Download A MEMBERSHIP FUNCTION SOLUTION APPROACH TO FUZZY QUEUE WITH ERLANG SERVICE MODEL Author: V.Ashok Kumar

International Journal of Mathematical Sciences and Applications
Vol. 1 No. 2 (May, 2011)
Copyright © Mind Reader Publications
www.ijmsa.yolasite.com
A MEMBERSHIP FUNCTION SOLUTION APPROACH TO FUZZY QUEUE
WITH ERLANG SERVICE MODEL
V.Ashok Kumar**
** Department of Mathematics, Shinas College of Technology, Oman.
E-mail: [email protected]
ABSTRACT
This paper develops a non-linear programming approach to derive the membership
functions of the steady-state performance measures in Erlang service model where the arrival rate and
service rate are fuzzy numbers. The basic idea is based on Zadeh’s extension principle. Two pairs of
mixed integer Non-linear programs with binary variables are formulated to calculate the upper and
lower bonds of the system performance at possibility level . Using -cut approach FM/FEK/I fuzzy
queue can be reduced to a family of M/EK/I queue with different  cuts. Trapezoidal fuzzy numbers
are used to demonstrate the validity of the proposal. Numerical examples are solved successfully.
Keywords: Erlang service, Fuzzy number, Queue, Mixed Integer non-linear
Programming.
1.INTRODUCTION
Till now all probability queuing models studied have assumed poisson input and
exponential service times. In many practical situations, the exponential assumptions may be
rather limiting, especially the assumption concerning service times being distributed
exponentially.
Most of the related studies are based on traditional queuing theory, in that the
inter arrival times and service times are assumed to follow certain probability distribution.
However, in practice there are cases that these parameters may be obtained subjectively [5].
The fuzzy queues are much more realistic than the commonly used crisp queues [1,2,5].
Clearly when the arrival rate and service rate are fuzzy the performance measure
of the queue also is fuzzy as well. The basic idea is to apply Zadeh’s extension principle
[7,8,9]. Two pairs of mixed integer non-linear programming models are formulated to
calculate the lower and upper bounds of the -cut of the system performance measure. The
membership function of the system performance measure is derived analytically.
2. TRAPEZOIDAL FUZZY NUMBER
~
The trapezoidal fuzzy number is usually defined as A = [a2–d1, a2, a3, a3+d2]
~
The membership function for the Trapezoidal fuzzy number A = [a1, a2, a3, a4] is defined as
μ A~(S)
 (S – a 1 )/(a 2 – a 1 ) a 1  S  a 2

1
a2  S  a3
= 
(a – S)/(a – a ) a  S  a
4
3
3
4
 4
 A(x)
Core
Boundary
O
a1
Boundary
a2
a3
X
a4
Support
3. FUZZY QUEUE WITH ERLANG SERVICE
Consider a queuing system in which the customer arrive at a single-server facility
~
~
with arrival rate λ and service rate μ
~
~ is fuzzy in nature Erlang service
Where λ is fuzzy in nature poisson rate and μ
rate made up of K exponential phases. Customers are served according to a first-come-firstserved (FCFS) discipline and both the size of calling population and the system capacity are
infinite.
The parameters of the Erlang are K and  and the mean and variance are given by
E(x) 
1
1
and V(x) 
μ
Kμ 2
The relation of the Erlang to the exponential distributions allow us to describe the
queuing models where the service (or arrivals) may consist of series of identical phases. For
example, in performing a laboratory text, the lab technician must perform k steps, each taking
the same mean time (sas 1/k), with the times distributed exponentially. This is represented
in figure 1. The overall service function is Erlang type k with mean 1/. If the input process
is poisson, we would have M/EK/1 model.
883
Step 1 Step 2
Step K
1/K
1/K
1/K
Figure 1: Use of Erlang for Phased Service
~
~ are approximately
In this model the group arrival rate λ and service rate μ
~
known and can be represented by convex fuzzy sets. Note that a fuzzy set A in its universal
set Z is convex if μ A~ [φ z1  (1 – φ)z 2 ]  min [μ A~ (z1 ), μ A~ (z 2 )] Where μ A~ is its membership
function   [0,1] and z1, z2Z.
Let μ ~λ (x) and μ μ~ (y) are membership functions of arrival rate and service rate




~
~
~)
~ = y, μ ~ (y) y  S(μ
respectively. We have λ = x, μ ~λ (x) x  S( λ ) and μ
μ
~
~) are the supports of ~
~ which denote the universal set of arrival
where S( λ) and S(μ
λ and μ
rate and service rate respectively.
~
~~
~ are fuzzy number then the performance measure ~
Clearly when λ and μ
ρ (λ, μ
)
are also fuzzy as well. On the basis of Zadeh’s extension principle [7,8,9], the membership
function of the performance measure is defined as
~ ~ ~ (z) =
μ
~)
ρ (λ,μ


sup
min μ ~λ (x), μ μ~ (y) z  ρ(x, y)
x  X, y  Y
Without loss of generality let us assume that the performance measure of interest
is the expected number of customers in queue (Lq). From the knowledge of traditional
queuing theory [4], under the study-state conditions  = x/y<1, the expected number of
 (K  1)K (K  1)λ  1
~
customers in the queue Lq  
and membership function of Lq is
–

2μ  K
 2(μ – x)
~ ~ (z) =
μ
Lq

min μ ~λ (x), μ μ~ (y) z 
x  X, y  Y, z  Z

sup
 (K  1)λ (K  1)λ  1 
 2(μ – λ) – 2μ  K  ........(1)
 

By using Little’s formula [1] in the same manner, the waiting time in queue
Wq 
K 1
λ
~
and the membership function of Wq is
2K μ(μ – λ)
~ (z) =
μW
q

K 1
λ 
min μ ~λ (x), μ μ~ (y) z  
–

x  X, y  Y, x  Y
μ(μ – λ)  
 2K

sup
Similarly W = Wq +
.......(2)
1
and L = W
μ
4. THE SOLUTION PROCEDURE:
One approach to construct the membership function of μ ~ρ(~λ,μ~) is on the basis of
~
~ as
deriving -cuts of μ ~ρ(~λ,μ~) . Denote -cuts of λ and μ



max
 min
 = x αL , x αU = 
x μ ~λ (x)  α ,
x, μ ~λ (x)  α
xX
x X
=
y
L
U
α , yα

=  ymin
y μ
Y


~ (y)
λ



 ymax

y μ μ~ (y)  α
Y
α ,

.......(3)
.......(4)
These intervals indicate where the group arrival rate and service rate lie at possibility level .
Consequently, the FM/FEK/I queue can be reduced to a family of crisp M/EK/I queens with
different -level sets {/ 0 < 1}.
By the convexity of a fuzzy number [9], the bands of these intervals are functions
of  and can be obtained as
x αL = min μ ~–1 ( α) and x αU = max μ ~–1 ( α) ,
λ
λ
y αL = min μ μ~–1 ( α) and y αU = max μ μ~–1 ( α) respectively
Consider the membership function of the expected number of customers in the queue Lq.
According to (1), μ L~q ( α) is the minimum of μ ~λ ( x) and μ μ~ ( y) . We need either μ ~λ ( x)  α
and μ μ~ ( y)  α (or) μ ~λ ( x)  α and μ μ~ ( y)  α such that
 (K  1)λ (K  1)λ  1

to satisfy μ L~q (z)  α . To find the membership function μ L~q (z)
z  
–
2μ  K
 2(μ – λ)
we have to find the lower bound z αL and the upper bound z αU of the -cuts of μ L~q (z) . Since
the requirement of μ ~λ (x)  α can be represented by x  x αL
(or) x  x αU this can be
885
formulated as the constraint of x  β1 x αL  (1 – β1 )x αU , where 1=0 (or) 1. Similarly
μ μ~ ( y)  α can be formulated as the constraint y  β 2 y αL  (1 – β 2 )y αU , where 2=0 (or) 1.
Moreover from the definition of (3) and (4) x and y can be respectively

replaced by x  x αL , x dU
 and

y  y αL , y dU
 consequently, considering both of these two
cases above, the membership function μ L~q can be constructed via finding the lower bound
(Lq) αL and upper bound (Lq) αU .

L
L


U
U

We set (Lq) αL = min (Lq) α 1 , (Lq) α 2 , and (Lq) αU =max (Lq) α 1 , (Lq) α 2 , respectively
where
L
(Lq) α 1
=
 K 1 λ
(K  1)  1
x  y
λ 
–
x, y  R
2μ
K
 2 (μ – λ)
min
....... (5)
such that x = t 1 x αL  (1 – t 1 )x αU , y αL  y  y αU and t1 = 0 or 1
L
(Lq) α 2 =
 K 1 λ
(K  1)  1
x  y
λ 
–
x, y  R
2μ
K
 2 μ–λ
min
....... (6)
such that y = t 2 y αL  (1 – t 2 )y αU , x αL  x  x αU and t2 = 0 or 1
U
(Lq) α 1 =
max
 K 1 λ
(K  1)  1
λ 
x  y
–
x, y  R
2μ
K
 2 (μ – λ)
....... (7)
such that x= t 3 x αL  (1 – t 3 )x αU , y αL  y  y U and t3 = 0 or 1
U
(Lq) α 2
=
max
 K 1 λ
(K  1)  1
λ 
x  y
–
x, y  R
2μ
K
 2 μ–λ
....... (8)
such that y = t 4 y αL  (1 – t 4 )y αU , x αL  x  x αU and t4 = 0 or 1
where x αL  y αL . From the knowledge of calculus, a unique minimum and a unique maximum
of the objective function of models (5), (6), (7) and (8) are assumed, which shows that the
lower bound (Lq) αL and upper bound (Lq) αU of the -cuts of Lq can be found by solving
these four models.
~
According to Zadeh’s extension principle, Lq defined in (1) is a Fuzzy number
that possesses convexity [6,9]. Therefore for two values of 1 and 2 such that 0<2<1 1,
we have (Lq) αL  (Lq) αL and (Lq) αU  (Lq) αU . In other words (Lq) αL is non-decreasing with
1
1
2
2
respect to  and (Lq) αU is non-decreasing with respect to . This property assures the
~
convexity of Lq . Consequently, the membership function μ L~q (z) can be obtained from the
solutions of models (5), (6), (7) and (8).
If both (Lq) αL and (Lq) αU are invertible with respect to , then a left shape

function L S (z)  (Lq) αL

–1
a right shape function can be obtained. From LS(z) and RS(z) the
membership function μ L~q is constructed as
 L S (z), (Lq) αL  0  z  (Lq) αL  

μ L~q (z) =  1,
(Lq) αL  1  z  (Lq) αU  1
R (z), (Lq) U  1  z  (Lq) U  0
α
α
 S
…….(9)
Since the above performance measures are described by membership functions,
they conserve completely all fuzziness of arrival rate and service rate.
5.NUMERAL EXAMPLE
Consider a centralized parallel processing system in which the service consists of
two phases. Both the arrival rate and service rate are trapezoidal fuzzy numbers represented
~
~ = [13,14,15,16] per minute, respectively. The system manager wants
by λ = [2,3,4,5] and μ
to evaluate the performance measures of the system such as the expected number of
customers in the queue.
The confidence interval at  are [2+, 5–] & [13+, 16–]
~
(Lq) αL
~
(Lq) αL is invertible
=
3α 2  12 α  12
4α 2 – 92 α  448
.................. (10 a)
887

=
0,
(92z+12)± 36 z 2  6 z  0
0
384 10752
,
14336 14336
z
=
z
= .027 or 0.75
(92z  12)  36 z 2  6z
1
2(4z – 3)
  
z
(92z  12)  36 z 2  6z
2(4z – 3)
=
240 24
,
320 320
= .75 or .075
~
(Lq) αL
=
~
(Lq) αU =
(92z  12)  36 z 2  6z
2(4z – 3)
3α 2 – 30α  75
8α 2  136α  104)
~
(Lq) αU is invertible

– (136z  30)  72 z 2  3z
=
2(8z – 3)
 0,
– (136z  30)  72 z 2  3z  0
 0,
z = .375 or .1802
1,
– (136z  30)  72 z 2  3z
1
2(8z – 3)
1,
z =.375 or .086
.027 z 
............ (10 b)
– (136z  36)  72 z 2  3z
~
(Lq) αU =
2(8z – 3)
.086 z .375
From the inverse function of (Lq) αL and (Lq) αU the membership function of
Lq is defined as follows:
 (92z  12) – 36 z 2  6z

2(4z – 3)

~
1
 μ Lq (z) = 
 – (136z  30)  72 z 2  3z

2(8z – 3)

Wq
=
Wq L =
z =
.075  z  .086
.086  z  .375
K 1
λ
.
2K μ(μ – λ)
3
λ
.
4 μ(μ – λ)
3α  6
2
8α – 184α – 896
=
(184z  3)  5184z 2  1296z  9
16z
0,
(184z  3)  5184z 2  1296z  9 0
z=
.027  z  .075
192
&z=0
28672
z = .0066 and z = 0
1,
(184z  3)  5184z 2  1296z  9
1
16z
z = 0 and z = 0.0125
(184z  3)  5184z 2  1296z  9
~L
Wq =
16z
.0066  z  0.0125
889
WqU =
=
(5 – α)
3
4 (13  α)[(13  α) – (5 – α)]
(15 – 3 α)
8α 2  136 α  416
WqU is invertible
– (136z  3)  5184z 2  1296 z  9
 =
16z
0, – (136z  3)  5184z 2  1296 z  9 
z = 0 and z = 0.0361
  1,
– (136z  3)  5184z 2  1296 z  9

16 z
z = 0 and z = 0.0214
 (184z  3)–


~ (z) =  1
μW
q

 –(136z  3) 

5184z 2  1296z  9
16z
.0066  z  .0125
.0125  z  .0214
5184z 2  1296z  9
16z
.0214  z  .036
α – cuts of arrival rate, service rate, queue length and waiting time in queue

x αL
x αU
y αL
y αU
(Lq) αL
(Lq) αU
(Wq) αL
(Wq) αU
0.0
2.0
5.0
13.0
16.0
.0268
.1803
.0067
.0361
0.1
2.1
4.9
13.1
15.9
.0301
.1676
.0072
.0342
0.2
2.2
4.8
13.2
15.8
.0338
.1558
.0077
.0325
0.3
2.3
4.7
13.3
15.7
.0378
.1448
.0082
.0305
0.4
2.4
4.6
13.4
15.6
.04196
.1345
.0087
.0293
0.5
2.5
4.5
13.5
15.5
.0465
.1250
.0093
.0278
0.6
2.6
4.4
13.6
15.4
.0514
.1101
.0099
.0264
0.7
2.7
4.3
13.7
15.3
.0567
.1077
.0105
.0250
0.8
2.8
4.2
13.8
15.2
.0624
.0999
.0111
.0238
0.9
2.9
4.1
13.9
15.1
.0685
.0926
.0119
.0226
1.0
3.0
4.0
14.0
15.0
.0750
.1161
.0125
.0214
Performance measure of Lq
Performance measure of Wq
891
6. CONCLUSION
The Erlang family of probability distributions provides much more modeling
flexibility than the exponential. In front, the exponential is a special Erlang, namely, type 1.
In many practical situations where observed data might not bear out the exponential
distribution assumption, the Erlang can provide greater flexibility by being better able to
represent the real world. The other reason why Erlang is useful in queuing analysis is its
relation to the exponential distribution with the Markorian property.
REFERENCE
1.
J.J. Buckley, Elementary Queuing Theory based on Possibility Theory, Fuzzy Set
System 37 (1990), 43-52.
2.
J.J. Buckley, T. Feuring, Y. Hasaki, Fuzzy Queuing Theory Resisted, Int, J.
Uncertain Fuzzy 9 (2001), 527-537.
3.
P. Fortemps, M. Roubens, Ranking and Defuzzification Method based on area
Compensation, Fuzzy Sets System 82 (1996), 319-330.
4.
D. Gross, C.M. Haris, Fundamentals of Queuing Theory, Third Ed., Wiley, New York,
1998.
5.
J.B. Jo, Y. Tsujimura, M.Gon, G. Yamazaki, Performance Evaluation of Network
Models based on Fuzzy Queuing System, JPn. J. Fuzzy Theory System 8 (1996), 393408.
6.
A. Karfmann, Introduction to the Theory of Fuzzy Subsets, Vol. I, Academic Press,
New York, 1975. 6
7.
R.R. Yovgav, A Characterisation of the Extension Principle, Fuzzy Sets and Systems
18 (1986), 205-217. 7
8.
L.A. Zardeh, Fuzzy Sets as a Basis for a Theory of Possibility, Fuzzy sets and Systems
1 (1978), 3-28.
9.
H.J. Zimmarmann, Fuzzy Set Theory and its Applications, Fourth ed., Kluwar
Academic, Boston, 2001.