Admission control in a parcel delivery system with account of the finite capacity of the warehouse at the pick-up point

Parcels delivering via pick-up points have become very popular due to the fast development of online marketplaces, customer-to-customer markets, and the use of parcel lockers as the last mile delivery solution. An adequate modelling of such delivery systems requires the creation of novel queueing models. In this paper, the process of delivering is described as the processing of a parcel in two service areas. The first area corresponds to processing until the parcel's arrival to the second area. The service in the second area corresponds to the storage of the parcel in a warehouse until it is picked up by a customer. The capacity of the warehouse is finite. A parcel arriving when the warehouse is full is lost. To enhance the efficiency of the system operation, threshold-type control by parcel admission at the first area is applied. Such a system is analysed under quite general assumptions about the parcel arrival flow. The analysis is implemented via consideration of the suitably constructed multidimensional continuous-time Markov chain. The dependence of the main performance measures of the delivery system on the warehouse capacity and the admission threshold are established and numerically illustrated. An optimisation problem is formulated and solved.