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 decomposition 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 considerably 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.
Highlights
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
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 size with no vacation and additional system size due to vacation
In this paper an attempt has been made to derive stochastic decomposition result for such types M/G/1 unreliable queue with generalized vacations with two types of general heterogeneous services under repeated service policy in order to unify the results of class of these types of vacation models for unreliable server with delayed repair
Summary
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. Considerable attention have been paid to study M/G/1 type of queueing system, in which the server provides two types of general heterogeneous service and optional repeated service policy In such a model an arriving unit has an option to choose any one of the two types of services before the service starts. More recently Kalita and Choudhury (2018, 2019[a,b]) generalized this type of batch arrival unreliable queueing model under different vacation policies. In this paper an attempt has been made to derive stochastic decomposition result for such types M/G/1 unreliable queue with generalized vacations with two types of general heterogeneous services under repeated service policy in order to unify the results of class of these types of vacation models for unreliable server with delayed repair.
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