###### Abstract  
In the context of sustainable development, this paper rationalises the outbound process of a four-way shuttle system with a focus on their modelling, performance evaluation and configuration using a parallel operation strategy to reduce resource waste, thus achieving sustainable development. The parallelism of the hoist and shuttle is innovatively incorporated into the four-way shuttle system, so the modelling content is divided into parallel and serial types. In the parallel operation strategy model, a separation-aggregation queueing network model is constructed, and the open-loop queueing network is innovatively solved using the maximum entropy method. In the serial operation strategy model, a semi-open-loop queuing network is constructed and solved using the geometric matrix method. By varying different parameters, the accuracy of the model is verified by Arena simulation with an error range of 10% or less, and the error of the system performance index calculation is reduced by 20% compared with the existing methods. Setting up 18 different sizes of shuttle systems provided a better performance than a single serial-operation strategy through the addition of parallel strategies, with an average reduction of 12.6% in the system response time and a minimum reduction of 1.8%. The conclusions of this paper were verified on the basis of an arithmetic case analysis.  
automated warehouses; four-way shuttle systems; queuing network model; analytical and numerical modelling; performance analysis +
Footnote †: journal: Applied  
For the study of dense storage systems, the establishment of a system model is an essential optimisation step. Scholars have mostly used the queuing network model; however, for complex shuttle systems, a single open-loop or closed-loop queuing network cannot meet the modelling needs under multilayer, multishuttle and multilift conditions. Therefore, it is also necessary to study the system's performance in order to establish a model that reflects the actual operation mode of the system. At present, studies of storage system shuttle and hoist operation strategies mostly use a single serial strategy; however, parallel operation strategies have been employed in real scenarios, and have a significant positive impact on the system performance. AVS/RS systems, AS/RS systems and SBS/RS systems have been developed to analyse the impact of the operating strategy on the operational efficiency of the whole system [17; 18; 19]. It can be observed from the available literature that dense storage systems have been widely studied by scholars, and the research has mostly been centred on AVS/RS. The research directions of some scholars have mostly focused on system scheduling, storage strategy and sorting order optimisation, while other scholars have focused on system modelling, system performance evaluation and system configuration. In addition to there being less research on four-way shuttle systems, the research has the following limitations: (1) it mostly presents single operation strategies for the shuttle and hoist in the system; (2) it mostly presents single-level and single-aisle analyses; and (3) generality is ignored when building queuing networks [20; 21; 22].  
Therefore, this study focuses on a four-way shuttle system with a parallel operation strategy; adopts system performance indices, such as equipment utilisation, system response time and external queue waiting time; establishes a separation-aggregation queuing network model and a SOQN model; and designs an algorithm based on the Coxian distribution and the MGM solution. The performance variation of the system is analysed under different operation strategies [23; 24; 25; 26].  
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