## The Waiting Time Distribution for the M/M/m Queue (2003)

Venue: | In Proceedings of IEE Communications |

Citations: | 2 - 0 self |

### BibTeX

@INPROCEEDINGS{Chan03thewaiting,

author = {Wah-chun Chan and Yi-Bing Lin},

title = {The Waiting Time Distribution for the M/M/m Queue},

booktitle = {In Proceedings of IEE Communications},

year = {2003}

}

### OpenURL

### Abstract

This paper presents a novel method for the calculation of the waiting time distribution function for the M/M/m queue. It is shown that the conditional waiting time obeys an Erlang distribution with rate mµ, where µ is the service rate of a server. Explicit closed form solution is obtained by means of the probability density function of the Erlang distribution. It turns out that the derivation of the result is very simple. The significance of Khintchine's method and its close relation to our method is pointed out. It is also shown that the waiting time distribution can be obtained from Takacs's waiting time distribution for the G/M/m queue as a special case. This reveals some insight into the significance of Takacs's more general but rather complex result.

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Citation Context ... function H(t) = ( 1 e t ; if t 0; 0; if ts0; (2) which is independent of f n g. This queueing system is known as the Erlang delay system which has been investigated intensively in the literature [12=-=, 9, 1, 11, 4]-=-. The main concern of the waiting time is its stochastic behavior and the determination of its distribution function. Khintchine [9] presented a very thoughtful approach and obtained an analytical clo... |

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Citation Context ... function H(t) = ( 1 e t ; if t 0; 0; if ts0; (2) which is independent of f n g. This queueing system is known as the Erlang delay system which has been investigated intensively in the literature [12=-=, 9, 1, 11, 4]-=-. The main concern of the waiting time is its stochastic behavior and the determination of its distribution function. Khintchine [9] presented a very thoughtful approach and obtained an analytical clo... |

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Citation Context ...=M=m queue Takacs investigated the multiple server queueing process for Palm input and Exponential service times. He obtained a more general but very complex formula for the waiting time distribution [14]. We quote his formula as follows: Pr[W t] = 1 Ae m(1 !)t 1 ! ; t 0 (20) where ! is the only root of the functional equation ! = (m(1 !)) (21) in the unit circle, where (s) = Z 1 t=0 e st dF (... |

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Citation Context ... time distribution function, which gives a new insight into the waiting time for the Erlang delay system. For a more general study of the G/G/m queue, the reader may refer to the following references =-=[8, 10, 6, 5, 7, 13, 3, 2]-=-. Note that in order to study the properties of the associated queueing process, these G/G/m studies ended up with the problem of solving integral equations, and no analytic closed form solutions such... |

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Citation Context ... time distribution function, which gives a new insight into the waiting time for the Erlang delay system. For a more general study of the G/G/m queue, the reader may refer to the following references =-=[8, 10, 6, 5, 7, 13, 3, 2]-=-. Note that in order to study the properties of the associated queueing process, these G/G/m studies ended up with the problem of solving integral equations, and no analytic closed form solutions such... |

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1 | Modeling Wireless Local Loop with General Call Holding Times and Finite Number of Subscribers
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Citation Context ...iterature in the way as presented in this paper. Also, our approach provides much simpler conditional waiting time expression to be governed in performance modeling of wireless telephone network. See =-=[15]-=-. Acknowledgement We would like to thank the anonymous reviewers. Their comments have significantly improved the quality of this paper. This work was sponsored in part by MOE Program for Promoting Aca... |