Analysis of semi-open queueing networks using lost customers approximation with an application to robotic mobile fulfilment systems

We consider a semi-open queueing network (SOQN), where one resource from a resource pool is needed to serve a customer. If on arrival of a customer some resource is available, the resource is forwarded to an inner network to complete the customer’s order. If no resource is available, the new customer waits in an external queue until one becomes available (“backordering”). When a resource exits the inner network, it is returned to the resource pool. We develop a new solution approach. In a first step we modify the system such that new arrivals are lost if the resource pool is empty (“lost customers”). We adjust the arrival rate of the modified system such that the throughputs in all nodes of the inner network are pairwise identical to those in the original network. Using queueing theoretical methods, in a second step we reduce this inner network to a two-station system including the resource pool. For this two-station systems, we invert the first step and obtain a standard SOQN which can be solved analytically. We apply our results to storage and delivering systems with 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.

Modeling and performance analysis of shuttle-based compact storage systems under parallel processing policy

Short response time for order processing is important for modern warehouses, which can be potentially achieved by adopting appropriate processing policy. The parallel processing policy have advantages in improving performance of many autonomous storage and retrieval systems. However, researchers tend to assume a sequential processing policy managing the movement of independent resources in shuttle-based compact storage systems. This paper models and analyses a single-tier of specialized shuttle-based compact storage systems under parallel processing policy. The system is modeled as a semi-open queueing network with class switching and the parallel movement of shuttles and the transfer car is modeled using a fork-join queueing network. The analytical model is validated against simulations and the results show our model can accurately estimate the system performance. Numerical experiments and a real case are carried out to compare the performance of parallel and sequential processing policies. The results suggest a critical transaction arrival rate and depth/width ratio, below which the sequential processing policy outperforms the parallel processing policy. However, the advantage of sequential processing policy is decreasing with the increasing of shuttle number, transaction arrival rate and depth/width ratio. The results also suggest an optimal depth/width ratio with a value of 1.75 for minimizing the expected throughput time in the real system. Given the current system configurations, the parallel processing policy should be considered when the number of shuttles is larger than 2 or the transaction arrival rate is larger than 24 per hour.

Performance measures of parallel tandem open queueing network

Purpose The authors consider parallel four-state tandem open queueing network. The queue capacity is infinite. Passenger arrival rate is Poisson distribution and service rate is exponential distribution. The queue is constructed in the form of tandem queue, and each and every queue of tandem queue is single server (M/M/1) queue. In tandem queue, passengers will leave the system once they receive service from both the states. The purpose of this paper is to provide performance analysis for four-state tandem open queue network, and a governing equation is formulated with the help of transition diagram. Using Burke theorem, the authors formulated equation for average number of passenger in the system, average waiting time of passenger in the system, average number of passenger in the queue and average waiting time of passenger in the queue. Design/methodology/approach This paper used Burke’s theorem. Findings In this paper, performance analysis is done for parallel four-state tandem open queueing network and performance measure solved using Burkes theorem formula. K. Sreekanth et al. has done performance analysis for single tandem queue with three states. In this paper, the authors have done performance analysis for two tandem queues parallel with four states. This four-state tandem open queueing network is suitable for real world applications. This paper can extend for more number of service states and multi-server states according to the application, and in such case, the authors have to prove and explain with numerical examples. This analysis is more useful for the applications such as airports, railway stations, bus-stands and banks. Originality/value In this paper, parallel four-state tandem open queueing network and performance measure has been solved using Burke’s theorem formula.

Solving Open Queueing Network Problem

Queueing networks may be considered as a group of inter-connected nodes (say ) where each node represents a service facility of some kind with servers at the node ( ). We propose to apply 1st the method given by J. R. Jackson and then we would like to solve the same problem by the method of J. D. C. Little.