Reliability and Sensitivity Analysis of a Batch Arrival Retrial Queue with k-Phase Services, Feedback, Vacation, Delay, Repair and Admission

Document Type : Original Article


1 Department of Statistics, Faculty of Statistics, Mathematics and Computer, Allameh Tabataba’i University, Tehran, Iran

2 Department of Statistics, Faculty of Statistics, Mathematics and Computer, Allameh Tabataba’i University, Tehran, Iran


Queueing theory is a way for real-world problems modeling and analyzing. In many processes, the input is converted to the desired output after several successive steps. But usually limitations and conditions such as Lack of space, feedback, vacation, failure, repair, etc. have a great impact on process efficiency. This article deals with the modeling the steady-state behavior of an M^X/G/ 1 retrial queueing system with k phases of service. The arriving batches join the system with dependent admission due to the server state. If the customers find the server busy, they join the orbit to repeat their request. Although, the first phase of service is essential for all customers, any customer has three options after the completion of the i-th phase (i=1,2,…,k). They may take the (i+1)-th phase of service with probability θ_i, otherwise return the orbit with probability p_i or leave the system with probability (〖1-p〗_i-θ_i). Also, after each phase, the probabilistic failure, delay, repair and vacation is considered. In this article, after finding the steady-state distributions, the probability generating functions of the system and orbit size have been found. Then, some important performance measures of the system have been derived. Also, the system reliability has been defined. Eventually, to demonstrate the capability of the proposed model, the sensitivity analysis of performance measures via some model parameters (arrival/retrial/vacation rate) in different reliability levels have been investigated in a specific case of this model. Additionally, for optimizing the performance of system, some technical suggestions are presented.


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Volume 3, Issue 2
Summer and Autumn 2020
Pages 27-40
  • Receive Date: 17 December 2020
  • Revise Date: 20 December 2020
  • Accept Date: 26 December 2020