Fuzzy Queueing Network Models of Computing ...

Fuzzy Queueing Network Models of Computing Systems Johannes L uthi and G unter Haring

Abteilung Advanced Computer Engineering Institut fur Angewandte Informatik und Informationssysteme, Universitat Wien Lenaugasse 2/8, A-1080 Wien, Austria Internet: fluethi,[email protected], http://www.ani.univie.ac.at

Abstract Performance engineering of computing systems (software as well as hardware systems) which integrates performance modeling with the various phases of design and implementation has become an important and popular issue. However, especially in early phases of design and development, exact values for all model parameters are often unknown, leading to uncertainties in the model parametrization. For example, the analyst may have to construct a model based on information such as \the mean service demand at device A will be about 30ms". Considering uncertainties in performance modeling and evaluation of computer and communication systems has been recognized to be of signicant importance. A popular mathematical approach with a sophisticated theoretical background is the use of fuzzy numbers to model systems characterized by uncertain parameters. A fuzzy number is represented by a set of real numbers and an associated membership function. Based on techniques used in interval arithmetic, the basic arithmetical operations are dened for fuzzy numbers allowing to adapt existing algorithms which use real numbers as input parameters to evaluate systems characterized by fuzzy parameters. For many situations, queueing network models are a popular approach in the modeling of computer and communication systems. We propose to replace single value parameters of open as well as closed queueing network models which exhibit uncertainties by fuzzy numbers. The adaptation of existing evaluation techniques for the analysis of queueing network models such as the well-known mean value analysis algorithm to fuzzy parameters is presented. Furthermore, the discussion of asymptotic results for these models is generalized to queueing network models parametrized with fuzzy numbers.

1 Introduction Modern computer and communication systems require e ective tools for predicting their performance and for analyzing their behavior. Since performance tuning of existing systems is usually more dicult and expensive, integrating performance modeling with various phases of design and implementation has recently become a popular issue 16]. Performance engineering is important for software as well as for hardware systems. Performance prediction is also essential for the capacity planning of computer systems 12]. However, especially in early phases of system design, the input parameters for mathematical models of computer and communication systems are not always known to the analyst leading to uncertainties in workload characterization. It is of signicant importance to represent such parameter uncertainties allowing the characterization of the corresponding uncertainties also in the predicted performance measures. Such uncertainties may be handled by trying various parameter combinations which is a cost intensive and insecure treatment. Another possibility to handle parameter uncertainties is to characterize

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model parameters as intervals and to use interval arithmetic or related techniques for the model evaluation. Using intervals as input parameters for models of computing systems is introduced in 10] and 11]. The adaptation of the well-known mean value analysis (MVA) algorithm for separable closed queueing network models (QNM) to handle interval parameters is presented in 9]. Interval arithmetic to compute performance guarantees for client-server systems is used in 17]. The adaptation of various analysis techniques for closed QNMs to histogram-based workload characterization which is also apt to approximate parameter distributions is proposed in 6], 7], and 8]. These interval- and histogram-based parameter modeling approaches are useful in a variety of situations. However, in early phases of system design, on one hand detailed information such as parameter distributions or probabilities for parameter histograms may be hard to estimate. On the other hand, the choice of the appropriate width of an input parameter interval is often dicult. Small intervals may produce more useful results, but the risk that the actual input parameter lies outside the chosen interval may be relatively high. Wide input intervals are of lower risk but may produce very wide performance measure intervals. Thus, it is desirable to integrate varying interval widths for the characterization of a parameter with varying risks in one model. Fuzzy numbers (FN) which are the arithmetical application of fuzzy sets (as opposed to fuzzy logic which is currently of higher popularity) allow such a parameter characterization with a continuum of parameter intervals and associated risk levels 4]. Using mathematical models with FNs as input parameters has recently become popular in other application areas (see for example 2]). Fuzzy numbers also allow the representation of so-called linguistic variables, i.e. of the linguistic characterization of parameters. A special class of FNs, the so-called triangular FNs, is of special convenience for parameter representation in performance modeling during early stages of system design. A triangular FN is characterized by only three values: one dening a single value estimate, the other two representing the widest possible interval for the parameter. TFNs integrate a conventional model with a continuum of interval parametrized models, allowing for a visualization of various degrees of uncertainty in easy-to-read diagrams. The evaluation of models parametrized with FNs is quite convenient as long as the interval evaluation of the model is possible. Based on the evaluation of so-called -cut intervals of the input parameter FNs, an existing interval algorithm can easily be adapted to handle FNs as input parameters. Using queueing networks for modeling system performance is a well-known and popular technique (see e.g. 5] for an introduction to the analysis of computer systems using queueing network models). In this paper, we consider analysis of single class queueing network models (QNM) and its adaptation to fuzzy input parameters. However, the techniques proposed in this work may also be applied to multiclass QNMs or to other modeling approaches. Moreover, in many situations in which a multiclass queueing network is more appropriate, a single class analysis may serve as a cost e ective rst order approximation. The paper is organized as follows: In the next section we present a short introduction to fuzzy sets and to arithmetic with fuzzy numbers. The adaptation of solution techniques for open QNMs is proposed in Section 3. For open QNMs, analytic expressions for the solutions are reported. In Section 4, the adaptation of the MVA algorithm for closed QNMs to fuzzy input parameters is presented. Because of the iterative structure of the MVA algorithm a numerical technique is proposed to compute the corresponding fuzzy performance measures. In Section 5, asymptotic results and considerations on the corresponding bottleneck analysis are presented. Section 6 presents a summary of the results and our conclusions.

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Fuzzy set D 1

Dα’ = [18,24]

μ (x) D

0,8 0,6

α-cuts

α D = [15,28]

0,4

α ’ = 0.8

α = 0.4

0,2 0 0

5

10

15

20

25

30

x

Figure 1: Example of a convex and normal fuzzy set (i.e. a fuzzy number).

2 Fuzzy Sets and Fuzzy Numbers 2.1 Basic Denitions Only a short introduction to the theory of fuzzy sets and fuzzy numbers is presented here. More detailed information can be found for example in the books 4], 19], or 20].

Denition 1 (Fuzzy set) Given a set X , a fuzzy set A in X is dened as a map A : X ! 0 1]:

A is also called the membership function of A. For any x 2 X , A (x) is called the membership degree of x in A. Additionally, we denote the set of all fuzzy sets in X by F (X ).

Denition 2 (Support) The support of a fuzzy set A is dened as the set of arguments where the membership function A is greater than zero: supp(A) = fx 2 X j A (x) > 0g:

Denition 3 ( -cut) The set of elements of a fuzzy set A whichs membership degree is at least , is called -cut of A. We denote the -cut of A by A :

A = fx 2 X j A (x)  g:

The value

is also called a risk level. Analogously, the strong -cut of A is dened as:

A  = fx 2 X j A(x) > g: 0

Denition 4 (Height) The height of a fuzzy set A is dened as the supremum of its membership function:

hgt(A) = sup A (x): x X 2

In this paper, with the exception of the fuzzy set of potential bottlenecks in Section 5, we consider only fuzzy sets in IR. Two important properties for the interpretation of fuzzy sets as fuzzy numbers are normality and convexity: 11/3

Denition 5 (Normality) A fuzzy set A in IR is called normal if its height equals one: hgt(A) = 1:

Denition 6 (Convexity) Let A be a fuzzy set in IR. A is called a convex fuzzy set if for any x1 x2 2 IR and arbitrary  2 0 1] it holds that: A (x1 + (1 ; )x2)  minfA (x1) A (x2 )g:

From this denition it follows that for a convex fuzzy set A , given any arbitrary 2 0 1], the corresponding -cut is an interval of real numbers: A = a  a ]. An example of a normal and convex fuzzy set in IR is depicted in Figure 1.

Denition 7 (Fuzzy number) A convex and normal fuzzy set is called a fuzzy number (FN).

2.2 Operations on Fuzzy Sets The two most important application areas for fuzzy sets are fuzzy logic and fuzzy modeling. The rst relies on the adaptation of set operations for fuzzy sets, while the latter employs generalizations of arithmetic operations to adapt existing modeling and solution techniques to handle FNs as input parameters for mathematical models. Fuzzy logic is the more popular application area of fuzzy theory. However, also fuzzy mathematical models in engineering and management science are becoming more and more popular 4]. Since in this paper set operations on fuzzy numbers are not the primary focus, we only give a very short introduction and concentrate on the generalization of arithmetic operations on FNs in more detail.

2.2.1 Set Operations on Fuzzy Sets Various concepts on how to generalize the basic set operations such as union, intersection, and complementation, have been proposed. Here we present the most popular one which is based on the minimum and maximum of the corresponding membership functions. Note, that based on so-called t-norms, there is a continuum of other possibilities for dening set operations with fuzzy sets 20].

Denition 8 (Set operations) Given fuzzy sets A and B in X , we dene: Intersection: A B (x) = minfA (x) B (x)g Union: A B (x) = maxfA (x) B (x)g Complementation: Ac (x) = 1 ; A (x): \



Note that w.r.t. the union and intersection operators, these denitions are consistent with the respective operations on the -cuts of the operands. I.e., for all 2 0 1] it holds that: (A \ B ) = A \ B  

and

(A  B ) = A  B  : 11/4

2.2.2 Arithmetic Operations with FNs The main focus of this contribution is the adaptation of analysis techniques for conventional models to handle models parametrized with FNs. Since such solution techniques usually involve computations with the input parameters, it is important to be able to perform the respective arithmetic operations and functions on FNs as well. The typical parameters of our application domain (queueing network models) are positive. Thus, in the following, we do not consider FNs A where 0 2 supp(A). This restriction allows to ignore the corresponding exceptions for the division and reciprocal operators. The most important concept for the generalization of real functions to fuzzy arguments is the so-called extension principle introduced by Zadeh 18]:

Denition 9 (Extension principle) Given the function  : X n ! Y , the extension of  is dened as:

b : F (X )n ! F (Y ) b (1  : : : n )(y ) = sup

y=(x1 :::xn )

minf1 (x1) : : : n (xn )g:

Since it is usually clear from the context whether the original function or its extension is meant, we will often denote a function or an operation and its extension by the same name or symbol. The extension principle can directly be applied to the basic arithmetic operations f+ ;  = min maxg: Addition: Subtraction: Multiplication: Reciprocal: Division:

A+B (z ) = sup minfA(x) B (y)g = sup minfA (x) B(z ; x)g x IR

z=x y

x IR

z=x y

x IR 0

z=x=y

x IR

2

A B (z ) = sup minfA(x) B (y)g = sup minfA (x) B (x ; z )g ;

;

A B (z) = sup minfA (x) B(y )g = 



2

sup minfA (x) B (z=x)g

2

nf g

A;1 (z) = A (1=z ) A=B (z) = sup minfA (x) B (y )g = sup minfA (x) B(x=z )g

Maximum: max AB (z ) = f

Minimum:

z=x+y

g

min AB (z) = f

g

sup

z=max xy f

sup

z=min xy f

g

g

minfA (x) B (y )g

2

minfA (x) B (y )g:

2.2.3 Interval Arithmetic and Fuzzy Numbers For practical reasons, it is often more convenient to perform arithmetic operations with FNs via arithmetic operations on the corresponding -cuts. Since a FN is a convex fuzzy set (per denition), the -cuts of a FN are intervals of real numbers. Thus, we provide a short introduction to interval arithmetic. More detailed discussions about interval arithmetic can be found for example in the books 1], 13], and 14].

Denition 10 (Interval) A real interval is a set of the form X = x x] = fx 2 IR j x x xg:

The set of all real intervals is denoted by IIIR.

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Denition 11 (Hull) If S is a nonempty bounded subset of IR, the interval

utS = inf(S ) sup(S )]

is called the hull of S. The hull is the tightest interval enclosing the set S .

Denition 12 (Interval extension) For a real function  : IRn ! IR, the extension to interval arguments is dened as:

(X1 : : : Xn) =

tuf(x1 : : : xn) j xi 2 Xi i = 1 : : : ng:

Using this denition, given intervals A = a a] and B = b b], for the basic arithmetic operations it follows that: Addition: A+B Substraction: A;B Multiplication: AB Division: A=B Maximum: maxfA B g Minimum: minfA B g

= = = = = =

a + b a + b] a ; b a ; b] minfab ab ab abg maxfab ab ab abg] A  1=b 1=b] if 0 2= B maxfa bg maxfa bg] minfa bg minfa bg]:

If all endpoints of the operands are positive, the multiplication and the division operations are reduced to:

A  B = ab ab] and A=B = a=b a=b]:

A disadvantageous e ect of interval arithmetic that arises with the multiple occurrence of a parameter in an arithmetic expression is the so-called dependency problem 3]. The dependency problem causes that the interval evaluation of such an expression may produce overly wide interval results. For example, given the interval X = x x], the expression X ; X evaluates to x ; x x ; x] instead of 0 0]. This is the case because every occurrence of an input parameter is treated independently. In some cases this problem can be dealt with by rewriting the expression in a way such that every parameter occurs only once. Furthermore, often monotonicity properties of an expression can be exploited to avoid the e ect of the dependency problem. If an expression is monotonic w.r.t. all input parameters, its interval extension can be computed using only the endpoints of the input argument intervals. However, in general, specialized techniques which are usually of higher computational complexity, like global optimization or interval splitting 11] have to be applied to obtain the interval extension of an arbitrary arithmetic expression. Using interval arithmetic, the arithmetic operations with FNs as arguments can also be dened via the respective interval operations on the -cuts. Given FNs A and B , it holds that for all 2 0 1] and all binary operations 2 f+ ;  = max ming: (A B ) = A B  :

However, if arithmetic on FNs is reduced to interval arithmetic on -cuts, the dependency problem arising with the multiple occurrence of input parameters in an arithmetic expression has to be taken into consideration.

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2.3 Triangular Fuzzy Numbers A class of FNs which is of special convenience in the modeling as well as in the evaluation process are the so-called triangular fuzzy numbers.

Denition 13 (Triangular FN) A triangular fuzzy number (TFN) is a FN whose membership function is of triangular shape. It is dened by a triplet (a b c) such that:

8> >> 0 >> >< x ; a  A (x) = > b ; a >> c ; x  >> c ; b >: 0

x < a a x b b x c x > c:

We also use the short notation A = TFN (a b c).

TFNs are of special convenience in the modeling of system parameters. The analyst needs only to supply a single value and an interval for every parameter. The value can be interpreted as the best guess for the parameter, and the interval serves as a guarantee for the respective parameter. I.e., only the -cuts for the two extreme risk level situations have to be provided. TFNs are also relatively convenient for numerical treatment: the result of an addition or subtraction of two TFNs is also of the same type. This does not hold for the other arithmetic operations. However, in many situations, the results can be approximated by TFNs 4]. An extension to the class of TFN are the so-called trapezoidal FNs, where the lowest as well as the highest risk level is characterized by an interval. Trapezoidal FNs have got similar convenient modeling as well as evaluation properties as TFNs and are discussed in more detail in 4].

3 Open Queueing Network Models An open single class QNM consisting of K service stations (queueing as well as delay devices) is characterized by the service demand parameters dk , k = 1 : : : K , and an arrival rate . In conventional models, these are parameters that have to be supplied as exact mean values. We denote the utilization of device k by uk , the corresponding device response time by rk , and the number of customers at device k by qk . The total response time and the total population is denoted by r and q , respectively. For models parametrized with FNs, these performance measures can be calculated using the solution formulae depicted in Figure 2 (see for example 5]). As discussed in Section 1, especially in early phases of system design, it can be useful to characterize the parameters of a QNM by FNs. Consider FNs Dk for the device service demand parameters, and the FN  to represent uncertainty in the arrival rate. We denote the corresponding -cuts by Dk = dk  dk ] and by  =    ]. The analysis of a model characterized by FNs as input parameters yields FNs also for the performance measures of that model. The utilization of device k is denoted by the FN Uk (for delay centers, the utilization is interpreted as the average number of jobs at that device). The fuzzy response time at device k is denoted by the FN Rk , the average queue length FN for device k is denoted by Qk , and the respective sums are denoted by R and Q .

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for k  1 to K do begin uk = 8dk

>< dk (delay centers) rk = > dk : 1 ; u (queueing centers) 8>k < uk (delay centers) qk = rk = > uk : 1 ; u (queueing centers)

end K r = P rk k=1 q = r

k

Figure 2: Solution algorithm for open QNMs with conventional parameters. We can now use the arithmetic operations dened for FNs to compute the membership functions as well as the -cuts of the FNs for the performance measures of interest. The utilization of device k is characterized by the product of the fuzzy arrival rate  and the fuzzy device parameter Dk : Uk (z ) = sup minf (x) Dk (z=x)g: x IR 2

Application of interval arithmetic to the -cuts of the input parameters yields the -cuts of the FNs for the device utilizations: Uk =   Dk = dk  dk]: Since the response time of a delay center is characterized by the respective service demand, it does not require any additional computation. The expression for the response time of a queueing device can be expressed in a more compact way if it is rewritten such that every input parameter occurs only once: (1) rk = 1=d 1;  : k Using the denitions for FN arithmetic presented in the previous section, the FN expression for the response time of queueing device k is computed step by step as follows. The FN 1=Dk is: 1=Dk (z ) = Dk (1=z): Thus, the FN 1=Rk = 1=Dk ;  is: 1=Dk (z) = sup minf1=Dk (x)  (x ; z)g ;

x IR

= sup minfDk (1=x)  (x ; z )g: 2

x IR 2

Thus, the membership function for the FN describing the response time of queueing device k computes to: Rk (z) = 1=Dk (1=z ) = sup minfDk (1=x)  (x ; 1=z)g: ;

x IR 2

Since the average number of jobs at a delay center equals its utilization, no additional computation is necessary in this case. Analogously to the response time expression for a queueing device (see Equation (1)), the expression for the number of jobs at queueing device k can be rewritten to: 11/8

qk = 1=(d ) ; 1 : k

Step by step application of FN arithmetic yields the FN characterizing the number of jobs at queueing device k:    Qk (z ) = sup min  (x) Dk x(1=z1 + 1) : x IR Note that the membership function of the response time as well as of the queue length for device k can also be derived by direct application of the extension principle to the respective conventional expressions. The corresponding expressions for the -cuts of these performance measure FNs are: Rk = Dk=(1 ; Uk) = dk=(1 ;  dk ) dk=(1 ; dk )] (2) 2

and

Qk = Uk=(1 ; Uk) = dk=(1 ;  dk )  dk =(1 ;  dk)]:

(3)

In this case there are no diculties because of the dependency problem since all occurrences of the service demand parameter in Equations (2) and (3) have got an increasing e ect on the respective expression. The FN expression for the total response time is constructed as the fuzzy sum of the respective device performance measures:

R(z) = sup fRk (xk )g: P min k z= k xk

Finally, the total customer population is computed as the fuzzy product of the arrival rate  and the total response time R :

Q (z ) = sup minf (x) R(z=x)g: x IR 2

The corresponding -cuts can be computed as the interval sums of the respective device performance measure -cuts:

X R = Rk = k

"X

and

Q = 



R =

k

dk=(1

" X  

k

;

X dk ) dk =(1 k

dk=(1

;

;

#

 dk)

X  dk)  dk=(1 k

;



#

dk )

:

Since the expressions for the membership functions of the performance measures are usually hard to evaluate, in practice, a numerical evaluation using -cuts is used. This technique is described in more detail in Section 4 along the lines of the evaluation of closed QNMs with FN parameters.

4 Closed Queueing Network Models 4.1 Fuzzy Mean Value Analysis (MVA) Consider a closed QNM consisting of K service centers (delay and queueing devices in arbitrary combination) with mean service demands dk , k = 1 : : : K , an optional terminal think time z , and N customers in the system. A separable closed QNM can be solved using the well-known 11/9

for k  1 to K do qk (0)  0 for n  1 to N do begin ( centers) for k  1 to K do rk (n)  ddk (1 + q (n ; 1)) (delay (queueing centers) k k P r(n)  j rj (n) x(n)  z +nr(n) for k  P 1 to K do qk (n)  x(n)rk (n) q (n)  j qj (n) end Figure 3: Exact single class MVA with conventional parameters. The depicted algorithm computes the performance measures for n = 1 : : : N customers. mean value analysis (MVA) algorithm 15]. The single class MVA algorithm is listed in Figure 3. The notation of the performance measures is the same as for open QNMs with the addition that the system throughput is denoted by x(n). In analogy to the fuzzy parametrization of open QNMs in Section 3 we consider FNs Dk as service demand parameters and a fuzzy terminal think time Z . Because of the dependency problem and the iterative structure of the MVA algorithm it can not be adapted directly to FNs as input parameters. However, numerical solutions of the membership functions of the performance measures can be computed by interval evaluation of the MVA algorithm using -cuts of the fuzzy parameters as input intervals. For such a numerical evaluation, the interval 0 1] is partitioned into m + 1 risk-levels i = i=m, i = 0 : : : m. Using the i -cut intervals Dki of the service demands and Z i of the terminal think time as input intervals for an interval MVA algorithm, intervals for all performance measures of interest can be computed. For example, we denote the throughput interval obtained at risk level i by X i = xi  xi ]. These performance measure intervals are i -cuts of the corresponding fuzzy performance measures. Thus, the endpoints xi and xi of the interval X i can be used as supportive points for the numerical representation of the fuzzy throughput X : X (xi ) = X (xi ) = i : Due to its iterative type and the dependency problem, the interval adaptation of the MVA algorithm requires special considerations. An ecient interval MVA algorithm relying on monotonicity properties of closed separable QNMs is proposed in 9]. Since the interval evaluation relying on monotonicity properties only doubles the computational e ort as compared to the conventional MVA, the computational complexity of one interval evaluation is O(NK ). Thus, the complexity of the numerical fuzzy MVA with m -level steps is O(mNK ). In the algorithm presented in this section, the risk levels are chosen in an equidistant way. This numerical fuzzy evaluation technique can be improved using an adaptive choice of risk levels i .

4.2 Example We demonstrate the proposed technique with results for a small closed QNM consisting of a CPU and two disks depicted in Figure 4. In this example we use TFNs for all service demand parameters. The average service demand at disk 1 is estimated to be 100msec. However, it is almost certain that the actual mean service demand is not exactly 100msec. Thus, the risk level for this expert 11/10

Disk

1

CPU Disk

2

membership degree

Figure 4: A small example QNM. 1

D

0.8

D

disk 1

disk 2

D cpu

0.6 0.4 0.2 0 0

20

40

60

80

100

120

140

msec

Figure 5: TFN parameters for the example QNM. guess is = 1. Suppose that there is reason to bound the service demand of disk 1 by the interval 90msec 140msec]. Since we assume that the actual parameter lies within this interval almost certainly, it serves as a strong -cut for the risk level = 0 (i.e. it is the support of the FN service demand). This information motivates the use of a TFN to characterize the service demand of disk 1: Ddisk1 = TFN (90 100 140). Analogously, let the expert guess for the service demand at disk 2 be given as 120msec. Assume that this value is bounded by the interval 80msec 130msec], which is a more optimistic bound as compared to the characterization of disk 1. Given this input, the service demand at disk 2 can be modeled as the TFN Ddisk2 = TFN (80 120 130). Finally, let the service demand at the CPU be characterized by the FN Dcpu = TFN (5 10 15). The jobs are of batch type, i.e. the terminal think time is zero. Where not stated explicitly, the unit of measure for all service time parameters is msec. The TFN parameters for our example are depicted in Figure 5. This example QNM with TFN input parameters is analyzed by numerical evaluation using -cuts as discussed above. 100 i values are used to obtain approximate results for the membership functions of all performance measures. Figure 6 shows the fuzzy performance measures for our example. The fuzzy parametrized QNM is evaluated with N = 5 and with N = 100 customers in the network. The visualization of a continuum of interval parametrized models with nested input intervals w.r.t. decreasing risk level provides a good overview of the relation between varying risk levels and expected performance indices. As it is discussed in more detail in Section 5, the asymptotic behavior of the system throughput as well as of the response time is determined basically by Dmax = maxk Dk . This e ect can be observed in Figure 6 (e) and (f), depicting the high load results for these performance measures. An interesting e ect with high load can also be found for the device performance measures (i.e. queue lengths and device response times). As it can be seen from the graphs of the fuzzy parameters for the two disks depicted in Figure 5, for risk levels > 0:75 disk 2 is the unique bottleneck of the system. With lower risk levels, both disks are potential bottlenecks. This e ect is the explanation for the shape of the membership functions of the queue lengths and disk response times for high load, depicted in Figure 6 (g) and (h).

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Throughput, N=100

1

membership degree

membership degree

Throughput, N=5

0.8 0.6 0.4 0.2 0 0.006

0.007

0.008

(a)

0.009

0.01

0.011

1 0.8 0.6 0.4 0.2 0 0.006

0.012

0.007

0.008

membership degree

membership degree

1

0.6 0.4 0.2 0 600

(b)

700

800

0.4 0.2

9000

10000

12000

Queue lengths, N=100 Q

disk 1

disk 2

0.6 0.4 0.2

Q

1

Q

disk 1

disk 2

0.8 0.6 0.4 0.2 0

0

1

2

(c)

3

4

0

20

R

R

disk 1

80

100

Response times, N=100 membership degree

1

60 jobs

Response times, N=5

0.8

40

(g)

jobs

membership degree

11000

(f)

0

disk 2

0.6 0.4 0.2 0

R

1

R

disk 1

0.8

disk 2

0.6 0.4 0.2 0

0

(d)

14000

0.6

msec

membership degree

membership degree

0.8

13000

1

Queue lengths, N=5 Q

0.012

0.8

0 8000

900

msec

1

0.011

Response time, N=100

0.8

500

0.01

jobs/msec

Response time, N=5

400

0.009

(e)

jobs/msec

100

200

300 msec

400

500

600

0

(h)

2500

5000

7500

10000

12500

msec

Figure 6: Fuzzy performance measure results given two load situations (N = 5 and N = 100) for the example QNM.

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membership degree

0.8

Dmax

0.6

Bdisk

0.4

Bdisk 2

1

0.2 0 70

90

110

130

150

msec

Figure 7: Maximum service demand FN Dmax and bottleneck shares of the two disks for the example parameter set.

5 Asymptotic and Bottleneck Analysis In conventional models, the asymptotic behavior of the system is determined by the maximum service demand dmax = maxk dk as follows: nlim x(n) = 1=dmax and nlim r(n)=n = dmax. The asymptotic analysis for interval parametrized models is presented in 9]. This asymptotic analysis can be generalized to evaluate models with FN service demands Dk . We denote the FN of maximum service demands by Dmax = maxk Dk . The corresponding membership function is: !1

!1

Dmax (z) = sup min fDk (xk )g: k z=maxk xk

Applying the asymptotic results derived for interval models 9] to the -cuts of a fuzzy model, we conclude that the asymptotic behavior of the fuzzy throughput and the normalized fuzzy response time is determined by the FN Dmax as follows:   and R =N ! D  (N ! 1) 8 2 0 1] X  ! 1=Dmax max =) X ! 1=Dmax  and R=N ! Dmax  (N ! 1): The share Bk of device k in the maximum service demand Dmax is characterized by the intersection Bk = Dk \ Dmax . The corresponding membership function is:

Bk (x) = minfDk (x) Dmax (x)g: Note that Bk is a fuzzy set which is not necessarily a FN, because in general it is not normal. Using these bottleneck shares of the device parameters, the set of potential bottlenecks dened in 9] can be generalized to the fuzzy set of potential bottleneck devices:

B (k) = sup Bk (x) = hgt(Bk ): x

For every device k, this fuzzy set characterizes the highest risk level with which k is a potential bottleneck. On the other hand, given a risk level , the -cut of B corresponds to the set of potential bottlenecks obtained from interval analysis using the -cuts of the service demands. Figure 7 illustrates the construction of the maximum service demand FN Dmax (thick black line), and the bottleneck shares for the two disks Bdisk1 (grey line) and Bdisk2 (dashed line) of our example QNM. The fuzzy set B of potential bottlenecks of our example is given as: B (cpu) = 0, B (disk1) = 0:75, and B (disk2) = 1. This means that disk 2 is a potential system bottleneck independently from the risk level under consideration, disk 1 is a potential bottleneck only if risk levels < 0:75 are considered, and the CPU can be ignored w.r.t. bottleneck analysis of this model.

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Comparing Dmax in Figure 7 to the response time FN for N = 100 customers in the system in Figure 6 (f), we can see that Dmax serves as a good approximation for the normalized response time R(N )=N . Analogously, the throughput for high load depicted in Figure 6 (e) is approximated by 1=Dmax.

6 Conclusions and Future Work Conventional analytic models of computer and telecommunication systems accept single mean value parameters as input and produce single mean performance measures as output. However, uncertainties in parameter values which are typical in early stages of design and implementation make these techniques ine ective. As an extension to interval-based workload characterization, we propose the representation of parameters which exhibit uncertainties by FNs. FNs can be interpreted as a continuum of intervals with associated risk levels. TFNs are especially well-suited for parameter estimation in early modeling phases since only parameter values for the two extreme risk level situations have to be supplied by the analyst. Using TFNs, conventional single value parametrized models are integrated with a continuum of interval-based models, which have been proposed for modeling uncertainties in previous work. As an application example for the fuzzy modeling of computing systems, the adaptation of solution techniques for single class QNMs is presented in this paper. The solution formulae for open QNMs can directly be adapted to fuzzy parameters. Due to its iterative structure, the adaptation of the MVA algorithm for closed QNMs requires special considerations. Based on the interval MVA evaluation, a numerical technique is presented which allows the computation of the corresponding fuzzy performance measures. Also asymptotic analysis including the fuzzy identication and characterization of system bottlenecks is proposed. The fuzzy MVA and bottleneck analysis techniques are demonstrated with the analysis of a small example QNM. It is illustrated how the behavior of performance measures with varying risk levels can be visualized in a convenient way, which allows the analyst to estimate the inuence of varying degrees of uncertainty on the output of the model in a relatively cost e ective way. Only single class QNMs are considered in this work. However, the proposed parameter modeling and evaluation techniques may also be applied to other modeling approaches such as multiclass QNMs or task graph models. The numerical evaluation algorithm for fuzzy parametrized models proposed in Section 4 uses an equidistant partition of the risk levels to obtain an approximate representation of the fuzzy performance measures. An adaptive algorithm which automatically chooses appropriate risk levels such that more accurate estimates of the output FNs can be computed while simultaneously decreasing the computational e ort is a subject of future work. Also, the fuzzication of the number of jobs N in closed QNMs will be considered in future research.

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