Mathematics and Statistics 8(2): 225-232, 2020 DOI: 10.13189/ms.2020.080219 http://www.hrpub.org Stochatic Decomposition Result of an Unrelaible Queue with Two Types of Services Gautam Choudhury 1 , Akhil Goswami 2 , Anjana Begum 1,* , Hemanta Kumar Sarmah 2 1 Mathematical Sciences Division, Institute of Advanced Study in Science and Technology, Paschim Boragaon, Guwahati 781035, Assam, Indias 2 Department of Mathematics,Gauhati University, Assam, India Received December 4, 2019; Revised February 13, 2020; Accepted February 18, 2020 Copyright ©2020 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of the Creative Commons Attribution License 4.0 International License Abstract The single server queue with two types of heterogeneous services with generalized vacation for unreliable server have been extended to include several types of generalizations to which attentions has been paid by several researchers. One of the most important results which deals with such types of models is the “Stochastic Decomposition Result”, which allows the system behaviour to be analyzed by considering separately distribution of system (queue) size with no vacation and additional system (queue) size due to vacation. Our intention is to look into some sort of united approach to establish stochastic decompo- sition result for two types of general heterogeneous service queues with generalized vacations for unreliable server with delayed repair to include several types of generalizations. Our results are based on embedded Markov Chain technique which is consid- erably a most powerful and popular method wisely used in applied probability, specially in queueing theory. The fundamental idea behind this method is to simplify the description of state from two dimensional states to one dimensinal state space. Finally, the results that are derived is shown to include several types of generalizations of some existing well- known results for vacation models, that may lead to remarkable simplification while solving similar types of complex models. Keywords Stochastic Decomposition, Heterogenous Service, Generalized Vacation, Markov Chain 1 Introduction Vacation models are characterized by the fact that the idle time of the server may be utilized for some other secondary jobs, for instance to serve the customers in other systems. Allowing the server to take vacations makes the queueing models more realistic and flexible in studying real world queueing situations. Applications of such models arise naturally in call centres with multitask employees, telecommunication and computer networks, production and quality control problems etc. Literature on vacation models is growing very rapidly and it has become the subject matter of current research due to its numerous applications in many real life situations. Instead of reviewing those again, it will be more convenient to refer the readers to survey papers by Doshi (1986), Teghem (1986), Medhi (1997) and also the monograph of Takagi (1990) for more information as well as complete set of references. One of the most remarkable result that concerned with these types of models is “Stochastic Decomposition Result”, which allows the system behaviour to be analyzed by considering separately distribution of system (queue) size with no vacation and additional system (queue) size due to vacation. The decomposition property for M/G/1 queue has been observed by Graver (1962), Cooper (1970), and Furhmann (1984). However, for generalized vacations the decomposition result was first established by Fuhrmann and Cooper (1985). Shanthikumar (1988) relaxed some of the assumptions considered by Fuhrmann and Cooper (1985). Harris and Marchal (1988) extended Fuhrmann and Cooper (1985) to the state dependent case. The distributional form of Little’s law for decomposition result has been discussed by Keilson and Servi (1990). Doshi (1990) obtained the decomposi- tion formula for virtual workload process. Miyazawa (1994) discussed certain generalizations of decomposition formula for the single server queue with two levels of service rates, which include vacation models as special cases. CITE THIS PAPER [1] Gautam Choudhury , Akhil Goswami , Anjana Begum , Hemanta Kumar Sarmah , "Stochatic Decomposition Result of an Unrelaible Queue with Two Types of Services," Mathematics and Statistics, Vol. 8, No. 2, pp. 225 - 232, 2020. DOI: 10.13189/ms.2020.080219.