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Analytically simple solution to discrete-time queue with catastrophes, balking and state-dependent service

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Abstract

A queuing system with catastrophes, balking and state-dependent service has gained importance in the recent past due to its applications in various fields like communication system, health care system, production system and computer science. Keeping this in mind, the current paper analyses Geo/Geo/1 queuing system with catastrophes, balking and state-dependent service. Whenever a catastrophe occurs at the system, all customers are forced to leave the system immediately. Two different service rates, depending on the critical value of number of customers in the system (denoted by r), have been used. If a customer on arrival finds other customers in the system, it either decides to enter the system or balks with a constant probability. The expression for steady state probability vector of system size for two models i.e. Late Arrival System with Delayed Access (LAS-DA) and Early Arrival System (EAS) has been found via rate element R using matrix geometric technique. The expressions for probability generating function of number of customers in the system and some performance measures have also been derived. A numerical study has been performed to show the effect of various values of parameters on performance measures. A cost function has also been presented and impact of varying values of various parameters on it has been studied. The model has also been solved for two particular cases of r i.e. when r = 1 and when r = 2. The study reveals that the optimum value of r is 1. Hence, it can be concluded that fast service rate should be given when the number of customers in the system are 2 or more for both the models. The comparative study of two models reveals that LAS-DA model is better than EAS model. The study also exhibits that when the concept of balking and state dependent service are added to Geo/Geo/1 with catastrophe model then there is increase in cost in both the models (LAS-DA and EAS).

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References

  • Abou El-Ata MO (1991) The state dependent queue: M/M/1/N with reneging and general balk functions. Microelectron Reliab 31(5):1001–1007

    Article  Google Scholar 

  • Abou El-Ata MO, Kotb KAM (1992) Linearly dependent service rate for the queue: M/M/1/N with general balk functions, reflecting barrier, reneging and additional service for longer queue. Microelectron Reliab 32(12):1693–1698

    Article  Google Scholar 

  • Abou El-Ata MO, Shawky AI (1992) The single-server Markovian over flow queue with balking, reneging and an additional server for longer queues. Microelectron Reliab 32(10):1389–1394

    Article  Google Scholar 

  • Abou El-Ata MO, Al-Seedy RO, Kotb KAM (1993) Transient solution of the state-dependent queue: M/M/1/N with balking and reflecting barrier. Microelectron Reliab 33(5):681–688

    Article  Google Scholar 

  • Ackere V, Ninios P (1993) Simulation and queuing theory applied to a single-server queue with advertising and balking. J Oper Res Soc 44:407–414

    Article  MATH  Google Scholar 

  • Al-Seedy RO (1996) Analytical solution of the state-dependent Erlangian queue: M/Ej/1/N with balking. Microelectron Reliab 36(2):203–206

    Article  Google Scholar 

  • Al-Seedy RO, Kotb KAM (1991) Transient solution of the state-dependent queue: M/M/1 with balking. Adv Modell Simul 35:55–64

    Google Scholar 

  • Ammar SI, Helan MM, Al-Amri FT (2013) The busy period of an M/M/1 queue with balking and reneging. Appl Math Model 37(22):9223–9229

    Article  MathSciNet  MATH  Google Scholar 

  • Ancker CJ Jr, Gafarian AV (1963a) Some queuing problems with balking and Reneging: I. Oper Res 11:88–100

    Article  MathSciNet  MATH  Google Scholar 

  • Ancker CJ Jr, Gafarian AV (1963b) Some queuing problems with balking and Reneging: I. Oper Res 11:928–937

    Article  MathSciNet  MATH  Google Scholar 

  • Antonis E, Demetris F (2008) Alternative approaches for the transient and analysis of Markov chains with catastrophes. J Stat Theory Pract 2(2):183–197

    Article  MathSciNet  MATH  Google Scholar 

  • Artalejo JR, Gomez-Corral A (1999) Computation of the limiting distribution in queueing systems with repeated attempts and disasters. Oper Res 33(3):371–382

    MathSciNet  MATH  Google Scholar 

  • Atencia I, Moreno P (2004) The discrete time Geo/G/1 queues with negative customers and disasters. Comput Oper Res 31:1537–1548

    Article  MATH  Google Scholar 

  • Athreya KB, Kaplan N (1976) Limit theorems for a branching process with disasters. J Appl Prob 13:466–475

    Article  MathSciNet  MATH  Google Scholar 

  • Bae J, Kim S (2010) The stationary workload of the G/M/1 queue with impatient customers. Queueing Syst 64(3):253–265

    Article  MathSciNet  MATH  Google Scholar 

  • BartoszynskiBhuler WJ, Chan W, Pearl DK (1989) Population process under the influence of disasters occurring independently of population size. J Math Biol 27:179–190

    MathSciNet  Google Scholar 

  • Boots N, Tijms H (1999) An M/N/C queue with impatient customers. TOP 7(2):213–220

    Article  MathSciNet  MATH  Google Scholar 

  • Boudali O, Economou A (2012) Optimal and equilibrium balking strategies in the single server markovian queues with catastrophes. Eur J Oper Res 218:708–715

    Article  MathSciNet  MATH  Google Scholar 

  • Brockwell PJ (1985) The extinction time of a birth, death and catastrophe process and of a related diffusion model. Adv Appl Prob 17:42–52

    Article  MathSciNet  MATH  Google Scholar 

  • Brockwell PJ (1986) The extinction time of a general birth and death process with catastrophes. J Appl Prob 23:851–858

    Article  MathSciNet  MATH  Google Scholar 

  • Brockwell PJ, Gani J, Resnick SI (1982) Birth, immigration and catastrophe processes. Adv Appl Prob 14(4):709–731

    Article  MathSciNet  MATH  Google Scholar 

  • Bura GS, Bura RN (2015) Time dependent analysis of a queueing system incorporating the effect of environment, catastrophe and restoration. J Reliab Stat Stud 8(2):29–40

    Google Scholar 

  • Chakravarthy SR (2009) A disaster queue with Markovian arrivals and impatient customers. Appl Math Comput 214(1):48–59

    MathSciNet  MATH  Google Scholar 

  • Chandrasekaran VM, Saravanarajan MC (2012) Transient and Reliability analysis of M/M/1 feedback queue subject to catastrophes, several failures and repairs. Int J Pure Appl Math 77(5):605–625

    MATH  Google Scholar 

  • Chang I, Krinik A, Swift RJ (2007) Birth-multiple catastrophe processes. J Stat Plann Inference 137:1544–1559

    Article  MathSciNet  MATH  Google Scholar 

  • Chen H, Li J, Tian N (2009) The GI/M/1 queue with phase-type working vacations and vacation interruption. J Appl Math Comput 30:121–141

    Article  MathSciNet  MATH  Google Scholar 

  • Choudhary A, Medhi P (2011) Balking and reneging multi-server Markovianqueueing system. Int J Math Oper Res 3(4):377–394

    Article  MathSciNet  Google Scholar 

  • Dharmaraja S, Kumar R (2015) Transient solution of a queueing model with heterogeneous severs and catastrophes. Opsearch 52(4):810–826

    Article  MathSciNet  MATH  Google Scholar 

  • Di Crescenzo A, Giorno V, Nobile AG, Ricciardi M (2003) On the M/M/1 queue with catastrophes and its continuous approximation. Queueing Syst 43:329–347

    Article  MathSciNet  MATH  Google Scholar 

  • Doshi BT, Jangerman D (1986) An M/G/1 queue with class dependent balking (reneging). In: Boxaman OJ, Cohen JW, Tijms HC (eds) Teletraffic analysis and computer performance evaluation. Elsevier Science Publications, North Holland

    Google Scholar 

  • Drekic S, Woolford DG (2005) A pre-emptive priority queue with balking. Eur J Oper Res 164(2):387–401

    Article  MATH  Google Scholar 

  • Economou A, Fakinos D (2003) A continuous-time Markov chain under the influence of a regulating point rocess and applications in stochastic models with catastrophes. Eur J Oper Res 149:625–640

    Article  MATH  Google Scholar 

  • Gani J, Swift RJ (2007) Death and birth-death and immigration processes with catastrophes. J Stat Theory Pract 1:39–48

    Article  MathSciNet  MATH  Google Scholar 

  • Garg D (2013) Performance analysis of the number of times a single-server queue reaches its capacity in time t under catastrophes and restorations. Am J Oper Res 3(3):75–82

    Google Scholar 

  • Gaver DP, Lehoczky JP (1976) Gaussian approximations to service problems: a communication system example. J Appl Probab 13:768–780

    Article  MathSciNet  MATH  Google Scholar 

  • Goswami V (2014) Analysis of discrete-time multi-server queue with balking. Int J Manage Sci Eng 9(1):21–32

    Google Scholar 

  • Haghighi AM, Medhi J, Mohanty SG (1986) On a multi-server Markovianqueueing system with balking and reneging. Comput Oper Res 13(4):421–425

    Article  MathSciNet  MATH  Google Scholar 

  • Haight FA (1957) Queueing with balking. Biometrika 44(3–4):360–369

    Article  MathSciNet  MATH  Google Scholar 

  • Hanson FB, Tuckwell HC (1997) Population growth with randomly distributed jumps. J Math Biol 36:169–187

    Article  MathSciNet  MATH  Google Scholar 

  • Indra and Rajan, V. (2017) Queuing Analysis of Markovian queue having two heterogeneous servers with catastrophes using Matrix Geometric technique. Int J Stat Syst 12(2):205–212

    Google Scholar 

  • Ioannidis S, Jouini O, Economopoulos AA, Vassilis S, Kouikoglou VS (2010) Control policies for single-stage production systems with perishable inventory and customer impatients. Ann Oper Res 209(1):115–138

    Article  MATH  Google Scholar 

  • Jain G, Sigman K (1996) A Pollaczek-Khintchine formula for M/G/1 queues with disasters. J Appl Prob 33:1191–1200

    Article  MathSciNet  MATH  Google Scholar 

  • Jeykumar S, Gunasekaran P (2017) An analysis of discrete queue with disaster and single vacation. Int J Pure Appl Math 113(6):82–90

    Google Scholar 

  • Kaplan N, Sudbury A, Nelson TS (1975) A branching process with disasters. J Appl Prob 12:47–59

    Article  MathSciNet  MATH  Google Scholar 

  • Kotb KAH, Akhdar M (2018) Feedback of a non- truncated erlangqueueing system with balking and retention of reneged customers. Appl Comput Math 7(2):40–49

    Article  Google Scholar 

  • Krinik A, Mortensen C (2007) Transient probability functions of finite birth-death processes with catastrophes. J Stat Plan Inference 137:1530–1543

    Article  MathSciNet  MATH  Google Scholar 

  • Kumar R (2017) Transient solution to the M/M/c queuing model with balking and catastrophes. Croat Oper Res Rev CRORR 8:577–591

    Article  MathSciNet  MATH  Google Scholar 

  • Kumar BK, Arivudainambi D (2000) Transient solution of an M/M/1 queue with catastrophes. Comput Math Appl 40:1233–1240

    Article  MathSciNet  MATH  Google Scholar 

  • Kumar R, Sharma S (2018) Transient analysis of an M/M/C queue with balking and retention of reneged customers. Commun Stat Theory Methods 47(6):1318–1327

    Article  MATH  Google Scholar 

  • Kumar R, Sumeet KS (2012) A n M/M/1/N queueing model with retention of reneged customers and balking. Am J Oper Res 2(1):1–5

    MathSciNet  Google Scholar 

  • Kumar BK, Krishnamoorthy A, PavaiMadheswari S, SadiqBasha, (2007) Transient analysis of a single server queue with catastrophes, failures and repairs. Queueing Syst 56(3–4):133–141

    Article  MathSciNet  Google Scholar 

  • Kumar BK, Anantha Lakshmi SR, Anbarasu S, PavaiMadheswari S (2014) Transient and steady-state analysis of queueing systems with catastrophes and impatient customers. Int J Math Oper Res 6(5):523–549

    Article  MathSciNet  MATH  Google Scholar 

  • Kyriakidis EG (1994) Stationary probabilities for a simple immigration birth-death process under the influence of total catastrophes. Statist Prob Lett 20:239–240

    Article  MathSciNet  MATH  Google Scholar 

  • Kyriakidis EG (2001) The transient probabilities of the simple immigration-catastrophe process. Math Sci 26:56–58

    MathSciNet  MATH  Google Scholar 

  • Lee C (2000) The density of the extinction probability of a time homogeneous linear birth and death process under the influence of a randomly occurring disasters. Math Biosci 164:93–102

    Article  MathSciNet  MATH  Google Scholar 

  • Lee DH, Yang WS, Park HM (2011) Geo/G/1 queues with disasters and repair times. Appl Math Model 35(4):1561–1570

    Article  MathSciNet  MATH  Google Scholar 

  • Liau PY (2011) A queueing model with balking index and reneging rate. Int J Serv Oper Manag 10(1):1–12

    MathSciNet  Google Scholar 

  • OkanIsguder H, Kaya C (2012) Analysis of a tandem queue with heterogeneous servers subject to catastrophes. Sci Res Essays 7(47):4076–4080

    Google Scholar 

  • Pakes AG (1987) Limit theorems for the population size of a birth and death process allowing catastrophes. J Math Boil 25:307–325

    Article  MathSciNet  MATH  Google Scholar 

  • Pakes AG (1988) The supercritical birth, death and catastrophe process: Limit theorems on the set of non-extinction. J Math Boil 26:405–420

    Article  MATH  Google Scholar 

  • Park HM, Yang WS, Chae KC (2009) The Geo/G/1 queue with negative customers and disasters. Stoch Model 25(4):673–688

    Article  MathSciNet  MATH  Google Scholar 

  • Park HM, Yang WS, Chae KC (2010) Analysis of GI/Geo/1 queue with disasters. Stoch Anal Appl 28(1):44–53

    Article  MathSciNet  MATH  Google Scholar 

  • Pazgal AI, Radas S (2008) Comparison of customer balking and reneging behaviour to queueing theory predictions: An experimental study. Comput Oper Res 35(8):2537–2548

    Article  MATH  Google Scholar 

  • Rao SS (1968) Queueing with balking and reneging in M/G/1 system. Metrika 12(1):173–188

    MATH  Google Scholar 

  • Sankanarayanan G, Krishnamoorthy A (1978) Branching processes with immigration subjected to disasters. J Math Phys Sci 12:165–176

    MathSciNet  MATH  Google Scholar 

  • Shanmugasundaram S, Chitra S (2016) Study on M/M/2 transient queue with feedback under catastrophic effect. Int J Comput Eng Res 6(10):18–25

    Google Scholar 

  • Sharma RR, Rai RC, Mishra A (1993) Optimal bus service on express basis in the case of balking and reneging. Eur J Oper Res 66:113–123

    Article  MATH  Google Scholar 

  • Sharma YK, Sharma GC, Jain M (2017) Analysis of a queueing model with balking for buffer sharing in ATM. Int J Eng Res Dev 13(9):1–11

    Google Scholar 

  • Shawky AI (1997) The single-server machine interference model with balking, reneging and an additional server for longer queues. Microelectr Reliab 37:355–357

    Article  Google Scholar 

  • Singh CJ, Jain M, Kumar B (2014) Analysis of Mx/G/1 queueing model with balking and vacation. Int J Oper Res 19(2):154–173

    Article  MathSciNet  MATH  Google Scholar 

  • Sophia S (2018) Transient analysis of a single server queue with chain sequence rates subject to catastrophes, failures and repairs. Int J Pure Appl Math 119(15):1969–1978

    Google Scholar 

  • Stirzaker D (2001) Disasters. Math Sci 26:59–62

    MathSciNet  MATH  Google Scholar 

  • Stirzaker D (2006) Process with catastrophes. Math Sci 31:107–118

    MathSciNet  MATH  Google Scholar 

  • Stirzaker D (2007) Process with random regulation. Prob Eng Inform Sci 21:1–17

    Article  MathSciNet  MATH  Google Scholar 

  • Sudhesh R, Priya RS, Lenin RB (2017) Transient analysis of a single server discrete-time queue with system disaster. RAIRO-Oper Res 51(1):123–134

    Article  MathSciNet  MATH  Google Scholar 

  • Swift RJ (2000) A simple immigration-catastrophe process. Math Sci 25:32–36

    MathSciNet  MATH  Google Scholar 

  • Swift RJ (2001) Transient probabilities for a simple birth-death-immigration process under the influence of total catastrophes. Int J Math Math Sci 25:689–692

    Article  MathSciNet  MATH  Google Scholar 

  • Switkes J (2004) An unbiased random walk with catastrophe. Math Sci 29:115–121

    MathSciNet  MATH  Google Scholar 

  • Towsley D, Tripathi SK (1991) A single server priority queue with server failures and queue flushing. Oper Res Lett 10(6):353–362

    Article  MathSciNet  MATH  Google Scholar 

  • Uma S, Manikandan P (2017) Single server bulk queueing system with three stage heterogeneous service compulsory vacation and balking. Int J Sci Res Publ 7(4):300–306

    Google Scholar 

  • Vijayalakshmi T, Thangaraj V (2016) Transient analysis of a fluid queue driven by a chain sequenced birth and death process with catastrophes. Int J Math Oper Res 8(2):164–184

    Article  MathSciNet  MATH  Google Scholar 

  • Vinodhini GAF, Vidhya V (2016) Computational analysis of queues with catastrophes in a multiphase random environment. Math Prob Eng 2016:7

    Article  MathSciNet  MATH  Google Scholar 

  • Wang KH, Chang YC (2002) Cost analysis of a finite M/M/R queueing system with balking, reneging and server breakdowns. Math Methods Oper Res 56(2):169–180

    Article  MathSciNet  MATH  Google Scholar 

  • Yechiali U (2007) Queues with system disasters and impatient customers when system is down. Queueing Syst 56:3–4

    Article  MathSciNet  MATH  Google Scholar 

  • Yi XW, Kim JD, Choi DW, Chae KC (2007) The Geo/G/1 queue with disasters and multiple working vacations. Stoch Model 23(4):537–549

    Article  MathSciNet  MATH  Google Scholar 

  • Yue D, Li C, Yue W (2006) The matrix geometric solution of the M/Ek/1 queue with balking and state-dependent service. Nonlinear Dyn Syst Theory 6(3):295–308

    MathSciNet  MATH  Google Scholar 

  • Yue D, Yue W, Sun Y (2005).Analysis of an M/M/c/N queue with balking, reneging and server vacations, In: Proceedings of the sixth international symposium on operations research and its applications, Xinjiang, China, pp 128–143

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  1. Department of Computer Applications, Gujranwala Guru Nanak Institute of Management and Technology, Civil Lines, Ludhiana, India

    Vijay Rajan Lumb

  2. Department of Statistics and Operational Research, Kurukshetra University, Kurukshetra, Haryana, India

    Indra Rani

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  1. Vijay Rajan Lumb
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Correspondence to Vijay Rajan Lumb.

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Lumb, V.R., Rani, I. Analytically simple solution to discrete-time queue with catastrophes, balking and state-dependent service. Int J Syst Assur Eng Manag 13, 783–817 (2022). https://doi.org/10.1007/s13198-021-01342-1

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