An M/ G/1 queue with two phases of service subject to the server breakdown and delayed repair

• Published in: Applied mathematical modelling Vol. 33; no. 6; pp. 2699 - 2709
• Main Authors: Choudhury, Gautam, Tadj, Lotfi
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Inc 01.06.2009; Elsevier
• Subjects: 60 K 25 (Primary); 90 B 22 (Secondary); Applied sciences; Delay time; Exact sciences and technology; Operational research and scientific management; Operational research. Management science; Queuing theory. Traffic theory; Random breakdown; Reliability theory. Replacement problems; Repair time; Second optional service and reliability; Stationary queue size distribution; Second optional service and reliability; 90 B 22 (Secondary); Random breakdown; Stationary queue size distribution; Delay time; 60 K 25 (Primary); Repair time; Performance evaluation; Probabilistic approach; Generating function; M G 1 queue; Two phase service; Joint distribution; Modeling; Steady state; Waiting time; Stationary condition; Queue length; Busy period; Occupation time; Laplace transformation; Stieltjes transformation; Reliability; Repair; Delayed time
• DOI: 10.1016/j.apm.2008.08.006
• ISSN: 0307-904X

This paper deals with the steady-state behaviour of an M/ G/1 queue with an additional second phase of optional service subject to breakdowns occurring randomly at any instant while serving the customers and delayed repair. This model generalizes both the classical M/ G/1 queue subject to random breakdown and delayed repair as well as M/ G/1 queue with second optional service and server breakdowns. For this model, we first derive the joint distributions of state of the server and queue size, which is one of chief objectives of the paper. Secondly, we derive the probability generating function of the stationary queue size distribution at a departure epoch as a classical generalization of Pollaczek–Khinchin formula. Next, we derive Laplace Stieltjes transform of busy period distribution and waiting time distribution. Finally, we obtain some important performance measures and reliability indices of this model.