Global Journal of Pure and Applied Mathematics (GJPAM) ISSN 0973-1768 Volume 11,Number 1 (2015)
© Research India Publications ::: http://www.ripublication.com
MULTI-SERVER FUZZY QUEUEING
MODEL USING DSW ALGORITHM
S. Shanmugasundaram
Assistant Professor,
B.Venkatesh
Assistant Professor,
Department of Mathematics,
Government Arts College, Salem-7, India
E-mail: Sundaramsss@hotmail.com .
Department of Mathematics
Sona College of Technology, Salem-5, India
E-mail: venkatbvs11@gmail.com
Abstract- In this paper we study DSW algorithm in
II. DESCRIPTION OF THE SYSTEM
fuzzy set for multiserver queueing system. DSW
algorithm is one of the approximate methods based on
the α-cut representation of fuzzy sets in a standard
interval analysis. Numerical example is also given.
We consider a traditional queueing system with
multi server(C), calling population is infinite and queue
discipline is first in first out.
i.e) ( M/M/C :
KEYWORDS: interarrival time, service time,
membership function, fuzzy number, DSW algotithm ,
standard interval analysis.
/FIFO ).
Arrival rate and service rate are fuzzy numbers denoted
by
I . INTRODUCTION

and
 .
The inter arrival time (A) & service
times (S) are represented by the following fuzzy sets.
The aim of all investigations in
queueing theory is to get the main performance measures
of the system: number of customers in the system,
average waiting time in system, etc.
 A
 S
In traditional queueing theory the inter arrival
times and service times are required to follow certain
distributions. In real life, the arrival rate and service rate
are represented by linguistic terms such as high, low or
moderate can be best described by fuzzy sets.
Where X and Y are the crisp universal sets of the inter
arrival time and service time.
The membership functions of A and S are
Fuzzy queueing model have been developed by
researchers Li and Lee[2 ], Chen[4] using Zadeh’s
extension principle[1]. Traditional queueing model will
be more realistic if it is converted into fuzzy queueing
model.
 A
 S
Where
and
are crisp sets using -cuts. Using
different levels of confidence intervals , the interarrival
time and service time are represented.
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Global Journal of Pure and Applied Mathematics (GJPAM) ISSN 0973-1768 Volume 11,Number 1 (2015)
© Research India Publications ::: http://www.ripublication.com
The membership function P(A,S) is constructed.
III. INTERVAL ANALYSIS ARITHMETIC
Let I1 and I2 be two interval numbers defined by ordered
pairs of real numbers with lower
and upper bounds.
I1 = [a,b] , a
where z1
L(z1 )
I2 = [c,d] , c
z 2 z 3 z4 and
R(z4 ) = 0.
Define a general arithmetic property with the symbol *,
An approximate method of extension is propagating
where * = [+, -, ×, ÷] symbolically
fuzziness for continuous valued mapping determined the
the operation.
membership functions for the output variables.
I1 * I2= [a,b] * [c,d]
represents another interval. The interval calculation
II. a. THE (FM/FM/C) : ( /FIFO) QUEUE
In this model we consider the
depends on the magnitudes and signs of the element a, b,
first-in first-out
c, d.
discipline and consider an infinite source population. The
[a, b] + [c, d] = [a + c, b + d]
inter arrival time and the service time follow exponential
distributions .
[a, b] − [c, d] = [a − d, b − c]
The expected number of customers in the system
[a, b] . [c, d]
[a, b] ÷ [c, d]
= [min (ac, ad, bc, bd) ,
max (ac, ad, bc, bd ) ]
=[a,b].
 [c, d]
The average waiting time in the system
The expected number of customers in the queue
IV. DSW ALGORITHM
DSW(Dong, Shah and Wong) is one of the approximate
methods make use of intervals at various α - cut levels in
defining membership functions. It uses full α-cut intervals
The average waiting time of a customer in the queue
in a standard interval analysis.
The DSW algorithm simplifies manipulation of the
extension principle for continuous valued fuzzy variables,
Where
such as fuzzy numbers defined on the real line.
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Global Journal of Pure and Applied Mathematics (GJPAM) ISSN 0973-1768 Volume 11,Number 1 (2015)
© Research India Publications ::: http://www.ripublication.com
Any continuous membership function can be
represented by a continuous sweep of α-cut interm from
α = 0 to
α = 1. Suppose we have single input mapping
given by
y = f (x) that is to be extended for
membership function
for the selected α cut level.
The DSW algorithm [3] consists of the following steps:
1. Selected a α cut value where 0 ≤ α ≤ 1.
2. Find the intervals in the input membership functions
that correspond to this α.
3. Using standard binary interval operations ,compute the
interval for the output membership function for the
selected α- cut level.
4. Repeat steps 1 -3 for different values of α to complete
a α- cut representation of the solution.
Where
V. NUMERICAL EXAMPLE
Where x = [11+α , 14–α] &
Consider a FM/FM/C queue where both arrival rate and
y = [7+α, 10–α]
service rate are fuzzy numbers represented by

= [11 12 13 14] ,
 = [ 7
8 9 10] and C=3.
The interval of confidence at possibility level α as
[11+α , 14–α] and [7+α, 10–α].
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Global Journal of Pure and Applied Mathematics (GJPAM) ISSN 0973-1768 Volume 11,Number 1 (2015)
© Research India Publications ::: http://www.ripublication.com
TABLE: The α-cuts of Ls , Lq,, Ws, Wq at α values
.
Fig:1 Ls
Fig:2 Lq
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Global Journal of Pure and Applied Mathematics (GJPAM) ISSN 0973-1768 Volume 11,Number 1 (2015)
© Research India Publications ::: http://www.ripublication.com
Fig:3
Ws
Fig:4
Wq
Using MATLAB we perform α – cuts of arrival rate
waiting time in the system Ls falls between 1.4367
and service rate and fuzzy expected number of jobs in
and2.0938, and it will never fall below 1.1225 or
queue at eleven distinct α levels: 0, 0.1, 0.2, 0.3, …1.
exceed 4.6184. The above information will be very
Crisp intervals for fuzzy expected number of jobs in
useful for designing a queueing system.
queue at different possibilistic α levels are presented
.
in table. The performance measures such as expected
CONCLUSION
number of jobs in the system (Ls), expected length of
queue (Lq), expected waiting time of job in queue
Fuzzy set theory has been applied in many fields
(Wq) and expected waiting time of job in the system
particularly in queueing system, it provide broader
(Ws) also derived in table.
application in many fields. When the inter arrival
time and service time are fuzzy variables, according
The α – cut represent the possibility that these four
DSW
performance measure will lie in the associated range.
algorithm, the performance measures such as the
Specially, α = 0 the range, the performance measures
average system length, the average waiting time, etc.,
could appear and for
α = 1 the range,the
will be fuzzy. In this numerical example, we
performance measures are likely to be. For example,
illustrate that the values of Ls ,Lq, Ws, and Wq in the
while these four performance measures are fuzzy, the
interval [1.4367 , 2.0938],
most likely value ofthe expected queue length Lq
[0.1197,0.1611], [0.0086, 0.0361]
falls between 0.1033 and 0.4688 and its value is
respectively.
impossible to fall outside the range of 0.0225 and
efficiency of the DSW algorithm.
2.6184; it isdefinitely possible that the expected
49
Numerical
[0.1033,0.4688 ],
example
are stationary
shows
the
Global Journal of Pure and Applied Mathematics (GJPAM) ISSN 0973-1768 Volume 11,Number 1 (2015)
© Research India Publications ::: http://www.ripublication.com
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Global Journal of Pure and Applied Mathematics (GJPAM) ISSN 0973-1768 Volume 11,Number 1 (2015)
© Research India Publications ::: http://www.ripublication.com
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