The Single Server Queue with Catastrophes and Geometric Reneging

Abstract

We consider the single server queue with catastrophes that occur according to a Poisson process. The catastrophes force all present customers to abandon the system and render the server inoperative. Then a repair time is set on till the server becomes ready to serve again the customers. In the meanwhile the customers are accumulated according to their arrival process. Recently, several authors have investigated the reneging behavior in such systems and other related models when the customers become impatient because of the failure of the server. Two kinds of reneging have been considered: independent and binomial. In the case of independent reneging, each customer has its own patience time and abandons the system as soon as it expires. In the case of binomial reneging, the abandonment opportunities occur according to a certain point process and then all present customers decide simultaneously but independently whether they will abandon the system or not. In the present paper, we complement these studies by considering the case of geometric reneging. This case arises when the abandonment opportunities occur according to a certain point process and the customers decide sequentially whether they will leave the system or not. We derive explicit expressions and computational schemes for various performance descriptors, concerning the number of customers in system, the sojourn time of a customer, the duration and the maximum number of customers in a busy period.