##
**Analysis of batch arrival queue with two stages of service and phase vactions.**
*(English)*
Zbl 1311.60107

Summary: We study a batch arrival queueing system of phase vacation with two stages of service based on a Bernoulli schedule. A single server provides essential service to all arriving customers with service time following a general distribution. After two stages of service completion, the server leaves for phase one vacation of random length with probability \(p\) or continues to stay in the system with probability \(1-p\). As soon as the completion of phase one vacation, the server undergoes phase two vacation. On completion of two heterogeneous phases of vacation the server returns back to the system. The vacation times are assumed to be general. The server is interrupted and the service interruption follows an exponential distribution. The arrivals follow a Poisson distribution. Using a supplementary variable technique, the Laplace transforms of the time-dependent probabilities of the system state are derived. From this we deduce the steady state results. We also obtain the average queue size and average waiting time.

### MSC:

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

60K30 | Applications of queueing theory (congestion, allocation, storage, traffic, etc.) |

90B22 | Queues and service in operations research |

### Keywords:

batch arrival queue; Bernoulli schedule; random breakdown; steady state; average waiting time
PDF
BibTeX
XML
Cite

\textit{S. Maragatha Sundari} et al., Missouri J. Math. Sci. 26, No. 2, 189--205 (2014; Zbl 1311.60107)

Full Text:
Euclid

### References:

[1] | Y. Baba, On the \(M^{[x]}/G/1\) queue with vacation time , Operations Research Letters, 5 (1986), 93-98. · Zbl 0595.60094 |

[2] | G. Chodhury, An \(M^{[x]}/G/1\) queueing system with a set up period and a vacation period , Questa, 36 (2000), 23-28. |

[3] | M. Cramer, Stationary distributions in a queueing system with a vacation times and limited service , Queueing Systems, 4 (1989), 57-68. · Zbl 0664.60095 |

[4] | N. Igaki, Exponential two server queue with N-policy and general vacation , Queueing Systems, 10 (1992), 279-294. · Zbl 0786.60115 |

[5] | B. Krishnakumar, A. Vijayakumar, and D. Arividainambi, An M/G/1 retrial queueing system with two phase service and preemptive resume , Annals of Operations Research, 113 (2002), 61-79. · Zbl 1013.90032 |

[6] | Y. Levi and U. Yechilai, An M/M/s queue with server vacations , Infor., 14 (1976), 153-163. |

[7] | K. C. Madan, On a single server with two stage heterogeneous service and binomial schedule server vacations , Egyptian Statistical Journal, 40.1 (2000), 39-55. |

[8] | K. C. Madan and R. F. Anabosi, A single server queue with two types of service, Bernoulli schedule server vacations and a single vacation policy , Pakistan Journal of Statistics, 19 (2003), 331-442. · Zbl 1129.60319 |

[9] | K. C. Madan and G. Chodhury, An \(M^{[x]}/G/1\) queue with Bernoulli vacation schedule under restricted admissibility policy , Sankhaya, 66 (2004), 172-193. |

[10] | K. C. Madan and G. Chodhury, A two stage arrival queueing system with a modified Bernoulli schedule vacation under N-policy , Mathematical and Computer Modeling, 42 (2005), 71-85. · Zbl 1090.90037 |

[11] | K. C. Madan, F. A. Maraghi, and K. D. Dowman, Batch arrival queueing system with random breakdowns and Bernoulli schedule server vacations having general vacation time distribution , Information and Management Sciences, 20 (2009), 55-70. · Zbl 1168.90390 |

[12] | K. C. Madan, An M/G/1 queue with second optional service , Queueing Systems, 34 (2000), 37-46. · Zbl 0942.90008 |

[13] | K. C. Madan and A. Z. Abu-Dayyeh, On a single server queue with optional phase type server vacations based on exhaustive deterministic service and a single vacation policy , Applied Mathematics and Computation, 149 (2004), 723-734. · Zbl 1041.60070 |

[14] | S. Maragathasundari and S. Srinivasan, Analysis of M/G/1 feedback queue with three stage multiple sever vacation , Applied Mathematical Sciences, 6 (2012), 6221-6240. · Zbl 1264.90082 |

[15] | S. Maragathasundari and S. Srinivasan, Analysis of M/G/1 queue with triple stage of service having compulsory vacation and service interruptions , Far East Journal of Mathematical Sciences, 69 (2012), 61-80. · Zbl 1266.90079 |

[16] | S. Maragathasundari and S. Srinivasan, Three phase M/G/1 queue with Bernoulli feedback and multiple server vacation and service interruptions , International Journal of Applied Mathematics and Statistics, 33.3 (2013), 55-70. |

[17] | S. Maragathasundari and S. Srinivasan, Analysis of M/G/1 feedback queue with multi stage and multiple server vacation , International Conference on Mathematical Sciences and Computer Engineering, (ICMSCE 2012, MALAYSIA), 90-96. |

[18] | S. Maragathasundari and S. Srinivasan, Analysis of transient behaviour of M/G/1 queue with single vacation , International Journal of Pure and Applied Mathematics, 76 (2012), 149-156. |

[19] | S. Maragathasundari and S. Srinivasan, Multi phase M/G/1 queue with Bernoulli feedback and multiple server vacation , International Journal of Computer Applications, 52.1 (2012), 18-23. |

[20] | S. Maragathasundari and S. Srinivasan, Retrial queue with phase vacations , European Journal of Scientific Research, 93 (2012), 226-235. |

[21] | S. Maragathasundari and S. Srinivasan, Analysis of M/M/1 queue for ATM facility , Global Journal of Theoretical and Applied Mathematics Sciences, 2 (2012), 41-46. |

[22] | S. Maragathasundari and S. Srinivasan, M/M/C queueing model for waiting time of customers in Bank sectors , International Journal of Mathematical Sciences and Applications, 1.3 , (2011), 1569-1575. · Zbl 1266.90123 |

[23] | S. Maragathasundari and S. Srinivasan, Time dependent solution of a non Markovian Queue with triple stage of service having compulsory vacation and service interruptions , International Journal of Computer Applications, 41.7 (2012), 37-43. |

[24] | S. Maragathasundari, S. Srinivasan, and A. Ranjitham, A batch arrival non-Markovian queue with three types of service , International Journal of Computer Applications, 83.5 (2013), 43-47. |

[25] | S. Maragathasundari, S. Srinivasan, and A. Ranjitham (2014), Batch arrival queueing with two stages of service , Int. Journal of Math. Analysis, 8 , 247-258. |

[26] | V. Thangaraj and S. Vanitha, Single server feedback queue with two types of service having general distribution , International Mathematical Forum, 5 (2010), 15-33. · Zbl 1203.60139 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.