De invloed van een prioriteitsregeling op de gemiddelde wachttijd

F.W. Steutel

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Abstract

The effect of priorities on the average waitingtime. A waitingtime process is described in terms of customers arriving at a single counter. In the first part of the article the formulae of A. C obham [1] and [2] for the case of two priorities are applied to the following situation: customers arrive in such a way that the interarrival times s are mutually independent and exponentially distributed with mean ¿-1. Servicetimes s are mutually independent and independent of the interarrival times. The distributionfunction of s is indicated by B (s), the mean of s by µ. The customers are divided in two groups, group 1 consisting of the customers having servicetimes cµ, group 2 of those having servicetimes > cµ, c being a positive number. Servicing takes place as follows: priority is given to customers of group 1 i.e. service of a customer of group 2 starts only if no customers of group 1 are present, customers of the same group are served in the order of their arrival, servicing of a customer is not interrupted. Keeping ¿µ <1 the mean waitingtime e¿* for an arbitrary customer is then calculated by means of Cobham's formulae and compared with the mean waitingtime e¿, in case that no priorities are given. Under hypothesis of differentiability of B (s>is the (unique) value of c which minimizes e¿*. It is shown that always c*> 1. In the second part a situation is considered which has been treated by T. E. Phipps [7]. In this situation the customer having the smallest servicetime of all customers present when the counter becomes free is served first. The mean waitingtime e¿** in this case is compared with e¿ and e¿*. Some examples and graphs are given.
LanguageEnglish
Pages503-513
Number of pages11
JournalStatistica Neerlandica
Volume13
Issue number4
DOIs
StatePublished - 1959

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title = "De invloed van een prioriteitsregeling op de gemiddelde wachttijd",
abstract = "The effect of priorities on the average waitingtime. A waitingtime process is described in terms of customers arriving at a single counter. In the first part of the article the formulae of A. C obham [1] and [2] for the case of two priorities are applied to the following situation: customers arrive in such a way that the interarrival times s are mutually independent and exponentially distributed with mean ¿-1. Servicetimes s are mutually independent and independent of the interarrival times. The distributionfunction of s is indicated by B (s), the mean of s by µ. The customers are divided in two groups, group 1 consisting of the customers having servicetimes cµ, group 2 of those having servicetimes > cµ, c being a positive number. Servicing takes place as follows: priority is given to customers of group 1 i.e. service of a customer of group 2 starts only if no customers of group 1 are present, customers of the same group are served in the order of their arrival, servicing of a customer is not interrupted. Keeping ¿µ <1 the mean waitingtime e¿* for an arbitrary customer is then calculated by means of Cobham's formulae and compared with the mean waitingtime e¿, in case that no priorities are given. Under hypothesis of differentiability of B (s>is the (unique) value of c which minimizes e¿*. It is shown that always c*> 1. In the second part a situation is considered which has been treated by T. E. Phipps [7]. In this situation the customer having the smallest servicetime of all customers present when the counter becomes free is served first. The mean waitingtime e¿** in this case is compared with e¿ and e¿*. Some examples and graphs are given.",
author = "F.W. Steutel",
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De invloed van een prioriteitsregeling op de gemiddelde wachttijd. / Steutel, F.W.

In: Statistica Neerlandica, Vol. 13, No. 4, 1959, p. 503-513.

Research output: Contribution to journalArticleAcademicpeer-review

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T1 - De invloed van een prioriteitsregeling op de gemiddelde wachttijd

AU - Steutel,F.W.

PY - 1959

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AB - The effect of priorities on the average waitingtime. A waitingtime process is described in terms of customers arriving at a single counter. In the first part of the article the formulae of A. C obham [1] and [2] for the case of two priorities are applied to the following situation: customers arrive in such a way that the interarrival times s are mutually independent and exponentially distributed with mean ¿-1. Servicetimes s are mutually independent and independent of the interarrival times. The distributionfunction of s is indicated by B (s), the mean of s by µ. The customers are divided in two groups, group 1 consisting of the customers having servicetimes cµ, group 2 of those having servicetimes > cµ, c being a positive number. Servicing takes place as follows: priority is given to customers of group 1 i.e. service of a customer of group 2 starts only if no customers of group 1 are present, customers of the same group are served in the order of their arrival, servicing of a customer is not interrupted. Keeping ¿µ <1 the mean waitingtime e¿* for an arbitrary customer is then calculated by means of Cobham's formulae and compared with the mean waitingtime e¿, in case that no priorities are given. Under hypothesis of differentiability of B (s>is the (unique) value of c which minimizes e¿*. It is shown that always c*> 1. In the second part a situation is considered which has been treated by T. E. Phipps [7]. In this situation the customer having the smallest servicetime of all customers present when the counter becomes free is served first. The mean waitingtime e¿** in this case is compared with e¿ and e¿*. Some examples and graphs are given.

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