In this paper, open queuing networks with Poisson arrivals and single-server infinite buffer queues are considered. Unlike traditional queuing models, customers are served (with exponential service time) in batches, so that the nodes are non-work-conserving. The main contribution of this work is the design of an efficient algorithm to find the batch sizes which minimize the average response time of the network. As preliminary steps at the basis of the proposed algorithm, an analytical expression of the average sojourn time in each node is derived, and it is shown that this function, depending on the batch size, has a single minimum. The goodness of the proposed algorithm and analytical formula were verified through a discrete-event simulation for an open network with a non-tree structure.

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We consider a closed queuing network with batch service and movements of customers in continuous time. Each node in the queueing network is an infinite capacity single server queueing system under a RANDOM discipline. Customers move among the nodes following a routing matrix. Customers are served in batches of a fixed size. If a number of customers in a node is less than the size, the server of the system is idle until the required number of customers arrive at the node. An arriving at a node customer is placed in the queue if the server is busy. The batсh service time is exponentially distributed. After a batсh finishes its execution at a node, each customer of the batch, regardless of other customers of the batch, immediately moves to another node in accordance with the routing probability. This article presents an analysis of the queueing network using a Markov chain with continuous time. The qenerator matrix is constructed for the underlying Markov chain. We obtain expressions for the performance measures. Some numerical examples are provided. The results can be used for the performance analysis manufacturing systems, passenger and freight transport systems, as well as information and computing systems with parallel processing and transmission of information.

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In this article, we study closed queuing networks with batch services of customers. Each node in the queueing network is an infinite capacity single-server queueing system under a RANDOM discipline. Customers move among the nodes following a routing matrix. We assume queueing systems in the network operate under the general batch service rule. The lower and upper bounds for the batch size are given. The batch service time is exponentially distributed. We presents an analysis of the queueing network using a Markov chain with continuous time. The qenerator matrix is constructed for the underlying Markov chain. We obtain expressions for the performance measures. In addition, we consider an unreliable case and propose an approximation. Some numerical examples are provided. The results can be used for the performance analysis manufacturing systems, production lines, trucking, ship locks.