Abstract
Queueing theory is implemented for modeling and analyzing actual conditions in industries and real-world problems. In many cases, the input is converted to the desired output after several successive steps. Lack of space, feedback and vacation are the main characters of these processes. This article deals with the modeling and analyzing the steady-state behavior of an \(M/G/1\) retrial queueing system with first essential and \(k-1\) optional phases of service. Also, the probabilistic feedback to orbit at each phase and Bernoulli vacation at the end of \(k\)-th phase may occur in this system. If the customers find the server busy or on vacation, they join to the orbit. In this article, after finding the probability generating functions of the system and orbit sizes, some important performance measures are found. Also, the system reliability is defined. Eventually, to demonstrate the capability of the proposed model, the sensitivity analysis of cost indices and performance measures via some model parameters (arrival/retrial/vacation rate) in different reliability levels are investigated in two applicable examples. Additionally, for optimizing the performance of the system, some technical suggestions are presented.
Similar content being viewed by others
Data Availability
Based on the assumptions of provided examples, the used data are simulated by R software.
Code Availability
The examples are solved by programming in R software. These codes are available. Also, the presented figures are reproducible and can be submitted if necessary.
Notes
Probability generating functions.
First come first serve.
References
Achcar JA, Piratelli CL (2013) Modeling quality control data Weibull distributions in the presence of a change point. Int J Adv Manuf Technol 66:1611–1621
Arivudainambi D, Godhandaraman P (2012) A batch arrival retrial queue with two phases of service, feedback and K optional vacations. Appl Math Sci 6(22):1071–1087
Artalejo JR (1999) Accessible bibliography on retrial queues. Math Comput Model 30:1–6
Atencia I, Moreno P (2003) A queueing system with linear repeated attempts, Bernoulli schedule and feedback. Top 11(2):285–310
Azhagappan A (2019) Transient behavior of a Markovian queue with working vacation variant reneging and awaiting server. TOP 27:351
Birnbaum ZW, Esary JD, Saunders SD (1961) Multi-component systems and structures and their reliability. Technometrics 3(1):55–77
Bouchentouf AA, Cherfaoui M, Boualem M (2019) Performance and economic analysis of a single server feedback queueing model with vacation and impatient customers. Opsearch 56:300–323
Choudhury G, Deka K (2009a) An M/G/1 retrial queueing system with two phases of service subject to the server breakdown and repair. Qual Technol Quant Manag 65:714–724
Choudhury G, Deka K (2009b) An MX/g/1 unreliable retrial queue with two phases of service and Bernoulli admission mechanism. Appl Math Comput 215(3):936–949
Choudhury G, Deka K (2010) A batch arrival retrial queueing system with two phases of service and service interruption. Comput Math Appl 59(1):437–450
Choudhury G, Ke JC (2012) A batch arrival retrial queue with general retrial times under Bernoulli vacation schedule for unreliable server and delaying repair. Appl Math Model 36:255–269
Choudhury G, Paul M (2005) A two phase queuing system with Bernoulli feedback. Inf Manag Sci 16:35–52
Dept. of Def. of USA (1998) Electronic reliability design handbook. MIL-HDBK-338B, USA
Dept. of Def. of USA (2003) Unmanned aerial vehicle reliability study. Office of the Secretary of Defence, USA
Dept. of NAVY of USA (NSWC) (2006) Handbook of reliability prediction procedures for mechanical equipment. Bethesda, Maryland, 20817–5700, USA
Falin GI, Templeton JGC (1997) Retrial queues. Chapman & Hall, New Jersey
Jain M, Bhagat A (2016) MX/G/1 retrial vacation queue for multi-optional services, phase repair and reneging. Qual Technol Quant Manag 13(3):63–288
Jeganathan K, Kathiresan J, Anbazhagan N (2016) A retrial inventory system with priority customers and second optional service. Opsearch 53:808–834
John FSh, James MTh, Donald G, Carl MH (2018) Fundamentals of queueing theory. Wiley Series in Probability and Statistics
Ke JC (2007) Operating characteristic analysis on the MX/G/1 system with a variant vacation policy and balking. Appl Math Model 31:1321–1337
Ke JC, Chang FM (2009) Modified vacation policy for M/G/1 retrial queue with balking and feedback. Comput Ind Eng 57(1):433–443
Kumar A (2020) Single server multiple vacation queue with discouragement solve by confluent hypergeometric function. J Ambient Intell Human Comput. https://doi.org/10.1007/s12652-020-02467-0
Kumar BK, Kumar AV, Arivudainambi D (2002a) An M/G/1 retrial queueing system with two phase service and preemptive resume. Ann Oper Res 113:61–79
Kumar BK, Madheswari S, Kumar AV (2002b) The M/G/1 retrial queue with feedback and starting failures. Appl Math Model 26:1057–1075
Li W, Shi D, Chao X (1997) Reliability analysis of M/G/1 queueing system with server breakdowns and vacations. J Appl Probab 34:546–555
Rao SH, Vemuri VK, Kumar BS, Rao TS (2017) Analysis of two-Phase queueing system with impatient customers, server breakdowns and delayed repair. Int J Pure Appl Math 115(4):651–663
Senthikumar R, Arumuganathan M (2008) On the single server batch arrival queue with general vacation time under Bernoulli schedule and two phases of heterogenous service. Qual Technol Quant Manag 5:145–160
Shahkar GH, Badamchizadeh A (2006) On M/(G1, G2, …, Gk )/V/1(BS). Far East J Theor Stat 20(2):151–162
Sharma R (2014) Mathematical analysis of queue with phase service: an overview. Adv Oper Res 2014(1):19
Shekhar C, Raina AA, Kumar A, Lgbal J (2017) A survey on queues in machining system: progress from 2010 to 2017. Yugoslav J Oper Res 17(4):391–413
Som BK, Seth S (2018) M/M/c/N queueing systems with encouraged arrivals, reneging, retention and feedback customers. Yugoslav J Oper Res 28:6–6
Tang Y (1997) A single server M/G/1 queueing system subject to breakdowns: some reliability and queueing problems. Microelectron Reliab 37:315–321
Wang J, Li J (2009) A single server retrial queue with general retrial times and two phase service. J Syst Sci Complex 22:291–302
Wang J, Cao J, Li Q (2001) Reliability analysis of the retrial queue with server breakdowns and repairs. Queueing Syst 38:363–380
Funding
None.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
All authors declare that they have no conflict of interest.
Rights and permissions
About this article
Cite this article
Abdollahi, S., Rad, M.R.S. & Farsi, M.A. Reliability and Sensitivity Analysis of Retrial Queue with Optional k-Phases Services, Vacation and Feedback. Iran J Sci Technol Trans Sci 45, 1361–1374 (2021). https://doi.org/10.1007/s40995-021-01101-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40995-021-01101-8