Fuzzy multiobjective routing model in B-ISDN - Semantic Scholar

Computer Communications 21 (1998) 1571–1584

Fuzzy multiobjective routing model in B-ISDN Emad Aboelela, Christos Douligeris* Department of Electrical and Computer Engineering, University of Miami, P.O. Box 248294, Coral Gables, FL 33124-0640, USA Received 14 October 1997; received in revised form 6 April 1998; accepted 7 May 1998

Abstract Routing algorithms are required to guarantee the various quality-of-service (QoS) characteristics requested by the wide range of applications supported by Broadband Integrated Services Digital Networks (B-ISDNs). In this paper the routing problem is formulated as a fuzzy multiobjective optimization model. The fuzzy approach allows for the inclusion and evaluation of several criteria simultaneously. The proposed model takes into consideration the balancing of the load in the network to avoid link saturation and hence the possibility of congestion. The efficiency and applicability of the model are tested under different load conditions by studying several measures of performance. 䉷 1998 Published by Elsevier Science B.V. All rights reserved. Keywords: Computer networks routing; QoS routing; Multimedia internetworking; Fuzzy logic; Multicriteria decision making

1. Introduction In traditional data networks, routing protocols usually characterize the network with a single metric such as hop count or delay. Shortest-path algorithms are then used in these networks for path computation [1]. The subjectivity of quality-of-service (QoS) requirements of the diverse traffic classes in Broadband Integrated Services Digital Networks (B-ISDNs) and the complex trade-offs among them make it difficult to define an appropriate unique routing metric. Moreover, given the distinct characteristics of the various traffic classes, the same metric is not universally applicable. Hence, a new routing paradigm that emphasizes searching for an acceptable path satisfying various QoS requirements is needed for integrated communication networks. Such a paradigm will affect not only the call blocking probability and connection setup delay, but also the utilization of network resources as well. However, it is known that such a routing problem involving two or more additive or multiplicative QoS parameters in any possible combination is NP-complete [2, 3]. Hence, a number of heuristic algorithms have been proposed recently to solve this problem [3, 4, 5]. In the heuristic approaches the complexity of the QoS routing algorithms is reduced by choosing a subset of QoS parameters — there are no dynamic routing methods that use a complete set of QoS parameters to determine a route for multimedia data flows [5]. One possible approach is to * Corresponding author: e-mail: [email protected]

define a function and generate a single metric from multiple parameters. For example, Wang and Crowcroft [3] proposed a composition rule by choosing the bottleneck bandwidth and propagation delay as the routing metrics. Iwata et al. [4] used a criterion which optimizes the network resource utilization of paths satisfying the multiple QoS requirements of users, while limiting both the call blocking probability and the connection setup delay to sufficiently low values. In this paper, we view the routing problem in B-ISDNs as a multiobjective problem. Principal to multiobjective optimization is the concept of an ‘‘efficient solution’’, where any improvement of one objective can only be achieved at the expense of another. The fuzzy approach can be used as an effective tool for quickly obtaining a good compromise solution. Since Zadeh initiated the fuzzy set theory in 1965 [6], fuzzy logic has been applied to problems that are either difficult to tackle mathematically or where the use of fuzzy logic provides improved performance. Many design and control problems in communication systems are indeed well suited for the analysis using fuzzy logic because of their characteristics of having multiple performance criteria, some of which are often conflicting. The fuzzy logic approach, gaining recognition as a tool for handling the imprecise nature of design-making environments without undue simplification, can be used efficiently as a solution technique over other major methods in multiobjective optimization. Chanas [7] stated the concept of finding a means of extending the classical algorithms to solve the shortestroute problem to the case of fuzzy data. Some articles in the literature have investigated applying the fuzzy logic

0140-3664/98/$ - see front matter 䉷 1998 Published by Elsevier Science B.V. All rights reserved. PII: S0140-366 4(98)00223-0

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approach to routing in communication networks. Chemouil et al. [8] developed a fuzzy routing system applied to a model of the French long-distance telephone network. They applied a fuzzy controller that is used for fuzzy adaptive routing. This controller determines the availability of all the paths, the quality of all routes and decides the best route to be used for routing the current traffic. They compared the performance of methods usually applied to adaptive traffic routing and the fuzzy control approach, showing that the fuzzy control approach could provide an effective framework for robust control of traffic routing in communications networks. In our approach we formulate the routing model as a fuzzy multiobjective model [9, 10]. The fuzzy approach is not only beneficial in transforming a nonlinear objective function to a linear one with the gain of having a simpler and faster solution to the model, but it also allows the user and network provider to negotiate more flexible and mutually beneficial contracts. The proposed model challenge is to find routes for flows through paths that are not hideously expensive, according to the required QoS, and do not penalize the other flows already existing or expected to arrive in the network. Having as an objective the balancing of the load over the network links contributes to reaching a good level of stability and fairness, which by themselves are important attributes of routing functions [11, 12]. The model is built in a modular and flexible manner so that it can be applied as a dynamic, semi-dynamic or static routing tool. The paper is organized as follows. In Section 2, we describe the QoS-based routing problem. In the same section, we also present an overview of related works and an elaboration of the benefits of utilizing fuzzy logic to solve this problem. In Section 3, we present the proposed fuzzy logic approach as well as the routing model. The model is tested and the results are analyzed in Section 4. Section 5 concludes the paper and discusses potential future work. 2. QoS routing in B-ISDN 2.1. Problem statement Broadband Integrated Services Digital Networks (BISDNs) support a wide range of applications with diverse traffic characteristics such as video, audio and bulk data transfers. Routing algorithms are required to guarantee the several QoS requested by these traffic classes. Without an efficient QoS routing algorithm, the network may fail to find a route and reject a request for a call connection, even though there are enough resources available to establish that call successfully. The QoS routing problem addresses the issue of finding a path from source node s to destination node t such that: 1. the QoS requirements, which are diverse and applicationdependent, are satisfied; 2. minimum resources are consumed; and

3. the probability of congestion is decreased by balancing the load in the network [12]. Points 2 and 3 above depend on the routing paradigm and can be considered as objects to be optimized. Point 1 represents the constraints to be met. Each constraint is associated with a metric in the network. The more metrics we have the more accurately the performance objectives of the network are represented but the harder the problem to be solved. The different metrics in communication networks can be divided into three categories according to the composition rules of metrics. Let d(i,j) be a metric for link (i,j) and p ˆ (s,i,j,…,k,t) be a path that connects source s with destination t. Wang and Crowcroft [3] summarized the different composition rules as follows. 1. Metric d is additive if: d…p† ˆ d…s; i† ⫹ d…i; j† ⫹ … ⫹ d…k; t†

…1†

ex: delay, delay jitter and cost. 2. Metric d is multiplicative if: d…p† ˆ d…s; i† × d…i; j† × … × d…k; t†

…2†

ex: loss probability. 3. Metric d is concave if: d…p† ˆ min‰d…s; i†; d…i; j†; …; d…k; t†Š:

…3†

ex: bandwidth. Wang and Crowcroft [3] proved that the problem of finding a path subject to constraints on two or more additive and multiplicative metrics in any possible combination is NPcomplete. So, heuristic approaches are often the only candidate solutions for this problem. 2.2. Related work A number of heuristic algorithms have recently been proposed to solve the routing problem with multiple QoS parameters. One approach uses an optimal routing algorithm based on a single cost aggregated by a combination of weighted QoS parameters. However, the solutions in this case are sensitive to the aggregating weights, and there is no clear guideline on how the weights should be chosen. The fallback routing algorithm [13] addressed the problem of determining these weights. This approach tries to find a path based on the single cost of weighted link metrics and, if it fails, it tries to find another path by changing the values of the weights iteratively so as to get more possible paths, until an appropriate path that guarantees QoS parameters can be found. The PNNI routing and signaling protocol specification [14] describes the framework approach of a routing algorithm, consisting of two stages. The first stage involves finding a path from a list of precalculated routes, and the second stage involves finding a path by an on-demand calculation excluding the blocked links detected during the first stage from the whole known link state information.

E. Aboelela, C. Douligeris / Computer Communications 21 (1998) 1571–1584

Similarly, Iwata et al. [4] proposed a guideline for a QoS routing algorithm, specifying how to actually find a path under constraints of QoS parameters. The algorithm takes an adaptive and iterative path search approach similar to the fallback routing algorithm [15], while being compliant with the PNNI routing protocol specifications [14]. It tries to find a path optimized for one of several link metrics, instead of optimizing for a single cost of weighted QoS parameters. The path obtained is evaluated further as to whether it can guarantee the other user QoS requirements or not. If no path can be obtained, it chooses another link metric for optimization, and tries to find another path optimized for this metric iteratively, until an appropriate guarantee QoS can be found or until all the attempts are exhausted and the call is blocked. Vogel et al. [5] argued that many end-to-end parameters in today’s networks are not completely independent from each other (e.g., if the load increases, delay, delay jitter or loss increase too). So, they selected the following set from the known QoS specifications: throughput, delay and error rate. A parameter set (t,d,e) is assigned to each network segment, where t is the currently available throughput in Kb/s, d is the current delay in ms, and e is the current error rate as a multiple of 10 ⫺6. For any candidate routes ttotal is the minimum t among all links in the route. dtotal is the aggregated sum of all ds of the links in the route. Finally, if L is theQ setÿ of
all links in the route: etotal ˆ 1 ⫺ ᭙l僆L 1 ⫺ el . A QoS (t,d,e) of a route is assumed to be sufficient for a demand (td,dd,ed) if: t ⱖ td, d ⱕ dd and e ⱕ ed. In case of more than one route satisfying the demand, a selection of the preferred route follows a simple rule depending on the degree of availability of each parameter (e.g., the availability of the throughput parameter ˆ t/td). Wang and Crowcroft [3] showed that trying to optimize any two or more of delay, delay jitter, cost or loss probability in any combination as metrics results in an NP-complete problem. They argued that the only feasible combination, to avoid NP-completeness, are bandwidth and one of the four (delay, loss probability, cost and delay jitter). They believed that, for the majority of applications, delay is comparatively more important than the others. In their paper, the bottleneck bandwidth and the propagation delay were chosen as the routing metrics. The routing algorithm presented in that paper depends on eliminating all links that do not meet the bandwidth requirement by setting their delay to ∞. Then, Dijkstra’s algorithm was used to find the minimum delay path. 2.3. The need for fuzzy logic The previous heuristics are, in one form or another, a search in lists of candidate paths ordered according to a different criterion. The selection of the appropriate criterion depends on the application requirements from the candidate metrics. These requirements are considered to be crisp

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values logically compared with the available values of the corresponding metrics. Gavish and co-workers [13, 16] formulated an optimization model to solve the routing problem in a single-commodity network. Their objective function was mainly to minimize the queuing end-to-end delay in the network. They used Lagrangean relaxation and subgradient optimization techniques for their proposed nonlinear model. For multicommodity networks, formulating the problem as a multiobjective optimization model is always avoided not only because of the expected nonlinear model but also because of the two types of inaccuracy incorporated in multiobjective optimization problems. One is the ambiguity inherited in the nature of the parameters in the problem, and the other is the fuzzy goals for each of the objective functions. For handling and tackling such kinds of imprecision or vagueness, it is not hard to imagine that the conventional multiobjective optimization approaches, such as a deterministic or even a probabilistic approach, may not be the most applicable techniques to be used for this problem. Hence, multiobjective optimization under imprecision or fuzziness seems to be promising practically and applicable for dealing with decision-making problems such as routing problems. The proposed routing model is formulated as a fuzzy multiobjective optimization model [9]. Linear fuzzy membership functions are chosen to represent the goals. The fuzzy decision method introduced by Bellman and Zadeh [17] is used to formulate the model. We can summarize the benefits of applying the fuzzy multiobjective optimization technique to solve the problem of QoS routing in B-ISDNs in the following points. • Fuzzifying the objectives increases the feasible solution space with the gain of avoiding the high probability of infeasible solutions as is the case in crisp multiobjective optimization models. In other words, the fuzzy approach is an effective tool for quickly obtaining a good compromise solution. • Recalling the imprecision or fuzziness inherent in human judgments, representing the required QoS parameters as a fuzzy goal results in more realistic problem representations since the network manager avoids specifying a crisp value for his/her goals. • Using linear membership functions for the QoS parameters contributes in avoiding the nonlinear nature of some requirements. For example, if one of the goals is to balance the load all over the network links, the deviation of the links utilization has to be minimized. Using a simple linear fuzzy membership function for link utilization, the nonlinearity of the deviation formula can be avoided. • QoS goals are normalized, by their membership functions, to a number between 0 and 1. Now aggregation of weighted QoS parameters into a single cost parameter is more meaningful by avoiding the aggregation of heterogeneous quantities. Each QoS parameter can be

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controlled separately by reshaping its membership function.

3. The fuzzy approach description In this study we consider the problem of optimal route selection in computer communication networks in which nodes, links, link capacities and external traffic load are given. Each traffic is categorized into a specific commodity service. Each category of commodity services has its own predefined QoS requirements. A set of communicating source/destination pairs is defined. Messages are transmitted from source to destination through intermediate nodes and links along fixed routes that are determined at the time of network definition. The objective of this study is to select the set of routes for a multicommodity flow representing connections with different QoS requirements in a network with predefined sets of routes between the communicating pairs of nodes. These routes have to satisfy the delay requirement of the message to be routed as well as ‘‘minimize’’ the probability of link congestion. The objective of ‘‘minimizing’’ the probability of congestion is achieved by: • having balanced utilized links; • selecting, as much as possible, routes with smaller numbers of hops; and • taking into consideration the likelihood of load congestion in the network links. We formulate the routing model as a fuzzy multiobjective model [9]. The fuzzy decision used is the one proposed by Bellman and Zadeh [17]. The model is tested with two metrics: bottleneck bandwidth and delay. The proposed approach is generic and can easily be applied to different routing metrics. Our particular choice of bandwidth and delay is based on the following. Wang and Crowcroft [3] argued that the feasible combinations as routing metrics are bandwidth and one of the other four (delay, loss probability, cost and delay jitter). They argued that, for the majority of applications, delay is comparatively more important than the others (loss probability, cost and delay jitter). Thus they chose bottleneck bandwidth and propagation delay as the routing metrics. The bottleneck bandwidth is the minimum of the residual bandwidth of all links on the path. Even though delay has two basic components, queuing delay and propagation delay, they considered only propagation delay since queuing delay is already reflected in the bandwidth metric. In this way, they made sure that the two metrics are not interdependent. Also, Gerla et al. [18] argued that the queuing delay is not a very meaningful optimization variable in ATM networks where it is much less than the propagation delay. They used the queuing delay only as an indirect measure of buffer overflow probability (to be minimized). In other computational studies it has been shown that it typically makes little difference whether the cost

function used in routing includes the queuing delay or the much simpler form of the link utilization. So, use of bottleneck bandwidth and propagation delay as metrics is a compromise between complexity and optimality. Fuzzification of objectives may result in capabilities of negotiating better and more flexible traffic contracts between users and service providers. It will enable users to present their traffic in more relaxed terms — users rarely know their exact traffic requirements and are usually willing to negotiate some of its parameters. Service providers, on the other hand, will be able to handle a larger number of users more flexibly and without the large burden introduced by the need to continuously and accurately estimate traffic. These problems have been identified and well documented in the ATM literature. For example, with regard to the definition of the service characteristics of the sources, Rathgeb [19] states that: ‘‘Another problem is caused by the inaccuracies and uncertainties in the knowledge about relevant parameters, like the mean bit rate, in the establishment of the call’’. These inaccuracies are amplified by the delay variation introduced in the network and significantly affect the instantaneous mean bit rate, which is used in most policing functions, as well as the peak bit rate [20, 21]. Moreover, ‘‘it has to be recognized that the set of policing parameters proposed by CCITT in recommendation I.311, namely average cell rate, peak cell rate, and duration of peak is not sufficient to completely describe the behavior of ATM traffic sources. Furthermore not all these characteristics may be known at call set-up with the required accuracy and some of them may be modified before the cells reach the policing function…’’ [19].

3.1. Notations and definitions The following notation is used in the proposed model: the set of the communicating source/destination pairs in the network. S the set of the commodity services to be transmitted in the network (voice, data, video, etc.). L the set of all links in the network. R the set of candidate routes. This set is provided by a route-generation algorithm (e.g., double-sweep algorithm). A route is characterized by the ordered set of links (from source to destination) in the route. Rp the set of candidate routes for communicating pairs p. Obviously we have: P

Y Rp ˆ R

᭙p僆P

Rp1 傽 Rp2 ˆ f V Cl El

p1 苷 p2 and ᭙p1; p2 僆 P

…4†

the average number of links in the routes that belong R. the capacity of link l in bits per second (bps). the current reserved bandwidth in link l.

E. Aboelela, C. Douligeris / Computer Communications 21 (1998) 1571–1584

Dl

the fixed delay (propagation delay, processing delay, etc.) through link l. M the length, in hops, of the longest route in R. lps ⬅ lrs᭙r 僆 Rp the message arrival rate into the communicating pairs p from the commodity service s in bps. It equals the arrival rate from this commodity in any of the candidate routes for communicating pairs p. yrl an indicator function which is one if link l is used in route r and zero otherwise. xrs a decision variable which is one if route r is selected for the routing of a message from commodity service s and zero otherwise. crs a fixed cost assigned to each route r for service s. It can be the number of links (hops) in this route. grs an estimated congestion degree assigned to each route r. It is a function of the number of candidate routes sharing the links of the given path. Using the above notation the following hold: • The total arrival rate in bps from commodity service s in the network: X

Ts ˆ

lps

…5†

p僆P

• The total arrival rate in bps from all commodity services in the network: Tˆ

X

Ts ˆ

X X

lps

…6†

p僆P s僆S

s僆S

• The total arrival rate in bps from commodity service s in link l according to the routing decision: Als ˆ

Xÿ

yrl lrs xrs




…7†

r僆R

• The total arrival rate in bps from all commodity services in link l according to the routing decision: Al ˆ

X

Als ˆ

s僆S

X

yrl ×

r僆R

X

[22]: Ds ˆ

1 X A × Dl Ts l僆L ls

3.2. Fuzzy membership functions In engineering applications of fuzzy logic, membership functions are, for the most part, associated with terms that appear in the antecedents or consequents of rules. The most commonly used shapes for membership functions are triangular, trapezoidal, piecewise linear and Guassian [23]. These membership functions provide simple formulas and computational efficiency and have been used extensively, especially in real-time implementations. It needs to be noted here that several studies of fuzzy systems have shown that the choice of membership functions does not dramatically change the behavior of the system. For most applications, particularly real-time ones, the end-to-end delay is one of the most important QoS requirements. Traffic types usually have exact upper bounds on the allowed delays, but in most cases the traffic application can compromise within a range around the required delay. A fuzzy membership function can be used to formulate this situation. Assume that it is required to have the delay Ds in the vicinity of the value ds. Fig. 1 depicts a possible shape of the fuzzy membership function ms(Ds). It shows that the highest membership value ( ˆ 1) is for Ds ˆ ds. Delays are allowed up to a maximum value dsmax but with a fast dropping membership function. A requirement for strict delay limits can also be accommodated if dsmax ˆ ds. In some cases it is required not to decrease the minimum delay more than a specific value dsmin. In these cases dsmin has to be a specific positive value. In most cases, there is no restriction on the minimum delay. In these cases dsmin can be assigned a value ⫺∞ to have ms(Ds) ˆ 1 for Ds ⬍ ds.

…8†

s僆S

• The utilization of link l: Ul ˆ

Al ⫹ El Cl

…9†

• The average networkwide delay for commodity service s

…10†

• The average networkwide delay for all commodity services: 1 X 1 X Ts × Ds ˆ A × Dl …11† Dˆ T s僆S T l僆L l

!

lrs xrs

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Fig. 1. Fuzzy membership function for delay.

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Balancing the utilization all over the network’s links is beneficial to avoid link saturation and subsequently decrease the possibility of congestion [11, 12]. Hence, having a balanced load all over the network links will guarantee low queuing delay and low buffer overflow probability as well. If Uavg represents the average utilization of the links in the network, then the membership function shown in Fig. 2 can be used to have, as much as possible, the utilization of each link (Ul) in the vicinity of Uavg. In the figure we show a case where an unutilized link (Ul ˆ 0) is acceptable with a specific degree (0.2 in the figure). This results in an equivalent representation (to keep the triangularity of the membership function) of Umin ⬍ 0. In many cases, the main objective is not to increase the utilization of each link over Uavg. In these cases Umin is set to ⫺ ∞ and hence ml(Ul) ˆ 1 for Ul ⬍ Uavg. In the next section, we shall discuss how Uavg can be predicted from the network topology and average loads.

3.3. The optimization model In this section we use the notation and relationships of Section 3.1 and the fuzzy membership functions of Section 3.2 to formulate the routing problem as a multiobjective program, which ultimately becomes a nonlinear zero–one problem. The techniques from fuzzy programming are used to relax the problem with the gain of having a simple and faster solution to the model, that associates a fuzzy membership function with the QoS parameters [9]. The fuzzy decision problem introduced by Bellman and Zadeh [17] has as a goal the maximization of the minimum value of the membership functions of the objectives to be optimized. Accordingly, the fuzzy optimization model can be represented as a multiobjective programming problem as follows:  ÿ
 ÿ
max : min ms Ds and min ml Ul ᭙s 僆 S and ᭙l 僆 L Such that X

Al ⱕ Cl ᭙l 僆 L

xrs ˆ 1 ᭙p 僆 P and ᭙s 僆 S

…P1†

r僆Rp

as follows: max w1 ·Z1 ⫹ w2 ·Z2 ÿ
Such that Z1 ⱕ ms Ds ᭙s 僆 S ÿ
Z2 ⱕ ml Ul ᭙l 僆 L Al ⱕ Cl ᭙l 僆 L X

xrs ˆ 1 ᭙p 僆 P and ᭙s 僆 S

r僆Rp

xrs ˆ 0 or 1 ᭙r 僆 R and ᭙s 僆 S

Z1 ; Z2 ⱖ 0

where w1 and w2 are the weighting coefficients assigned to the delay and utilization objectives respectively. These factors can play the role of priority levels where their relative values reflect their significance to the network manager. It needs to be noted here that this linearization is done in the membership function space (values between 0 and 1). Traditional linearization techniques in multiobjective programming suffer from the scaling factor problem — what is the ratio between w1 and w2 when one objectives differs from the other by orders of magnitude? From Figs. 1 and 2, we have: ÿ
ms Ds ˆ ts ˆ 0 Ds ⱕ dsmin ÿ
D ⫺ dsmin ms Ds ˆ ts ˆ s ds ⫺ dsmin

dsmin ⬍ Ds ⱕ ds

ÿ
d ⫺ Ds ms Ds ˆ ts ˆ smax ds ⬍ Ds ⱕ dsmax dsmax ⫺ ds ÿ
ms Ds ˆ ts ˆ 0 Ds ⬎ dsmax ÿ
Ul ⫺ Umin ml Ul ˆ tl ˆ Uavg ⫺ Umin

…12†

Ul ⱕ Uavg

ÿ
1 ⫺ Ul ml Ul ˆ tl ˆ Uavg ⬍ Ul ⱕ 1 1 ⫺ Uavg ÿ
ml Ul ˆ tl ˆ 0 Ul ⬎ 1 The optimization model can then be rewritten as follows: max w1 ·Z1 ⫹ w1 ·Z2

xrs ˆ 0 or 1 ᭙r 僆 R and ᭙s 僆 S In (P1), the objective is to maximize the minimum membership function of all traffic services (classes) delays as well as the membership function of the utilization of all links in the network. The constraints are to guarantee that no link has a load exceeding its capacity and only one path is chosen for each service/pair. One of the possible ways of scalarizing the multiobjective programming problem is the weighted maxmin method [9] that is used in the proposed model. By introducing the auxiliary variables Z1 and Z2, the problem is to optimize the weighted sum of these variables

…P2†

Fig. 2. Fuzzy membership function for the utilization.

E. Aboelela, C. Douligeris / Computer Communications 21 (1998) 1571–1584

Such that Z1 ⱕ ts

᭙s 僆 S

Z2 ⱕ tl

᭙l 僆 L

D ⫺ dsmin ts ⱕ s ds ⫺ dsmin

᭙s 僆 S

ts ⱕ

dsmax ⫺ Ds dsmax ⫺ ds

᭙s 僆 S

tl ⱕ

Ul ⫺ Umin Uavg ⫺ Umin

᭙l 僆 L

tl ⱕ

1 ⫺ Ul 1 ⫺ Uavg

᭙l 僆 L

X

xrs ˆ 1

᭙p 僆 P and ᭙s 僆 S

xrs ˆ 0 or 1

᭙r 僆 R and ᭙s 僆 S

ts ⱖ 0

᭙s 僆 S

tl ⱖ 0

᭙l 僆 L

…P3†

r僆Rp

Z1 ; Z2 ⱖ 0 In this model the capacity constraint (Al ⱕ Cl) has been ignored since its effect is automatically brought into play by the constraints of ml. So, the capacity constraints are handled as penalty functions embedded in the utilization formula. Fratta et al. [24] defined the balanced network from the perspective of the requirements of the communicating source/destination pairs. Assuming that our network is a balanced network, the current objective is to have homogeneous loads in all links in the network. Actually having a balanced load all over the network links will guarantee low queuing delay and low buffer overflow probability as well. A balanced load can be achieved by equally utilize the links in the network. The average utilization of any link (Uavg) can be calculated as follows: 1 X 1 X Al ⫹ E l Ul ˆ …13† Uavg ˆ 兩L兩 l僆L 兩L兩 l僆L Cl A relaxation is needed to simplify the model. Without loss of generality, we will assume that Cl is constant all over the links. So,
1 X 1 1 Xÿ Ul ˆ A l ⫹ El …14† Uavg ˆ 兩L兩 l僆L 兩L兩 Cl l僆L Also, we can have the following approximation for the previous summation: Xÿ
A l ⫹ El ˆ T × V …15† l僆L

where V, as defined before, is the average number of links in the routes connecting all communicating pairs in the network. T is the total load in the network from all commodity services.

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So, now we have Ds (10) and Ul (9) as a linear expression in the zero–one variable x. Also, Uavg is a constant. This makes the proposed optimization model a zero–one linear program. Even though maximizing the minimum value of the membership function in fuzzy optimization models is widely used, in the case of the routing problem and the proposed model this objective is not enough. For example, in many cases there is a bottleneck link that must be used by many routes. This link will be overloaded, resulting in a poor or small membership utilization function. This low membership value, with no way of being improved, will make the overall optimization process stop seeking for more improvement in the issue of balancing the utilization of other links in the network (because there is no longer any way to maximize the minimum utilization membership function). To solve this problem, other terms may be added to the objective function. Sommer and Pollastschek [25] proposed the adoption of the add operator for aggregating the fuzzy membership functions (fuzzy goals). In our model, the new added terms are the normalized weighted aggregation of the membership functions for all traffic services delay and links utilization. Now we can make sure that the optimization model will maximize all membership functions as well as the minimum of them. This will have the effect of having homogeneously distributed traffic in the whole network. In the model (P4) the normalized weighted summation of the fixed cost assigned to each candidate route, crs, has also been added to the objective function to be minimized. The fixed cost can be the length of the route in hops. This will help avoid using longer routes just to balance the utilization. The weighting coefficient for this term is w3. As we see, the model is supposed to route loads coming into different communicating pairs in the network. Such kind of optimization models need a long run time to be solved using traditional optimization tools, especially for large networks. So, one solution to this problem is to divide the set of communicating pairs into smaller subsets. Instead of solving the problem in one stage (for the whole set of communicating pairs), the model is used to solve the problem in stages. Each stage involves the solution of the problem for a specific subset of communicating pairs. During each stage the optimization model tries to keep the utilization, using the fuzzy membership function ml(Ul), in the vicinity of Uavg. From eqn (14), Uavg can be estimated for the network in its final state (all pairs are communicating with the required traffic loads.) Solving the problem in stages results in suboptimal solutions that can be enhanced by adding another term in the objective function. This term is the normalized weighted summation of the estimated congestion degree of each candidate route, grs. This estimated degree can be calculated from the list of all candidate routes known in advance between all communicating pairs of nodes. For a specific route, it reflects the number of other routes fighting to share

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its links. This term is weighted by w2, the same as the utilization terms, because it helps in avoiding congestion, but it can easily be given another weight. w X w X ts ⫹ 2 tl max w1 ·Z1 ⫹ w2 ·Z2 ⫹ 1 兩S兩 ᭙s 兩L兩 ᭙l ⫺

X X w3 w2 crs ·xrs ⫺ grs ·xrs M·兩P兩·兩S兩 ᭙r;᭙s M·兩P兩·兩S兩 ᭙r;᭙s Z1 ⫺ ts ⱕ 0 ᭙s 僆 S

Such that ÿ

Z2 ⫺ tl ⱕ 0 ᭙l 僆 L



1 X X ÿ d1 yrl lrs xrs ⱕ ⫺dsmin ᭙s 僆 S ds ⫺ dsmin × ts ⫺ Ts l僆L r僆R



1 X X ÿ dl yrl lrs xrs ⱕ dsmax ᭙s 僆 S dsmax ⫺ ds × ts ⫹ Ts l僆L r僆R  
1 1 1 X Xÿ × T × V ⫺ Umin × tl ⫺ yrl lrs xrs × C C 兩L兩 l l s僆S r僆R ÿ

El ⫺ Umin ᭙l 僆 L Cl  
1 1 1 X Xÿ × T × V × tl ⫹ y l x 1⫺ × Cl Cl s僆S r僆R rl rs rs 兩L兩 ⱕ

ⱕ1⫺ X

4.2. Performance measures The following are the performance measures reported by testing the case study:

El ᭙l 僆 L Cl

xrs ˆ 1 ᭙p 僆 P and ᭙s 僆 S

r僆Rp

xrs 僆 …0; 1† ᭙r 僆 R and ᭙s 僆 S tl ⱖ 0 ᭙l 僆 L

definition of the traffic load. Traffic types can be classified into three categories [16]. First, we have low-latency traffic which consists of small messages, each sent as small number of packets. The key performance measure of such traffic is the end-to-end per packet delay. Second, continuous-rate traffic consists of a continuous traffic stream with a certain intrinsic rate where the application does not benefit from bandwidths higher than this rate; e.g., video traffic. Finally, high-bandwidth traffic consists of transfers of large blocks of data; e.g., file transfer. The key performance measure is the average throughput. Following Ma et al. [26], our evaluation is based on distinguishing two loads: low-latency (LL) and high-bandwidth (HB) traffic. The continuous-rate traffic is thus included in the low-latency traffic load. This classification distinguishes two traffic classes, each of them having compatible measures of performance from a routing perspective. The high-bandwidth traffic service will have 90% of the load assigned to each communicating nodes pair.

ts ⱖ 0 ᭙s 僆 S

Z1 ; Z2 ⱖ 0 …P4†

4. Tests and results

• Throughput as a function of the load: the number of bytes/s delivered successfully to the destination nodes as a percentage of the total load. • Link utilization: the fraction of the capacity used by the transferred load. • Load deviation: a measure of how the load is distributed all over of the network links. It is the standard deviation of the links utilization. • Average propagation delay (ms): the propagation delay experienced by traffic. • Average number of hops/path: for all paths used to route the load during the measurement interval, this is the average number of hops of these paths.

The following metrics have been used in [5] as a comparison basis for routing techniques:

4.3. A test model for qualitative comparison

1. minimum number of hops; 2. maximum throughput; and 3. minimum delay.

The network shown in Fig. 3 is used to test the proposed model and provide initial qualitative results. In this network, BW stands for the current residual bandwidth in Mbps and D stands for the propagation delay of the link in ms. Assume

In the proposed model, testing the effect of the different levels of the weighting coefficients w1, w2 and w3 will include implicitly the effect of the previous metrics individually (by setting one or more coefficients to zero) and with different levels of combinations between these metrics. The analysis will show the effect of these combinations on the performance measures. 4.1. Traffic load Testing the proposed optimization model requires the

Fig. 3. Test network.

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Fig. 4. Sample network (distances in miles).

that we have two traffic loads T1 and T2. For T1, dsmin, ds and dsmax are 0 ms, 12 ms and 25 ms, respectively. The corresponding values for T2 traffic are 0 ms, 25 ms and 50 ms, respectively. T1 has a load of 10 MB and T2 has a load of 90 MB. It is required to route T1 and T2 from node 1 to node 3. The minimum hops and minimum delay algorithms select the path 1–3 to route both traffic T1 and traffic T2. For the proposed model, the path 1–3 is selected for T1 and the other path, 1–2–3, is selected for T2. It is obvious that the proposed model chooses for each traffic the path that satisfies the delay requirement and at the same time tries to balance the load over the network links as well. 4.4. Quantitative comparison The previous model has been applied to the network in Fig. 4. This topology has been used in computational experiments in [16]. The CPLEX娃 mixed integer optimizer has been used to solve the model. All distances are in miles and the capacity of all links is assumed to be 622 Mbps. The propagation delay is assumed to be 10 ms per 1000 mile. Fig. 5 shows the three groups of nodes that communicate with each other in this study. Each node is allowed to communicate with any other node not in its group. So, we have 40 pairs of communicating nodes. These nodes are chosen to cover the whole network. The double-sweep algorithm [27, 28] is used to allocate the 12 shortest (in distance) paths between each of the communicating pairs. As discussed before, the load is supposed to be from two traffic services, LL traffic and HB traffic. The test has been done with different loads into each pair ranging from 40 Mbps to 140 Mbps, 10% of the load being from LL traffic

and 90% from HB traffic. For LL traffic, dsmin, ds and dsmax are chosen to be ⫺ ∞ ms, 12.5 ms and 25 ms, respectively. The corresponding values for HB traffic are ⫺ ∞ ms, 25 ms and 50 ms, respectively. These values reflect the statistics gathered from the topology used by testing the minimum, average and maximum lengths of randomly chosen routes. Figs. 6–14 show the results of applying the proposed model on the previous case study by changing the different parameters. Current Internet routing protocols (e.g., BGP, OSPF, RIP) use ‘‘shortest-path routing’’, i.e., routing that is optimized for a single arbitrary metric, administrative weight or hop count [29]. Figs. 9–14 show comparisons between applying the recommended fuzzy model (FZ) and two other single metric models, one using hop count (HC) as a metric and the other using delay (DL) as a routing metric. For the current test, the weighting coefficients are assigned equal values of 1; i.e., W ˆ [1 1 1]. 4.4.1. Impact of segment size In Figs. 6 and 7, the proposed model (FZ) has been used to route a load of 60 Mbps between the communicating pairs. The pairs are divided into segments of size N (number

Fig. 5. Communicating pairs.

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Fig. 6. Effect of the segment size on load deviation and average utilization: using FZ and load ˆ 60 MB/node.

of source/destination pairs per segment). The model has been tested for different values of N (5, 10, 15 and 20). From the figures increasing the segment size has little effect on the performance. This is because, while solving the model for the first segments, the degree of estimated congestion is taken into consideration as well as the updated load in the links. Fig. 8 shows the effect of segment size on the number of iterations (an indicator of run time) used by the optimizer to solve the model. It is clear that solving the problems using a small segment size will not affect the results dramatically while decreasing the run time dramatically. In our case, having 10 pairs per segment (25% of the whole pairs)

seems to be a reasonable choice. This choice compromises between being a small value and offering good performance. In Fig. 9, for a load of 70 Mbps per each pair, the throughput as a function of the load is represented. Using FZ we get the highest throughput for all segment sizes. This is due to the fact that balancing of the load in all links as a goal while solving for the first segments will help in avoiding congestion in routing the later segments. 4.4.2. Impact of traffic load Fig. 10 shows the throughput for different load values ranging from 60 Mbps to 140 Mbps per each communicating pair. From this figure, for a low load value, the

Fig. 7. Effect of the segment size on traffic delay: using FZ and load ˆ 60 MB/node.

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Fig. 8. Effect of the segment size on number of iterations used for solving the model: using FZ and load ˆ 60 MB/node.

Fig. 9. Effect of segment size on throughput for different W values: load ˆ 70 MB/node.

Fig. 10. Effect of different loads/pair on the throughput for different W values: segment size ˆ 10.

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Fig. 11. Effect of different loads/pair on the load deviation for different W values: segment size ˆ 10.

three modes of operation deliver the whole load (100% throughput). For higher loads (in this case between 70 Mbps and 110 Mbps), FZ has the highest throughput. The intuition is that with higher load, resources become more scarce, and the recommended mode will outperform the other modes because of its strategy of balancing the load in the network which decreases the likelihood of links’ congestion. As the load becomes very high (greater than 120 Mbps), the performance of the three modes becomes poor and almost the same. The reason is that with very high load, all links are likely to be congested. Fig. 11 shows that FZ has the lowest load deviation among the other protocols (DL and HC). DL and HC protocols have almost the same performance because they do not take into consideration the balancing of the load among the network links. In this figure, 10 communicating pairs per segment are used.

Figs. 12 and 13 represent the effect of the different mode of operations in the delay of the two traffic classes using different load (40, 50, and 60 Mbyte). These low loads are used to make the comparison meaningful, where — for these low loads — FZ, DL and HC deliver the same throughput (100%). Of course, DL has the lowest delay among the three modes. One of the objectives of the recommended FZ model is to have the delay of both traffic classes in the vicinity of their ds value. That is why, for FZ, the delay of LL traffic is low while that for HB traffic is high as required. The effect of using the three routing techniques on the average path length (in hops) is shown in Fig. 14. As expected, HC has the lowest average number of hops in the routes as it is its main function. DL has the worst values for different loads while the recommended FZ model comes in the middle. The previous results show that the proposed model, FZ, has a better performance. Without a great loss in performance level, it allows the communicating pairs to be segmented with the benefit of decreasing the run time. Balancing the load in all links, by fairly distributing the load all over the network links, helps in increasing the throughput and avoiding congestion as well. Also, the chosen routes meet the delay requirements for each traffic class.

5. Conclusion and future work In this study, we proposed a fuzzy optimization model for routing in B-ISDN. The proposed model challenge is to find routes for flows through paths that are not hideously expensive, according to the required QoS, and do not penalize the other flows already existing or expected to arrive in the network. Fuzzifying the objectives increases the feasible solution space with the gain of avoiding the high probability of infeasible solutions as in the case of crisp multiobjective

Fig. 12. Effect of different loads/pair on the delay of LL traffic for different W values: segment size ˆ 10.

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Fig. 13. Effect of different loads/pair on the delay of HB traffic for different W values: segment size ˆ 10.

Fig. 14. Effect of different loads/pair on the average path length for different W values: segment size ˆ 10.

optimization models. In other words, the fuzzy approach is an effective tool for quickly obtaining a good compromise solution. The proposed model takes into consideration the estimated congestion as well as the current load in the links. This allows for solving the problem in multiple stages. Each stage is for a segment of communicating nodes. The results showed that using a small segment size will not affect the results dramatically while it will decrease the run time dramatically. The model is analyzed in terms of performance under different routing scenarios. We obtained good improvements in performance compared with the traditional routing techniques while maintaining a sufficient low processing overhead. Throughput has been increased and the probability of congestion has been decreased by balancing the load all over the network links.

Current work is addressing the testing of the model using a simulation model to generate real-time load. The fairness in serving different traffic classes will be shown as another benefit of applying the proposed model. We plan to solve the model by using the generalized network model (mathematical perspective) that has been successfully applied to solve large-scale zero-one or mixed integer programming. In this case, the proposed model can be tested as a dynamic routing tool.

Acknowledgements This work has been partially supported by NSF under grant no. 9810534.

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[22] M. Schwartz, Telecommunication Networks: Protocols, Modeling and Analysis, Addison-Wesley, Reading, MA, 1987. [23] J.M. Mendel, Fuzzy logic systems for engineering: a tutorial, Proceedings of the IEEE 83 (3) (1995) 345–377. [24] L. Fratta, M. Gerla, L. Kleinrock, The flow deviation method: an approach to store-and-forward communication network design, Networks 3 (2) (1973) 97–133. [25] G. Sommer, M.A. Pollastschek, A fuzzy programming approach to an air pollution regulation problem, in: R. Trappl, G.J. Klir, L. Ricciardi (Eds.), Progress in Cybernetics and Systems Research, 1978, pp. 303– 323. [26] Q. Ma, P. Steenkiste, H. Zhang, Routing high-bandwidth traffic in max–min fair share networks, in: ACM SIGCOMM’96 (California, USA), 1996, pp. 206–217. [27] J.R. Evans, E. Minieka, Optimization Algorithms For Networks and Graphs, Marcel Dekker, Inc., New York, 1992. [28] D.R. Shier, On algorithms for finding the k-shortest paths in a network, Networks 9 (3) (1979) 195–214. [29] D. Zappala, D. Estrin, S. Shenker, Alternative path routing and pinning for interdomain multicast routing, USC Computer Science Technical Report #97-655, USC, 1997. Emad Aboelela was born in Alexandria, Egypt, and received B.Sc. and M.Sc. degrees in computer science and automatic control from Alexandria University, Egypt, in 1990 and 1993, respectively. Since 1994 he has been a teaching assistant at the Department of Electrical and Computer Engineering, University of Miami, Florida, where he is currently a Ph.D. candidate. His research interests are in applying smart systems techniques (fuzzy logic, neural networks and expert systems) in computer communication networks (routing, bandwidth allocation and management).

Christos Douligeris was born in Melissopetra Arcadia. Greece, received his Diploma in Electrical Engineering from the National Technical University of Athens in 1984, and M.S., M.Phil. and Ph.D. degrees from Columbia University in 1985, 1987 and 1990, respectively. Since August 1989 he has been with the Department of Electrical and Computer Engineering at the University of Miami, where he is currently an Associate Professor. He is the Associate Director for Engineering of the Ocean Pollution Research Center and has served in technical program committees of several conferences. His main technical interests lie in the areas of performance evaluation of high-speed networks, congestion and flow control, neurocomputing in networking and information management for emergency response operations. He is a technical editor of the IEEE Communications Magazine, a technical editor of IEEE Communications Magazine Interactive, a senior member of the IEEE, and a member of INFORMS (formerly ORSA) and the Technical Chamber of Greece.