- Email: [email protected]
Fuzzy assignment of customers for a queueing network
FUZZY ASSIGNMENT OF CUSTOMERS FOR A QUEUEING NETWO...
14th World Congress of IFAC
K-3e-11-2
Copyright© 1999 IFAC 14th Triennial World Congress~ Beijing, P.R. China
FUZZY ASSIGNMENT OF CUSTOMERS FOR A QUEUEING NETWORK Runtong Zhang*
Yannis A. Phil1is~
*College ofEconomics and Administration Northern Jiaotong University Beijing 100044 People's Republic ofChina E-mail: [email protected]
Xiaomin Zhu*
~ Department ofProduction Engineering and .1Uanagemenl
Technical University
oj~Crete
Chania 73100, Greece E-mail: [email protected]
Abstract: This paper considers a queueing network with two parallel heterogeneous servers. Each server has its own queue and customers arrive at each queue according to independent Poisson processes. Servjce times are independent and exponentiaJly distributed. When a customer arrives at queue 1, the customer can be transferred to queue 2 by paying an assignment cost. Each customer in queue pays a holding cost per unit time. The objective is to dynamically determine the optimal assignment policy so as to minimize the average cost. In this paper, a novel approach is presented. using fuzzy control to solve the problem. Copyright© 1999 IFAC
Key words: fuzzy control, queueing networks, path planning
comprehensive discussion on optimal control of queueing networks can be found in the survey papers of Stidham and Weber (1993). Recently,
1. INTRODUCTION This paper considers an optimal control problem for a parallel queueing network with rnro heterogeneous servers using the fuzzy logic technique (Driankov et. al~~ 1993). The proposed model is an ac.yclic. queueing network consisting of two parallel queues. Each queue has its own arriving customers and one stream of customers upon arrival can be transferred to another queue. The system objective is to dynamically determine the optimal assignment policy so as to minimize the average cost. This is a simple model of computer or communication networks, and has been examined by Koyanagi and Kawai (1985). However, their ""ork investigates only the structure whereas the present work specifies the optimal policy.
Zhang and Phillis (1999, 1999a, 1999b, 1999c) proposed a new method using fuzzy logic to solve queueing control problems and show via simulation that the new method generalizes existing solutions and also efficiently solves problems intractable with classical methods. This approach signals a departure from classical techniques. Different criteria of individual or social optimization are faced, when we deal with problems of admission and routing control. The fonner criterion depends on the customer' s o"~{n benefits, while the latter views the system's performance as a whole. It is believed that the policy imp)emented by self-interested individuals does not lead in general to the best social outcome. Such questions have attracted considerable interest
Optimal control of queueing systems has been extensively investigated in the literature. A
5422
Copyright 1999 IFAC
ISBN: 0 08 043248 4
FUZZY ASSIGNMENT OF CUSTOMERS FOR A QUEUEING NETWO...
among econonlists and operations researchers. For discussions on individual vs. socially optimal
14th World Congress of IFAC
;Queue I L~ 1 Departures
Class 1 .Arrivals .11
----/.........
prob]ems~
see the review paper (Stidham and W eber~ 1993). Research along this line is continuing. Optimal policies for individual
1I11~
....
1It--G
...
I
1
\
----~-........
customers are often easy to obtain and take on simple and explicit fonns, while socially optimal
Class 2 ..A.rrivals .L1
Queue 2
Server 2 Departures
policies which are of greater practical importance, Figure 2: Queueing network
often defy simple analysis. This paper uses a socially optimal criterion and use the term '~optimar' in the sense of "socially optimaP'. This
applications. To transform the fuzzy output into an
usable
paper detennines socially optimal with the aid of individually optimal policies and hence, the fuzzy logic approach also contributes towards exploring the relationships between individually and socially optimal policies.
the
height
method
of
2. PROBLEM DESCRIPTION The queueing network considered is shown in Figure 2. This network consists of two heterogeneous servers
At each decision epoch, the fuzzy controller makes a decision by emulating a skilled human operator.
in parallel called server i) i~l )2, each with its own queue of unlimited capacity. There are two classes
Specifically, based on the current state of the system, an inference engine equipped with a fuzzy rule base determines an on·line decision to adjust the system behavior in order to guarantee that the system is optimal in the sense of minimizing a given cost. This is thejUzzy logic controller (Figure 1).
of customers. Customers of class i, i~l )2, join queue i to be served by server i. The length of queue i is observable. However, a customer of class 1\ upon arrival) can be transferred to queue 2 by paying an assignment cost C!I ifhe expects to incur a lower cost by this transfer. It is assumed that customers of class i arrive at queue i in a Poisson stream with constant rate Ai. Successive services by
In this paper, all of the membership functions for the fuzzy sets are chosen to be triangular. This is because the parametrjc~ functional descriptions of triangular membership functions are the most economic ones. In addition, it has been proven that such membership functions can approximate most other membership functions. To present the fuzzy roles, ZO, PS, PM, PB are used to indicate the "zero1)~ "positive smaH", "positive medium" and "positive bigH unless otherwise explained.
server
i
are
independent
and
exponentially
distributed with mean 11 fli regardless of the class of customer, where Aj
Queuelng systems are simulated and controlled in C
language. Mamdani ilnplication is used to represent "if-then'~
rules., because it is precise, cornputationally simple and fits various practical The Queueing Syst~m
one,
countIng.
The fuzzy control method aims at finding optimal policies in computationally efficient ways. l~hroughout this paper, the term "po1icy~) is used in the sense of ' 'deterministic stationary policy'~~
Jurivals
crisp
defuzzification is used because this method is also simple, fast, and has the advantage of weio-ht . 0
lengths. However the explicit determination of the optimal policy remains an open problem.
Departures
3. ARCHITECTURE OF THE FUZZY LOGIC
CONTROLLER Without loss of generality, the decision epochs at which arriving customers of class 1 are assigned c,an be restricted to their arrival time. The state of the system at the decision epochs can be described by si=O)1,2, ... , where Si is the number of customers
Figure 1: The fuzzy queueing control system
5423
Copyright 1999 IFAC
ISBN: 0 08 043248 4
FUZZY ASSIGNMENT OF CUSTOMERS FOR A QUEUEING NETWO...
14th World Congress of IFAC
257
PS
in queue i~ Therefore SI+S2 is the total number of queueing customers in the system~ An arriving customer of class 1 can be transferred to queue 2 by paying an assignment cost. It is optimal for this customer to join queue 2 only when his expected holding cost in queue 1 is not smaller than that in queue 2 plus the assignment cost, that is
PM
PE
O.
0
2
3
4
5
(a)
1
s=sl-h 1 - - - -
JJ 1
1
s2·h2·--~C
PS
(1)
O.
Figure 3: Membership functions
A fuzzy rule base is developed as in Table 1. We choose 4 fuzzy sets for each of these 3 inputs and this rule base consists of 4 3 =64 rules. The membership functions for the fuzzy inputs s are shown in Figure 3(a)~ for Az, they are shown in Figure 3(b), and for the fuzzy output d they are shown in Figure 3(c). The universe of discourse for the fuzzy input s is [0, +aJ) and fOT A.i~ i=] ,2, they are [0,6J. The universe of discourse for the fuzzy output d is [0,1]. The establishment of the fuzzy rule base relies on the following arguments: (i) Because of the previous discussion, the greater the difference between the expected holding costs in both queues, the easier it is to decide d=l for an arriving class 1 customer. (ii) A higher arrival rate of class 1 customers strengthens the decision d= 1. indeed, if the manager of the system knew that additional class 1 customers were arriving at a very high rate, which means that the length of queue I would increase very fast, he would like to transfer the present customer to queue 2. This is so because the difference bernreen the expected holding costs in both queues would also increase. (Hi) A higher arrival rate of class 2 customers weakens the decision d= t. A higher arrival rate of class 2 customers means that the difference between the expected holding costs in both queues is becoming slnaller. Based on propositions (i)-(Hi}, the fuzzy rule base is ready.
smaller than zero. Table 1: R pIe base
s
A.t
A,2
d
ZO
ZO
ZO
0
PS
ZO
ZO
PM PB
ZO ZO
ZO ZO
1 1
... ...
...
ZO
PB
PS PM
PH
PB
PB
PB
...
1
...
...
..... ...
PB PB
0
...
PB PB
O.
(c)
Formally speaking, the following parameters are chosen as fuzzy inputs; the difference, SE (-00,+00), between the expected holding costs in both queues given by (1), and the customer arrival rates AtE [O,,ut) of class i, i=1,2. The decision, d= I ,0, is the fuzzy output where 1 indicates that an arriving customer of class 1 is transferred to queue 2 and 0 that the customer is denied the transfer. Based on the previous discussions, when s is negative, an arriving class 1 customer is never allocated to queue 2. Therefore, a crisp rule is first written for the case SE(-O'J,O), if s is negative, then d is O. Henceforth, we will only focus on the cases where s is not
... ...
PB
J.12
In other words, the difference between the customerlg expected holding cost in both queues plays a key role in the course of decision making. This is way we approach the problem from the individually optimal criterion. To achieve the socially optimal goal; we should consider the effects of the policy on the whole system, or equivalently on the customers of both classes to arrive later. Specifically, if no additional customers arrive, the individually optimal behavior is also socially optimal, otherwise, usually it is not socially optimal. The arrival rates of both classes play an important role in bridging these two criteria.
....
PM
1 1
From expression (1) and the fuzzy rule in Table I, if PS and Al is ZO and A2 is 20) then d is 1:> it is concluded that PS for the input s in the fuzzy rule base with membership grade 1.0 is fixed at C. s is
1
5424
Copyright 1999 IFAC
ISBN: 0 08 043248 4
FUZZY ASSIGNMENT OF CUSTOMERS FOR A QUEUEING NETWO...
Optirnality is a topic that needs further elaboration. Finding an analytical expression for the optimal policy is generally difficult for nonlinear systems due to the lack of suitable analytical descriptions. This is one of the reasons why we use fuzzy control when we deal with nonlinear and complex systems. The price is the difficulty to explicitly prove optimality. An alternative method of analyzing optimality for fuzzy control systems is to remove the clear distinction between optimal and nonoptimal, that is allow a degree of optimality in [0,1]. We then have a linguistic statement of optimality such as H very optimal" or "optimal') or "somewhat optima]". The design methods of a fuzzy logic controller are based on the expert~s experience and knowledge or on a self-learning process which needs the experience of a human to . enable the fuzzy control system to attain good performance. Therefore, the evaluation criteria should focus on the designer's arguments for constructing fuzzy controllers follo\ving a human way of thinking, but not classical proofs. In a sense, no human is perfect, hence no fuzzy control system is '~very optimal" with grade 1. This is why fuzzy control is an approximation technique, which is also justified by the term ~'fllZZY". Throughout this paper, we equip the fuzzy controller with reasonable arguments, and therefore it can be called '~very optimal" with a high grade.
4. A NUMERICAL EXAMPLE
14th World Congress ofIFAC
(b) Two numbers M and .J.-~l are arbitrarily define as the limits for Si and S2 respectively, where the search is stopped. If these limits are too small, it runs the risk of missing the switching points, whereas if they are too large, we waste computational time, The proper size of these quantities is a matter of experience and experimentation. (c) The algorithm is started from an initial state S)=S2=O.
(d) According to the current Sl and S2, the fuzzy input s is first calcuJated. (e) Using the current s as well as the given A.] and A.2 as crisp inputs, the decision d is determined via fuzzification, fuzzy inference and de-fuzzification.
For example, let us assume that the current numbers of queueing customers of class 1 and 2 are sl=3 and s2=2. According to (1), it is obtained that s=3x2xl/l . . 2xlxl/l=4_, which should be scaled down to 1.6. It can be seen from Figure 3(a) that s corresponds to ZO with grade 0.47 and PS with grade 0.87 and PM with grade 0.2. In addition, A.)=O.8 corresponds to PS \vith grade 0,06 and PM with grade 0.73 and PB with grade 0.6, and 2 2=0.5 corresponds to PS with grade 0.67 and PM with grade 0.67, respectively. According to the fuzzy rule base (Table 1) and Mamdani implication which is associated with the min-operation, the fuzzy decisions d are formulated as follows. -
If s is ZO with grade 0.46 and A.] is PS with grade 0.06 and A.2 is PS with grade 0.67 then d is 0 with grade 0.06. Ifs is PS \\~ith grade 0.86 and Al is PS with grade 0.06 and A,2 is PS with grade 0.67, then d is 1 with grade 0.06. If s is PM with grade 0.2 and Al is PS with grade O~06 and ..1.2 is PS with grade 0.67, then d is 1 with grade 0.06. If s is ZO with grade 0.46 and Al is PM with grade 0.73 and A2 is PS with grade O.67~ then dis 1 with grade 0.46. t
The network shown in Figure 2 with arrival control to tv",'o queues in parallel is examined. The parameters are as follows: arrival rates ),,=0.8, A2=O.5, service rates jJI=J12= I, assignment cost C=5> and holding costs per customer per unit h.=2 and hi==l.
-
-
The optimal policy for d is determined from the architecture of the fuzzy logic contra Her (refer to Figure 1). Th e algorithm is outlined as fo lIows.
-
(a) According to the given infonnation, the scaling factors for the fuzzy inputs s, Al and ..1.2 in the rule base is determined.
~
If s is PM with grade 0.2 and }.. 1 is PM with grade 0.73 and A.2 is PM with grade 0.67, then d 1S 1 with grade 0.2. - If s is ZO with grade 0.46 and A) is PB with grade 0.6 and A2 is PM with grade 0.67, then d is 1 with grade 0.46. - I f s is PS with grade 0.86 and 2 1 is PB with grade 0.6 and A2 is PM with grade 0.67, then d is 1 with grade 0.6.
For this example~ C=5 which corresponds to PS for s with membership grade 1.0 in the rule base. From Figure 3(a), PS for s with Inembership grade 1.0 is 2 in the universe of discourse. Hence the scaling factor for s in the rule base is obtained which is Ps=O.4. Similarly, it is obtained that PA.L=PJ..2=6. All these calculations are automatically implemented by the computer.
5425
Copyright 1999 IFAC
ISBN: 0 08 043248 4
FUZZY ASSIGNMENT OF CUSTOMERS FOR A QUEUEING NETWO...
~
If s is PM with grade 0.2 and Al is PB with grade 0.6 and A..2 is PM with grade O~67, then d is 1 with grade 0.2.
Each of the two inputs sand A[ has 3 fuzzy sets whereas A2 has 2 fuzzy sets, hence 3x3 x2= 18 fuzzy decisions are fired as shown above. From Figure 3(c), the peak values of the decision dare e(l)=O, e(2)= 1, , e(l8)=1 and the heights of the decisions are .fi=O.06, h=O.06, ...... , Jis=O.2. By the height method of defuzzification, the crisp output d* is given by
~e(X) f d*
x
_x_;::_l- - -
~fx
Then the decision is d=l, which means that the arriving class 1 customer should be transferred to queue 2. (f) Plot the decision d in the two dimensional plane
of Sf and S2, (g) If the current s2=M, go to (h), otherwise let S2=S2+ 1 and go to (d), (h) If the current Sl=JV) the calculations stop, otherwise let SI=Sl+1 and go to (d). Each rule in the rule base expresses an optimal decision in a certain situation. Therefore the resulting decisions from all possible individual situations are optimal in the intuitive sense of fuzzy control. Following this algorithm~ the optimal control policy shown in Figure 4 is obtained, which is of the switchover structure in the two dimensional state space of SI and S2 proven by Koyanagi and Kawai (1985). SI
•
.5
•
•
•
• •
•
4
•
3
•
•
2
•
• •
•
•
•
•
•
•
•
•
• •
0
0
•
•
• •
•
• 0
0
0
0
0
0
0
0
0
0
0
0 Q
2
3
4
j
queue 1 is smaller than the length of queue 2. This is because, in this example, the holding cost per customer per unit time in queue 1 is greater than that in queue 2 and the arrival rate of class 1 customers is greater than the corresponding rate of class 2 customers.
5. CONCLUSIONS This paper considers the optimal assignment of arrivals to two heterogeneous servers in parallel by means offuzzy logic. Our results not only verify the switch structure of the optimal policy but also specify it.
(2)
x==l
6
14th World Congress of IFAC
6
1
8
9
In this paper, the socially optimal policy is determined with the aid of individually optimal behaviors. Our approach provides a ne~' brjdge between the socially and individually optimal criteria. It appears that this model can be extended to a variety of unsolved control problems with general distributions, which lead to non-Markovian systems, and analysis is rather hopeless. Finally~ the fuzzy logic techniques seem promising in controlling complex netvlorks. All these problems are topics for future research.
REFERENCES _Driankov, D. H. Hellendoom and M. _Reinfrank,
(1993). An Introduction to Fuzzy
C.~ontrol.
Springer-V erlag~ Berlin, N e\\' York. Hassin, R. (1985). On the optimality of first come last served queues. Econometrica, vol. 53, pp. 201-202,. Phillis, Y. A. and R. Zhang (1999) Fuzzy service rate control of queueing systems. To appear in IEEE Trans. Systems, Man} and Cybernetics in August 1999. Stidham, S. Jr. and R. Weber (1993) A survey of Markov decision models for control of networks of queues. Queueing Systems, vol. 13, pp. 291314.
$2
Figure 4: Optilnal policy The system pays an assignment cost for transferring a class 1 customer to queue 2, which suggests that this action should be rarely taken. However, we see from Figure 4 that an arriving class 1 customer can be transferred to queue 2 even when the length of
Zhang, R. and Y. A. Phillis (1999a) A fuzzy approach to the flow control problems. To appear in J Intelligent and Fuzzy Systems. Zhang, R. and Y. A. Phillis (1999b) Fuzzy control of two-station queuejng networks ,"vith two types of customers. To appear in J, Intelligent and Fuzzy Systelns. Zhang, R~ and Y. A. Phillis (1999c) Fuzzy control of queueing systems \\dth heterogeneous servers. To appear in IEEE Trans. Fuzzy Le:;ystems.
5426
Copyright 1999 IFAC
ISBN: 0 08 043248 4
14th World Congress of IFAC
K-3e-11-2
Copyright© 1999 IFAC 14th Triennial World Congress~ Beijing, P.R. China
FUZZY ASSIGNMENT OF CUSTOMERS FOR A QUEUEING NETWORK Runtong Zhang*
Yannis A. Phil1is~
*College ofEconomics and Administration Northern Jiaotong University Beijing 100044 People's Republic ofChina E-mail: [email protected]
Xiaomin Zhu*
~ Department ofProduction Engineering and .1Uanagemenl
Technical University
oj~Crete
Chania 73100, Greece E-mail: [email protected]
Abstract: This paper considers a queueing network with two parallel heterogeneous servers. Each server has its own queue and customers arrive at each queue according to independent Poisson processes. Servjce times are independent and exponentiaJly distributed. When a customer arrives at queue 1, the customer can be transferred to queue 2 by paying an assignment cost. Each customer in queue pays a holding cost per unit time. The objective is to dynamically determine the optimal assignment policy so as to minimize the average cost. In this paper, a novel approach is presented. using fuzzy control to solve the problem. Copyright© 1999 IFAC
Key words: fuzzy control, queueing networks, path planning
comprehensive discussion on optimal control of queueing networks can be found in the survey papers of Stidham and Weber (1993). Recently,
1. INTRODUCTION This paper considers an optimal control problem for a parallel queueing network with rnro heterogeneous servers using the fuzzy logic technique (Driankov et. al~~ 1993). The proposed model is an ac.yclic. queueing network consisting of two parallel queues. Each queue has its own arriving customers and one stream of customers upon arrival can be transferred to another queue. The system objective is to dynamically determine the optimal assignment policy so as to minimize the average cost. This is a simple model of computer or communication networks, and has been examined by Koyanagi and Kawai (1985). However, their ""ork investigates only the structure whereas the present work specifies the optimal policy.
Zhang and Phillis (1999, 1999a, 1999b, 1999c) proposed a new method using fuzzy logic to solve queueing control problems and show via simulation that the new method generalizes existing solutions and also efficiently solves problems intractable with classical methods. This approach signals a departure from classical techniques. Different criteria of individual or social optimization are faced, when we deal with problems of admission and routing control. The fonner criterion depends on the customer' s o"~{n benefits, while the latter views the system's performance as a whole. It is believed that the policy imp)emented by self-interested individuals does not lead in general to the best social outcome. Such questions have attracted considerable interest
Optimal control of queueing systems has been extensively investigated in the literature. A
5422
Copyright 1999 IFAC
ISBN: 0 08 043248 4
FUZZY ASSIGNMENT OF CUSTOMERS FOR A QUEUEING NETWO...
among econonlists and operations researchers. For discussions on individual vs. socially optimal
14th World Congress of IFAC
;Queue I L~ 1 Departures
Class 1 .Arrivals .11
----/.........
prob]ems~
see the review paper (Stidham and W eber~ 1993). Research along this line is continuing. Optimal policies for individual
1I11~
....
1It--G
...
I
1
\
----~-........
customers are often easy to obtain and take on simple and explicit fonns, while socially optimal
Class 2 ..A.rrivals .L1
Queue 2
Server 2 Departures
policies which are of greater practical importance, Figure 2: Queueing network
often defy simple analysis. This paper uses a socially optimal criterion and use the term '~optimar' in the sense of "socially optimaP'. This
applications. To transform the fuzzy output into an
usable
paper detennines socially optimal with the aid of individually optimal policies and hence, the fuzzy logic approach also contributes towards exploring the relationships between individually and socially optimal policies.
the
height
method
of
2. PROBLEM DESCRIPTION The queueing network considered is shown in Figure 2. This network consists of two heterogeneous servers
At each decision epoch, the fuzzy controller makes a decision by emulating a skilled human operator.
in parallel called server i) i~l )2, each with its own queue of unlimited capacity. There are two classes
Specifically, based on the current state of the system, an inference engine equipped with a fuzzy rule base determines an on·line decision to adjust the system behavior in order to guarantee that the system is optimal in the sense of minimizing a given cost. This is thejUzzy logic controller (Figure 1).
of customers. Customers of class i, i~l )2, join queue i to be served by server i. The length of queue i is observable. However, a customer of class 1\ upon arrival) can be transferred to queue 2 by paying an assignment cost C!I ifhe expects to incur a lower cost by this transfer. It is assumed that customers of class i arrive at queue i in a Poisson stream with constant rate Ai. Successive services by
In this paper, all of the membership functions for the fuzzy sets are chosen to be triangular. This is because the parametrjc~ functional descriptions of triangular membership functions are the most economic ones. In addition, it has been proven that such membership functions can approximate most other membership functions. To present the fuzzy roles, ZO, PS, PM, PB are used to indicate the "zero1)~ "positive smaH", "positive medium" and "positive bigH unless otherwise explained.
server
i
are
independent
and
exponentially
distributed with mean 11 fli regardless of the class of customer, where Aj
Queuelng systems are simulated and controlled in C
language. Mamdani ilnplication is used to represent "if-then'~
rules., because it is precise, cornputationally simple and fits various practical The Queueing Syst~m
one,
countIng.
The fuzzy control method aims at finding optimal policies in computationally efficient ways. l~hroughout this paper, the term "po1icy~) is used in the sense of ' 'deterministic stationary policy'~~
Jurivals
crisp
defuzzification is used because this method is also simple, fast, and has the advantage of weio-ht . 0
lengths. However the explicit determination of the optimal policy remains an open problem.
Departures
3. ARCHITECTURE OF THE FUZZY LOGIC
CONTROLLER Without loss of generality, the decision epochs at which arriving customers of class 1 are assigned c,an be restricted to their arrival time. The state of the system at the decision epochs can be described by si=O)1,2, ... , where Si is the number of customers
Figure 1: The fuzzy queueing control system
5423
Copyright 1999 IFAC
ISBN: 0 08 043248 4
FUZZY ASSIGNMENT OF CUSTOMERS FOR A QUEUEING NETWO...
14th World Congress of IFAC
257
PS
in queue i~ Therefore SI+S2 is the total number of queueing customers in the system~ An arriving customer of class 1 can be transferred to queue 2 by paying an assignment cost. It is optimal for this customer to join queue 2 only when his expected holding cost in queue 1 is not smaller than that in queue 2 plus the assignment cost, that is
PM
PE
O.
0
2
3
4
5
(a)
1
s=sl-h 1 - - - -
JJ 1
1
s2·h2·--~C
PS
(1)
O.
Figure 3: Membership functions
A fuzzy rule base is developed as in Table 1. We choose 4 fuzzy sets for each of these 3 inputs and this rule base consists of 4 3 =64 rules. The membership functions for the fuzzy inputs s are shown in Figure 3(a)~ for Az, they are shown in Figure 3(b), and for the fuzzy output d they are shown in Figure 3(c). The universe of discourse for the fuzzy input s is [0, +aJ) and fOT A.i~ i=] ,2, they are [0,6J. The universe of discourse for the fuzzy output d is [0,1]. The establishment of the fuzzy rule base relies on the following arguments: (i) Because of the previous discussion, the greater the difference between the expected holding costs in both queues, the easier it is to decide d=l for an arriving class 1 customer. (ii) A higher arrival rate of class 1 customers strengthens the decision d= 1. indeed, if the manager of the system knew that additional class 1 customers were arriving at a very high rate, which means that the length of queue I would increase very fast, he would like to transfer the present customer to queue 2. This is so because the difference bernreen the expected holding costs in both queues would also increase. (Hi) A higher arrival rate of class 2 customers weakens the decision d= t. A higher arrival rate of class 2 customers means that the difference between the expected holding costs in both queues is becoming slnaller. Based on propositions (i)-(Hi}, the fuzzy rule base is ready.
smaller than zero. Table 1: R pIe base
s
A.t
A,2
d
ZO
ZO
ZO
0
PS
ZO
ZO
PM PB
ZO ZO
ZO ZO
1 1
... ...
...
ZO
PB
PS PM
PH
PB
PB
PB
...
1
...
...
..... ...
PB PB
0
...
PB PB
O.
(c)
Formally speaking, the following parameters are chosen as fuzzy inputs; the difference, SE (-00,+00), between the expected holding costs in both queues given by (1), and the customer arrival rates AtE [O,,ut) of class i, i=1,2. The decision, d= I ,0, is the fuzzy output where 1 indicates that an arriving customer of class 1 is transferred to queue 2 and 0 that the customer is denied the transfer. Based on the previous discussions, when s is negative, an arriving class 1 customer is never allocated to queue 2. Therefore, a crisp rule is first written for the case SE(-O'J,O), if s is negative, then d is O. Henceforth, we will only focus on the cases where s is not
... ...
PB
J.12
In other words, the difference between the customerlg expected holding cost in both queues plays a key role in the course of decision making. This is way we approach the problem from the individually optimal criterion. To achieve the socially optimal goal; we should consider the effects of the policy on the whole system, or equivalently on the customers of both classes to arrive later. Specifically, if no additional customers arrive, the individually optimal behavior is also socially optimal, otherwise, usually it is not socially optimal. The arrival rates of both classes play an important role in bridging these two criteria.
....
PM
1 1
From expression (1) and the fuzzy rule in Table I, if PS and Al is ZO and A2 is 20) then d is 1:> it is concluded that PS for the input s in the fuzzy rule base with membership grade 1.0 is fixed at C. s is
1
5424
Copyright 1999 IFAC
ISBN: 0 08 043248 4
FUZZY ASSIGNMENT OF CUSTOMERS FOR A QUEUEING NETWO...
Optirnality is a topic that needs further elaboration. Finding an analytical expression for the optimal policy is generally difficult for nonlinear systems due to the lack of suitable analytical descriptions. This is one of the reasons why we use fuzzy control when we deal with nonlinear and complex systems. The price is the difficulty to explicitly prove optimality. An alternative method of analyzing optimality for fuzzy control systems is to remove the clear distinction between optimal and nonoptimal, that is allow a degree of optimality in [0,1]. We then have a linguistic statement of optimality such as H very optimal" or "optimal') or "somewhat optima]". The design methods of a fuzzy logic controller are based on the expert~s experience and knowledge or on a self-learning process which needs the experience of a human to . enable the fuzzy control system to attain good performance. Therefore, the evaluation criteria should focus on the designer's arguments for constructing fuzzy controllers follo\ving a human way of thinking, but not classical proofs. In a sense, no human is perfect, hence no fuzzy control system is '~very optimal" with grade 1. This is why fuzzy control is an approximation technique, which is also justified by the term ~'fllZZY". Throughout this paper, we equip the fuzzy controller with reasonable arguments, and therefore it can be called '~very optimal" with a high grade.
4. A NUMERICAL EXAMPLE
14th World Congress ofIFAC
(b) Two numbers M and .J.-~l are arbitrarily define as the limits for Si and S2 respectively, where the search is stopped. If these limits are too small, it runs the risk of missing the switching points, whereas if they are too large, we waste computational time, The proper size of these quantities is a matter of experience and experimentation. (c) The algorithm is started from an initial state S)=S2=O.
(d) According to the current Sl and S2, the fuzzy input s is first calcuJated. (e) Using the current s as well as the given A.] and A.2 as crisp inputs, the decision d is determined via fuzzification, fuzzy inference and de-fuzzification.
For example, let us assume that the current numbers of queueing customers of class 1 and 2 are sl=3 and s2=2. According to (1), it is obtained that s=3x2xl/l . . 2xlxl/l=4_, which should be scaled down to 1.6. It can be seen from Figure 3(a) that s corresponds to ZO with grade 0.47 and PS with grade 0.87 and PM with grade 0.2. In addition, A.)=O.8 corresponds to PS \vith grade 0,06 and PM with grade 0.73 and PB with grade 0.6, and 2 2=0.5 corresponds to PS with grade 0.67 and PM with grade 0.67, respectively. According to the fuzzy rule base (Table 1) and Mamdani implication which is associated with the min-operation, the fuzzy decisions d are formulated as follows. -
If s is ZO with grade 0.46 and A.] is PS with grade 0.06 and A.2 is PS with grade 0.67 then d is 0 with grade 0.06. Ifs is PS \\~ith grade 0.86 and Al is PS with grade 0.06 and A,2 is PS with grade 0.67, then d is 1 with grade 0.06. If s is PM with grade 0.2 and Al is PS with grade O~06 and ..1.2 is PS with grade 0.67, then d is 1 with grade 0.06. If s is ZO with grade 0.46 and Al is PM with grade 0.73 and A2 is PS with grade O.67~ then dis 1 with grade 0.46. t
The network shown in Figure 2 with arrival control to tv",'o queues in parallel is examined. The parameters are as follows: arrival rates ),,=0.8, A2=O.5, service rates jJI=J12= I, assignment cost C=5> and holding costs per customer per unit h.=2 and hi==l.
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The optimal policy for d is determined from the architecture of the fuzzy logic contra Her (refer to Figure 1). Th e algorithm is outlined as fo lIows.
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(a) According to the given infonnation, the scaling factors for the fuzzy inputs s, Al and ..1.2 in the rule base is determined.
~
If s is PM with grade 0.2 and }.. 1 is PM with grade 0.73 and A.2 is PM with grade 0.67, then d 1S 1 with grade 0.2. - If s is ZO with grade 0.46 and A) is PB with grade 0.6 and A2 is PM with grade 0.67, then d is 1 with grade 0.46. - I f s is PS with grade 0.86 and 2 1 is PB with grade 0.6 and A2 is PM with grade 0.67, then d is 1 with grade 0.6.
For this example~ C=5 which corresponds to PS for s with membership grade 1.0 in the rule base. From Figure 3(a), PS for s with Inembership grade 1.0 is 2 in the universe of discourse. Hence the scaling factor for s in the rule base is obtained which is Ps=O.4. Similarly, it is obtained that PA.L=PJ..2=6. All these calculations are automatically implemented by the computer.
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Copyright 1999 IFAC
ISBN: 0 08 043248 4
FUZZY ASSIGNMENT OF CUSTOMERS FOR A QUEUEING NETWO...
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If s is PM with grade 0.2 and Al is PB with grade 0.6 and A..2 is PM with grade O~67, then d is 1 with grade 0.2.
Each of the two inputs sand A[ has 3 fuzzy sets whereas A2 has 2 fuzzy sets, hence 3x3 x2= 18 fuzzy decisions are fired as shown above. From Figure 3(c), the peak values of the decision dare e(l)=O, e(2)= 1, , e(l8)=1 and the heights of the decisions are .fi=O.06, h=O.06, ...... , Jis=O.2. By the height method of defuzzification, the crisp output d* is given by
~e(X) f d*
x
_x_;::_l- - -
~fx
Then the decision is d=l, which means that the arriving class 1 customer should be transferred to queue 2. (f) Plot the decision d in the two dimensional plane
of Sf and S2, (g) If the current s2=M, go to (h), otherwise let S2=S2+ 1 and go to (d), (h) If the current Sl=JV) the calculations stop, otherwise let SI=Sl+1 and go to (d). Each rule in the rule base expresses an optimal decision in a certain situation. Therefore the resulting decisions from all possible individual situations are optimal in the intuitive sense of fuzzy control. Following this algorithm~ the optimal control policy shown in Figure 4 is obtained, which is of the switchover structure in the two dimensional state space of SI and S2 proven by Koyanagi and Kawai (1985). SI
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queue 1 is smaller than the length of queue 2. This is because, in this example, the holding cost per customer per unit time in queue 1 is greater than that in queue 2 and the arrival rate of class 1 customers is greater than the corresponding rate of class 2 customers.
5. CONCLUSIONS This paper considers the optimal assignment of arrivals to two heterogeneous servers in parallel by means offuzzy logic. Our results not only verify the switch structure of the optimal policy but also specify it.
(2)
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In this paper, the socially optimal policy is determined with the aid of individually optimal behaviors. Our approach provides a ne~' brjdge between the socially and individually optimal criteria. It appears that this model can be extended to a variety of unsolved control problems with general distributions, which lead to non-Markovian systems, and analysis is rather hopeless. Finally~ the fuzzy logic techniques seem promising in controlling complex netvlorks. All these problems are topics for future research.
REFERENCES _Driankov, D. H. Hellendoom and M. _Reinfrank,
(1993). An Introduction to Fuzzy
C.~ontrol.
Springer-V erlag~ Berlin, N e\\' York. Hassin, R. (1985). On the optimality of first come last served queues. Econometrica, vol. 53, pp. 201-202,. Phillis, Y. A. and R. Zhang (1999) Fuzzy service rate control of queueing systems. To appear in IEEE Trans. Systems, Man} and Cybernetics in August 1999. Stidham, S. Jr. and R. Weber (1993) A survey of Markov decision models for control of networks of queues. Queueing Systems, vol. 13, pp. 291314.
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Figure 4: Optilnal policy The system pays an assignment cost for transferring a class 1 customer to queue 2, which suggests that this action should be rarely taken. However, we see from Figure 4 that an arriving class 1 customer can be transferred to queue 2 even when the length of
Zhang, R. and Y. A. Phillis (1999a) A fuzzy approach to the flow control problems. To appear in J Intelligent and Fuzzy Systems. Zhang, R. and Y. A. Phillis (1999b) Fuzzy control of two-station queuejng networks ,"vith two types of customers. To appear in J, Intelligent and Fuzzy Systelns. Zhang, R~ and Y. A. Phillis (1999c) Fuzzy control of queueing systems \\dth heterogeneous servers. To appear in IEEE Trans. Fuzzy Le:;ystems.
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Copyright 1999 IFAC
ISBN: 0 08 043248 4