ORIGINAL PAPER Monte Carlo methods in fuzzy queuing theory Areeg Abdalla Æ James J. Buckley Published online: 25 October 2008 Ó Springer-Verlag 2008 Abstract In this paper, we consider an optimization problem in fuzzy queuing theory that was first used in web planning. This fuzzy optimization problem has no solution algorithm and approximate solutions were first produced by computing the fuzzy value of the objective function for only sixteen values of the fuzzy variables. We introduce our fuzzy Monte Carlo method, using a quasi-random number gener- ator, to produce 100,000 random sequences of fuzzy vectors for the fuzzy variables, which will present a much better approximate solution. Keywords Fuzzy queuing theory  Monte Carlo  Random fuzzy vectors 1 Introduction Let us first introduce the notation we will be using in this paper. We place a ‘‘bar’’ over a symbol to denote a fuzzy set. All our fuzzy sets will be fuzzy subsets of the real numbers so  U;  V ;  A; ... are all fuzzy subsets of the real numbers. We write  AðxÞ; a number in [0, 1], for the membership function of  A evaluated at x. An a-cut of  A; written as  A½a; is defined as fxj  AðxÞ ag; for 0 \ a B 1. We separately specify  A½0 as the closure of the union of all the  A½a; 0\a  1: A triangular fuzzy number  V is defined by three num- bers a \ b \ c where the base of the triangle is on the interval [a, c] and  V ðxÞ¼ 1 for x = b. The sides of  V are straight line segments. We write  V ¼ða=b=cÞ for trian- gular fuzzy number  V : A triangular shaped fuzzy number  U has base on an interval ½a; c;  UðbÞ¼ 1; but the sides are curves not straight lines. We write  U ða=b=cÞ for a tri- angular shaped fuzzy number  U: For triangular (shaped) fuzzy numbers the support is the interval [a, c]. Alpha-cuts of fuzzy numbers are always closed, bounded, intervals so we write this as  V ½a¼½v 1 ðaÞ; v 2 ðaÞ; for 0 B a B 1.  V ð  UÞ is non-negative if a C 0. All the fuzzy numbers used in this paper, except fuzzy profit starting in Sect. 3, will be non-negative. We will program our fuzzy Monte Carlo method in MATLAB (2006)(http://www.mathworks.com). This paper is based on, and expanded from, Chapters 11, 12 and 14 of Buckley (2004) which is about using fuzzy probabilities and fuzzy sets in web site planning. All the needed information needed from Buckley (2004) is inclu- ded in this paper so the reader does not need to obtain a copy of Buckley (2004). The queuing network considered in this paper is within a web site. For other papers/chapters in books, on this topic of fuzzy queuing theory, we refer the reader to Buckley (1990, 2003), Buckley et al. (2001, 2002) and the references in these papers/books. In the next section, we discuss the crisp queuing optimization problem and then we fuzzify the optimization problem in the Sect. 3. In Sect. 4, we present our fuzzy Monte Carlo method and how we will generate sequences of random fuzzy vectors. Our fuzzy Monte Carlo solution to the fuzzy queuing optimization problem is Sect. 5 and the last sec- tion has a summary and our conclusions. 2 Queuing model We will model the queuing system using the arrival rate k and the service rate l for any server. This is a common A. Abdalla  J. J. Buckley (&) Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294, USA e-mail: buckley@math.uab.edu A. Abdalla e-mail: abdalla@math.uab.edu 123 Soft Comput (2009) 13:1027–1033 DOI 10.1007/s00500-008-0376-y