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\title{Interval and Fuzzy Approaches to Queuing Systems} 
\author{Huang Chongfu}


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\markright{APIC'95, El Paso,
Extended Abstracts,
A Supplement to the international journal of {\rm Reliable
Computing}\ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ }

\maketitle

\auffil{The author is with the Management School, 
Beijing University of Aeronautics and Astronautics,
Beijing 100083, 
China.}

\section{Queuing Models Describe Many Real-Life Phenomena}
Queuing models adequately describe many real-life systems:
\begin{itemize}
\item {\it communication networks};
\item {\it management of the engineering team projects}; 
\item {\it transportation systems};
\item {\it flexible
 manufacturing systems};
\item {\it service systems}; 
\item etc.
\end{itemize}

\section{In Many Real-Life Cases, Queuing Models Lead To Explicit Or
Implicit Formulas}
In many real-life cases, we can apply queuing models (e.g.,
M/M/1) that are {\it simple} in the following sense: 
\begin{itemize}
\item either, we 
have an {\it explicit} formula $$y=f(x_1,...,x_n)$$ 
that describes the desired characteristics
$y$ of
the queue as a function of the model's parameters $x_1,...,x_n$, or
\item we have an {\it implicit} equation $$F(y,x_1,...,x_n)=0$$ from which, if
we know $x_i$, we can easily compute $y$.
\end{itemize}

\section{Parameters Of A Queuing Model Are Not Known Precisely}
In real life, the value $x_i$ of the parameters of the queuing model 
must be determined
from the experimental data. Therefore, these parameters are never
absolutely precise: instead of the actual values of these parameters,
we know only the {\it intervals} ${\bf x}_i=[\tilde
x_i-\Delta_i,\tilde x_i+\Delta_i]$ of possible values of
$x_i$, where $\tilde x_i$ is an estimate, and $\Delta_i$ is an
accuracy of this estimate. 

Traditionally, as an estimate $\tilde y$ 
for the desired characteristic $y$, we
take correspondingly either the value 
$\tilde y=f(\tilde x_1,...,\tilde x_n)$, or the solution of the
equation $F(\tilde y,\tilde x_1,...,\tilde x_n)=0$. 

\section{As A Result Of This Uncertainty, Estimated Values Of The
Desired Characteristics
Of The System May Differ From Their Actual Values}
Our numerical experiments 
show that if we are determining the parameters $x_i$ from
a small sample, then the values $\Delta_i$ may be large, and the
resulting deviation of $\tilde y$ from the actual value of $y$ may
also be large. 

\section{Interval Approach To Queuing Models}
In view of the above-mentioned difference, 
it is better, instead of an estimate $\tilde y$, to provide the
user with an {\it interval} $\bf y$ of possible values of $y$:
\begin{itemize}
\item For an {\it explicit} model, this interval can be computed by applying
interval computations as 
${\bf y}=f({\bf x}_1,...,{\bf x}_n)$.
\item For an {\it implicit} model, as $\bf y$, we take an 
enclosure of the set of all
solutions of the equation $F(y,{\bf x}_1,...,{\bf x}_n)=0$. 
\end{itemize}
We have applied these methods to several real-life systems.

\section{Fuzzy Approach}
In some cases, in addition to the interval ${\bf x}_i$, the experts
can supply us with the {\it degrees of possibility} $d_i(x_i)$ of different
values $x_i$ from this interval. As a result, 
we have a {\it fuzzy membership} function. In this case, in addition
to the interval $\bf y$, we can also compute the degrees of belief
$d(y)$ of different values of $y$:

Namely, for every level $\alpha\in
[0,1]$, we can do the following:
\begin{itemize}
\item for every $i$, we can find the interval ${\bf
x}_i(\alpha)$ that contains only those values $x_i\in {\bf x}_i$ that
are possible with degree of possibility $\ge\alpha$ (i.e., for which
$d_i(x_i)\ge\alpha$).
\item apply the above-described interval techniques 
to the resulting intervals ${\bf x}_i(\alpha)$, and thus find 
the interval ${\bf y}(\alpha)$ of values $y$ that are possible with
this degree of possibility.
\end{itemize}
We have designed a software for these computations.

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\end{thebibliography}


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