Optimal design and performance modelling of \(M/G/1/K\) queueing systems.

*(English)*Zbl 1112.90319Summary: Approximating the performance measures of \(M/G/1/K\) systems is a difficult, challenging, and important problem for applications in science and engineering. An approach based on a two-moment approximation of the process is presented and is contrasted with an embedded Markov chain approach, Gelenbe’s approach, simulation, and finally, the statistics of \(M/M/1/K\) systems. The closed form expressions for the different performance measures should be very handy. The use of the approximation in the performance modelling and design of \(M/G/1/K\) systems is also explored in order to demonstrate the practical usefulness of the concepts contained within the paper.

##### MSC:

90B22 | Queues and service in operations research |

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

PDF
BibTeX
XML
Cite

\textit{J. MacGregor Smith}, Math. Comput. Modelling 39, No. 9--10, 1049--1081 (2004; Zbl 1112.90319)

Full Text:
DOI

**OpenURL**

##### References:

[1] | Morse, P, Queues, inventories, and maintenance, (1958), Wiley |

[2] | Gross, D; Harris, C, Fundamentals of queueing theory, (1985), Wiley · Zbl 0658.60122 |

[3] | Tijms, H, Stochastic modeling and analysis, (1986), Wiley New York |

[4] | Cooper, R, Introduction to queueing theory, (1981), North-Holland |

[5] | Tijms, H, Heuristics for finite-buffer queues, Probability in the engineering and informational sciences, 6, 277-285, (1992) · Zbl 1134.90343 |

[6] | Schweitzer, P; Konheim, A, Buffer overflow calculations using an infinite-capacity model, (), 267-276 · Zbl 0373.60123 |

[7] | Sakasegawa, H; Miyazawa, M; Yamazaki, G, Evaluating the overflow probability using the infinite queue, Operations research, 39, 1238-1245, (1993) · Zbl 0792.60095 |

[8] | Tijms, H, Stochastic models: an algorithmic approach, (1994), Wiley New York · Zbl 0838.60075 |

[9] | Gaver, D; Shedler, G.S, Processor utilization in multi-programming systems via diffusion approximations, Operations research, 21, 569-576, (1973) |

[10] | Gaver, D; Shedler, G.S, Approximate models for processor utilization in multiprogrammed computer systems, SIAM J. of computing, 2/3, 183-192, (1973) · Zbl 0286.68034 |

[11] | Gelenbe, E, On approximate computer system models, Jacm, 22, 2, 261-269, (1975) · Zbl 0322.68035 |

[12] | Yao, D; Buzacott, J.A, Queueing models for a flexible machining station part 1: the diffusion approximation, Ejor, 19, 233-240, (1985) · Zbl 0553.90049 |

[13] | Kimura, T, Equivalence relations in the approximations for the M/G/s/s+r queue, Mathl. comput. modelling, 31, 10-12, 215-224, (2000) · Zbl 1042.60540 |

[14] | Kimura, T, A transform-free approximation for the finite capacity M/G/s queue, Operations research, 44, 6, 984-988, (1996) · Zbl 0879.90098 |

[15] | Kimura, T, Optimal buffer design of an M/G/s queue with finite capacity, Commun. statist.-stochastic models, 12, 1, 165-180, (1996) · Zbl 0846.60090 |

[16] | De Kok, A.G; Tijms, H, A two-moment approximation for a buffer design problem requiring a small rejection probability, Performance evaluation, 5, 77-84, (1985) |

[17] | Kelton, D; Sadowski, R; Sadowski, O, Simulation with arena, (2001), McGraw-Hill |

[18] | J. MacGregor Smith and F.R. Cruz, The buffer allocation problem for general finite buffer queueing networks, (submitted). · Zbl 1187.90087 |

[19] | J. MacGregor Smith and L. Kerbache, Buffer space allocation in general closed finite queueing networks, (submitted). · Zbl 0691.60088 |

[20] | Springer, M; Makens, P, Queueing models for performance analysis and selection of single station models, Ejor, 58, 123-145, (1991) · Zbl 0761.90045 |

[21] | Seelen, L.P; Tijms, H; Van Hoorn, M.H, Tables for multi-server queues, (1985), North-Holland · Zbl 0626.60090 |

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.