Warehouses recently face increasing stress imposed by a volatile customer demand and increasing customer expectations in terms of ever shorter order response times. In that respect, warehouses more and more offer same-day and next-day shipment conditions. However, same-day shipment promises are challenging to fulfil, especially as the order fulfilment process operates against fixed deadlines imposed by the predefined truck departure times. As a natural mitigation strategy, warehouses set a cutoff point and offer same-day shipment only to customers that order until the cutoff point, but next-day shipment to all customers ordering thereafter. Setting an appropriate cutoff point is challenging as it affects multiple facets of the service quality, such as the order response time and the service level. 
 
 In this paper, we study the design of cutoff-based shipment promises for stochastic deadline-oriented order fulfilment processes in warehouses. We present a discrete-time Markov chain model for exact steady-state performance analysis and propose two novel performance measures – α- and β-cutoff service level – for service level measurement in these systems. We numerically show the benefit of cutoff-based shipment promises. Even with a late cutoff point, there is a significant gain in the system performance. Furthermore, we find that warehouses should set the cutoff point such that it balances customer expectations in terms of service level and order response time. Finally, warehouses can improve their shipment promises when referring to β- instead of α-cutoff service level and by implementing measures reducing the utilisation and the variabilities of the order fulfilment process.