fuzzy-based network bandwidth design under demand ... - Springer Link

J Syst Sci Complex (2010) 23: 61–70

FUZZY-BASED NETWORK BANDWIDTH DESIGN UNDER DEMAND UNCERTAINTY∗ Lean YU · Wuyi YUE · Shouyang WANG

DOI: 10.1007/s11424-010-9272-5 Received: 5 May 2009 / Revised: 18 July 2009 c The Editorial Office of JSSC & Springer-Verlag Berlin Heidelberg 2010 Abstract In communication networks (CNs), the uncertainty is caused by the dynamic nature of the traffic demands. Therefore there is a need to incorporate the uncertainty into the network bandwidth capacity design. For this purpose, this paper developed a fuzzy methodology for network bandwidth design under demand uncertainty. This methodology is usually used for offline traffic engineering optimization, which takes a centralized view of bandwidth design, resource utilization, and performance evaluation. In this proposed methodology, uncertain traffic demands are first handled into a fuzzy number via a fuzzification method. Then a fuzzy optimization model for the network bandwidth allocation problem is formulated with the consideration of the trade-off between resource utilization and network performance. Accordingly, the optimal network bandwidth capacity can be obtained by maximizing network revenue in CNs. Finally, an illustrative numerical example is presented for the purpose of verification. Key words Communication networks, demand uncertainty, fuzzy set theory, network bandwidth design, network optimization.

1 Introduction Network bandwidth design under demand uncertainty is one of the most important issues in communication networks (CNs), which is closely related to network resource utilization, network performance stability as well as network revenue management. In many past studies, the optimization design of network bandwidth was usually formulated as a deterministic multicommodity flow (MCF) model, where traffic demand of each network channel was assumed to be a deterministic quantity[1−5] . However, traffic connections in CNs tend to be set up on demand dynamically with the rapid increase of network users. Due to such uncertain environment, the amount of network Lean YU Institute of Systems Science, MADIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China. Email: [email protected]. Wuyi YUE Department of Intelligence and Informatics, Konan University, Kobe 658-8501, Japan. Email: [email protected]. Shouyang WANG Institute of Systems Science, MADIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China. Email: [email protected]. ∗ This work is partially supported by the grants from the National Natural Science Foundation of China, the Knowledge Innovation Program of the Chinese Academy of Sciences, and the GRANT-IN-AID FOR SCIENTIFIC RESEARCH (No. 19500070) and MEXT.ORC (2004-2008), Japan.

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LEAN YU · WUYI YUE · SHOUYANG WANG

bandwidth design required for carrying demand must satisfy the dynamic traffic demands[6] . Consequently, the previous deterministic MCF methodologies may not necessarily be suitable for network bandwidth design under demand uncertainty. In the presence of uncertainty in network traffic demand, we cannot know the exact network traffic demand or network traffic load. It is therefore difficult for network service providers to design an optimal network bandwidth capacity for all possible traffic demands. On the one hand, if a large network bandwidth capacity is designed, then the possibility that the network bandwidth capacity is fully utilized will decrease. Furthermore, the overprovisioned network bandwidth capacity will lead to idle network resources and some extra network maintenance costs. On the other hand, to ensure effective utilization of designed network bandwidth capacity, the provisioned network bandwidth capacity should be small, but with a small capacity the network may not satisfy the possible traffic demands and thus increasing a risk of reduction of network total revenue. Also, less-provisioned network bandwidth capacity may depress network service performance in CNs such as network congestion or traffic jam. For these reasons, there is a need for network service providers to design an optimal network bandwidth capacity under the environment of traffic demand uncertainty[1] . In the past studies, the uncertain demand is usually treated as a stochastic variable to design an optimal network bandwidth capacity in CNs. Typical examples include Mitra and Wang[3,7−8] and Wu et al.[2,9−10]] . But in many practical applications, such as multimedia network design, the network demands are often considered as possibilistic situations where the demands usually vary within the confidence interval due to uncertain environment. In such cases, uncertain network demand can be reasonably treated as a fuzzy number corresponding to the confidence interval. Based on the fuzzification treatment for network demands, this paper proposes a fuzzy methodology to optimize network bandwidth capacity in CNs. In the proposed methodology presented in this paper, uncertain traffic demand is first handled as a fuzzy number using a fuzzification method. Then a fuzzy optimization methodology for the network bandwidth design is provided based on the fuzzy traffic demand. As a consequence, some important results about the optimal network bandwidth design are obtained in terms of the fuzzy optimization methodology. The main purpose of this paper is to design an optimal network bandwidth capacity so that the maximum network profits or maximum network revenues from serving traffic demands are obtained. The remainder of this paper is organized as follows. Section 2 provides some background knowledge about fuzzy set theory and system models of CNs. In Section 3, a fuzzy optimization methodology for network bandwidth design under demand uncertainty is formulated and some important results are reported. For illustration purpose, a simple numerical example is presented in Section 4. And some concluding remarks are drawn in Section 5.

2 Background In this section, some preliminaries on fuzzy set theory and system models of CNs are presented. 2.1 Basic Concepts of Fuzzy Number and Fuzzy Integral In order to apply fuzzy set theory to solve the network bandwidth design problem under demand uncertainty, some preliminary definitions about fuzzy number and fuzzy integral are presented. Interested readers can refer to [11] for more details about fuzzy sets theory.

FUZZY NETWORK BANDWIDTH DESIGN UNDER UNCERTAINTY

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is a fuzzy set of the real line R = (−∞, +∞) Definition 1 A triangular fuzzy number D whose membership function μD (B) has the following characteristics with −∞ < l < m < r < +∞: ⎧ B−l ⎪ ⎪ , l ≤ B ≤ m, ⎪ L(B) = ⎪ ⎨ m−l (1) μD (B) = ⎪ R(B) = r − B , m ≤ B ≤ r, ⎪ ⎪ r−m ⎪ ⎩ 0, otherwise. where L(B) and R(B) are the left-shape and right-shape functions, respectively, of the fuzzy l, m, and r are the left, middle and right values of defining different intervals for number D, fuzzy membership function. For example, as shown in Figure 1, when B ∈ [l, m], the fuzzy membership function is L(B); similarly, if B ∈ [m, r], the fuzzy membership function is R(B).

Figure 1 The membership function μD (B) of triangular fuzzy number D ∈ F, Definition 2 Let F be the family of fuzzy sets on the real number set R. For each D we have an α-cut or α-level set D(α) = {B|μD (B) ≥ α} = [D (α), D (α)] (0 ≤ α ≤ 1). l r be a fuzzy number whose membership function is given via Equation Definition 3 Let D of D can (1) and let λ ∈ [0, 1] be a predetermined parameter. The total λ-integral value Iλ (D) be defined as = (1 − λ)IL (D) + λIR (D), Iλ (D) (2) and IR (D) are the left and right integral values of defined, respectively as follows: where IL (D) = IL (D)

0

1

L−1 (α)dα,

= IR (D)

0

1

R−1 (α)dα,

(3)

where L−1 (α) and R−1 (α) are the inverse functions of L(B) and R(B), respectively. Actually, represents the area bounded by the left-shape function, the x-axis, the y-axis, and IL (D) represents the area bounded by the right-shape function, horizontal line μA = 1, and IR (D) the x-axis, the y-axis, and horizontal line μA = 1. Usually, the total λ-integral value of fuzzy number can be used to rank fuzzy numbers. Remark 1 The parameter λ ∈ [0, 1] in Equation (2) reflects decision-maker’s degree of optimism for market estimation, thus it is also called as “optimistic coefficient”. Usually a = IL (D) (λ = 0) and large λ indicates a high degree of optimism. In particular, I0 (D) I1 (A) = IR (A) (λ = 1) represent pessimistic and optimistic estimation viewpoints, respectively, = 0.5[IL (D) + IR (D)] (λ = 0.5) provides a comparison criterion to the moderately while I0.5 (A) optimistic decision-makers for fuzzy numbers[12] .


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2.2 Preliminaries on Communication Network Systems In CNs, a network system can usually be regarded as a collection of nodes and links. Similar to the previous descriptions in [1-2, 7, 9], let (N , L) be a CN system decomposed of nodes ni (ni ∈ N , 1 ≤ i ≤ N ) and links l (l ∈ L), where N is the total number of nodes and L is the total number of links in the CN system. For any link l, it has maximal network bandwidth capacity Bl max for serving traffic demands including voice, packet data, image and full-motion video. Let V be the set of all node pairs and n ∈ V denote an arbitrary node pair, where n = (ni , nj ) and ni , nj ∈ N. Usually, a link l between two nodes ni and nj can formulate a route r, but there may be more than one route to be routed for a node pair, we use R(n) to represent an admissible route set for n ∈ V. Denote the traffic load or traffic demand on an arbitrary node pair n = (ni , nj ) by dn , let bn be the amount of network bandwidth capacity provisioned to an arbitrary node pair n = (ni , nj ), andlet ξr (r ∈ R(n)) denote the amount of capacity provisioned on route r, then we have bn = ξr , where n ∈ V . r∈R(n)

In this paper, we consider the CN to be a whole system from a centralized view. Let B > 0 denote the amount of network bandwidth capacity provisioned in the CN system, then we have B = bn . Let D > 0 be the traffic demand in the whole CN system, then we n∈V have D = dn , which is characterized by a fuzzy number. Usually, a CN system can gain n∈V

its revenue by serving traffic demands to and from its users. Let a be the unit revenue by transmitting the traffic load or serving the traffic demands and c denote the unit cost for unit bandwidth capacity allocated in the CN system. If a network bandwidth capacity is designed to be too small to satisfy traffic demand, a unit penalty cost (i.e., actual and potential loss) s should be charged due to network congestion or traffic jam. If a network bandwidth capacity is initially designed to be too large, an extra unit maintenance cost h for idle bandwidth capacity should be considered in the design of the whole CN system. The objective of this paper is to design an optimal network bandwidth capacity for a planning CN system considering a trade-off between resource utilization and network performance. For this purpose, the current issue is how to design a reasonable network bandwidth capacity B to maximize the network revenue under the environment of demand uncertainty. In the next section, we come to solve this issue. To avoid unrealistic and trivial cases, we assume a > s > c > h > 0.

3 Fuzzy-Based Optimization Design for Network Bandwidth Capacity Under Demand Uncertainty In this section, we propose a fuzzy optimization methodology for the network bandwidth design in CN system and derive the optimal network bandwidth capacity under demand uncertainty by maximizing network profit. Let P (B, D) denote the total profit function by transmitting messages in CNs, where B is the network bandwidth capacity and D is the traffic demand. Assume that the traffic demand with a membership function described in Equation is uncertain, a triangular fuzzy number D (1) is used to describe the uncertain traffic demand. In order to obtain the maximum profit, the current problem is how to design a suitable network bandwidth capacity to satisfy all possible traffic demands so that the total network profit is maximized. Suppose that network bandwidth B is given, then the total profit function of the network can be formulated as = aD − cB − h max{0, B − D} − s max{0, D − B}, P (B, D)

(4)


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is the total profit function in the fuzzy sense associated with a fuzzy demand where P (B, D) D, a and c are the unit revenue for serving traffic demands and the unit cost for each network bandwidth capacity allocation, respectively. h is the unit maintenance cost for idle network bandwidth capacity. s is the unit penalty cost for each unsatisfied traffic demand or the actual and potential loss caused by network congestion or traffic jam, and B is the network bandwidth capacity, which is a decision variable in this problem. We would like to know, in the presence of demand uncertainty, how to design an optimal network bandwidth capacity to gain the maximum profit in the CN systems. is dependent From Equation (4), it is easy to find that the total profit function P (B, D) on the uncertain demand D. Thus, the total profit function P (B, D) is also a fuzzy number, that is, D and P(B, D) have which has the same membership grade as the fuzzy demand D, the same shape of the membership function, as illustrated in Figure 1. From Definition 1 and According to Definition 2, we let P (α) denote the α-cut of P(B, D). Figure 1, we are easy to find that there are two typical scenarios with the consideration of the values of traffic demand D. In this paper, when the market estimation for traffic demand is optimistic, it is suitable for the network bandwidth capacity to select the right-shape function R(B) (m ≤ B ≤ r). On the contrary, if the market estimation is a pessimistic scenario, selecting left-shape function L(B) (l ≤ B ≤ m) as the range of designing network bandwidth capacity is suitable. According to the two scenarios, we have the following two propositions. Proposition 1 If the traffic demand is estimated to be a pessimistic scenario, then the can be represented as α-cut of P (B, D) ⎧ [aL−1 (α) − cB − h(B − L−1 (α)), aR−1 (α) − cB − s(R−1 (α) − B)], ⎪ ⎪ ⎨ 0 ≤ α ≤ L(B), P (α) = (5) −1 −1 −1 [aL (α) − cB − s(L (α) − B), aR (α) − cB − s(R−1 (α) − B)], ⎪ ⎪ ⎩ L(B) ≤ α ≤ 1. Proof In the pessimistic scenario, the network bandwidth capacity B lies between l and m, the membership grade μP is the same as L(·). If α ≤ L(B), then the lower bound of the is aL−1 (α) − cB − h(B − L−1 (α)) because the network bandwidth capacity is α-cut of P(B, D) greater than the traffic demand with an amount (B − L−1 (α)). Also, the upper bound of the is aR−1 (α) − cB − s(R−1 (α) − B) because the network bandwidth capacity α-cut of P (B, D) does not satisfy the traffic demand. If α ≥ L(B), then the network bandwidth capacity is always insufficient for the traffic demand defined in the α-cut. Thus, the lower bound of α-cut is aL−1 (α) − cB − s(L−1 (α) − B) and the upper bound of the α-cut of P (B, D) of P (B, D) −1 −1 is aR (α) − cB − s(R (α) − B). Thus, when the network bandwidth capacity follows a can be described as Equation (5). left-shape function L(B), the α-cut of P (B, D) Likewise, when the network bandwidth capacity B lies between m and r, a similar proposition can be obtained, as shown below. Proposition 2 If the traffic demand is estimated to be an optimistic scenario, then the can be represented as α-cut of P (B, D) ⎧ [aL−1 (α) − cB − h(B − L−1 (α)), aR−1 (α) − cB − s(R−1 (α) − B)], ⎪ ⎪ ⎨ 0 ≤ α ≤ R(B), P (α) = (6) −1 −1 −1 [aL (α) − cB − h(B − L (α)), aR (α) − cB − h(B − R−1 (α))], ⎪ ⎪ ⎩ R(B) ≤ α ≤ 1. Proof In the optimistic scenario, the network bandwidth capacity B lies between m and r, the membership grade μP is the same as R(·). If α ≤ R(B), then the lower bound of the α-cut


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is aL−1 (α)− cB − h(B − L−1(α)) because the network bandwidth capacity is greater of P(B, D) is aR−1 (α) − cB − than the traffic demand. Also, the upper bound of the α-cut of P (B, D) −1 s(R (α) − B) because the network bandwidth capacity is smaller than the traffic demand. If α ≥ R(B), then the network bandwidth capacity is always greater than the traffic demand in is aL−1 (α) − cB − h(B − L−1 (α)) and the α-cut. Thus the lower bound of α-cut of P(B, D) is aR−1 (α) − cB − h(B − R−1 (α)). Thus when the the upper bound of the α-cut of P(B, D) can be network bandwidth capacity follows a right-shape function R(B), the α-cut of P (B, D) described as Equation (6). Now, the main task is to derive optimal network bandwidth capacity B ∗ from the α-cut in a CN system. As previously mentioned, the total profit P(B, D) is a fuzzy of P (B, D) number, it can be ranked by the existing ranking methods for ranking fuzzy numbers. The is the optimal network bandnetwork bandwidth capacity with the maximum profit P (B, D) width capacity to be designed. In the past studies, there were many ranking methods for fuzzy number ranking[13] . However, most of the methods require the explicit form of the membership functions of all fuzzy numbers to be ranked, which is impossible in some cases. The method of Yager[14] , which is later modified by Liou and Wang[15] , does not require knowing the knowledge of the membership functions and can thus be applied. Using the existing ranking methods for instance in Definition 3, we have the following theorem. Theorem 1 If the uncertain traffic demand is fuzzified into a triangular fuzzy number, then the optimal network bandwidth capacity B ∗ satisfies the following equation: λR(B ∗ ) − (1 − λ)L(B ∗ ) = 2λ −

2(s − c) . s+h

(7)

Proof According to Definition 3 and Equations (2), (3), (5), and (6), we can calculate the i.e., corresponding total λ-integral value of P (B, D), Iλ (P) = (1 − λ)I L (P ) + λIR (P ) L(B) {[aL−1 (α) − cB − h(B − L−1 (α))] + [aR−1 (α) − cB − s(R−1 (α) − B)]}dα = (1 − λ) 0 1 +(1 − λ) {[aL−1 (α) − cB − s(L−1 (α) − B)] + [aR−1 (α) − cB − s(R−1 (α) − B)]}dα R(B) L(B) +λ {[aL−1 (α) − cB − h(B − L−1 (α))] + [aR−1 (α) − cB − s(R−1 (α) − B)]}dα 0 1 +λ {[aL−1 (α) − cB − h(B − L−1 (α))] + [aR−1 (α) − cB − h(B − R−1 (α))]}dα R(B) L(B) L−1 (α)dα = (1 − λ) 2(s − c)B − B(s + h)L(B) + (a + h) 0

+(a − s) +(a − s)

1

L−1 (α)dα + (a − s)

L(B) R(B) 0

R−1 (α)dα + (a + h)

1

0

R−1 (α)dα + λ [−2(c + h)B + B(s + h)R(B)

1

R(B)

L−1 (α)dα+(a + h)

0

1

L−1 (α)dα.

(8) we can derive the optimal network bandUsing the above total λ-integral value of P (B, D), width capacity with the fuzzy demand. The first order derivative of Iλ (P ) with respect to B is


67

given as follows: ∂I(P) = (1 − λ)[2(s − c) − (s + h)L(B)] + λ[−2(c + h) + (s + h)R(B)]. ∂B

(9)

The second order derivative of Iλ (P) with respect to B is given as follows: ∂ 2 I(P ) = −(1 − λ)(s + h)L (B) + λ(s + h)R (B). ∂B 2

(10)

Since s and h are larger than zero, λ ∈ [0, 1], L(·) is an increasing function with L (B) > 0, R(·) is an decreasing function with R (B) < 0, and thus Equation (10) is negative and therefore can arrive at the the second condition of optimization is met. This indicates that P (B, D) ∗ maximum at optimal point B . From Equation (9), we can obtain Equation (7). If the L(B ∗ ) and R(B ∗ ) satisfy the definition of Equation (1), we have the following corollary. Corollary 1 If L(B ∗ ) = (B ∗ − l)/(m − l) and R(B ∗ ) = (r − B ∗ )/(r − m), then the optimal network bandwidth capacity B ∗ is B∗ = Proof

(1 − λ)l(r − m) + λr(m − l) + [2(s − c) − 2λ(s + h)](m − l)(r − m)/(s + h) . (1 − λ)(r − m) + λ(m − l)

(11)

According to the Equations (1) and (7), we have λ

r − B∗ B∗ − l 2(s − c) − (1 − λ) = 2λ − . r−m m−l s+h

(12)

By reformulation, the optimal network bandwidth capacity B ∗ can be represented as Equation (11). In terms of different optimistic coefficients λ, we have the following three theorems (Theorems 2–4) and three corollaries (Corollaries 2–4). Theorem 2 If decision-makers of network service providers have an optimistic market estimation, then the optimal network bandwidth capacity B ∗ with the fuzzy traffic demand can be calculated by 2(c + h) B ∗ = R−1 , f or (c + h) ≤ (s − c). (13) s+h Proof According to the Remark 1 in Section 2, we have λ = 1. Using Equation (7) and λ = 1, we have the following equation: R(B ∗ ) = 2 −

2(c + h) 2(s − c) = . s+h s+h

(14)

From Figure 1, it is easy to find that Equation (14) should lie between 0 and 1 so that the optimal network bandwidth capacity B ∗ lies between m and r. Since c, s, h are larger than zero, the 2(c + h)/(s + h) is always positive. The requirement of 2(c + h)/(s + h) ≤ 1 implies (c + h) ≤ (s − c). Hence the optimal network bandwidth capacity B ∗ is easily calculated, as shown in Equation (13). If the R(B ∗ ) satisfies the definition of Equation (1), we have the following corollary. Corollary 2 If R(B ∗ ) = (r − B ∗ )/(r − m), then the optimal network bandwidth capacity ∗ B is 2(c + h) (r − m), for (c + h) ≤ (s − c). (15) B∗ = r − s+h


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Combining R(B ∗ ) = (r − B ∗ )/(r − m) and Equation (14), the following equation 2(c + h) r − B∗ = . (16) R(B ∗ ) = r−m s+h By reformulation, the optimal network bandwidth capacity B ∗ can be represented as Equation (15). Actually, according to Equation (1) and the definition of α-cut of the fuzzy number, we have D(α) = [L−1 (α), R−1 (α)] = [l + α(m − l), r − α(r − m)]. From D(α), we can easily obtain the optimal network bandwidth capacity B ∗ as shown in Equation (15). In addition, if we use λ = 1 into Equation (11), the same results can be obtained. Besides the optimistic estimation, other two theorems (Theorems 3 and 4) and corollaries (Corollaries 3 and 4) for pessimistic estimation and moderately optimistic estimation can be obtained, respectively. Since the proofs of these theorems and corollaries are very similar to the proofs of Theorem 2 and Corollary 2, their proofs are omitted for space consideration. Theorem 3 If decision-makers of network service providers have a pessimistic market estimation, then the optimal network bandwidth capacity B ∗ with fuzzy traffic demand is 2(s − c) , f or (c + h) ≥ (s − c). (17) B ∗ = L−1 s+h Proof holds

Corollary 3 If L(B ∗ ) = (B ∗ − l)/(m − l), then the optimal network bandwidth capacity B can be described as ∗

B∗ = l +

2(s − c) (m − l), f or (c + h) ≥ (s − c). s+h

(18)

Theorem 4 If decision-makers have a moderately optimistic market estimation, then the optimal network bandwidth capacity B ∗ with fuzzy traffic demand satisfies the following expression: 2(s − 2c − h) , f or (3s − 4c) ≥ h and (4c + 3h) ≥ s. (19) L(B ∗ ) − R(B ∗ ) = s+h Corollary 4 If L(B ∗ ) = (B ∗ − l)/(m − l) and R(B ∗ ) = (r − B ∗ )/(r − m), then the optimal network bandwidth capacity B ∗ is B∗ = m +

2(s − 2c − h) (m − l)(r − m), f or (3s − 4c) ≥ h and (4c + 3h) ≥ s. (s + h)(r − l)

(20)

Combining the above three corollaries (Corollaries 2–4), the optimal network bandwidth capacity can be calculated as ⎧ 2(s − c) ⎪ ⎪ l+ (m − l), for (c + h) ≥ (s − c) and λ = 0 ⎪ ⎪ s+h ⎪ ⎨ 2(s − 2c − h) B∗ = (m − l)(r − m), for (3s − 4c) ≥ h, (4c + 3h) ≥ s and λ = 0.5 m+ ⎪ (s + h)(r − l) ⎪ ⎪ ⎪ ⎪ ⎩ r − 2(c + h) (r − m), for (c + h) ≤ (s − c) and λ = 1. s+h (21) For illustration purpose, a numerical example is given in the next section.

4 An Illustrative Numerical Example In this section, we explore the effectiveness of the proposed fuzzy-based optimization methodology for network bandwidth design under demand uncertainty with a numerical example.


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Considering that a communication network (CN) service provider would like to design a new CN system for multimedia services, the current problem that he/she faced is how to determine a suitable network bandwidth capacity to gain maximum revenue from serving different traffic demands. From the previous investigation, the unit revenue from serving demand in the CN system is 20 (i.e., a=20), the unit construction cost for each channel is 6 (i.e., c=6), the unit maintenance cost for extra network bandwidth capacity is 2 (i.e., h=2) as an example. But the market demand for a new CN is unknown. In terms of experts’ estimation from the field of communication network systems, the demand follows a triangular fuzzy number with the form = (80, 100, 140), that is, l = 80, m = 100, and r = 140. of D Due to the different market demands, the potential loss or the unit penalty cost caused by network congestion or traffic jam will be different. In the pessimistic estimation, the unit penalty cost and the potential loss is set to 8 (i.e., s=8) because less market demand will lead a small potential loss. In the moderately optimistic estimation, the potential loss is set to 12 (i.e., s=12). In the optimistic estimation, the potential loss is assumed to be 16 (i.e., s=16). Based on the above theorems and corollaries presented in Section 3, the corresponding computational results are reported in Table 1. Note that in Table 1, λ denotes the optimistic coefficient, θ is the condition verification, and B ∗ is the optimal network bandwidth capacity. Table 1 Optimal network bandwidth with different optimistic coefficients θ B∗ c + h = 6 + 2 = 8, s − c = 8 − 6 = 2, (c + h) ≥ (s − c) 88.00 3s − 4c = 36 − 24 = 12, 4c + 3h = 24 + 6 = 30, h = 2, s = 12, 0.5 96.19 (3s − 4c) ≥ h, (4c + 3h) ≥ s 1.0 c + h = 6 + 2 = 8, s − c = 16 − 6 = 10, (c + h) ≤ (s − c) 104.44 λ 0.0

As can be seen from Table 1, the following three interesting findings can be summarized. 1) In terms of the computational results from the above theorems and corollaries, the corresponding optimal design for network bandwidth capacity is easily made and implemented. 2) There is a great impact of optimistic coefficients on optimal network bandwidth design. With the increase of optimistic coefficients, the network bandwidth capacity should be increased to satisfy possible network traffic demand. But with the increase of market optimistic degree, the increment volume of network bandwidth capacity is decreasing. This indicates the network bandwidth is fully utilized with the increase of network traffic demand. 3) It should be noted that the θ condition should be satisfied before calculation. That is, the before-mentioned experimental results in Table 1 should be based on the prerequisites or basic requirements of computations.

5 Conclusion Remarks In this paper, a fuzzy methodology was proposed to optimize the network bandwidth design under demand uncertainty in communication networks. Through fuzzification processing for uncertain demands, we can obtain the optimal network bandwidth capacity based on different optimistic coefficients in terms of market estimation. For illustration purpose, a simple numerical example was used to verify the effectiveness of the results about the optimal bandwidth capacity design. The experimental results reveal that these obtained theorems and corollaries can be easily applied to other practical network bandwidth design problems with demand uncertainty in communication networks.

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Acknowledgements The authors would like to thank the guest editors and the three anonymous reviewers for their valuable comments and constructive suggestions to the paper. Their comments have improved the quality of the paper immensely. References [1] L. Yu, W. Yue, and S. Wang, Network bandwidth design under uncertainty, Memoirs of Konan University, Intelligence & Informatics Series, 2008, 1(1): 91–98. [2] J. Wu, W. Yue, and S. Wang, Stochastic model and analysis for capacity optimization in communication network, Computer Communications, 2006, 29(12): 2377–2385. [3] D. Mitra and Q. Wang, Stochastic traffic engineering for demand uncertainty and risk-aware network revenue management, IEEE Transactions on Networks, 2005, 13(2): 221–233. [4] D. Mitra and K. Ramakrishnan, A case study of multiservice multipriority traffic engineering design for data networks, in Proc. IEEE GLOBECOM, 1999, 1077–1083. [5] S. Suri, M. Waldvogel, D. Bauer, and P. Warkhede, Profile-based routing and traffic engineering, Computer Communications, 2003, 26(4): 351–365. [6] K. Meesublak, Network design under demand uncertainty, Doctoral Dissertation, University of Pittsburgh, USA, 2009. [7] D. Mitra and Q. Wang, Stochastic traffic engineering with applications to network revenue management, in Proc. IEEE INFOCOM, 2001, 1300–1309. [8] D. Mitra and Q. Wang, Risk-aware network profit management in a two-tier market, in Proc. 18th International Teletraffic Congress, 2003. [9] J. Wu, W. Yue, and S. Wang, Stochastic traffic engineering design and optimization for multimedia communication networks, IEICE Technical Report, 2005, 104(564): 19–24. [10] J. Wu, W. Yue, and S. Wang, Optimization of bandwidth allocation in communication networks with penalty cost, Lecture Notes in Computer Science, 2005, 3516: 539–547. [11] D. Dubois and H. Prade, Evaluating Consumer Loans Using Neural Networks, Academic Press, New York, 1980. [12] L. Li, S. N. Kabadi, and K. P. K. Nair, Fuzzy models for single-period inventory problem, Fuzzy Sets and Systems, 2002, 112(3): 273–289. [11] S. J. Chen and C. L. Hwang, Fuzzy Multiple Attribute Decision Making: Methods and Applications, Springer-Verlag, Berlin, 1992. [14] R. R. Yager, A procedure for ordering fuzzy subsets of the unit interval, Information Sciences, 1981, 24(2): 143–161. [15] T. S. Liou and M. J. Wang, Ranking fuzzy numbers with integral values, Fuzzy Sets and Systems, 1992, 50(3): 247–255.