Unreliable multiserver queueing system with modified vacation policy.

*(English)*Zbl 1397.90105Summary: This paper presents modeling and analysis of Markovian multiserver queue under (\(e, d\)) single vacation policy with server breakdowns and repairs. In such a system with \(c\) servers, when some (say \(d < c\)) servers become idle at a service completion time, then \(e(\leq d)\) servers take single vacation only. During idle as well as busy state, the servers are subject to accidental breakdowns. We formulate the model as a quasi-birth-and-death (QBD) process. By employing matrix geometric method, some stationary performance measures after service completion are established. Furthermore, the direct search algorithm is used to simultaneously determine the optimal number of idle, vacationing and total number of servers at minimum cost. Numerical illustration is facilitated to explore the effect of the number of servers, vacation rate, failure and repair rates on the performance measures of interest. Furthermore, the numerical results obtained are compared by using soft computing approach based on adaptive neuro fuzzy inference system (ANFIS).

##### MSC:

90B22 | Queues and service in operations research |

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

90C70 | Fuzzy and other nonstochastic uncertainty mathematical programming |

##### Keywords:

multiserver; modified vacation; (\(e,d\)) policy; servers breakdown; matrix geometric method; cost analysis; neuro-fuzzy approach##### Software:

MOSEL
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\textit{C. Bhargava} and \textit{M. Jain}, Opsearch 51, No. 2, 159--182 (2014; Zbl 1397.90105)

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##### References:

[1] | Artalejo, JR; Orlovsky, DS; Dudin, AN, Multi-server retrial model with variable number of active servers, Comp. Indus. Engg., 48, 273-288, (2005) |

[2] | Chan, R; Lee, S; Sun, H, Boundary value methods for transient solutions of queueing networks with variantvacation policy, J. Comp. Appl. Math., 236, 3948-3955, (2012) · Zbl 1246.65020 |

[3] | Chao, X; Zhao, YQ, Analysis of multi-server queues with station and server vacations, Euro. J. Oper. Res., 110, 392-406, (1998) · Zbl 0947.90023 |

[4] | Chaudhry, ML; Zhao, YQ, Transient solutions of some multiserver queueing systems with finite spaces, Inter. Trans. Oper. Res., 6, 161-182, (1999) |

[5] | Choudhury, G, A batch arrival queue with a vacation time under single vacation policy, Comp. Oper. Res., 29, 1941-1955, (2002) · Zbl 1010.90010 |

[6] | Choudhury, G; Deka, M, A single server queueing system with two phases of service subject to server breakdown and Bernoulli vacation, Appl. Math. Model., 36, 6050-6060, (2012) · Zbl 1349.90213 |

[7] | Cornelius, T., Leondes: Fuzzy logic and expert systems applications, Vol. 6 of Neural Network Systems Techniques and Applications. Academic Press, San Diego (1998) · Zbl 0943.68159 |

[8] | Dimitriou, I, A mixed priority retrial queue with negative arrivals, unreliable server and multiple vacations, Appl. Math. Model., 37, 1295-1309, (2013) · Zbl 1351.90075 |

[9] | Gharbi, N; Ioualalen, M, GSPN analysis of retrial systems with server breakdowns and repairs, Appl. Math. Comp., 174, 1151-1168, (2006) · Zbl 1156.68319 |

[10] | Gharbi, N; Ioualalen, M, Numerical investigation of finite-source multiserver systems with different vacation policies, J. Comp. Appl. Math., 234, 625-635, (2010) · Zbl 1188.65009 |

[11] | Gray, WJ; Wang, PP; Scott, M, A queueing model with multiple types of server breakdowns, QTQM, 1, 245-255, (2004) |

[12] | Hsieh, YC; Andersland, MS, Repairable single server systems with multiple breakdown modes, Micro. Reliab., 35, 309-318, (1995) |

[13] | Jain M. and Priya K.: Transient analysis of photonic networks, Proc. Natl. Conf. Inf. Technol. Oper. Res. Comp. 3-28 (2006) |

[14] | Jain, M; Upadhyaya, S, Synchronous working vacation policy for finite-buffer multiserver queueing system, Appl. Math. Comp., 217, 9916-9932, (2011) · Zbl 1220.65011 |

[15] | Ke, JC, Batch arrival queues under vacation policies with server breakdowns and startup/closedown times, Appl. Math. Model., 31, 1282-1292, (2007) · Zbl 1278.90093 |

[16] | Ke, JC; Wang, KH, Vacation policies for machine repair problem with two spares, Appl. Math. Model., 31, 880-894, (2007) · Zbl 1137.90422 |

[17] | Labzovski, SN; Mehrez, A; Frenkel, IB, The a priori vacation probability in the M/G/1 single vacation models, Math. Comp. Simul., 54, 183-188, (2000) |

[18] | Liu, GS, Three \(m\)-failure group maintenance models for \(M\)/M/N unreliable queueing service systems, Comp. Indus. Engg., 62, 1011-1024, (2012) |

[19] | Madan, KC; Dayyeh, WA; Taiyyan, F, A two server queue with Bernoulli schedules and a single vacation policy, Appl. Math. Comp., 145, 59-71, (2003) · Zbl 1101.60345 |

[20] | Morozov, E; Fiems, D; Bruneel, H, Stability analysis of multiserver discrete-time queueing systems with renewal-type server interruptions, Perf. Eval., 68, 1261-1275, (2011) |

[21] | Neuts, MF, The probabilistic significance of the rate matrix in matrix geometric invariant vectors, J. Appl. Prob., 17, 291-296, (1980) · Zbl 0424.60091 |

[22] | Neuts, M.F.: Matrix geometric solutions in stochastic models-an algorithmic approach. Dover Publications, New York (1981) |

[23] | Neuts, MF; Lucantoni, DM, A Markovian queue with N servers subject to breakdownand repairs, Manag. Sci., 25, 849-861, (1979) |

[24] | Omey, E; Gulck, SV, Maximum entropy analysis of the M\^{}{X}/M/1 queueing system with multiple vacations and server breakdowns, Comp. Indus. Eng., 54, 1078-1086, (2008) |

[25] | Sztrik, J; Gál, T, A recursive solution of a queueing model for a multi-terminal system subject to breakdowns, Perf. Eval., 11, 1-7, (1990) · Zbl 0697.60092 |

[26] | Takagi H.: Fusion technology of fuzzy theory and neural networks-survey and future directions. Proc. International Conference on Fuzzy Logic and Neural Networks, Japan 13-26 (1990) |

[27] | Tian, N; Zhang, ZG, A two threshold vacation policy in multiserver queueing systems, Euro. J. Oper. Res., 168, 153-163, (2006) · Zbl 1077.90016 |

[28] | Vinod, B, Unreliable queueing system, Comp. Oper. Res., 12, 323-340, (1985) · Zbl 0608.90030 |

[29] | Wartenhorst, P, N parallel queueing systems with server breakdown and repair, Euro. J. Oper. Res., 82, 302-322, (1995) · Zbl 0905.90080 |

[30] | Wu, J; Lian, Z, A single-server retrial G-queue with priority and unreliable server under Bernoulli vacation schedule, Comp. Indus. Eng., 64, 84-93, (2013) |

[31] | Xu, X; Zhang, ZG, Analysis of multi-server queue with a single vacation (e, d)-policy, Perf. Eval., 63, 825-883, (2006) |

[32] | Yang, X; Alfa, AS, A class of multiserver queueing system with server failures, Comp. Indus. Eng., 56, 33-34, (2008) |

[33] | Zhang, ZG; Tian, N, An analysis of queueing systems with multi-task servers, Euro. J. Oper. Res., 56, 375-389, (2004) · Zbl 1056.90041 |

[34] | Zhang R., Phillis Y.A. and Kouikoglou V.S.: Ä fuzzy control of queueing systems. Springer-Verlag London Limited (2005) · Zbl 1064.60004 |

[35] | Zreikat, AI; Bolch, G; Sztrik, J, Performance modelling of nonhomogeneous unreliable multiserver systems using MOSEL, Comp. Math. Appl., 46, 293-312, (2003) · Zbl 1095.68013 |

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