Loss systems in a random environment

We consider a single server system with infinite waiting room in a random environment. The service system and the environment interact in both directions. Whenever the environment enters a prespecified subset of its state space the service process is completely blocked: Service is interrupted and newly arriving customers are lost. We prove an if-and-only-if-condition for a product form steady state distribution of the joint queueing-environment process. A consequence is a strong insensitivity property for such systems. We discuss several applications, e.g. from inventory theory and reliability theory, and show that our result extends and generalizes several theorems found in the literature, e.g. of queueing-inventory processes. We investigate further classical loss systems, where due to finite waiting room loss of customers occurs. In connection with loss of customers due to blocking by the environment and service interruptions new phenomena arise. We further investigate the embedded Markov chains at departure epochs and show that the behaviour of the embedded Markov chain is often considerably different from that of the continuous time Markov process. This is different from the behaviour of the standard M/G/1, where the steady state of the embedded Markov chain and the continuous time process coincide. For exponential queueing systems we show that there is a product form equilibrium of the embedded Markov chain under rather general conditions. For systems with non-exponential service times more restrictive constraints are needed, which we prove by a counter example where the environment represents an inventory attached to an M/D/1 queue. Such integrated queueing-inventory systems are dealt with in the literature previously, and are revisited here in detail

Queues in a random environment

Exponential single server queues with state dependent arrival and service rates are considered which evolve under influences of external environments. The transitions of the queues are influenced by the environment's state and the movements of the environment depend on the status of the queues (bi-directional interaction). The structure of the environment is constructed in a way to encompass various models from the recent Operation Research literature, where a queue is coupled e.g. 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 separable systems as upper and lower bound

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

We consider a semi-open queueing network (SOQN), where a customer requires exactly one resource from the resource pool for service. If there is a resource available, the customer is immediately served and the resource enters an inner network. 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 and waits for another customer. In this paper, we present a new solution approach. To approximate the inner network with the resource pool of the SOQN, we consider a modification, where newly arriving customers will decide not to join the external queue and are lost if the resource pool is empty "lost customers". We prove that we can adjust the arrival rate of the modified system so that the throughputs in each node are pairwise identical to those in the original network. We also prove that the probabilities that the nodes with constant service rates are idling are pairwise identical too. Moreover, we provide a closed-form expression for these throughputs and probabilities of idle nodes. To approximate the external queue of the SOQN with backordering, we construct a reduced SOQN with backordering, where the inner network consists only of one node, by using Norton's theorem and results from the lost-customers modification. In a final step, we use the closed-form solution of this reduced SOQN, to estimate the performance of the original SOQN. 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, analyse its stability and determine the minimal number of robots for such systems using the results from the first part