*Article* **Analytic and Computational Analysis of GI/M***<sup>a</sup>***,** *<sup>b</sup>***/***c* **Queueing System**

**Mohan Chaudhry and Jing Gai \***

> Department of Mathematics and Computer Science, Royal Military College of Canada, Kingston, ON K7K 7B4, Canada

**\***Correspondence: jing.gai@rmc.ca

**Abstract:** Bulk-service queueing systems have been widely applied in many areas in real life. While single-server queueing systems work in some cases, multi-servers can efficiently handle most complex applications. Bulk-service, multi-server queueing systems (compared to well-developed single-server queueing systems) are more complex and harder to deal with, especially when the inter-arrival time distributions are arbitrary. This paper deals with analytic and computational analyses of queuelength distributions for a complex bulk-service, multi-server queueing system GI/M*<sup>a</sup>*, *<sup>b</sup>*/*<sup>c</sup>*, wherein inter-arrival times follow an arbitrary distribution, *a* is the quorum, and *b* is the capacity of each server; service times follow exponential distributions. The introduction of quorum *a* further increases the complexity of the model. In view of this, a two-dimensional Markov chain has to be involved. Currently, it appears that this system has not been addressed so far. An elegant analytic closed-form solution and an efficient algorithm to obtain the queue-length distributions at three different epochs, i.e., pre-arrival epoch (p.a.e.), random epoch (r.e.), and post-departure epoch (p.d.e.) are presented, when the servers are in busy and idle states, respectively.

**Keywords:** queues; bulk service; multi-server; Markov chain; quorum

**MSC:** 60-08; 60J27
