add images and md files from sonja otten

topics/Queues_in_an_interactive_random_environment.md

+# Queues in an interactive random environment
+
+### Working Groups: aa
+
+### Collaborators (MAT): kkruse, sotten
+
+### Collaborators (External): [Prof. Dr. Hans Daduna](https://www.math.uni-hamburg.de/home/daduna/), [Dr. Ruslan Krenzler](https://www.researchgate.net/profile/Ruslan-Krenzler)
+
+## Description
+
+We consider exponential single server queues with state dependent arrival and service rates which evolve under in uences of external environments. The transitions of the queues are in uenced by the environment's state and the movements of the environment depend on the status of the queues (bi-directional interaction). The environment is constructed in a way to encompass various models from the recent Operations Research literature, where a queue is coupled with an inventory or with reliability issues. With a Markovian joint queueing-environment process we prove separability for a large class of such interactive systems, i.e. the steady state distribution is of product form and explicitly given. The queue and the environment processes decouple asymptotically and in steady state.
+
+For non-separable systems we develop ergodicity criteria via Lyapunov functions. By examples we show principles for bounding throughputs of non-separable systems by throughputs of two related separable systems as upper and lower bound.
+
+
+![Queues in a random environment](/img/Queues-in-random-environment.png)
+
+
+## References
+
+[1] Queues in a random environmentlois Theory. S. Otten, R. Krenzler, H. Daduna, K. Kruse. Available at [arXiv:2006.15712](https://arxiv.org/abs/2006.15712
+)
+
+[2] Integrated Models for Performance Analysis and Optimization of Queueing-Inventory-Systems in Logistic Networks. S. Otten. Available at [ediss.sub.hamburg](https://ediss.sub.uni-hamburg.de/handle/ediss/7668
+)

topics/Semi-open_queueing_networks_and_robotic_mobile_fulfillment_systems.md

+# Semi-open queueing networks and robotic mobile fulfillment systems
+
+### Working Groups: aa
+
+### Collaborators (MAT): kkruse, sotten
+
+### Collaborators (External): [Prof. Dr. Hans Daduna](https://www.math.uni-hamburg.de/home/daduna/), [Dr. Ruslan Krenzler](https://www.researchgate.net/profile/Ruslan-Krenzler), [Prof. Dr. Lin Xie](https://www.leuphana.de/institute/iis/personen/lin-xie.html)
+
+## Description
+
+We consider a semi-open queueing network (SOQN), where a customer requires one resource from the resource pool for service. If there is a resource available,  the resource enters an inner network to complete the customer's order. If there is no resource available, the new customer has to wait in an external queue until one becomes available "backordering". When a resource exits the inner network, it is returned to the resource pool.
+
+We present a new solution approach. To approximate the inner network with the resource pool, we consider a  odification, where newly arriving customers are lost if the resource pool is empty. We adjust the arrival rate of the modified system so that the throughputs in each node are pairwise identical to those in the original network. To approximate the external queue of the SOQN with  ackordering, we construct a reduced SOQN with backordering by using results from the lost-customers modification.
+
+We apply our results to robotic mobile fulfilment systems (RMFSs). Instead of sending pickers to the storage area to search for the ordered items and pick them, robots carry shelves with ordered items from the storage area to picking stations. We model the RMFS as an SOQN to determine the minimal number of robots.
+
+
+
+![Overview of the models](/img/Lost-customers-approximation-of-SOQN.jpg)
+
+
+## References
+
+[1] Lost-customers approximation of semi-open queueing networks with backordering -- An application to minimise the number of robots in robotic mobile fulfilment systems. S. Otten, R. Krenzler, L. Xie, H. Daduna, K. Kruse. Available at [arXiv:1912.01782](https://arxiv.org/abs/1912.01782
+)