Abstract
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.
Highlights
INTRODUCTIONQueueing models are widely used for system performance evaluation and prediction for different classes of real systems
There is a large number of works focused on queueing networks where customers are served one at a time [1–3]
There are a lot of systems where customers are served in batches
Summary
Queueing models are widely used for system performance evaluation and prediction for different classes of real systems. The most natural models for this systems are queueing networks with batch service and movements. Discrete-time open queueing networks with batch service were considered in [8–10]. The networks are realistic and practical for modeling wireless sensor networks, ATM, slotted ALOHA It is worth mentioning, there is a product form stationary distribution for the queueing networks. Daduna [9] extends product form results for discrete-time open queueing networks to include availability of unreliable nodes and state dependent arrival rates. A queueing network with triggered concurrent batch arrivals and batch services was considered in [11]. This triggered event may either be the addition of a batch of customers to the network, or the removal of a batch of customers from the network For this network, its stationary distribution has a product form. We consider a closed continuous-time queuing network with batch service and movements of customers. A section of conclusions commenting the main research contributions of this paper is presented
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