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`Mathematical Problems in EngineeringVolume 2015, Article ID 173938, 12 pageshttp://dx.doi.org/10.1155/2015/173938`
Research Article

## A Discrete-Time Queue with Preferred Customers and Partial Buffer Sharing

1Department of Statistics and Financial Mathematics, School of Science, Nanjing University of Science and Technology, Nanjing, Jiangsu 210094, China
2School of Science, Guangxi University for Nationalities, Nanning, Guangxi 530006, China
3Nanjing University of Science and Technology, Nanjing, China

Received 16 April 2015; Revised 26 June 2015; Accepted 7 July 2015

Academic Editor: Francesco Soldovieri

Copyright © 2015 Shizhong Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We analyze a discrete-time Geo/Geo/1 queueing system with preferred customers and partial buffer sharing. In this model, customers arrive according to geometrical arrival processes with probability . If an arriving customer finds the server idle, he begins instantly his services. Otherwise, if the server is busy at the arrival epoch, the arrival either interrupts the customer being served to commence his own service with probability (the customer is called the preferred customer) or joins the waiting line at the back of the queue with probability (the customer is called the normal customer) if permitted. The interrupted customer joins the waiting line at the head of the queue. If the total number of customers in the system is equal to or more than threshold , the normal customer will be ignored to enter into the system. But this restriction is not suitable for the preferred customers; that is, this system never loses preferred customers. A necessary and sufficient condition for the system to be stable is investigated and the stationary distribution of the queue length of the system is also obtained. Further, we develop a novel method to solve the probability generating function of the busy period of the system. The distribution of sojourn time of a customer in the server and the other indexes are acquired as well.