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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 71 http://sites.google.com/site/ijcsis/ 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) 72 http://sites.google.com/site/ijcsis/ 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 73 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 10, No. 4, 2012 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)) 74 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 10, No. 4, 2012 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 75 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 10, No. 4, 2012 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). [1]. Medhi, J. (1991): Stochastic models in queuing theory, [20]. Shukla, D., Jain Saurabh, Singhai Rahul and Agarwal R.K. Academic Press Professional, Inc., San Diego, CA. [2009 f]: A Markov chain model for the analysis of round robin [2]. Medhi, J. (1992): Stochastic Processes, Ed.4, Wiley Eastern scheduling scheme, International Journal of Advanced Limited (Fourth reprint), New Delhi. Networking and Applications (IJANA), vol. 01, no. 01, pp. 01- [3]. Chen, D.X. and Mark, J.W. (1993): A fast packet switch 07. shared concentration and output queuing, IEEE Transactions [21]. Shukla, D., Thakur S. and Deshmukh Arvind [2009 g]: State on Networking, vol. 1, no. 1, pp. 142-151. probability analysis of Internet traffic sharing in computer [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. Journal of Computer Science. Vol. 15, Issue No. 02, Pp. 34- [22]. Shukla, D., Tiwari, Virendra, and Thakur, S. (2010 a): 42. Effects of disconnectivity analysis for congestion control in 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, Network, International Journal of Advanced Networking and Issue No. 09, pp. 94-99. 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. [29]. Shukla, D., Tiwari, V., Thakur, S. and Deshmukh, A.K. [41]. Shukla, D., Gangele, Sharad, Verma, Kapil and Trivedi, [2010 a]: Two call based analysis of internet traffic sharing, Manish, [2011]: Two-Call Based Cyber Crime Elasticity International Journal of Computer and Engineering (IJCE), Analysis of Internet Traffic Sharing In Computer Network, 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. network, Journal of Global Research in Computer Science, Vol.1, no. 4. pp. 14-21. [44]. Shukla, D., Verma, Kapil and Gangele, Sharad, [2012]: Least [33]. Shukla, D. and Thakur, Sanjay [2010 e ]: Iso-share Analysis Square Curve Fitting in Internet Access Traffic Sharing in Two 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. Telecommunications, 4(27), pp. 16-22. [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 Computer Network in two Market Environment, Journal of in internet traffic sharing in computer network, International Global research in Computer Science (JGRCS) Vol.2, No. 6, Journal of Computer Science and Telecommunications, 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 78 http://sites.google.com/site/ijcsis/ ISSN 1947-5500