Transient analysis of a Markov queueing model with feedback, discouragement and disaster.

*(English)*Zbl 1456.60241Summary: The transient model for a Markovian feedback queue with system disaster and customer impatient has been investigated. After service completion, dissatisfied customers can feedback to the system to get another service. During the service, the system may suffer disaster failure and consequently lose all the customers present in the system. After the occurrence of a disaster, the system immediately goes under the repair process. During the repair, the newly arriving customers may get discouraged and balk without joining the queue. Upon arrival in the down state of the system, the arriving customers activate their individual timer. A customer waiting in the queue departs and never comes back if the timer run out before the system gets repaired. The time-dependent system size distribution is formulated analytically by applying the techniques of probability generating function along with continued fractions. The computational results are presented in graphical and tabular form to examine the variation of system descriptors on various performance indices.

Reviewer: Reviewer (Berlin)

##### MSC:

60K25 | Queueing theory (aspects of probability theory) |

90B22 | Queues and service in operations research |

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\textit{M. Jain} and \textit{M. Singh}, Int. J. Appl. Comput. Math. 6, No. 2, Paper No. 31, 14 p. (2020; Zbl 1456.60241)

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##### References:

[1] | Jain, G.; Sigman, K., A Pollaczek-Khintchine formula for M/G/1 queues with disasters, J. Appl. Probab., 33, 1191-1200 (1996) · Zbl 0867.60082 |

[2] | Yechiali, U., Queues with system disasters and impatient customers when system is down, Queueing Syst., 56, 195-202 (2007) · Zbl 1124.60076 |

[3] | Chakravarthy, S., A disaster queue with Markovian arrivals and impatient customers, Appl. Math. Comput., 214, 48-59 (2009) · Zbl 1170.60330 |

[4] | Lee, DH; Yang, WS; Park, HM, Geo/G/1 queues with disasters and general repair times, Appl. Math. Model., 35, 1561-1570 (2011) · Zbl 1217.90073 |

[5] | Ammar, SI; Rajadurai, P., Performance analysis of preemptive priority retrial queueing system with disaster under working breakdown services, Symmetry, 11, 419 (2019) · Zbl 1423.90058 |

[6] | Kumar, BK; Krishnamoorthy, A.; Madheswari, SP; Basha, SS, Transient analysis of a single server queue with catastrophes, failures and repairs, Queueing Syst., 56, 133-141 (2007) · Zbl 1124.60073 |

[7] | Kumar, BK; Vijayakumar, A.; Sophia, S., Transient analysis for state-dependent queues with catastrophes, Stoch. Anal. Appl., 26, 1201-1217 (2008) · Zbl 1153.60394 |

[8] | Ammar, SI, Transient analysis of an M/M/1 queue with impatient behavior and multiple vacations, Appl. Math. Comput., 260, 97-105 (2015) · Zbl 1410.90045 |

[9] | Parthasarathy, PR; Sudhesh, R., Exact transient solution of a discrete time queue with state-dependent rates, Am. J. Math. Manag. Sci., 26, 253-276 (2006) · Zbl 1244.90068 |

[10] | Sudhesh, R.; Raj, LF; Krishna, PV; Babu, MR; Ariwa, E., Computational analysis of stationary and transient distribution of single server queue with working vacation, Global Trends in Computing and Communication Systems, 480-489 (2012), Heidelberg: Springer, Heidelberg |

[11] | Sudhesh, R.; Savitha, P.; Dharmaraja, S., Transient analysis of a two-heterogeneous servers queue with system disaster, server repair and customers’ impatience, TOP, 25, 179-205 (2017) · Zbl 1364.60124 |

[12] | Sudhesh, R., Transient analysis of a queue with system disasters and customer impatience, Queueing Syst., 66, 95-105 (2010) · Zbl 1197.60087 |

[13] | Vinodhini, GAF; Vidhya, V., Transient solution of a multi server queue with catastrophes and impatient customers when system is down, Appl. Math. Sci., 8, 4585-4592 (2014) |

[14] | Parthasarathy, PR; Selvaraju, N., Transient analysis of a queue where potential customers are discouraged by queue length, Math. Probl. Eng., 7, 433-454 (2001) · Zbl 1008.60098 |

[15] | Altman, E.; Yechiali, U., Analysis of customers’ impatience in queues with server vacations, Queueing Syst., 52, 261-279 (2006) · Zbl 1114.90015 |

[16] | Kumar, R.; Sharma, S., Transient solution of a two-heterogeneous servers’ queuing system with retention of reneging customers, Bull. Malays. Math. Sci. Soc., 42, 223-240 (2019) · Zbl 1406.90026 |

[17] | Yue, D.; Zhang, F., A discrete-time Geo/G/1 retrial queue with J-vacation policy and general retrial times, J Syst. Sci. Complex., 26, 556-571 (2013) · Zbl 1292.90094 |

[18] | Luo, C.; Tang, Y.; Li, C., Transient queue size distribution solution of Geom/G/1 queue with feedback—a recursive method, J. Syst. Sci. Complex., 22, 303-312 (2009) · Zbl 1300.90011 |

[19] | Choi, BD; Kim, YC; Lee, YW, The M/M/c retrial queue with geometric loss and feedback, Comput. Math. Appl., 36, 41-52 (1998) · Zbl 0947.90024 |

[20] | Kumar, BK; Madheswari, SP; Vijayakumar, A., The M/G/1 retrial queue with feedback and starting failures, Appl. Math. Model., 26, 1057-1075 (2002) · Zbl 1018.60088 |

[21] | Jain, M.; Rani, S.; Singh, M.; Deep, K.; Jain, M.; Salhi, S., Transient analysis of Markov feedback queue with working vacation and discouragement, Performance Prediction and Analytics of Fuzzy, Reliability and Queuing Models, 235-250 (2019), Singapore: Springer, Singapore |

[22] | Lorentzen, L.; Waadeland, H., Continued Fractions with Applications. Studies in Computational Mathematics (1992), Amsterdam: Elsevier, Amsterdam · Zbl 0782.40001 |

[23] | Ancker, CJ Jr; Gafarian, AV, Some queueing problems with balking and reneging: I, Oper. Res., 11, 88-100 (1963) · Zbl 0109.36604 |

[24] | Ancker, CJ Jr; Gafarian, AV, Some queuing problems with balking and reneging: II, Oper. Res., 11, 928-937 (1963) · Zbl 0123.35805 |

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