Thrivikraman, Ramanatha
(1963)
*Equilibrium delay distribution for queues with random service.*
MSc(R) thesis, University of Glasgow.

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## Abstract

The problem which the thesis discusses is that of determining the probability of delay of a demand (customer) in a queueing system in which service is random, i.e. on the completion of a service-time the server obtains the next customer for service by choosing at random from among those waiting. The system is assumed to be in statistical equilibrium arrivals are assumed to follow the Poisson distribution and two distinct assumptions regarding service-time are made, (i) that it follows the negative exponential distribution, (ii) that it is constant. For the case of negative exponential service-time, the work of a number of authors is reviewed: (i) Molina (1927), who derived the equilibrium state probabilities of the system; (ii) Mellor (1942), who was the first to discuss the actual delay distribution, but whose treatment of the problem is incorrect; (iii) Vaulot (1946), who formulated the problem correctly and gave a fundamental differential-difference equation, which he used to find the delay distribution as a Maclaurin series; (iv) Palm (1946), who, independently of Vaulot and almost simultaneously with him, derived the fundamental equation, and discussed methods (involving generating functions) by which it might be solved, the determination of the general form of the distribution by means of the first two moments, and the question of numerical computation; (v) Pollaczek (1946), who used Laplace transforms end contour integration to find an exact expression for the delay distribution function, but In a form too complicated for actual computation; (vi) Riordan (1953), who, in cm attempt to check numerical values obtained by means of a differential analyzer, found a method of evaluating exactly the moments of the distribution, and used them to approximate to the distribution function by a sum of a few exponentials, thus obtaining numerical values comparatively easily; (vii) Le Roy (1937), who discussed the problem In matrix notation and used an approximating process similar to Riordan's. The case in which the number of places in the queue is finite does not appear to have been discussed, and in the next section, which is new, the modifications to the state probabilities and to the fundamental equation for this case are given. The results of act vial solution of the equation, by means of the Sirius digital computer, for 20, 40 and 60 places In the queue are given, and their relation to the results for an unrestricted queue cure discussed. The case of constant service-time has received comparatively little attention, and the section dealing with this first reviews the work of Crommelin (1932), who derived equations satisfied by the equilibrium state probabilities and also obtained an expression for a generating function of these probabilities, and of Burke (1959), gave a very clear analysis of the problem and obtained actual numerical values for the delay distribution, but only for the case of one server. Burke's work appears to be capable of extension, and in the next section, which is new, it is shown that his methods can be used In the case of two servers. There seems to be no record of a Monte Carlo investigation of the constant service-time case, and in the following section, which is also new, the method by which such an investigation was carried out, by means of the Sirius computer, for one and for two servers is described. It is shown that for one server good agreement with Burke's results was obtained. Finelly, it is pointed out that although Burke's methods can probably be extended to more than two servers, Monte Carlo methods offer an easier way of dealing with this problem, and there seems to be no serious difficulty in using them to analyse not only larger systems but also cases in which more realistic assumptions are made regarding the arrival and service-time distributions.

Item Type: | Thesis (MSc(R)) |
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Qualification Level: | Masters |

Additional Information: | Adviser: A J Howie |

Keywords: | Statistics |

Date of Award: | 1963 |

Depositing User: | Enlighten Team |

Unique ID: | glathesis:1963-73106 |

Copyright: | Copyright of this thesis is held by the author. |

Date Deposited: | 14 Jun 2019 08:56 |

Last Modified: | 14 Jun 2019 08:56 |

URI: | https://theses.gla.ac.uk/id/eprint/73106 |

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