Optimization of Queuing Theory Based on Vague Environment

Optimization of Queuing Theory Based on Vague Environment

Verónica Andrea González-López (Department of Statistics, University of Campinas, Campinas, Brazil), Ramin Gholizadeh (Department of Statistics, University of Campinas, Campinas, Brazil) and Aliakbar M. Shirazi (Department of Statistics, University of Campinas, Campinas, Brazil)
Copyright: © 2016 |Pages: 26
DOI: 10.4018/IJFSA.2016010101


Waiting lines or queues are commonly occurred both in everyday life and in a variety of business and industrial situations. The various arrival rates, service rates and processing times of jobs/tasks usually assumed are exact. However, the real world is complex and the complexity is due to the uncertainty. The queuing theory by using vague environment is described in this paper. To illustrate, the approach analytical results for M/M/1/8 and M/M/s/8 systems are presented. It optimizes queuing models such that the arrival rate and service rate are vague numbers. This paper results a new approach for queuing models in the vague environment that it can be more effective than deterministic queuing models. A numerical example is illustrated to check the validity of the proposed method.
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1. Introduction

In everyday life we experience a lot of irritating instances of queues, a phenomenon which is more associated with urban and dense communities. Examples include visible lines in traffic jams, airport check-in desks and supermarkets as well as invisible queues such as voices calls and wireless channels. Queuing theory, therefore, has a long history and has been used to solve practical problems in manufacturing, communication and many other fields. This theory was initiated by the Danish engineer, Erlang (1909) when he was studying the behavior of telephone networks. Jacksons (1954) studied the behavior of a queue system containing phase type service. Reich (1957) discussed the waiting times when queues are in tandem.

In the real world there are vaguely specified data values in many applications such as sensor information. Fuzzy set theory has been proposed to handle such vagueness by generalizing the notion of membership in a set. Essentially, in a fuzzy set each element is associated with a point value that is selected from the unit interval [0,1], and is termed the grade of membership in the set. A vague set, as well as Intuitionistic fuzzy set are further generalizations of a fuzzy set. Instead of using point-based membership in fuzzy sets interval-based membership is used in a vague set.

Gau and Buehrer (1993) define vague sets and Chen and Tan (1994) discussed about handling multicriteria fuzzy decision-making problems based on vague set theory. Chen (2003) presented a method to analyze system reliability using the vague set theory where the reliabilities of system components are represented by vague sets. Taheri et al. (2011) developed a method for analyzing system reliability by using Bayesian under the vague environment and Gholizadeh et al. (2012) studied system reliability and availability using vague sets. Lu et al. (2005) made a more detailed comparison between vague sets and intuitionistic Fuzzy Sets from various perspectives such as algebraic properties, graphical representations and practical applications.

Bellman and Zadeh (1970) and Zadeh (1978) introduced the concept of fuzziness so that imprecise information could be handled in decision making problems, and fuzzy set theory is often recurred to the situations where imprecise and uncertainty needs to be modelling. Fuzzy queuing models have been described by researchers such as Prade (1980), Li and Lee (1989), Buckley (1990), Negi and Lee (1992), Jo et al. (1997), Kao et al. (1999), Buckley et al. (2001), Chen (2005, 2006), Ke and Lin (2006) and Brak et al. (2012).

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