at any nodes. In the second regime, in which the accumulation of currents over time exceeds the expected value by a large amount (severe large deviation), we find that the large-deviation current distribution is sensitive to interaction details, and there is a long-time accumulation of particles (condensation) at some nodes. The transition between the two regimes can be described as a dynamical second order phase transition. We illustrate these ideas using the simple, yet non-trivial, example of a single node with feedback.Comment: 26 pages, 5 figure