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De invloed van een prioriteitsregeling op de gemiddelde wachttijd *
Author(s) -
Steutel F. W.
Publication year - 1959
Publication title -
statistica neerlandica
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.52
H-Index - 39
eISSN - 1467-9574
pISSN - 0039-0402
DOI - 10.1111/j.1467-9574.1959.tb01025.x
Subject(s) - mathematics , group (periodic table) , order (exchange) , customer service , combinatorics , mean value , statistics , service (business) , business , marketing , physics , finance , quantum mechanics
Summary 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 λ ‐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 λμ < 1 the mean waitingtime εω* for an arbitrary customer is then calculated by means of Cobham's formulae and compared with the mean waitingtime εω, in case that no priorities are given. Under hypothesis of differentiability of B (s> is the (unique) value of c which minimizes εω*. 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 εω** in this case is compared with εω and εω*. Some examples and graphs are given.