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We consider a semi-open queueing network (SOQN), where a customer requiresexactly one resource from the resource pool for service. If there is a resourceavailable, the customer is immediately served and the resource enters an innernetwork. If there is no resource available, the new customer has to wait in anexternal queue until one becomes available ("backordering"). When a resourceexits the inner network, it is returned to the resource pool and waits foranother customer. In this paper, we present a new solution approach. Toapproximate the inner network with the resource pool of the SOQN, we consider amodification, where newly arriving customers will decide not to join theexternal queue and are lost if the resource pool is empty "lost customers". Weprove that we can adjust the arrival rate of the modified system so that thethroughputs in each node are pairwise identical to those in the originalnetwork. We also prove that the probabilities that the nodes with constantservice rates are idling are pairwise identical too. Moreover, we provide aclosed-form expression for these throughputs and probabilities of idle nodes.To approximate the external queue of the SOQN with backordering, we construct areduced SOQN with backordering, where the inner network consists only of onenode, by using Norton's theorem and results from the lost-customersmodification. In a final step, we use the closed-form solution of this reducedSOQN, to estimate the performance of the original SOQN. We apply our results torobotic mobile fulfilment systems (RMFSs). Instead of sending pickers to thestorage area to search for the ordered items and pick them, robots carryshelves with ordered items from the storage area to picking stations. We modelthe RMFS as an SOQN, analyse its stability and determine the minimal number ofrobots for such systems using the results from the first part.

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Lost-customers approximation of semi-open queueing networks with  backordering -- An application to minimise the number of robots in robotic  mobile fulfilment systems