Computers and Industrial Engineering Vol. 25, Nos 1--4, pp. 143-146, 1993 0360-8352/9356.00+0.00 Printed in Great Britain. All rights reserved Copyright © 1993 Pergamon Press Ltd

A Delay Model of Queueing Network System Based on Fuzzy Sets Theory

Jung Bok JO', Yasuhiro TSUJIMURA", Mitsuo GEN", and Genji YAMAZAKI"

"Dept. of Engineering Management, Tokyo Metropolitan Institute of Technology, I-Iino City, Tokyo 191, JAPAN

"Dept. of Industrial and Systems Engineering, Ashikaga Institute of Technology, Ashikaga, 326, JAPAN

ABSTRACT

The goal of our research is the delay analysis of queueing model for data networks using fuzzy sets theory. We propose an fuzzification of M/M/1 queueing system. We also apply fuzzy sets theory to the open central server network model with the fuzzy queues. Thus, we represent the delay analysis and performance of open central server network model based on fuzzy sets theory.

INTRODUCTION

Many mathematical models have been developed to solve queueing network problems. Most of these assume

probability distribution for arrivals and services. But, when we develop a new system or some information about an existing system is unavailable, we are forced to employ subjective possibility or probability. In some situations the data can be possibilistic rather than stochastic. In these cases, we apply to fuzzy number for the arrival and service rates of queueing network system.

The problem of fuzzy queues has been analyzed by Prade[1] and Li and Lee[5] through the use of the

extension principle. Buekley[2] considers elementary queueing systems, with multiple parallel servers with finite or infinite system capacity and calling source, whose arrivals and departures are restricted by arbitrary possibility distributions.

Recently Negi and Lce's contribution[4] was shown that their approach can utilize the advantages of both the fuzzy and probability approaches to make the model more realistic and less restrictive. Their approach for F/F/1 system is a single value simulation of a fuzzy variable by the use of two random variables.

We introduce fuzzy sets theory into Little's law. By

using fuzzy Little's law we can compute the delay analysis for the fuzzy M/M/1 queue when arrival and service rates are fuzzy variables. We also apply fuzzy sets theory to the open central server model[6] with a fuzzy queue. Thus, we show the delay analysis of open

central server network model based on fuzzy queueing system. All comparisons of fuzzy numbers are assumed to be made by Kanfmann and Gupta[10] methodology using linear ordering of fuzzy numbers.

PRELIMINARIES

The following notations are used throughout this paper:

N(t) : number of customers in the system at time t

~(t) : number of customers who arrived in the interval[O,t] : average number of customers in the system

"~' : average customer waiting time in the system

~o : average number of customers waiting in queue

"~/Q : average customer waiting time in queue ~. : possibility of n customers in the system

where ~ denotes a triangular fuzzy number(TFN). We represent an interval of confidence of TFN .~ for all a - cu t level, that is, for a ~ [0,1] A ( a ) = [AI<"),A2¢'~) ]. If a = 1, AI Ca) is equal to A2 ~").

We wish to find F,(t) = Poss[N(t) = n], (1)

for n = 0,1,2... and for t = 1,2 ..... One unit of time, be it one second, one minute, etc., represented by t = 1. We assume that we have the initial possibility distribution

~(0) = (Po(0), F~(0),....). (2) We also wish to compute

when this limit exists, which will be called the steady- state possibility distribution.

Dubois and Prade[3] have defined the fuzzifying function and integration of a fuzzifying function over a crisp interval, as follows:

Definition 1 Let X and Y be universes and P(Y) be the set of all

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144 Proceedings of the 15th Annual Conference on Computers and lndgstrial Engineering

fuzzy sets in Y(power set). ~' : X -+ P'(Y) is a mapping

'~ is a fuzzifying function iff #~,)(y) = /z,~(x,y), V (x,y) e X x y

where /2fffx,y ) is the membership function of a fuzzy relation. Definition 2 The crisp set of elements that belong to the fuzzy set

at least to the degree a E [0,1] is called the a - leve l set:

A,, = { x e X l UA(x) > a }. Definition 3 Let T(x) be a fuzzifying function from [a,b] C_ R to R

such that V x E [a,b] ~'(x) is a fuzzy number. And f , - (x) and f / ( x ) are a - leve l curves of a fuzzifying

function "~(x). The integral of '~(x) over [a,b] is then defined to be the fuzzy set as follows:

f.t~(x, ,~= {(f'c: (,,:)~+ff,: <,,:, .:z,,,,,,)}. Fuzzy Little's law

We introduce fuzzy sets theory into Little's law[7-10]. The fuzzy number of customers in the system observed

up to time t is as follows:

# t : --~ f[#(~) d~ (4)

which we call the time average of N ( r ) up to time t, where a fuzzifying function l~(v ) and a integration of fuzzifying function are defined by definition 1, 3, respectively. Naturally, N(t) changes with the time t, but in many systems of interest, ~(t) tend to a steady- state ~ as t increases, that is,

A~ = lira #t' (5)

In this case, we call N the steady-state fuzzy time average of ~ ( r ) . It is also natural to view

£(t) I t : (6) t

as the time average fuzzy arrival rate over the

interval[0,t]. The steady-state fuzzy arrival rate is defined as

I : lim I t (7) t-m

assuming that the limit exist. The fuzzy time average of the customer delay up to time t is similarly defined a s

1

that is, the fuzzy average time spent in the system per customer up to time t. The steady-state fuzzy time

average customer delay is defined as

# : lim #t (9)

assuming that the limit exists. Therefore, we have

# # = ~ . (z0)

If eq.(lo) is represented a-cut level, we have

¥ ee[o,1]

[.1'., e'] w(.) ° ' l' m )

Thus, wc can compute the ~ with interval confidence as follows:

¥ ~e[0,1]

Similar formulas exist for 1~ o and Wo, that is

#o : ~. (z3) A

of

And we have

¥ ,e[o,1], N0({t) = r~(e}W(e) ~,(It)W, (~}1 t 2 ~{~, 1 02 s , (t4)

T H E F U Z Z Y M / M / I Q U E U E I N G S Y S T E M

Consider now the steady-state possibilities

/~ = lira /~.{t). £-m

Note that during any time interval, the total number of transitions from state n to n+l must differ from the total number of transitions from n+l to n by at most 1. Thus asymptotically, the frequency of transitions from n to n+l is equal to the frequency of transitions from n+l to n. Equivalently, the possibility that the system is in state n and makes a transition to n+l in the next transition interval is the same as the possibility that the system is in state n+l and makes a transition

to n, that is, ~.~ 6+006 ) = P.+,~'6+o(6) (15)

By taking the limit in this equation as 6 ~0, wc obtain

Eq.(16) can be also written as P'n+l = ~ ' . , (17)

where

X p = .~ (le)

It follows that

If P'<I, the possibilities P. arc all positive, so (19)

Jo et al.: Queueing Network System 145

(20)

Combining eq.(19) and eq.(20), we finally obtain

a,,O

We can now calculate the average number of customers in the system in steady-state:

D'O n'O D'O

- {l-p) 2 (l-p) "

(22)

The fuzzy average delay per customer is given by fuzzy Litfle's law,

~7 = --N = ~ (23) X ~ (Z-p) "

If eq.(23) is represented a -cu t level, we have

V ~e[o,1],

[ 1)I" ] W(a) : i ~ ( , ) "~'.) (.)' ~(.)_~(.).(~) '

L,.2 -A2 1)1 "1 " : 1'2

The fuzzy average delay time in queue ~Q, is the fuzzy average delay W less the fuzzy average service time 1/~, so

1~= ~ Z X(1-p) - "~' (24)

A very useful interpretation is to view the quantity P

as the fuzzy utilization factor of the qucueing system.

FUZZY QUEUEING NETWORK MODEL

Now, for an application of the proposed fuzzy queueing model, we consider that the open central server system, shown in Fig. 1. The central server is, of course, the CPU. Lavenberg and Sauer in Lavenberg[ll] use the open central server qucueing model to study memory management of a multiprogramming system. They were trying to optimize

the performance of operating system. The open central server model can be used to model a system, such as a transaction processing system in which the system is not limited by the multiprogramming level. As shown in Fig. 1, there are K-1 of the I/O devices, each with its own queue, and each exponentially distributed with average service rate ~k (k = 2,3,...,K).

N Circulating Program

I/0 Devicea

Fig. 1. Open central server system.

The CPU was assumed to have an exponential distribution. The queue discipline is assumed to be first come first serve (FCFS) for the I/O devices. Upon completion of a CPU service, a customer exits the system with possibility ~s or enters service at I/O

device k with possibility '~k (k = 2,3,...,K). Upon completion of an I/O service, the customer returns to the CPU queue for another cycle. An incoming arrow leading to the CPU queue is added to indicate the arriving traffic with average rate '~'. The arrival pattern is assumed to have an exponential interarrival time. We assume the system is not overload, so the throughput is also *~. Therefore, Jackson's theorem can be applied to the system as stated in the following algorithm.

Algorithm Lavenberg

Consider the open central server system described above. Suppose we are given the average arrival rate ~', the mean total resource requirement, ~k. And 'Nk is average number of customer at the k-th device. Then we use Jackson's theorem to calculate the performance measures of the system as follows:

Step 1 [Set the demands, Dk]

/~t, k = 1,2 . . . . . K.

Step 2 [Calculate device performance] For k = 1,2,...,K, calculate

~k 5, ~ -- x- a,, *, = ~ , ~d ~, o ~---~, .

Step 3 [Compute system performance measures] Set average delay time (average time in system) to

K

146 Proceedings of the 15th Annual Conference on Computers and Industrial Engineering

Set average number of customer in the system to

V ~e[o,l] N(,,) : p.~'~,~('~,~.f'~('q.

The bottleneck device is device j, where j is the integer for which ~j = ~ .~ , where

The maximum possible throughput is given by

x :_.~__ ~ Dmx

Now, we show the following numerical example[6] to illustrate fuzzy Lavenberg's algorithm.

NUMERICAL EXAMPLE

processed is

A 7= ( 4.718, 6.417, 7.999 ).

The bottleneck is the third disk driver, so the TFN of maximum possible throughput, '~m~, is (0.244, 0.25, 0.256) transactions per second.

CONCLUSION

The goal of this paper is the delay analysis of queueing model for data networks using fuzzy sets theory. We proposed a fuzzy M/M/1 system that is applied fuzzy Little's law. we represented the delay analysis and performance of open central server network model based on fuzzy sets theory. Therefore we can get more realistic solution when some data of network are ambiguous. This analysis was much simple and required less computation.

We investigate a transaction processing computer system that can be modeled as an open central server model with one CPU and three disks. We find that the TFN of average transaction arrival rate, '~ is (0.18, 0.2, 0.21) transactions per second with an TFN of average CPU service requirement of (2.8, 3, 3.2) seconds per transaction. The TFNs of average I/O service requirements are (0.9, 1, 1.1), (1.8, 2, 2.1),and (3.9, 4, 4.1) seconds, respectively.

Let us step through Lavenberg's algorithm. In Step 1 we set D1 = (2.8, 3, 3.2), D2 = (0.9, 1, 1.1), ~3 = (1.8, 2, 2.1), and D4 = (3.9, 4, 4.1) seconds. In Step 2 we compute N1 = (1.016, 1.5, 2.049), ~2 = (0.193, 0.25, 0.3), N3 = (0.479, 0.667, 0.789), and N4 = (2.356, 4, 6.194) and P'k, "~'k (k = 1, 2, 3, 4) as the following Table 1.

Table 1 The server utilizations and the response times.

k ~rk ffk 1 (0.504, 0.6, 0.672) (5.645, 7.5, 9.756) 2 (0.162, 0.2, 0.231) (1.074, 1.25, 1.43) 3 (0.324, 0.4, 0.441) (2.663, 3.333, 3.757) 4 (0.702, 0.8, 0.861) (13.087, 20, 29.496)

In Step 3, from the Table 1 we calculate the average transaction response time

= ( 22.469, 32.083, 44.439 ) seconds,

and the average number of transactions being

REFERENCE

1. H.M. Prade, An Outline of Fuzzy or Possibilistic Models for Oueueing Systems, In : Fuzzy sets, pp.147-153(P.P. Wang and S.K. Chang, editors), Plenum Press(1980). 2. J.J. Buckley, Elementary Queueing Theory Based on Possibility Theory, Fuzzy Sets and Systems 37, pp.43- 52(1990) 3. D. Dubois and H. Prade, Fuzzy Sets and Systems : Theory and Applications, Plenum Press(1980). 4. D.S. Negi and E.S. Lee, Analysis and simulation of fuzzy queues, Fuzzy Sets and Systems 46, pp.321- 330(1992). 5. R.J. Li and E.S. Lee, Analysis of Fuzzy Queues, Computers and Mathematics with Applications 17, pp.1143-1147(1989). 6. A.O. Allen, Probability, Statistics, and Queueing Theory with Computer Science Applications, 2nd ed., Academic Press(1990) 7. H.J. Zimmermann, Fuzzy Set Theory and Its Applications, 2nd ed., Kluwer Academic Publishers(1991). 8. D.P. Bertsekas, Data Networks, 2nd ed., Prentice-Hall Inc(1992). 9. R.W. Wolff, Stochastic Modeling and the Theory of Queues, Prentice-Hall Inc(1989). 10. A. Kaufmann and M.M. Gupta, Fuzzy Mathematical Models in Engineering and Management Science, Van Nostrand Reinhold(1988). 11. S.S. Lavenberg, ed., Computer Performance Modelling Handbook, Academic Press(1983).