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VOL. 1, NO. 1, November 2011 ISSN XXXX-XXXX ARPN Journal of Science and Technology ©2011-2012 ARPN Journals. All rights reserved. http://www.ejournalofscience.org Plant Stress Gene Database: A collection of plant genes responding to stress condition Ratna Prabha1,2, Indira Ghosh3 , Dhananjaya P. Singh 4 1 2 3 Project Trainee, School of Information Dean, Professor Senior Scientist, Technology, School of Information Technology, National Bureau of Agriculturally JNU, New Delhi- 110067, India. JNU, New Delhi- 110067, India Important Microorganisms, Email: ratna.bioinfo@gmail.com Kushmaur, 2 Current Address: Senior Research Fellow, Mau Nath Bhanjan -275101, U. P., India National Bureau of Agriculturally Important Microorganisms, Kushmaur, Mau Nath Bhanjan -275101, U. P., India ABSTRACT Network traffic monitoring is an important way for network performance analysis and monitor. The present article explores how to build the basic model of network traffic analysis based on Queuing Theory. In the present work, two queuing models (M/M/1): ((C+1)/FCFS) and (M/M/2): ((C+1)/FCFS) have been applied to determine the forecast way for the stable congestion rate of the network traffic. Using this we can obtain the network traffic forecasting ways and the stable congestion rate formula. Combining the general network traffic monitor parameters, we can realize the estimation and monitor process for the network traffic rationally. Keywords: Network traffic, Queuing Theory, stable congestion rate 1. INTRODUCTION Adopting Queuing Theory to estimate the network traffic, it becomes the important ways of network Network traffic monitoring is an important way for performance prediction, analysis and estimation and, network performance analysis and monitor. The present through this way, we can imitate the true network, it is analysis seeks to explore how to build the basic model of useful and reliable for organizing, monitoring and network traffic analysis based on Queuing Theory [1]. defending the network. Using this, we can obtain the network traffic forecasting ways and the stable congestion rate formula, combining 2. THE MATHEMATICAL MODEL OF THE the general network traffic monitor parameters. QUEUING THEORY Consequently we can realize the estimation and monition process for the network traffic rationally. In network communication, from sending, transferring Queuing Theory, also called random service theory, is to receiving data and the proceeding of the data coding, a branch of Operation Research in the field of Applied decoding and sending to the higher layer, in all these Mathematics. It is a subject which analyze the random process, we can find a simple queuing model. According regulation of queuing phenomenon, and builds up the to the Queuing Theory, this correspond procedure can be mathematical model by analyzing the date of the network. abstracted as Queuing theory model [2], like fig. 1. Through the prediction of the system, we can reveal the Considering this kind of simple data transmitting system regulation about the queuing probability and choose the satisfies the queue model [3]. optimal method for the system. Nq Ts TN λ' TJ TD TC λ 1 VOL. 1, NO. 1, November 2011 ISSN XXXX-XXXX ARPN Journal of Science and Technology ©2011-2012 ARPN Journals. All rights reserved. http://www.ejournalofscience.org Figure 1: The abstract model of communication process From the above fig. 1, In model M/M/1, the two M represent the : Sending rate of the sender. sending process of the sender and the receiving process TN : Transportation delay time. of the receiver separately. They both follow the Markov : Arriving speed of the data packets Process [4], also keep to Poisson Distribution, while the Nq : Quantity of data packets stored in the number 1 stands for the channel. buffer (temporary storage). Let N(t)=n be the length of the queue at the : Packets rate which have mistake in sending moment of t. So the probability of the queue whose length is n, be from receiver i.e., lost rate of the receiver. Ts : Service time of data packets in the server Pn =Prob [N(t)= n] where Ts=TJ+TD+TC In this model, TJ : Decoding time n = Rate of arrival into the state n TD : Dispatching time µn =Rate of departure from the state n TC : Calculating time or evaluating time or handling time. We have the transition rate diagram as follows 3. Model-1: The Queuing model with one server (M/M/1):((C+1)/FCFS) Figure 2: State transition diagram The system of differential difference equation is d d dt {Pn (t )} n Pn (t ) n Pn (t ) n1 Pn1 (t ) n1 Pn1 (t ) dt Pn (t ) Pn1 (t ) Pn1 (t ) ( ) Pn (t ) for , for n≥1 (1) n≥1 And d P0 (t ) 0 P0 (t ) 1 P1 (t ) for And d P0 (t ) P0 (t ) P1 (t ) for dt dt n=0 (2) n=0 In model M/M/1, we let Here, λ is considered as the arrival rate while μ as the n And n service rate. Where λ and µ are constants. In the steady state equation Then eqs.(1) and (2) reduces to Lt Pn (t ) Pn t d and Lt {Pn (t )} 0 t dt 2 VOL. 1, NO. 1, November 2011 ISSN XXXX-XXXX ARPN Journal of Science and Technology ©2011-2012 ARPN Journals. All rights reserved. http://www.ejournalofscience.org Hence, from eqs. (3) and (4) when t we get Hence L (7) 0 Pn 1 Pn 1 ( ) Pn for n≥1 1 (5) Also L Since, and 0 P0 P1 This implies P1 ( ) P0 If N q denotes the average volume of the buffers data From eq.(5) when n=1, we get packets then ( ) P1 P0 P2 2 Nq L (8) Therefore, P2 ( ) 2 P0 1 Also In general, Pn ( ) n P0 2 Nq ( ) or, Pn n P0 where Using the Little’s law we have Ts and Here, is called server utilization factor or traffic (9) intensity. Using eq.(9), eq.(8) reduces to We know, P n 0 n 1 Nq 2 1 Also, Pn n P0 This implies (1 Ts ) N q (Ts ) 2 Therefore, Pn n P0 n 0 n 0 Or, ( Ts ) 2 Ts N q N q 0 , ( ' ) (10) or, 1 P0 n n 0 The above equation eq.(10) provides the relation between following parameters Consequently, P0 1 , where < 1 T s = Service time Hence, Pn n (1 ) , n=0, 1, 2,… . Sending rate N q Quantity of data (6) packets stored in the buffer If we know any two variables, it is easy to gain the Suppose, L stands for the length of the queue under the numerical value of the third one. steady state condition. It includes the average volume of So, these three variables are key parameters for all the data packets which enter the processing module measuring the performance of the transmission system. and store in the buffer. 4. QUEUING THEORY AND THE L nPn n (1 ) n n 0 n 1 NETWORK TRAFFIC MONITOR (1 ) n n 4.1.Forecasting the network traffic using n 1 Queuing Theory The network traffic is very common [5]. The system will be in worse condition, when the traffic becomes 3 VOL. 1, NO. 1, November 2011 ISSN XXXX-XXXX ARPN Journal of Science and Technology ©2011-2012 ARPN Journals. All rights reserved. http://www.ejournalofscience.org under extreme situation, in which leads to the network be lost. When the arriving time of group timeouts, the congestion [6]. There are a great deal of research about group has to resend. Suppose, the group’s average monitoring the congestion at present ,besides, the 1 documents which make use of Queuing Theory to waiting time is . We identify Pi (t) to be the arrival research the traffic rate appear more and more. For forecasting the traffic rate, we often test the data disposal probability of the queue length for the routers group at function of the router used in the network. the moment of t, supposing the queue length is i: P(t) = (P0 (t), P1 (t), . . . , Pn−1 (t) ), i = 0,1, . . . ,C+1. Considering a router’s arrival rate of data flow in groups is , and the average time which the routers use Then the queuing system of the router’s date groups satisfies simple Markov Process [7], according to 1 Markov Process, we can find the diversion strength of to dispose each group is , the buffer of the routers is matrix of model 1 as follow: C, if a certain group arrives, the waiting length of the queue in groups has already reached, so the group has to 0 0 0 0 ( ) 0 0 0 0 ( 2 ) 2 0 0 0 0 ( 3 ) 0 0 Q 0 0 0 0 0 0 0 0 0 ( C ) C 0 0 0 0 4.2 Network Congestion Rate For k ≥ 1 Network congestion rate is changing all the time [8]. The instantaneous congestion rate and the stable congestion rate are often used to analysis the Pk 1 (t t ) = Prob { k number of data packets network traffic in network monitor. The instantaneous present in the system at time t } Prob { rate AC (t ) is the congestion rate at the moment of t. The no data packet arrival in time ( ∆t) } Prob { no data AC (t ) can be obtained by solving the system length of packet departure in time ∆t } the queue’s probability distributing, which is called PC 1 (t ) . + Prob { (k-1) number of data packets present in the system at time t } Let, Pk (t ) (k=0,1,. . .,C+1) to be the arrival probability Prob { 1 data packet arrival in time ( ∆t) } of the queue length for the routers group at the moment Prob { no data packet departure in time ∆t } + of t by considering the queue length is k. Prob { (k+1) number of data packets present in the Then, the queuing system of the router’s date system at time t } groups satisfies simple Markov Process. According to Prob {no data packet arrival in time ( ∆t) } Markov Process, Pk (t ) satisfies the following system of differential difference equations. Prob { 1 data packet departure in time ∆t }+… Let, Pk (t t ) Pk (t ){1 k t o(t )}{1 k t o(t )} Pk (t ) = Prob { k number of data packets present in the system in time t } and Pk 1 (t ) = Prob { k number of data Pk 1 (t ){k 1t o(t )}{1 k 1t o(t )} packets present in the system in time (t + ∆t) } Case 1: 4 VOL. 1, NO. 1, November 2011 ISSN XXXX-XXXX ARPN Journal of Science and Technology ©2011-2012 ARPN Journals. All rights reserved. http://www.ejournalofscience.org + Dividing both sides by t and taking limit as t 0 , Pk 1 (t ){1 k 1t o(t )}{ k 1t o(t )} + we obtain o(t ) Pk (t t ) Pk (t ) (k k ) Pk (t )t Pk 1 (t )k 1t Pk 1 (t )k 1t o(t ) d {P0 (t )} P0 (t ) P (t ) 1 dt Dividing both sides by t and taking limit as t 0 (since, k k and k ) d {Pk (t )} (k k ) Pk (t ) k 1 Pk 1 (t ) dt Case 3: o(t ) k 1 Pk 1 (t ) , since lim 0 For k=C+1, we have (11) t t PC 1 (t t ) = Prob { (C+1 ) no. of data packets Here, in state k, data packets arrival is ( k ) present in the system at time (t+∆t )} i.e. k k = Prob { C no. of data packets present in time t } prob {1 data packet arrival in time ∆t } Prob { no data packet departure in time ∆t } Also, in state k, data packet departure is + Prob { (C+1) no of data packets present in time t } i.e. k Prob { no data packet departure in time ∆t }+… PC (t ){C t o(t )}{1 C t o(t )} PC 1 (t ){1 C 1t o(t )} o(t ) Hence, eq.(11) reduces to d {Pk (t )} ( k ) Pk (t ) { (k 1) }Pk 1 (t ) Pk 1 (t ) PC 1 (t t ) PC 1 (t ) PC (t )C t C 1 PC 1 (t )t o(t ) dt (12) where k=1,2,…,C Dividing both sides by t and taking limit as Case 2: t 0 we get d For k=0, we have {PC 1 (t )} PC (t )C C 1 PC 1 (t ) dt P0 (t+∆t) = Prob { no data packet present in the system d at time (t+∆t) } {PC 1 (t )} ( C ) PC (t ) PC 1 (t ) , dt = Prob { no data packet present in time t } since C C Prob { no data packet arrival in time ∆t } + Prob {one data packet present in time t } Prob { no data packet arrival in time ∆t } By solving this differential equation system, we get the Prob { one data packet departure in time ∆t } instantaneous congestion rate +… A0(t) as A0 (t ) P1 (t ) (1 e ( )t ) P (t ){1 0t o(t )} P1 (t ){1 1t o(t )}{ 1t o(t )} o(t ) = 0 The instantaneous congestion rate can not be P0 (t t ) P0 (t ) 0 P0 (t ) P (t ) 1 o(t ) 1 used to measure the stable operating condition of the system, so we must obtain the stable congestion rate of the system. The so-called stable congestion rate means, it 5 VOL. 1, NO. 1, November 2011 ISSN XXXX-XXXX ARPN Journal of Science and Technology ©2011-2012 ARPN Journals. All rights reserved. http://www.ejournalofscience.org will not change with the time changing, when the system Under steady state condition, eqs.(15),(16) and (17) works in a stable operating condition. The definition of transform to following balance equations. the stable congestion rate is ( k ) Pk (t ) { (k 1) }Pk 1 (t ) Pk 1 (t ) AC lim AC (t ) for k=1, 2, 3, … ,C (18) t Considering, P lim P(t ) as the distributing t 0 P0 (t ) P (t ) for k=0 1 of the stable length of the queue and C as the buffer of the router, the stable congestion rate can be obtained in (19) two ways: firstly, we obtain the instantaneous congestion rate, then find its limit. According to its definition, it can 0 ( C ) PC (t ) PC 1 (t ) for k=C+1 be obtained with the distributing of the length of the (20) queue. Secondly, according to the Markov Process, we know that the distributing of the stable length of queue The above system of steady state equations can be can be obtained through system of steady state written in matrix from as equations. PQ 0 From eq.(12), eq.(13) and eq.(14), we have the system of C 1 differential difference equations as follows and P 1 i 0 i d {Pk (t )} ( k ) Pk (t ) { (k 1) }Pk 1 (t ) Pk 1 (t ) where P ( P0 , P ,..., PC 1 ) 1 dt (15) and for k=1, 2, 3, … ,C 0 0 0 0 ( ) 0 0 0 d 0 ( 2 ) 2 0 0 {P0 (t )} P0 (t ) P (t ) 1 for k=0 0 0 ( 3 ) 0 0 dt Q 0 0 0 0 0 (16) 0 0 0 0 ( C ) C 0 d 0 0 0 {PC 1 (t )} ( C ) PC (t ) PC 1 (t ) for k=C+1 dt For C= 0, (17) From eq.(19), we have According to some properties of Markov process, we P0 P1 (21) know that Also, P0 P 1 1 (22) Pi (t ) (i=0,1,2,…,C+1) satisfies the above Solving (21) and (22) we get differential equation. P1 Here, P(t ) [ P0 (t ), P (t ),..., PC1 (t )] 1 Hence, P0 (0) 1, P (0) 0, P2 (0) 0,..., PC 1 (0) 0 1 P(0) [ P0 (0), P1 (0),..., PC 1 (0)] A0 P1 (23) For C=1 For steady state equation, 0 P0 P1 dP (t ) (24) lim Pk (t ) Pk and lim k 0 t t dt 0 ( ) P1 P0 P2 (25) 6 VOL. 1, NO. 1, November 2011 ISSN XXXX-XXXX ARPN Journal of Science and Technology ©2011-2012 ARPN Journals. All rights reserved. http://www.ejournalofscience.org 0 ( ) P P2 1 (26) From eq.(29), we have Also, P0 P P2 1 ( ) 1 P2 k From eq.(23), we get From eq.(31), we have P0 P1 k ( 2 ) P3 P2 Therefore, P0 k , P k 1 ( )( 2 ) k From eq.(25), we have 2 P2 ( )k , since, P k (27) From eq.(32), we have 1 2 k ( ) 2 ( ) ( )( 2 ) Using eq.(26), we obtain Hence, k[ ] 1 ( )( 2 ) A2 P2 ( ) ( ) ( )( 2 ) 2 k ( ) ( ) For C=3, yielding From eq.(27) yields 0 ( ) P1 P0 P2 (33) 0 ( 2 ) P2 ( ) P1 P3 P2 ( ).. (34) ( ) ( ) 0 ( 3 ) P3 ( 2 ) P2 P4 Hence, (35) 0 P0 P1 (36) ( ) 0 ( 3 ) P3 P4 (37) A1 P2 ( ) ( ) From eq.(36), we obtain For C=2, P0 k and P1 k 0 P0 P1 (28) From eq.(33) 0 ( ) P1 P0 P2 (29) ( ) P2 k From eq.(34) 0 ( ) P ( 2 ) P2 P3 ( )( 2 )k 1 (30) P3 2 0 ( 2 ) P2 P3 (31) From eq.(37), we have Also, P0 P P2 P3 1 ( 3 ) 1 (32) P4 P3 From eq.(28), we obtain P0 k and P1 k 7 VOL. 1, NO. 1, November 2011 ISSN XXXX-XXXX ARPN Journal of Science and Technology ©2011-2012 ARPN Journals. All rights reserved. http://www.ejournalofscience.org ( )( 2 )( 3 )k per unit time. The overall service rate can be obtained in 3 two situations. If there are n numbers of data packets are present in the system. Also, P0 P P2 P3 P4 1 1 3 Case-1 k ( ) ( ) ( )( 2 ) ( )( 2 )( 3 ) For n < 2 3 2 There will be no queue. Therefore (2-n) server will Hence , remain idle and the combined service rate will be ( )( 2 )( 3 ) n n , 1 n 2 A3 P4 ( ) ( ) ( )( 2 ) ( )( 2 )( 3 ) 3 2 Case-2 On the analogy of this, we conclude that, the stable For n 2 Then, all the servers will be busy. So, maximum (n- congestion rate is 2) ( C 1 ) number of data packets present in the queue. The combined service rate will be AC PC 1 1 ( C ) AC 1 { (C 1) }(1 AC 1 ) AC 2 n 2 , n 2 , for C 2 Hence, combining Case-1 and Case-2, we have 5. THE QUEUING MODEL WITH n for all n 0 ADDITIONAL ONE SERVER (M/M/2) n n for 1 n 2 : ((C+1)/FCFS) n 2 , n 2 In this model, number of servers or 0 0 , n 0 channels is two and these are arranged in parallel. Here, 1 , n 1 arrival distribution is Poisson distribution with mean rate per unit time. The service time is exponentional with mean rate per unit time. Each server is identical i.e. each server gives identically service with mean rate Figure 3: State transition rate diagram The steady state equations are P0 P1 for n=0 (38) ( ) P1 P0 2P2 for n 1 (39) 8 VOL. 1, NO. 1, November 2011 ISSN XXXX-XXXX ARPN Journal of Science and Technology ©2011-2012 ARPN Journals. All rights reserved. http://www.ejournalofscience.org { (n 1) }Pn1 2Pn1 ( n ) Pn 2Pn for 1 n C (40) ( C ) PC 2PC 1 for k C 1 (41) The above system of steady state balance equations can be written in matrix form as PQ 0 C 1 and P 1 i 0 i where P ( P0 , P ,..., PC 1 ) 1 and 0 0 0 0 ( ) 0 0 0 0 2 ( 2 2 ) 2 0 0 0 0 2 ( 2 3 ) 0 0 Q 0 0 0 2 0 0 0 0 0 0 ( 2 C ) C 0 2 2 0 0 0 From eq.(49), we obtain Hence ( )( 2 ) ( ) A2 P3 P2 k since, P0 k , P k 4 ( ) ( )2 ( )( 2 ) 2 2 1 From eq.(51), we get ( 2 ) For C = 3 P3 P2 2 P0 P1 (53) ( )( 2 ) ( ) P1 P0 2P2 P3 k , Using the value 4 2 (54) ( 2 2 ) P2 ( ) P1 2P3 of P2 (55) ( 3 2 ) P3 ( 2 ) P2 2P4 Since, P0 P P2 P3 1 1 (56) ( ) ( 2 )( ) ( 3 ) P3 2 P4 or, k[ ] 1 2 4 2 (57) 4 2 From eq.(53), we have k 4 2 ( ) ( )2 ( )( 2 ) P0 k and P1 k From eq.(54) 9 VOL. 1, NO. 1, November 2011 ISSN XXXX-XXXX ARPN Journal of Science and Technology ©2011-2012 ARPN Journals. All rights reserved. http://www.ejournalofscience.org 2P2 ( ) P1 P0 2 AC PC 1 1 ( 2 C ) AC 1 { (C 1) }(1 AC 1 ) AC 2 2 ( ) k for C 2 P2 k , 2 2 6. CONCLUSION ( ) k This research paper cites the analysis of the 2 network traffic model through Queuing Theory. In the From eq.(55), we get present analysis, we describe that how we can make a queuing model on the basis of queuing theory and subsequently we derive the estimation after analyzing 2P3 ( 2 2 ) P2 ( ) P1 the network traffic through queuing theory models. In ( 2 2 )( )k the present work two queuing models (M/M/1): 2P3 ( )k ((C+1)/FCFS) and (M/M/2):((C+1)/FCFS) have been 2 applied. These two models are used to determine the forecast way for the stable congestion rate of the ( 2 2 ) network traffic. Using the Queuing Theory models, it is ( )k [ 1] 2 convenient and simple way for calculating and monitoring the network traffic properly in the network communication system. We can monitor the network k ( )( 2 ) efficiently, in the view of the normal, optimal and or 2 even for the high overhead network management, by monitoring and analyzing the network traffic rate. k ( )( 2 ) P3 Finally, we can say that network traffic rate can have an 4 2 important role in the network communication system. From eq.(57) ( 3 ) REFERENCES P4 P3 2 [1] John N. Daigle, 2005, Queueing Theory with k ( )( 2 )( 3 ) Applications to Packet Telecommunication, ISBN: 0-387-22857-8, Springer, Boston, USA. 8 3 [2] Vern Paxson, Sally Floyd, 1997, Why We Don’t Know How To Simulate The Internet. In Also, P0 P P2 P3 P4 1 1 Proceedings of the 1997 Winter Simulation Conference, ed. S. Andradóttir, K. J. Healy, D. H. Withers, and B. L. Nelson, USA:ACM. ( ) ( )( 2 ) ( )( 2 )( 3 ) k[( ) ] 1 [3] Ren Xiangcai, Xiong Qibang, 2002, “An Application 2 4 2 8 3 of Mobile Agent for IP Network Traffic Management,” Computer Engineering, 2002-11. 8 3 [4] Li Da-Qi, Shen Jun-Yi, and Zhou Jiang-liang, 2007, k 3 “Queuing Theory Supervising K-Means Clustering 8 ( ) 4 2 ( ) 2( )( 2 ) ( )( 2 )( 3 ) Algorithm and ITS Appllication in Optimized Design of TTC Network,” Journal of Astronautics, 28 (3), pp. 752-756. Hence, [5] Wang Pei-Fa, Zhang Shi-wei, Li Jun, 2005, “The ( )( 2 )( 3 ) Application and Achievement of SVG in Network A3 P4 3 Netflow Monitor Field,” Chinese Journal of 8 ( ) 4 2 ( ) 2 ( )( 2 ) ( )( 2 )( 3 ) Semiconductors, 22(4), pp. 162-165. [6] Wang Ting, Wang Yu, 2007, “Survey on a Queue Theory Based Handover Scheme for UAVS On the analogy of this, we conclude that, the stable Communication Network,” Chinese Journal of congestion rate is Sensors and Actuators, 2007, 04. [7] Gunther, N., 1998, The Practical Performance Analyst, McGraw-Hill Inc., New York. 10 VOL. 1, NO. 1, November 2011 ISSN XXXX-XXXX ARPN Journal of Science and Technology ©2011-2012 ARPN Journals. All rights reserved. http://www.ejournalofscience.org [8] Han Jing, Guo Fang, Shi Jin-Hua, 2007, “Research based on human immune algorithm,” Microcomputer on the traffic monitoring of the distributed network Information, 2007-18. 11

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