The $$M^{X}/M/c$$ M X / M /
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Arabian Journal of Mathematics
R E S E A R C H A RT I C L E
Amina Angelika Bouchentouf · Abdelhak Guendouzi
The M X /M/c Bernoulli feedback queue with variant multiple working vacations and impatient customers: performance and economic analysis
Received: 12 July 2018 / Accepted: 6 June 2019 © The Author(s) 2019
Abstract The present paper deals with an M X /M/c Bernoulli feedback queueing system with variant multiple working vacations and impatience timers which depend on the states of the servers. Whenever a customer arrives at the system, he activates an random impatience timer. If his service has not been completed before his impatience timer expires, the customer may abandon the system. Using certain customer retention mechanism, the impatient customer can be retained in the system. After getting incomplete or unsatisfactory service, with some probability, each customer may comeback to the system as a Bernoulli feedback. Using the probability generating functions (PGFs), we derive the steady-state solution of the model. Then, we obtain useful performance measures. Moreover, we carry out an economic analysis. Finally, numerical study is performed to explore the effects of the model parameters on the behavior of the system. Mathematics Subject Classification
60K25 · 68M20 · 90B22
1 Introduction Vacation queueing models with impatient customers are very helpful in providing basic framework for efficient design and study of diverse practical situations including telephone switchboard, inventory problems with perishable goods, computer and communication network, data/voice transmission, manufacturing system, etc. In recent past, vacation queueing models have been widely studied. Doshi [3], Takagi [17], and Tian and Zhang [18] are excellent survey works on the subject. An extensive amount of the literature is available on queueing models with server vacation and batch arrival and can be found in Madan and AI-Rawwash [13], Wang et al. [20], Haridass and Arrumuganathan [5], etc. Working vacation queues with customer impatience have attracted the interest of many researchers. Altman and Yechiali [1] treated the infinite-server queueing model with system’s additional tasks and impatient customers. Perel and Yechiali [14] considered a two-phase Markovian random environment with impatient customers. Working vacation queueing model with customer impatience has been analyzed by Yue et al. [22]. Then, Zhang et al. [25] presented an equilibrium balking strategies in Markovian queues with working vacations. Sun et al. [16] gave the equilibrium and optimal behavior of customers in Markovian queues with multiple working vacations. Goswami [4] analyzed a queueing model with impatient customers with Bernoulli A. A. Bouchentouf (B) Department of Mathematics, Mathematics Laboratory, Djillali Liabes University of Sidi Bel Abbes, B. P. 89, 22000 Sidi Bel Abbés, Algeria E-mail: [email protected] A. Guendouzi Laboratory of Stochastic Models, Statistic and Applications, Dr Moulay Tahar University of Saida, En-Nasr, B. P. 13
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