Curve Fitting Approximation in Internet Traffic Distribution in Computer Network in Two Market Environment by ijcsiseditor

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									                                                                 (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                                                         Vol. 10, No. 4, 2012



 Curve Fitting Approximation in Internet Traffic
Distribution in Computer Network in Two Market
                  Environment

          Diwakar Shukla                                       Kapil Verma                                       Sharad Gangele
   Deptt. of Maths and Statistics                      Deptt. of Computer Science                          Deptt. of Computer Science
  Dr. H.S. Gour Central University                    M.P.Bhoj (Open) University,                          M.P.Bhoj (Open) University,
         Sagar, M.P., India.                               Bhopal, M.P., India.                                Bhopal, M.P, India
  diwakarshukla@rediffmail.com                        B.T. Institute of Research and                       sharadgangele@gmail.com
                                                    Technology, Seronja, Sagar, M.P.
                                                     Kapil_mca100@rediffmail.com



Abstract— The Internet traffic sharing problem has been studied              used to generate model based data and least square curve
by many researchers using a Markov chain model. The market                   fitting approach is applied.
situations are also responsible for determining the traffic share.
The market prime location has better chance to capture the users                                     II. A REVIEW
proportion. Using Markov chain model one can established                     The stochastic process has been used by many scientists and
mathematical relationship among the system parameters and                    researchers for the purpose of statistical modeling whose
variables. If the relationship is complicated than it is difficult to        detailed description is in Medhi (1991, 1992). Chen and Mark
predict about the output variable when input variables are                   (1993) discussed the fast packet switch shared concentration
known. This paper presents least square curve fitting approach               and output queueing for a busy channel. Humbali and Ramani
to simplify and present the complicated relationship into a simple
                                                                             (2002) evaluated multicast switch with a variety of traffic
linear relationship. This methodology is in use for the case of
traffic sharing under Markov chain model with two operators
                                                                             patterns. Newby and Dagg (2002) have a useful contribution
and two market environments. The coefficient of determination is             on the optical inspection and maintenance for stochastically
used as a tool to judge the accuracy of line fitting between two             deteriorating system. Dorea et al. (2004) used Markov chain
prime system variables. Graphical study is performed to support              for the modelling of a system and derived some useful
the findings.                                                                approximations. Yeian and Lygeres (2005) presented a work
                                                                             on stabilization of class of stochastic different equations with
Keywords- User behavior, Transition Probability Matrix (TPM),                Markovian switching. Shukla et al. (2007 a) advocated for
Markov Chain Model (MCM), Coefficient of Determination (COD),                model based study for space division switches in computer
Confidence Interval.                                                         network. Francini and Chiussi (2002) discussed some
                                                                             interesting features for QoS guarantees to the unicast and
                             I. INTRODUCTION                                 multicast flow in multistage packet switch. On the reliability
                                                                             analysis of network a useful contribution is by Agarwal and
The traffic pattern depends upon the market situation in the
                                                                             Lakhwinder (2008) whereas Paxson (2004) introduced some
city and an internet café in the prime place generates high
                                                                             of their critical experiences while measuring the internet
amount of users. If the same café is in remote area, the
                                                                             traffic. Shukla et al. (2009 a, b and c) presented different
customer arrival pattern shifts toward lower side. We come
                                                                             dimensions of internet traffic sharing in the light of share loss
across this of situation by the contribution of Naldi (2002) and
                                                                             analysis and comparison of method for internet traffic
Shukla et al. (2011). Most of authors quoted above have
                                                                             sharing. Shukla et al.(2009) have given rest state analysis in
shown the application of Markov chain model in defining the
                                                                             internet traffic distribution in multi-operator environment.
interrelationship between traffic sharing and blocking
                                                                             Shukla and Thakur (2009) discussed modeling of behavior of
probability. Their derived expressions are in polynomial order.
                                                                             cyber criminals when two internet operators are in market.
It is hard to specify the actual relationship in simple manner.
                                                                             Shukla et al. (2009) studied and discussed Markov chain
Shukla, Verma and Gangele (2012) discussed a methodology
                                                                             model for the analysis of round robin scheduling and derived
related to curve fitting with the same idea for the contributions
                                                                             state probability analysis of internet traffic sharing. Shukla et
of Shukla et al. (2011 a). The earlier expressions have been



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                                                                                                        ISSN 1947-5500
                                                                    (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                                                            Vol. 10, No. 4, 2012
al. (2010 a, b. c, d, e and f) have given some Markov Chain                     (b) After choosing a market, User in the cyber-café (shop),
model applications in view to disconnectivity factor, multi                     chooses the first operator Ou with probability p or to Ov with
marketing and crime based analysis. Shukla et al. (2010)                        (1-p).
presented index based internet traffic analysis of users by a
Markov chain model. Shukla et al. (2010 a, b, c and d)                          (c) Blocking probability experienced by the operator Ou are
discussed cyber crime analysis for multidimensional effect in                   L1 & L3 and by Ov are L2 & L4
computer network and internet traffic sharing. Shukla et
al.(2010) presented Iso-Share analysis of internet traffic                      (d) The connectivity attempts by user between operators are
sharing in presence of favoured disconnectivity. Shukla et al.                  on call-by-call basis, if the call for Ou is blocked in kth
(2011 a, b, c, d, e , f and g) discussed the elasticity property
and its impact on parameters of internet traffic sharing in                     attempt (k >O) then in (k + 1)th attempt user shifts to Ov. If
presence blocking probability of computer network specially                     this also fails, user switches to Ou in (k+2)th.
when two operators are in business competitions with each
other in a market. Shukla, Tiwari and Thakur (2011)                             (e) Whenever call connects through either of operators Ou or
presented analysis of internet traffic distribution for user                    Ov, we say system reaches to the state of success in n
behavior based probability in multi-market environment.                         attempts.
Shukla et al. (2011) presented analysis of user web browsing                    (f) User can terminate the attempt process which is marked as
for iso-browser share probability. Shukla et al. (2012) studied                 system to the abandon state Z at nth attempts with probability
least square curve fitting for Iso-failure in web browsing                      pA (either Ou or from Ov).
using Markov chain model. Shukla, Verma and Gangele
Presented least square based curve fitting in internet access
traffic sharing in two operator environment. Shukla, Verma                                                                                 1
and Gangele studied least square curve fitting application                                        M1                Market-I
under rest state environment in internet traffic sharing in                                                                                Z1
                                                                                                                    (1-p)
computer network.
                                                                                                                                                           (1-L2)
                                                                                                       p
                                                                                                               (1-L1)
                                                                                                                                      L1
           III. MARKOV CHAIN MODEL                 [As per Shukla et al.
(2011)]


Let {X (n), n ≥ 0} be a Markov chain model. As per Fig 3.1, let                                            O1                                                              O2
O1, O2, O3 and O4 be operators (ISP) in the two competitive                                 q
Market-I (M1) and Market-II (M2). User chooses a market                                                                                    L2
                                                                                                                                                                     L2 pA
first, then enters into a cyber-café situated there in, where                                                    L1 pA

computer terminals of different operators are available to
access the Internet. Operators are grouped as Ou (u=1,3) and                                                                               A
Ov (v=2,4) for market-I and market-II.                                          Users
State O1 : First operator in market-I,                                                                                  L3pA                                        L4pA
                                                                                                                                           L3
State O2 : Second operator in market-I,
                                                                                          (1-q)
State O3 : Third operator in market-II,
                                                                                                               O3                                                            O4
State O4 : Fourth operator in market-II,
State     Z1 : Success (link) in market-I(M1)                                                                                           L4

                                                                                                                    (1-L3)
State     Z2 : Success (link) in market- II (M2)                                                                                                                    (1-L4)
                                                                                                           p
State A : Abandon the attempt process.
                                                                                                                       (1-p)
                                                                                                                                               Z2
The X(n) stands for the state of random variable X at nth                                         M2
attempt of connectivity (n > 0) made by the user. Some                                                          Market - II                         1
underlying assumptions of the Markov chain model are:
(a) A User (or Customer or CU) first select the Market-I with
                                                                                                               FIGURE 3.1 : Transition Diagram of model.
probability q and Market-II with probability (1-q), (see Fig
3.1)




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                                                                                                                            ISSN 1947-5500
                                                                                                             (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                                                                                                     Vol. 10, No. 4, 2012
Fig.3.1 Explains the transition mechanism with transition probability matrix in (3.1)                                                          (1 − L2 ) q
                                                                                                                         p2 M 1 =                                 [(1 − p ) + pL1 (1 − p A )]...(5.2)
                                                         States                                                                            1 − L1 L2 (1 − p A ) 2
                                                             X(n)
                     O1       O2         O3           O4            Z1   Z2 A       M1 M2
                                                                                                                                          (1 − L1 )(1 − q )
                                                                                                                         p3 M 2 =                              [ p + (1 − p ) L2 (1 − p A )]...(5.3)
          O1         0        L1(1-pA ) 0            0              1- L1 0 L1 PA    0      0                                           1 − L1 L2 (1 − p A ) 2
          O2         L2(1-pA ) 0             0           0          1- L2 0    L2pA 0       0
          O3         0        0              0        L3(1-pA ) 0        1- L3 L3 pA 0      0                                                (1 − L2 )(1 − q )
          O4         0        0              L4(1- PA ) 0            0   1-L4 L4PA 0        0
                                                                                                                         p4 M =                                   [(1 − p ) + pL1 (1 − p A )]...(5.4)
                                                                                                                                    2
                                                                                                                                           1 − L1 L2 (1 − p A ) 2
X(n-1)    Z1         0        0              0           0           1    0     0        0 0
          Z2         0        0              0           0           0    1     0        0 0
                                                                                                                              VI. LEAST SQUARE FITTING OF STRAIGHT LINE
          A          0        0              0           0           0    0     1        0 0
          M1         p        1-p            0           0           0    0     0        0 0                             We have to approximate the relationship between parameter
          M2         0        0              p           1-p         0    0     0        0 0                                                               ∧
                                                                                                                         P1M and p through a straight line P1 M = a + b . L 1 where a
                                                                                                                                1                                                    1


                                                                                                                         and b are constants to be obtained by the method of least
                                                                                                                         square. For the ith observation pi we write the relationship as
                                                                                                                          ∧

            IV. SOME USEFUL RESULTS FOR nth                                                                              P1 M 1 i = a + b . L 1 i (i=1, 2, 3,…, n). The normal equations are
         CONNECTIVITY ATTEMPTS [Shukla et al. (2011)]                                                                     n                                  n
                                                                                                                                                                                ⎫
   Theorem 4.1: The nth step transitions probability for O2 in
                                                                                                                         ∑ P1 M1i = n.a + b∑ L1i                                ⎪
                                                                                                                         i =1                               i =1                ⎪
                        Market -1 is:                                                                                                                                           ⎬                    ...(6.1)
                                                                                                                                                                                ⎪
                                                                                                                                                      n                  n
 P[ X      (n)
                      = O2 ]       M1    =       q p (1 - p A )(1 - p A ) n -2 ( Even )                                  ∑P         1M1i   .L1i = a∑ L1i + b∑ L1i 2
                                                                                                                                                     i =1             i =1
                                                                                                                                                                                ⎪
                                                                                                                                                                                ⎭
 p[ X         (n )
                         = O2]                = q (1 - p ) (1 - p A ) n -1 ( O d d )
                                    M1
                                                                                                                         By solving the normal equations (5.1), the least square
                                                                                                                                                                   ∧ ∧
   Theorem 4.2: The nth step transitions probability for O3 in                                                           estimates of a and b are a, b :
                        Market-II is:
                                                                                                                             ⎧ n                 n
                                                                                                                                                                 ⎫
P[ X      (n)
                     = O3 ]   M2    = (1- q) (1- p ) L4 (1- p A ) (1- p A )                       n -2
                                                                                                         ( Even )        ∧   ⎪ n∑ P M1i L1i − (∑ P M1i )(∑ L1i ) ⎪
                                                                                                                             ⎪      1                   1
                                                                                                                                                                 ⎪
 p[ X      (n)
                          = O3]                  = (1 - q ) p (1 - p A )                 n -1
                                                                                                (O d d )                 b = ⎨ i =1     n
                                                                                                                                               i =1
                                                                                                                                                      n          ⎬                              ..... (6.2)
                                                                                                                             ⎪      n∑ L1i − (∑ L1i )            ⎪
                                     M   2
                                                                                                                                             2            2

  Theorem 4.3: The nth step transitions probability for O4 in                                                                ⎪
                                                                                                                             ⎩        i =1          i =1         ⎪
                                                                                                                                                                 ⎭
                        Market-II is:
 P[ X (n)
          = O4 ] M 2 = (1 - q ) p L3 (1 - p A ) (1 - p A ) n -2 ( Even )

 P[ X      (n)
                         = O4]     M 2       = (1 - q ) (1 - p )(1 - p A ) n -1 ( O d d )

                              V. LIMITING BEHAVIOUR

Let L1 be traffic share by the first operator and L2 be traffic
share by the second operator. Using Markov chain model &
Naldi (2002), Shukla et al. (2007) we can obtain the
expression of traffic sharing as:


                     (1 − L1 ) q
 p1 M 1 =                              [ p + (1 − p ) L2 (1 − p A )]...(5.1)
                1 − L1 L2 (1 − p A ) 2

∧   ⎧1 n        ∧ n     ⎫
a = ⎨ ∑ P M1i − b ∑ L1i ⎬
             1                                                                                   ...(6.3)                     Where n is the number of observations in sample of size n,
    ⎩ n i =1      i =1  ⎭                                                                                                     and resultant straight line is




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                                                                                                                                                                    ISSN 1947-5500
                                                                                              (IJCSIS) International Journal of Computer Science and Information Security,
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                                                                                                             P1M1 obtained through Markov chain model. The term
∧

           {   ∧
P M 1 = a + b .L1
 1
                   ∧

                           }                                                    ....(6.4)
                                                                                                              ∧           ∧      ∧
                                                                                                             P 1 M 1i = a + b . L1 i is the estimated by values of P1M 1 i
                                                                                                             given observation L1i. The coefficient of determination lies
                                                                                                             between 0 to 1. If the line is good fit then it is near to 1. We
The coefficient of determination (COD) as a measure of
                                                                                                             generate pair of values (L 1 , P1 M ) in tables (6.1, 6.2, and
good curve fitting is given in equations (6.5)                                                                                                               1

                                                                                                             6.3, 6.4, 6.5 and 6.6) by providing few fixed input
                                             2                                                               parameters.
       ∑ ⎛ P1 M 1i − P1 M1 ⎞
           ∧
         ⎜                 ⎟
C O D=   ⎝                 ⎠                                                        ...(6.5)
                   (                     )
                            2
       ∑ P1M1i − P1 M1
where L = 1            ∑   P1 M         is mean of original data of variable
       1
                   n               1i




                                                                                          ∧
                           Table 6.1 ( P1 M 1 by expression (6.1), P1                          M1
                                                                                                     by (6.4) with known pc, b, pq , and line in(6.4.1))

         Fixed                     L1              0.1          0.2             0.3                 0.4           0.5            0.6          0.7          0.8       0.9       COD
       parameter
     L2=0.2,p=0.3                  P M1          0.1502        0.1353       0.1199             0.1042         0.0880           0.0714      0.0543      0.0367      0.0186
                                   1
                                                                                                                                                                               0.9990
     q=0.4,pA=0.2
                                   ∧
                                   P M1
                                    1            0.1522        0.1358       0.1194             0.1029         0.0865           0.7009      0.5365      0.3721      0.2077


               ∧                        ∧                       ∧           ∧        ∧               ∧
               a = 0 .1 6 8 7;          b = − 0 .1 6 4 3 ; P 1 M 1 = a + b . L1 ;                   P1 M 1 = (0 .1 6 8 7 − 0 .1 6 4 3 . L1 )                        ...(6 .4 .1)

                                                                                          ∧
                           Table 6.2 ( P M 1 by expression (6.1), P
                                        1                          1                           M1    by (6.4) with known pc, b, pq , and line in,(6.4.2))
      Fixed                L1                                                                                                                                                 COD
                                             0.1          0.2           0.3               0.4              0.5           0.6            0.7          0.8          0.9
    parameter
    L2=0.2,p=0.5               P 1 0.1989 0.1777 0.1563 0.1346
                               1M                                                                         0.1128        0.0907         0.6839       0.0458       0.0230        0.9998
    q=0.4,pA=0.5
                               ∧
                               P M1 0.2003 0.1780 0.1560
                                1                                                        0.1340           0.1120        0.0900         0.0680       0.0460       0.0240


       ∧                           ∧                       ∧            ∧       ∧               ∧
       a = 0.2220;                 b = − 0.2199 ; P 1 M 1 = a + b . L1 ;                        P1 M 1 = (0.2220 − 0.2199. L1 )                                         ...(6.4.2 )




                                                                                          ∧
                           Table 6.3 ( P M 1 by expression (6.1), P
                                        1                          1                          M1    by (6.4) with known pc, b, pq , , and line in (6.4.3))




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       Fixed           L1                0.1           0.2              0.3                 0.4                  0.5               0.6           0.7            0.8         0.9         COD
     parameter
    L2=0.2,p=0.7        P1
                        1M               0.2589         0.2305          0.2021               0.1735                0.1449          0.1161         0.0872         0.0582     0.2919        0.9999
    q=0.4,pA=0.7
                        ∧
                        P M1
                         1
                                         0.2594         0.2307          0.2019               0.1732                0.1445          0.1158         0.0871         0.0584     0.0296

∧                  ∧                              ∧               ∧     ∧                   ∧
a = 0.2881; b = −0.2871; P1M1 = a + b .L1;                                                  P M1 = (0.2881 − 0.2871.L1 )
                                                                                             1                                                                              ...(6.4.3)

                                                                                                ∧
                       Table 6.4 ( P M 1 by expression (6.1), P
                                    1                          1                                    M1       by (6.4) with known pc, b, pq , , and line in (6.4.4))
       Fixed           L1                      0.1           0.2                0.3                 0.4                0.5           0.6          0.7            0.8         0.9        COD
     parameter
    L2=0.4,p=0.3        P1                 0.1935          0.1767           0.1589              0.1401             0.1201          0.0990        0.0766       0.0527        0.0272
                        1M
                                                                                                                                                                                         0.9955
    q=0.4,pA=0.2
                         ∧
                        P M1               0.1992          0.1782           0.1575              0.1386             0.1161          0.0954        0.0746       0.0539        0.0332
                         1



           ∧                         ∧                        ∧               ∧       ∧                  ∧
           a = 0.2197; b = −0.2071; P1M1 = a + b .L1;                                                    P M1 = (0.2197 − 0.2071.L1 )
                                                                                                          1                                                                       ...(6.4.4)

                                                                                                ∧
                       Table 6.5 ( P M 1 by expression (6.1), P
                                    1                          1                                    M1       by (6.4) with known pc, b, pq , , and line in (6.4.5))
        Fixed                L1                      0.1              0.2               0.3                  0.4             0.5           0.6          0.7           0.8         0.9    COD
      parameter
     L2=0.6,p=0.5            P M1               0.2375           0.2144             0.1905           0.1659             0.1405           0.1142        0.0871     0.0590     0.0300
                             1
                                                                                                                                                                                         0.9986
     q=0.4,pA=0.5
                                 ∧
                              P M1              0.2413           0.2154             0.1895           0.1636             0.1377           0.1183        0.0859     0.0600     0.0341
                               1



       ∧                     ∧                               ∧              ∧       ∧                    ∧
       a = 0.2672; b = −0.2591; P1M1 = a + b . L1;                                                   P M1 = (0.2672 − 0.2591.L1 )
                                                                                                      1                                                                      ...(6.4.5)

                                                                                                ∧
                       Table 6.6 ( P M 1 by expression (6.1), P
                                    1                          1                                    M1       by (6.4) with known pc, b, pq , , and line in (6.4.6))
        Fixed            L1                     0.1              0.2                0.3                  0.4             0.5               0.6          0.7           0.8         0.9    COD
      parameter
     L2=0.8,p=0.7           P 1                0.2799        0.2506             0.2209              0.1907              0.1601           0.1290        0.0975     0.0655     0.0330
                             1M
                                                                                                                                                                                         0.9997
     q=0.4,pA=0.7
                             ∧
                             P M1              0.2820        0.2512             0.2203              0.1894              0.1586           0.1277        0.1969     0.0660     0.0352
                              1

           ∧                         ∧                           ∧              ∧       ∧                    ∧
           a = 0.3129; b = −0.3085; P1M1 = a + b .L1;                                                    P M1 = (0.3129 − 0.3085.L1 )
                                                                                                          1                                                                       ...(6.4.6)



VII. CONFIDENCE INTERVAL




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The 100(1-   α ) percent confidence interval for a and b are                                                α⎫ ⎡                                    ⎤
                                                                                            ∧
                                                                                                ⎧                           n
                        ⎡
                        ⎢
                                                           ⎤
                                                           ⎥
                                                                                            b ± ⎨t( n − 2) , ⎬ .s ⎢
                                                                                                ⎩           2⎭ ⎢
                                                                                                                       ∑ (L           1i   − L1 ) 2 ⎥             ...(7.2)
∧
    ⎧           α ⎫           1                L1                                                                 ⎣        i =1                     ⎥
                                                                                                                                                    ⎦
a ± ⎨ t( n − 2 ) ⎬ .   s⎢       +                          ⎥       ...(7 .1)
    ⎩           2⎭      ⎢                                  ⎥
                                     n                                                                                          2
                                    ∑
                              n
                                                                                                                                                       α
                                                                                                                       ∧
                        ⎢                  ( L1 i − L1 ) 2 ⎥                                             ∑   ( Pi − Pi )
                        ⎣           i =1                   ⎦                                where s=                                 and t ( n − 2 )        is obtained from
                                                                                                             n − 2                                      2
where L1 = 1
                  n


           n
                 ∑L
                 i=0
                        1i   . The L1 = 4.5 from table (6.1-6.6)                                standard table. Take        α =0.05, n=9 then t7, 0.025=2.365




                                                Table: 7.1 Calculation of Confidence interval for a and b

        Fixed parameter                              Constant (a)                    Constant (b)                                   Confidence Interval

                                                       ∧                                ∧                          for a: (a=0.1653,                        a=0.1721)
    L2=0.2,p=0.3,q=0.4,pA=0.2                          a =0.1687                        b =-0.1643                for b: (b= -0.1616 , b=-0.1671)

    L2=0.2,p=0.5,q=0.4,pA=0.5                          ∧                                ∧                             for a: (a=0.2203, a=0.2237)
                                                       a =0.2220                        b =-0.2199                for b: (b=-0.2185 , b=-0.2212)
                                                                                        ∧
    L2=0.2,p=0.7,q=0.4,pA=0.7                          ∧                                b =-0.2871                    for a (a=0.2873 , a=0.2889)
                                                       a =0.2881                                                  for b: (b=-0.2865, b=-0.2878)

    L2=0.4,p=0.3,q=0.4,pA=0.2                          ∧                                ∧                             for a: (a=0.2103, a=0.2290)
                                                       a =0.2197                        b =-0.2071                    for b: (b=-0.1997, b=-0.2146)
    L2=0.6,p=0.5,q=0.4,pA=0.5                          ∧                                ∧                              for a: (a=0.2608, a=0.2737)
                                                       a =0.2672                        b =-0.2591                    for b: (b=-0.2539, b=-0.2642)
    L2=0.8,p=0.7,q=0.4,pA=0.7                          ∧                                ∧                             for a: (a=0.3094, a=0.3164)
                                                       a =0.3129                        b =-0.3085                    for b: (b=-0.3057,b=-0.3113)

                                                                                                                  ∧
                                                     a = 0.2464                                                  P1 M 1 = a + b ( L1 )
        Average Estimate                                                         b = − 0 .2 4 1 0
                                                                                                                  ∧
                                                                                                                 P1 M 1 = (0.2464 − 0.2410. L1 )




                                                                                   76                                      http://sites.google.com/site/ijcsis/
                                                                                                                           ISSN 1947-5500
                                                                  (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                                                          Vol. 10, No. 4, 2012

                 VIII. DISCUSSIONS:                                                [5]. Naldi, M. (2002): Internet access traffic sharing in a multi-user
                                                                                          environment, Computer Networks. Vol. 38, pp. 809-824.
The linear pattern between L1 and p1 M 1 is replaced by                            [6]. Newby, M. and Dagg, R. (2002): Optical inspection and
a direct equation of a straight line in the form                                          maintenance for stochastically deteriorating systems: average
                                                                  ∧
                                                                                          cost criteria, Jour. Ind. Statistical Associations. Vol. 40, Issue
 ∧        ∧    ∧
                         The least square estimates of a are                              No. 02, pp. 169-198.
P1 M 1 = a + b . L 1 .
                                                                                   [7]. Francini, A. and Chiussi, F.M. (2002): Providing QoS
                                                                          ∧               guarantees to unicast and multicast flows in multistage packet
0.1687, 0.2220, 0.2881, 0.2197, 0.2672, 0.3129 and b                                      switches, IEEE Selected Areas in Communications, vol. 20,
are -0.1643, -0.2199, -0.2871, -0.2071, -0.2591, -0.3085                                  no. 8, pp. 1589-1601.
respectively. The six possible equations of linear                                 [8]. Dorea, C.C.Y., Cruz and Rojas, J. A. (2004): Approximation
                                      ∧                                                   results for non-homogeneous Markov chains and some
                                                                                          applications, Sankhya. Vol. 66, Issue No. 02, pp. 243-252.
relationship between L1 and P M are
                             1              1                                      [9]. Paxson, Vern, (2004): Experiences with internet traffic
 ∧                                                                                        measurement and analysis, ICSI Center for Internet Research
P1 M 1 =(0.1687-0.1643.L 1 )                                                              International Computer Science Institute and Lawrence
 ∧
                                                                                          Berkeley National Laboratory.
P1 M 1 =(0.2220-0.2199.L1 )                                                        [10]. Yeian, C. and Lygeres, J. (2005): Stabilization of class of
                                                                                          stochastic differential equations with Markovian switching,
 ∧                                                                                        System and Control Letters. Issue 09, pp. 819-833.
P1 M 1 =(0.2881-2871.L 1 )                                                         [11]. Shukla, D., Gadewar, S. and Pathak, R.K. (2007 a): A
 ∧
                                                                                         stochastic model for space division switches in computer
P1 M 1 = ( 0 .2 1 9 7 − 0 .2 0 7 1 . L1 )                                                networks, International Journal of Applied Mathematics and
                                                                                         Computation, Elsevier Journals, Vol. 184, Issue No. 02,
 ∧                                                                                       pp235-269.
P1 M 1 = (0.2 672 − 0.2 591. L1 )                                                  [12]. Shukla, D. and Thakur, Sanjay, (2007 b) Crime based user
 ∧                                                                                       analysis in internet traffic sharing under cyber crime,
P1 M 1 = (0.3129 − 0.3085. L1 )                                                          Proceedings of National Conference on Network Security and
                                                                                         Management (NCSM-07), pp. 155-165, 2007.
The coefficients of determination (COD) in each case
are nearly 1 therefore the estimated values of a and b                              [13]. Shukla, D., Virendra Tiwari, M. Tiwari and Sanjay Thakur
                                                                                          [2007 c]: Rest State analysis of Internet traffic distribution in
are very close to the real values. The average equation                                   multi-operator environment published in the Journal of
of linear relationship over six values is                                                 management Information Technology (JMIT-09), Vol. 1, pp.
 ∧                           ∧                                                            72-82
P1 M 1 = a + b ( L 1 ) ;     P1 M 1 = ( 0 .2 4 6 4 − 0 .2 4 1 0 . L 1 )             [14]. Agarwal, Rinkle and Kaur, Lakhwinder (2008): On
                                                                                          reliability analysis of fault-tolerant multistage interconnection
                                                                                          networks, International Journal of Computer Science and
                                                                                          Security (IJCSS) Vol. 02, Issue No. 04, pp. 1-8.
                                                                                     [15].Shukla, D., Tiwari, Virendra, Thakur, S. and Deshmukh, A.
                         XI. CONCLUSION                                                   (2009 a):Share loss analysis of internet traffic distribution in
                                                                                          computer networks, International Journal of Computer Science
                                                                                          and Security (IJCSS), Malaysia, Vol. 03, issue No. 05, pp.
  The data is generated from the Markov chain model                                       414-426.
for P1M1 and L1 values. It is found that both of these                              [16]. Shukla, D., Tiwari, Virendra, Thakur, S. and Tiwari, M.
values are negatively correlated. The increasing value                                    (2009 b) :A comparison of methods for internet traffic sharing
of blocking probability reduces the traffic share in the                                  in computer network, International Journal of Advanced
                                                                                          Networking and Applications (IJANA).Vol. 01, Issue No.03,
first market. The average and best predicted relationship                                 pp.164-169.
     ∧
is P1 M = ( 0 .2 4 6 4 − 0 .2 4 1 0 . L1 ) which is useful for                      [17]. Shukla, D., Tiwari, V. and Kareem, Abdul, (2009 c) All
       1                                                                                  comparison analysis in internet traffic sharing using markov
quick decision making and calculation whereas the                                         chain model in computer networks, Georgian Electronic
                                                                                          Scientific       Journal:       Computer        Science      and
general relationship depends upon many model                                              Telecommunications. Vol. 06, Issue No. 23, pp. 108-115.
parameters. The coefficient of determination supports                               [18]. Shukla, D, Tiwari, M., Thakur, Sanjay and Tiwari,
the fact that the line fitting is good and robust. The                                    Virendra [2009 d]: Rest State Analysis in Internet Traffic
estimated values of P1M1 are very close to the true                                       Distribution in Multi-operator Environment, (GNIM's)
                                                                                          Research Journal of Management and Information
values showing the consistancy of the result.                                             Technology, Vol. 1, No. 1, pp. 72-82.
                                                                                   [19].Shukla, D. and Thakur, Sanjay [2009 e]: Modeling of
                                                                                         Behavior of Cyber Criminals When Two Internet Operators in
                             References                                                  Markets, Accepted for publication in ACCST Research Journal,
                                                                                         Vol. VIII, No. 3, July, (2009).
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[2]. Medhi, J. (1992): Stochastic Processes, Ed.4, Wiley Eastern                         scheduling scheme, International Journal of Advanced
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      shared concentration and output queuing, IEEE Transactions                   [21]. Shukla, D., Thakur S. and Deshmukh Arvind [2009 g]: State
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[4]. Hambali, H. and Ramani, A. K., (2002): A performance study                          network, International Journal of Advanced Networking and
      of at multicast switch with different traffics, Malaysian                          Applications (IJANA), vol. 1, issue 1, pp. 90-95.
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                                                                                         internet traffic sharing, National Conference on Research and




                                                                              77                                     http://sites.google.com/site/ijcsis/
                                                                                                                     ISSN 1947-5500
                                                                (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                                                        Vol. 10, No. 4, 2012

      Development Trends in ICT (RDTICT-2010), Lucknow                                 Internet Browser share Problem, International Journal of
      University, Lucknow.                                                             Advanced Research in Computer Science (IJARCS),Vol. 02,
[23].Shukla, D., Gangele, Sharad and Verma, Kapil, (2010 b):                           No. 04, pp.473-478.
      Internet traffic sharing under multi-market situations, Published        [36].Shukla, D., Gangele, Sharad, Verma, Kapil and Thakur,
      in Proceedings of 2nd National conference on Software                            Sanjay, (2011 c): A Study on Index Based Analysis of Users
      Engineering and Information Security, Acropolis Institute of                     of Internet Traffic Sharing in Computer Networking, World
      Technology and Research, Indore, MP, (Dec. 23-24,2010), pp                       Applied Programming (WAP), Vol. 01, No. 04, pp. 278-287.
      49-55.                                                                     [37]. Shukla, D., Tiwari, Virendra and Thakur, Sanjay [2011]
[24].Shukla, D., and Thakur, S. (2010 c): Stochastic Analysis of                       Analysis of Internet Traffic Distribution for User Behavior
      Marketing Strategies in internet Traffic, INTERSTAT (June                        Based Probability in Two Market Environment, International
      2010).                                                                           Journal of Computer Application (IJCA), Vol. 30, Issue No.
[25].Shukla, D., Tiwari, V., and Thakur, S., (2010 d): Cyber Crime                     08. pp. 44-51.
      Analysis for Multi-dimensional Effect in Computer Network,                 [38]. Shukla, D., Gangele, Sharad, Singhai, Rahul and Verma,
      Journal of Global Research in Computer Science(JGRCS), Vol.                      Kapil, (2011 d): Elasticity Analysis of Web Browsing
      01, Issue 04, pp.31-36.                                                          Behavior of Users, International Journal of Advanced
[26].Shukla, D., Tiwari V. and Thakur S. [2010 e]: User behavior                       Networking and Applications (IJANA), Vol. 03, No. 03,
      Based Probability Analysis of Internet Traffic Distribution in                   pp.1162-1168.
      Two market in Computer Networks, Kalpagam Journal of                       [39]. Shukla, D., Verma, Kapil and Gangele, Sharad, (2011 e):
      Cambridge Studies (KJCS)                                                         Re-Attempt Connectivity to Internet Analysis of User by
[27].Shukla, D., Tiwari V. and Thakur S. [2010 f]: Performance                         Markov Chain Model, International Journal of Research in
      Analysis for Two Call Attempt of rest State Based Traffic                        Computer Application and Management (IJRCM) Vol. 01,
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      Application (IJANA)                                                        [40]. Shukla, D., Gangele, Sharad, Verma, Kapil and Trivedi,
[28].Shukla, D. and Thakur, Sanjay [2010]: Index based Internet                        Manish, (2011 f): Elasticity variation under Rest State
       traffic sharing analysis of users by a Markov chain probability                 Environment In case of Internet Traffic Sharing in Computer
       model. , Karpagam Journal of Computer Science, vol. 4, no. 3,                   Network, International Journal of Computer Technology and
       pp. 1539-1545.                                                                  Application (IJCTA) Vol. 02, Issue No. 06, pp. 2052-2060.
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       [2010 a]: Two call based analysis of internet traffic sharing,                  Manish, [2011]: Two-Call Based Cyber Crime Elasticity
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       Vol. 1, No. 1, pp. 14-24.                                                       International Journal of Computer Application (IJCA) Vol.02,
[30].Shukla, D. and Singhai, Rahul [2010 b]: Traffic analysis of                       Issue 01, pp.27-38.
       message flow in three cross-bar architecture in space division            [42]. Shukla, D., Singhai, Rahul [2011]: Analysis of User Web
       switches, Karpagam Journal of Computer Science, vol. 4, no.                     Browsing Using Markov chain Model, International Journal of
       3, pp. 1560-1569.                                                               Advanced Networking and Application (IJANA), Vol. 02,
[31]. Shukla, D., Thakur, Sanjay and Tiwari, Virendra [2010 c]:                        Issue No. 05, pp. 824-830.
       Stochastic modeling of Internet traffic management,                      [43]. Shukla, D., Verma, Kapil and Gangele, Sharad, [2012]: Iso-
       International Journal of the Computer the Internet and                           Failure in Web Browsing using Markov Chain Model and
       Management, Vol. 18, no. 2 pp. 48-54.                                            Curve Fitting Analysis, International Journal of Modern
[32]. Shukla, D., Tiwari, Virendra and Thakur, Sanjay [2010 d]:                         Engineering Research (IJMER) , Vol. 02, Issue 02, pp. 512-
       Cyber crime analysis for multi-dimensional effect in computer                    517.
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       Vol.1, no. 4. pp. 14-21.                                                [44]. Shukla, D., Verma, Kapil and Gangele, Sharad, [2012]: Least
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       of Internet Traffic Sharing in Presence of Favoured                            Operator Environment, International Journal of Computer
       Disconnectivity,      GESJ:       Computer      Science      and               Application (IJCA), Vol.43(12), pp. 26-32.
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[34]. Shukla, D., Gangele, Sharad, Verma, Kapil and Singh,                     [45]. Shukla, D., Verma, Kapil and Gangele, Sharad, [2012]: Least
       Pankaja (2011 a): Elasticity of Internet Traffic Distribution                  square curve fitting applications under rest state environment
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       pp.6-12.                                                                       (IJCST), Vol. 03, Issue 05.
[35]. Shukla, D., Gangele, Sharad, Verma, Kapil and Singh,
       Pankaja (2011 b): Elasticities and Index Analysis of Usual




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