Analytical Comparison of Fairness Principles for Resource Sharing in Packet-Based Communication Networks by ijcsiseditor


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									                                                       (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                                            Vol. 8, No. 4, July 2010

   Analytical Comparison of Fairness Principles for
  Resource Sharing in Packet-Based Communication
                                             Yaser Miaji and Suhaidi Hassan

                         InterNetWorks Research Group, UUM College of Arts and Sciences
                             Universiti Utara Malaysia, 06010 UUM Sintok, MALAYSIA

ABSTRACT                                                              potential responsibility in structuring the fairness principle.
                                                                      Some applications require more sensitive pamper and care
Current Internet users are enormously increased and                   such as voice and interactive application such as video
application that they are using is magnificently bandwidth            conversation and so forth. The sensitivity of these
devoured. With this manner, Internet is no longer a fair and          applications significantly involved in fairness principle.
protective environment. The diversity in the Internet
applications required a reconsideration of the mechanisms
                                                                           Moreover, providing Quality of Service (QoS) is one
used to deliver each packet pass through a router in order to
provide better fairness and more protective place.                    big dimension which should be achieved if not fully at
Furthermore, the observer of the Internet packet could easily         least to the large extent. QoS requirements rhyme heavily
identify the purpose of the delay which is indeed caused by           with user and application requirements. Even though,
the queuing in the output buffer of the router.                       Service Providers (SP) is one potential dimension which
                                                                      tighten fairness principle, their requirements is highly
      Therefore, to reduce such delay for those sensitive             depend on financial matters.
applications such as real-time applications, scholars develop
many fairness principle which by turn could improve the                    Fairness principle is indeed, applied in routers or to
QoS and hence the fairness and the protection aspect. This            be more specific in the process of scheduling the
study highlight most famous fairness principles used in the
                                                                      transmission of the packets over a shared link. Fairness
literature and some other novel ideas in the concept of
fairness. The analytical comparison of these principles shows         principle should provide three primary function selection,
the weakness and the strength of each principle.                      promptness, and QoS consideration. Selection is the
Furthermore, it illuminates which fairness principle is more          basically which packet deserves to be transmitted.
appropriate in which environment.                                     Promptness means when the selected packet will be
                                                                      transmitted. QoS requires considering the delay, loss and
     Keywords-components; Fairness, max-min, proportional             error of overall network performance.
fairness, balanced, max-min charge
                                                                            Scholars, since the discovery of the sensitive and
                     1. INTRODUCTION                                  bandwidth hanger applications, dedicate their research in
                                                                      providing superior fairness and larger protection for these
Internet utilization in public and private sector is                  applications over others less sensitive. This paper
magnificently growing with extraordinary manner. The                  demonstrates most available and used fairness principles in
occupation of the World Wide Web is unpredictable over                scheduling packets depending on application sensitivity
time frame. Daily usage of the Internet resources with                and user usage. The rest of the paper is organized as
current scrambles in network access is hard to be estimated           following. Next section gives the state diagram of the
and hence the distribution of these resources is dynamic.             literature and brief information about the evolution of the
This dynamic behavior leads to vagueness in constructing              fairness principle. This is followed by thorough conceptual
the essential principle of fairness for resource utilization.         and analytical illustration of five fairness definition
                                                                      namely; max-min fairness, proportional fairness, utility
                                                                      fairness, balanced fairness, and max-min charge fairness.
     Furthermore, not only the dynamic attitude of the
                                                                      Section four compares and contrasts all six principles and
resource utilization is an issue, the behavior and the
                                                                      finally the conclusion and future works are drawn.
characteristics of the application itself also, play a

                                                                                                   ISSN 1947-5500
                                                       (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                                            Vol. 8, No. 4, July 2010

          2. MIND-MAP OF FAIRNESS LITERATURE                           vision of fairness by providing the principle of charge
                                                                       allocation rather than bandwidth allocation and it named as
In this section, related works to the fairness is presented in         max-min charge. Max-min charge is a new fairness
state diagram or min-map diagram to correlate and track                principle based on charge allocation instead of
the evolution of fairness principle. Exhibit 2.1 shows the             conventional bandwidth allocation.        Next       section
mind-map diagram which explains the evolution of                       presents all the above mentioned five fairness principles
fairness principle. In 1967, Kleinrock [1] published his               conceptually and analytically.
article in sharing one common resource. Although the
article is primarily designed for addressing this specific
issue from processor sharing prospective, it opened sites in
discussing fairness in networks since process sharing
environment shares some similarity with resource sharing
in the Internet or networks. Kleinrock then wrote his book
which consists of two volume in queuing systems [2, 3]. In
this book the essential ideas and explanation of max-min
fairness principle is been demonstrated with the aid of
mathematic. Jaffe [4] incorporates the max-min fairness
principle explicitly in network resource sharing. This
concept is been presented in data networks book written by
Bertsekas et al. [5].

     Nevertheless, the concept and regulations which rule
max-min fairness and lead to its result are not convenience
and does not provide the efficient fairness from Kelly
point of view [6, 7]. Consequently, he proposed an
alternative fairness principle named as proportional
fairness. This concepts is further developed by Massoulie
and Roberts [8]. Bothe principles; max-min and                                    Exhibit 2.1: Mind-map literature of fairness
proportional are further compared and thoroughly                                                principles
analyzed by Denda et al. [9]. However, the advocates of
proportional fairness has comprehensively illustrate the
principle in [10].                                                                     3. PRINCIPLES OF FAIRNESS

     Despite the success of the most famous principles;                Approaching an optimum fairness in shared elastic
max-min and proportional, they have some weaknesses                    environment such as the Internet is complicated and
which are discovered by Bonald and Proutiere [11].                     frustrated. As a consequence, different proposals have
Balanced fairness is their proposal which is inspired by               been drawn to accomplish the mission in several
Erlang [12] ideas, has different approaches. All three                 prospective. This section provides rigorous knowledge in
principles; max-min, proportional and balanced fairness                the most five adopted fairness notions. This
are presented in Bonland et al. paper [13]. Bonland has                comprehensive illustration will reach the conceptual and
provided some comparison using analytical demonstration.               analytical approach of each o these five notions. Next
Another fairness view is called utility fairness introduced            section compares and contrasts these five principles.
by Cao and Zegura [14]. Utility fairness has adopted the
concept of utility proposed in [15]. All the above                          Before the explanation of the five notions mentioned
mentioned fairness definition have been presented in [16]              earlier, a scenario of shared resource is been assumed. So,
by Hosaagrahara.                                                       let consider the following scenario. Consider a contended
                                                                       user n with demands                                   varies
     However,     these    four    principles;    max-min,             from one user to another. Those users are sharing the one
proportional, balanced and utility fairness are in principle           resource R. Additionally, each user is allocated a specific
correlated and based on bandwidth allocation with                      portion                                             of the
different approaches in determining the proper algorithm               resource R according to a policy P. There are two main
to chose the next packet in line. The entire principle of              stipulations for such allocation;
bandwidth allocation has been criticized in Briscoe article
[17]. Therefore, Miaji and Hassan in [18] proposed a new                  a.The resource which is allocated is finite and limited.

                                                                                                  ISSN 1947-5500
                                                         (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                                              Vol. 8, No. 4, July 2010

     b.There is no resource feedback from users’ side.                  user 4 and 5 will take equal resource allocation no matter
                                                                        what they demand for.
      Consequently, any policy abides by these two
conditions is said to be active and defined as follows                       Additionally, any user attempts to increase its
]61[:                                                                   allocation will result in decrease in the resource allocated
                                                                        to another. Furthermore, it could be obviously seen that the
      Definition 3.1: The policy P is said to be active if, for         attempts to increase the demand will not influence the
all possible demands D, it results in an allocation A such              decision of allocation [16].
                                                                              Exhibit 3.1 provides us with much information which
          1.                                                            has not been illustrated yet. The essential inspiration of
          2.                                                            max-min fairness is the Pareto superiority as well as Pareto
                                                                        efficiency which were suggested by Pareto [19, 20]. In fact,
                                                                        Pareto proposed his notion in political economic and it has
     Now, let establish the investigation in the five fairness
                                                                        two main concept; superiority and efficiency for two
                                                                        active allocation. Firstly, if we have to allocate
                                                                        to two different resources                ,      is considered
3.1 Max-min Fairness                                                    as Pareto superior with respect to         if      expands the
                                                                        allocation of at least one entity while not reducing the
Let first simplify the principle of ma-min fairness be the              allocation of any other entity; for instance, at least one user
following example. Let assume that there are buckets                    prefers      over     . In the case of exhibit 3.1, user 4
which are corresponding to the demand         of the users.             prefers to obtain 40 units over 50 units and no other user
Moreover, let assume that all buckets share the same tab                request it. This preference will affect other users [21].
which corresponds to the resource R. Therefore, since the
resource is limited and the buckets cannot, indeed, provide                  Secondly, an allocation is considered as Pareto
any resource enhancement which there is no other resource               optimal if it is active and Pareto superior to all other active
except the one which is shared as seen in exhibit 3.1.                  allocations. Indeed, Max-Min fairness shows its Pareto
                                                                        optimality and hence it is unique since it is the only notion
                                                                        which meets the conditions of the Pareto optimality [22].

                                                                             Now, let take the analytical vision of the notion of
                                                                        Max-Min fairness. So, let presume that               is the
                                                                        allocation dedicated for        with demand       in flow
                                                                        and      is the allocation specified for   with demand .
                                                                        If we assume the                then the following theorem
                                                                        could be deduced;

                                                                         Theorem 3.1:

                                                                         The Max-Min fairness is unique.


                                                                             Let     and      two users with demands  and
                                                                        respectively and the resource allocated for them is
        Exhibit 3.1: Users Share the same resource                                  respectively as well. So, if     then the
                                                                        allocation results could be;
     According to max-min principle no user will obtain
more than its demand and also, all not fully served users
will be equally allocated in term of the resource.
                                                                             Only first one is possible since the remaining two are
    Therefore, user 1, 2, and 3 will take exactly what they             not Max-Min fair.
demand since their demands is the lowest. In comparison,

                                                                                                     ISSN 1947-5500
                                                     (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                                          Vol. 8, No. 4, July 2010

      Moreover, consider    is the service received by ,            less throughput [13].
then if          that means this allocation is not Max-Min
fair because in Max-Min fair the following should be true:
         . Hence the following definition is true for
Max-Min fair:

    Definition 3.1:

A policy       is considered Max-Min fair if and only if
satisfies the following conditions [22, 23]:

      1- A is active;
      2- Any attempts to increase and allocation for
 specific user result in a decrease in another user with
 equal or less value.

    Therefore, Max-Min policy should have the following                           Exhibit 3.2 a: Max-Min fairness

    Property 3.1: No user gets resource allocation than
what it have been requested.

     Property 3.2: users with same demands will be
allocated similar resource.

     Property 3.3: Any increase in the demand will not
affect the allocation procedure.


The idea of the proportional fairness is, indeed, proposed
after the discovery of some gap in the fairness of                               Exhibit 3.2 b: Proportional fairness2
Max-Min. We will simplify this concept by illustrating a
wireless node example [10].                                              Nevertheless, this maximization will not be fair since
                                                                    those nodes with bad radio channel will suffer from
     It well known that the fairness goal is not to                 starvation. In Max-Min concept as shown in exhibit 3.2a,
maximize the overall throughout or the bit rate or increase         those nodes with bad radio channel will be allocated more
the efficiency, it rather to be fair in allocating the              bandwidth since the main aim of such principle is to
bandwidth in accordance to the current network status.              maximize the minimum. However, from proportional
From this sense, consider a constant 1 wireless network             fairness point of view, this solution is not the optimum.
where there are two status of a node either good or bad.
Therefore, in order to achieve high throughput and hence                 Therefore, there is a trade of between efficiency and
to maximize the bit rate or increase the efficiency, it is          fairness. Proportional fairness is trying to solve and hence
better to allocate more bandwidth, transmission power and           minimize this trade of by proposing the concept of
so on to those good nodes since the bad one will                    allocating bandwidth in proportion to charge [6, 7].
experience more loss and required more bandwidth with

        1 This situation is likely to be impossible                          2The width of the wireless communication
especially in the case of mobile wireless                           link corresponds to the bandwidth allocated to this
environment.                                                        specific user.

                                                                                                 ISSN 1947-5500
                                                       (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                                            Vol. 8, No. 4, July 2010

     Logarithmic approach has been dedicated to such
approach. So, proportional fairness concept proposed the
notion of price per unit used or shared (see exhibit 3.2b). If
we assume that user        is charge of an amount of
for unit shared. Therefore, in proportion to        this user
will be allocated     . As a consequence the problem of
maximization could be formed as following;

                                                                                         (a)                           (b)
      So, the allocation for each user is depend on the
amount it is charged. This gives some restriction in the
                                                                                    Exhibit 3.3: example of utility fairness
utilization of such concept which will be discussed later in
the analysis and comparison section.
                                                                            Cao [14] in his article proposed and proof the
                                                                       following theorem;
3.3 Utility Fairness

The concept of utility fairness is easily to be inferred from               If     is the allocated bandwidth for session,      is
its name. This notion is based on the utility or the                   the link capacity, the real utility function for session is
application. It basically, derives the bandwidth allocation               ,    is the error in the advertised bandwidth and     is
in accordance to the characteristic of the application to be           the difference between the utility achieved by session
transmitted through the link. Therefore, packets which has               and the allocation deserved by the same session then;
elastic or more tolerance in term of delay or loss or any
other specified criterion, are allocated bandwidth
depending on its specifications, behavior, and
characteristics [14].                                                       A quantitative measure of the error in utility
                                                                       allocation is given by such theorem which resulted from
     Therefore, in the case of the identical utility or packet         the inaccurate information. Moreover, it reveals that there
specification or in other words applications, packets will             is a strong relationship between the error of utility
be treated as in Max-Min fairness. On the other hand, as               allocated to an individual source and the accuracy of
the application or packets diverse in its characteristics or           advertised utility functions; nevertheless, it is not affected
manners, the allocation scheme will also, changed and is               by the number of sources sharing the same bottleneck link
highly depends on the utility.                                         and hence no harms from any exponential increase in the
                                                                       users side.
     To simplify the idea of utility life example is been
provided. Now, consider an apple which needs to be                     3.4 Balanced fairness
divided among three people fairly as in exhibit 3.3. The
simple and basic way is to allocate one third of this apple            The proper definition of balanced fairness is the unique
to each person equally as shown in exhibit 3.3a. However,              balanced allocation such that           belongs to the
this sort of division is considered unfair if the                      boundary f the capacity set in any state            If Φ
circumstances of the people are not equal.                             corresponds to the balance function, the following
                                                                       equation is true in any state
     So, now consider the first person is a child how will
any way, cannot eat more than a quarter of the apple. The                                                                        (3.1)
second person is in diet and he also, cannot eat more than a
quarter of the apple and the third is very hungry energetic
youth. Consequently, according to the utility as one half is               Therefore,      is recursively defined as the
allocated to the youth, quarter for the child and the last             minimum positive constant β such that the vector
quarter portion is allocated for the person in diet (see                                        belongs to .
exhibit 3.3b).
                                                                            Balanced fairness is a new notion of bandwidth
                                                                       allocation with the very gratifying property that flow level
                                                                       performance metrics are insensitive to detailed traffic

                                                                                                   ISSN 1947-5500
                                                       (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                                            Vol. 8, No. 4, July 2010

characteristics [24]. This is particularly important for data         misbehaved user. Some charges will be applied to such
network engineering since performance can be predicted                users and hence minimize the allocation.
from an estimate of overall traffic volume alone and is
independent of changes in the mix of user applications                        4.   COMPARISON OF FAIRNESS PRINCIPLES
                                                                      The current status of the Internet provides only best-effort
3.5 Max-Min charge                                                    service. Consequently, providing enhancement for traffic
                                                                      flows for bandwidth reservation purposes is almost absent,
Max-Min charge has taken new different vision of fairness             or to be more precise bounding delay and jitter is not up to
in packet switching networks. The authors claim that to               the expectation level or even not met. Moreover, any
provide better fairness and proper protection to any user in          further modification in the protocols to be able to adopt the
a common shared resource, some aggressive penalty                     concept of reservation high efficiency of Quality of
should applied for those who are maliciously use the                  Service (QoS) required a crucial modification in the core
sharing procedure [18].                                               of the Internet which is unachievable. These boundaries
                                                                      rigorously reduce the ability of flows to demand
     Let take the analogy of multiple buckets sharing one             guarantees from the Internet, and the capability of the
 fountain or resource as in exhibit 3.4. So, let consider             Internet to put forward and accomplish such guarantees.
 that      greedily attempts to gain more bandwidth by
 initiating several session with multiple connation and                    If these constraints taken in account, the most
 hence reserves more bandwidth than the others. Such                  appropriate notion to be considered is max-min fairness.
 manner could breaches both the protection of other users             The principle of proportional fairness necessitates flows to
 who indeed fairly be using the resource and the fairness             transmit information about their bandwidth requirements
 by making        get double service than the others.                 and reservations to each router along their rout. The
                                                                      principle of utility fairness is unclear in term of the
                                                                      specification of the utility function and it rather demands
                                                                      flows to convey their utility.

                                                                           Nevertheless, minimum information about the flows
                                                                      among all notions is required by the principle of max-min
                                                                      fairness; a flow has a demand of unity if it has a packet
                                                                      enqueued and has a demand of zero otherwise. This
                                                                      information is, indeed, promptly available to each router
                                                                      and therefore the max-min principle of fairness is the most
                                                                      amenable to implementation. Likewise, max-min fairness
                                                                      is presently the most accepted principle of fairness in the
                                                                      research community.

                                                                                           5.   CONCLUSION

                                                                      The nature of the Internet traffics is random and dynamic;
         Exhibit 3.4: analogy of max-min charge
                                                                      therefore, such behaviors should be taken in account once
                                                                      the issue of resource allocation is investigated. It has
     Nevertheless, Max-Min fairness has nothing to do                 proven that max-min fairness, proportional fairness and
regarding such issue since it concern about fairness among            balanced fairness provide stability to the network
flows and not users. However, Max-Min charge assigns a                particularly when the vector of traffic intensities depends
specific values        and parameters to each user. The               on the interior of the capacity set. It is also, proven that
following equation has been deducted to improve the                   balance property have not been met by max-min fairness
protection level:                                                     notion with the exception of the trivial case where the
                                                                      network condenses to a set of independent links. This
                                                      (3.2)           justifies the limitation of the analytical results for this
                                                                      allocation and strengthens the assumption that such results
                                                                      should not be excluded. Proportional fairness is not
     By this notion, the only user who will suffer from any           balanced either except in some specific cases.
increase in the demand or in queue number is the

                                                                                                  ISSN 1947-5500
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                                                                                                           Vol. 8, No. 4, July 2010

     Balanced fairness, conversely, is balanced by                         "Rate control for communication networks:
construction and consequently leads to a well-mannered                     shadow prices, proportional fairness and
queueing network. Nevertheless, it is lack of practical                    stability," Journal of the Operational
implementation; nonetheless, balanced fairness could to a                  Research society, vol. 49, pp. 237-252, 1998.
certain extent be contemplated as a mathematical tool
practical for the performance evaluation of more practical           [8]   L. Massoulie and J. Roberts, "Bandwidth
allocations like max-min fairness and proportional                         sharing: objectives and algorithms," 1999.
fairness.                                                            [9]   R. Denda, A. Banchs, and W. Effelsberg, "The
                                                                           fairness challenge in computer networks,"
     Max-min fairness, in contrast, could achieve much                     2000, pp. 208-220.
worse performance than balanced fairness and
proportional fairness. This statement is been drown since            [10] N. S. Walton, "Proportional fairness and its
max-min fairness is giving the absolute priority to flows                 relationship with multi-class queueing
with small bit rates. Therefore, in wireless communication,               networks," The Annals of Applied Probability,
such notion results in an inefficient allocation where flows              vol. 19, pp. 2301-2333, 2009.
that experience bad radio conditions expend most radio
                                                                     [11] T. Bonald and A. Proutiere, "Insensitive
resources. Proportional fairness and balanced fairness,
alternatively, are homothetic as a result the allocated                   bandwidth sharing in data networks,"
network resources will not rely on the radio conditions.                  Queueing systems, vol. 44, pp. 69-100, 2003.
Thus, in heterogeneous networks, such allocations are                [12] A. K. Erlang, "Solution of some problems in
much more client and robust than max-min.                                 the theory of probabilities of significance in
                                                                          automatic telephone exchanges," The Post
     Max-Min charge is new definition which required                      Office Electrical Engineers’ Journal, vol. 10,
more conceptual and analytical proof to take place in the                 pp. 189–197, 1918.
competition. However, its primary ideas demonstrate its
novelty. Finally, market action is required to improve these         [13] T. Bonald, L. Massoulié, A. Proutiere, and J.
less-developed notions which could result in enormous                     Virtamo, "A queueing analysis of max-min
improvement in the fairness and protection of the                         fairness, proportional fairness and balanced
scheduling.                                                               fairness," Queueing systems, vol. 53, pp.
                                                                          65-84, 2006.
REFERENCES                                                           [14] Z. Cao and E. W. Zegura, "Utility max-min: an
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[2]   L. Kleinrock, "Queueing systems, volume 1:                          Communications, IEEE Journal on, vol. 13, pp.
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[3]   L. Kleinrock, Queueing Systems: Volume 2:                      [16] M. Hosaagrahara, "A generalized framework for
      Computer Applications: John Wiley & Sons                            achieving     max-min     fairness:  theory     and
      New York, 1976.                                                     applications." vol. PhD: Drexel University, 2006, p.
[4]   J.    Jaffe,   "Bottleneck      flow   control,"
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      [legacy, pre-1988], vol. 29, pp. 954-962, 1981.                     religion,"  ACM       SIGCOMM           Computer
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[5]   D. P. Bertsekas, R. Gallager, and T. Nemetz,
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      Cliffs, NJ, 1987.
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[6]   F. P. Kelly, "Charging and rate control for                         Communications (NetCoM-2009), 2009, pp.
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[7]   F. P. Kelly, A. K. Maulloo, and D. K. H. Tan,                       Journal of Political Economy, pp. 485-502, 1897.

                                                                                                ISSN 1947-5500
                                                   (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                                        Vol. 8, No. 4, July 2010

[20] V. Pareto, R. Marchionatti, and F. Mornati,                  AUTHORS PROFILE
     Considerations on the fundamental principles of pure
     political economy: Routledge, 2007.
[21] E. Karipidis, N. D. Sidiropoulos, and Z. Q. Luo,
     "Quality of service and max-min fair transmit
     beamforming to multiple cochannel multicast
     groups," IEEE Transactions on Signal Processing,
     vol. 56, pp. 1268-1279, 2008.
[22] D. Chakrabarty, J. Chuzhoy, and S. Khanna, "On
     allocating goods to maximize fairness," 2009, pp.
                                                                                          Yaser Miaji          received the B.E.
     107-116.                                                     form Riyadh College of Technology, Saudi Arabia and M.E.
[23] A. Sridharan and B. Krishnamachari, "Maximizing              degrees, from University of New South Wales, Australia. in 1997
     network utilization with max–min fairness in                 and 2007, respectively.      He is [16]currently a doctoral
                                                                  researcher in Computer Science in the University Utara Malaysia.
     wireless sensor networks," Wireless Networks, vol.
                                                                  Previously, he works as a lecturer in the College of
     15, pp. 585-600, 2009.                                       Telecommunication and Electronic in Jeddah from 1998-2206.
[24] T. Bonald, A. Proutiere, J. Roberts, and J. Virtamo,         His research interest includes digital electronics, computer
     "Computational aspects of balanced fairness," 2003,          network, distributed system and genetic algorithm. He is a
                                                                  member of InternetWorks research group, IEEE, ACM ISOC and
     pp. 801–810.                                                 STMPE.

                                                                                     Suhaidi Hassan PhD SMIEEE is an
                                                                  associate professor in computer systems and
                                                                  communication networks and the Assistant Vice
                                                                  Chancellor of the Universiti Utara Malaysia’s College of
                                                                  Arts and Sciences. He received his PhD in Computing
                                                                  from University of Leeds in United Kingdom, MS in
                                                                  Information Science from University of Pittsburgh, PA and
                                                                  BS in Computer Science from Binghamton University,
                                                                  NY. He currently heads the InterNetWorks Research
                                                                  Group at the Universiti Utara Malaysia and chairs SIG
                                                                  InterNetWorks of the Internet Society Malaysia Chapter.

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