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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 Monitoring of Network Traffic based on Queuing Theory S. Saha Ray, P. Sahoo National Institute of Technology Department of Mathematics Rourkela‐769008, India Email: santanusaharay@yahoo.com, saharays@nitrkl.ac.in 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 Figure 1: The abstract model of communication process 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 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 Here, λ is considered as the arrival rate while μ as the d service rate. {Pn (t )} n Pn (t ) n Pn (t ) n 1 Pn 1 (t ) n 1 Pn 1 (t ) dt In the steady state equation , for n≥1 (1) Lt Pn (t ) Pn t d And P0 (t ) 0 P0 (t ) 1 P1 (t ) for d dt and Lt {Pn (t )} 0 t dt n=0 (2) In model M/M/1, we let Hence, from eqs. (3) and (4) when t we get n And n 0 Pn 1 Pn 1 ( ) Pn for n≥1 Where λ and µ are constants. (5) Then eqs.(1) and (2) reduces to and 0 P0 P1 d dt This implies P1 ( ) P0 Pn (t ) Pn1 (t ) Pn1 (t ) ( ) Pn (t ) for From eq.(5) when n=1, we get n≥1 (3) ( ) P1 P0 P2 d And P0 (t ) P0 (t ) P1 (t ) for dt Therefore, P2 ( ) 2 P0 n=0 (4) 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 Also In general, Pn ( ) n P0 2 Nq ( ) or, Pn n P0 where Using the Little’s law we have Here, is called server utilization factor or traffic Ts and intensity. (9) Using eq.(9), eq.(8) reduces to We know, P n 0 n 1 2 Nq Also, Pn n P0 1 This implies (1 Ts ) N q ( Ts ) 2 Therefore, P n0 n n0 n P0 Or, ( T s ) 2 T s N q N q 0 , or, 1 P0 n ( ' ) (10) n 0 The above equation eq.(10) provides the relation Consequently, P0 1 , where < 1 between following parameters Ts = Service time Hence, Pn n (1 ) , n=0, 1, 2,… . Sending rate N q Quantity of data (6) packets stored in the buffer Suppose, L stands for the length of the queue under the If we know any two variables, it is easy to gain the steady state condition. It includes the average volume of numerical value of the third one. all the data packets which enter the processing module So, these three variables are key parameters for and store in the buffer. measuring the performance of the transmission system. L nPn n n (1 ) 4. QUEUING THEORY AND THE n 0 n 1 NETWORK TRAFFIC MONITOR (1 ) n n 4.1.Forecasting the network traffic using n 1 Queuing Theory Hence L (7) The network traffic is very common [5]. The system 1 will be in worse condition, when the traffic becomes under extreme situation, in which leads to the network Also L Since, congestion [6]. There are a great deal of research about monitoring the congestion at present ,besides, the documents which make use of Queuing Theory to research the traffic rate appear more and more. For forecasting the traffic rate, we often test the data disposal If N q denotes the average volume of the buffers data function of the router used in the network. packets then Considering a router’s arrival rate of data flow in groups is , and the average time which the routers use 2 1 Nq L (8) to dispose each group is , the buffer of the routers is 1 C, if a certain group arrives, the waiting length of the queue in groups has already reached, so the group has to be lost. When the arriving time of group timeouts, the group has to resend. Suppose, the group’s average 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 1 Then the queuing system of the router’s date groups waiting time is . We identify Pi (t) to be the arrival satisfies simple Markov Process [7], according to Markov Process, we can find the diversion strength of probability of the queue length for the routers group at matrix of model 1 as follow: the moment of t, supposing the queue length is i: P(t) = (P0 (t), P1 (t), . . . , Pn−1 (t) ), i = 0,1, . . . ,C+1. 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 no data packet arrival in time ( ∆t) } Prob { no data packet departure in time ∆t } Network congestion rate is changing all the time [8]. The instantaneous congestion rate and the + Prob { (k-1) number of data packets present in the stable congestion rate are often used to analysis the system at time t } network traffic in network monitor. The instantaneous rate AC (t ) is the congestion rate at the moment of t. The Prob { 1 data packet arrival in time ( ∆t) } Prob { no data packet departure in time ∆t } + AC (t ) can be obtained by solving the system length of the queue’s probability distributing, which is called Prob { (k+1) number of data packets present in the PC 1 (t ) . system at time t } Prob {no data packet arrival in time ( ∆t) } Let, Pk (t ) (k=0,1,. . .,C+1) to be the arrival probability Prob { 1 data packet departure in time ∆t }+… of the queue length for the routers group at the moment of t by considering the queue length is k. Pk (t t ) Pk (t ){1 k t o(t )}{1 k t o(t )} Then, the queuing system of the router’s date groups satisfies simple Markov Process. According to Markov Process, Pk (t ) satisfies the following system of Pk 1 (t ){k 1t o(t )}{1 k 1t o(t )} differential difference equations. + Let, Pk (t ) = Prob { k number of data Pk 1 (t ){1 k 1t o(t )}{k 1t o(t )} + packets present in the system in time t } o(t ) and Pk 1 (t ) = Prob { k number of data packets present in the system in time (t + ∆t) } P(t t)P(t) (k k)P(t)t P1(t)k1t P1(t)k1t o(t) k k k k k Case 1: Dividing both sides by t and taking limit as t 0 For k ≥ 1 d {Pk (t )} ( k k ) Pk (t ) k 1 Pk 1 (t ) dt Pk 1 (t t ) = Prob { k number of data packets o(t ) k 1 Pk 1 (t ) , since lim 0 present in the system at time t } Prob { t t 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 Here, in state k, data packets arrival is ( k ) PC 1 (t t ) = Prob { (C+1 ) no. of data packets i.e. k k present in the system at time (t+∆t )} = Prob { C no. of data packets present in time t } Also, in state k, data packet departure is prob {1 data packet arrival in time ∆t } Prob { no data packet departure in time ∆t } i.e. k + Prob { (C+1) no of data packets present in time t } Prob { no data packet departure in time ∆t }+… Hence, eq.(11) reduces to P (t){ C o()}{C o()}P1(t){ C1 o()}o() t t 1 t t C 1 t t t d C {P (t)} ( k )P (t) { (k 1)}P1(t) P1(t) k k k k dt (12) P 1 (t t) P 1 (t) P (t)C t C1P 1 (t)t o(t) C C C C where k=1,2,…,C Case 2: Dividing both sides by t and taking limit as t 0 we get For k=0, we have d P0 (t+∆t) = Prob { no data packet present in the system {PC 1 (t )} PC (t )C C 1 PC 1 (t ) dt at time (t+∆t) } d = Prob { no data packet present in time t } {PC 1 (t )} ( C ) PC (t ) PC 1 (t ) , dt Prob { no data packet arrival in time ∆t } + Prob {one data packet present in time t } since C C Prob { no data packet arrival in time ∆t } Prob { one data packet departure in time ∆t } By solving this differential equation system, we get the +… instantaneous congestion rate A0(t) as P(t){ 0t o( )}P(t){ o( )}{1t o( )}o(t) 1 t 1 1 1t t t = 0 A0 (t ) P1 (t ) (1 e ( ) t ) P0 (t t ) P0 (t ) 0 P0 (t ) P1 (t ) 1 o(t ) The instantaneous congestion rate can not be used to measure the stable operating condition of the system, so we must obtain the stable congestion rate of Dividing both sides by t and taking limit as t 0 , the system. The so-called stable congestion rate means, it will not change with the time changing, when the system we obtain works in a stable operating condition. The definition of the stable congestion rate is d AC lim AC (t ) {P0 (t )} P0 (t ) P (t ) 1 t (13) dt Considering, P lim P (t ) as the distributing (since, k k and k ) t of the stable length of the queue and C as the buffer of the router, the stable congestion rate can be obtained in Case 3: two ways: firstly, we obtain the instantaneous congestion rate, then find its limit. According to its definition, it can For k=C+1, we have be obtained with the distributing of the length of the queue. Secondly, according to the Markov Process, we know that the distributing of the stable length of queue 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 can be obtained through system of steady state PQ 0 equations. C 1 From eq.(12), eq.(13) and eq.(14), we have the system of and P 1 i 0 i differential difference equations as follows where P ( P0 , P1 ,..., PC 1 ) d and {P (t)} ( k )P (t) { (k 1)}P 1(t) P 1(t) k k k k dt 0 0 0 0 (15) ( ) 0 0 0 0 ( 2 ) 2 0 0 for k=1, 2, 3, … ,C 0 0 ( 3 ) 0 0 Q d 0 0 0 0 0 {P0 (t )} P0 (t ) P1 (t ) for k=0 dt 0 0 0 0 ( C ) C 0 0 0 0 (16) d For C= 0, {PC 1 (t )} ( C ) PC (t ) PC 1 (t ) for k=C+1 From eq.(19), we have dt (17) P0 P1 (21) According to some properties of Markov process, we Also, P0 P1 1 (22) know that Solving (21) and (22) we get Pi (t ) (i=0,1,2,…,C+1) satisfies the above P differential equation. 1 Here, P (t ) [ P0 (t ), P1 (t ),..., PC1 (t )] Hence, A0 P1 (23) P0 (0) 1, P (0) 0, P2 (0) 0,..., PC 1 (0) 0 1 P(0) [ P0 (0), P1 (0),..., PC 1 (0)] For C=1 0 P0 P1 (24) For steady state equation, dPk (t ) 0 ( ) P1 P0 P2 (25) lim Pk (t ) Pk and lim 0 t t dt 0 ( ) P1 P2 (26) Under steady state condition, eqs.(15),(16) and (17) transform to following balance equations. Also, P0 P1 P2 1 ( k ) Pk (t ) { (k 1) }Pk 1 (t ) Pk 1 (t ) for k=1, 2, 3, … ,C (18) From eq.(23), we get P0 P 1 k 0 P0 (t ) P1 (t ) for k=0 (19) Therefore, P0 k , P1 k 0 ( C ) PC (t ) PC 1 (t ) for k=C+1 From eq.(25), we have (20) P2 ( )k , since, P k (27) The above system of steady state equations can be 1 written in matrix from as 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 Using eq.(26), we obtain 2 k ( ) 2 ( ) ( )( 2 ) k[ ] 1 Hence, ( )( 2 ) k A2 P2 ( ) ( ) ( ) ( ) ( )( 2 ) 2 From eq.(27) yields For C=3, yielding 0 ( ) P1 P0 P2 (33) P2 ( ).. 0 ( 2 ) P2 ( ) P1 P3 ( ) ( ) (34) Hence, 0 ( 3 ) P3 ( 2 ) P2 P4 (35) ( ) 0 P0 P1 (36) A1 P2 ( ) ( ) 0 ( 3 ) P3 P4 (37) For C=2, From eq.(36), we obtain 0 P0 P1 (28) P0 k and P1 k 0 ( ) P1 P0 P2 (29) From eq.(33) ( ) 0 ( ) P1 ( 2 ) P2 P3 P2 k (30) From eq.(34) 0 ( 2 ) P2 P3 (31) ( )( 2 )k P3 Also, P0 P1 P2 P3 1 (32) 2 From eq.(37), we have From eq.(28), we obtain ( 3 ) P0 k and P1 k P4 P3 From eq.(29), we have ( )( 2 )( 3 )k 3 ( ) P2 k Also, P0 P1 P2 P3 P4 1 From eq.(31), we have 3 k ( 2 ) 3()()2 ()(2)()(2)(3) P3 P2 ( )( 2 ) Hence , k 2 ()( 2)( 3) A P ()() ()(2)()( 2)( 3) 3 4 3 2 From eq.(32), we have 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 On the analogy of this, we conclude that, the stable There will be no queue. Therefore (2-n) server will remain idle and the combined service rate will be congestion rate is n n , 1 n 2 AC P 1 1 ( C )AC1 { (C1)}( AC1)AC2 C 1 Case-2 , for C 2 For n 2 Then, all the servers will be busy. So, maximum (n- 2) ( C 1 ) number of data packets present in the 5. THE QUEUING MODEL WITH queue. ADDITIONAL ONE SERVER (M/M/2) The combined service rate will be : ((C+1)/FCFS) n 2 , n 2 In this model, number of servers or Hence, combining Case-1 and Case-2, we have channels is two and these are arranged in parallel. Here, n for all n 0 arrival distribution is Poisson distribution with mean rate n n for 1 n 2 per unit time. The service time is exponentional with mean rate per unit time. Each server is identical i.e. n 2 , n 2 each server gives identically service with mean rate 0 0 , n 0 per unit time. The overall service rate can be obtained in 1 , n 1 two situations. If there are n numbers of data packets are present in the system. Case-1 For n < 2 Figure 3: State transition rate diagram The steady state equations are P0 P1 for n=0 (38) ( ) P1 P0 2P2 for n 1 (39) { (n 1) }Pn 1 2 Pn 1 ( 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 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 C 1 and P 1 i 0 i where P ( P0 , P1 ,..., PC 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 0 0 0 2 2 From eq.(49), we obtain P0 P1 (53) ( ) P2 k since, P0 k , P k ( ) P1 P0 2 P2 2 1 (54) ( 2 2 ) P2 ( ) P1 2P3 From eq.(51), we get (55) ( 2 ) ( 3 2 ) P3 ( 2 ) P2 2P4 P3 P2 (56) 2 ( 3 ) P3 2P4 ( )( 2 ) P3 k , Using the value (57) 4 2 of P2 From eq.(53), we have Since, P0 P1 P2 P3 1 P0 k and P1 k ( ) ( 2 )( ) From eq.(54) or, k[ ] 1 2 4 2 2P2 ( ) P1 P0 4 2 ( ) k k P2 k 4 2 ( ) ( ) 2 ( )( 2 ) 2 2 ( ) Hence k ( )( 2 ) 2 A2 P3 From eq.(55), we get 4 ( ) ( )2 ( )( 2 ) 2 2P3 ( 2 2 ) P2 ( ) P1 ( 2 2 )( )k 2P3 ( )k 2 For C = 3 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 ( 2 2 ) the network traffic through queuing theory models. In ( )k [ 1] the present work two queuing models (M/M/1): 2 ((C+1)/FCFS) and (M/M/2):((C+1)/FCFS) have been applied. These two models are used to determine the k ( )( 2 ) forecast way for the stable congestion rate of the network traffic. Using the Queuing Theory models, it is 2 convenient and simple way for calculating and monitoring the network traffic properly in the network k ( )( 2 ) communication system. We can monitor the network P3 efficiently, in the view of the normal, optimal and or 4 2 even for the high overhead network management, by From eq.(57) monitoring and analyzing the network traffic rate. ( 3 ) P4 P3 2 Finally, we can say that network traffic rate can have an important role in the network communication system. k ( )( 2 )( 3 ) 8 3 REFERENCES Also, P0 P1 P2 P3 P4 1 [1] John N. Daigle, 2005, Queueing Theory with Applications to Packet Telecommunication, ISBN: 0-387-22857-8, Springer, Boston, USA. () ()( 2) ()( 2)( 3) [2] Vern Paxson, Sally Floyd, 1997, Why We Don’t k[( ) ] 1 Know How To Simulate The Internet. In 2 42 83 Proceedings of the 1997 Winter Simulation Conference, ed. S. Andradóttir, K. J. Healy, D. H. 83 Withers, and B. L. Nelson, USA:ACM. 3 k 8 ()4()2)( 2 )()( 2 )( 3 ) 2 ( [3] Ren Xiangcai, Xiong Qibang, 2002, “An Application of Mobile Agent for IP Network Traffic Management,” Computer Engineering, 2002-11. [4] Li Da-Qi, Shen Jun-Yi, and Zhou Jiang-liang, 2007, Hence, “Queuing Theory Supervising K-Means Clustering ()( 2)( 3 ) Algorithm and ITS Appllication in Optimized A P 3 3 4 Design of TTC Network,” Journal of Astronautics, 8 () 4() 2)( 2 ) ()( 2 )( 3 ) 2 ( 28 (3), pp. 752-756. [5] Wang Pei-Fa, Zhang Shi-wei, Li Jun, 2005, “The On the analogy of this, we conclude that, the stable Application and Achievement of SVG in Network congestion rate is Netflow Monitor Field,” Chinese Journal of 2 Semiconductors, 22(4), pp. 162-165. A P 1 1 C C [6] Wang Ting, Wang Yu, 2007, “Survey on a Queue (2C)A 1 {(C1}( A 1)A 2 2 C ) 1 C C Theory Based Handover Scheme for UAVS , for C 2 Communication Network,” Chinese Journal of Sensors and Actuators, 2007, 04. 6. CONCLUSION [7] Gunther, N., 1998, The Practical Performance Analyst, McGraw-Hill Inc., New York. This research paper cites the analysis of the network traffic model through Queuing Theory. In the [8] Han Jing, Guo Fang, Shi Jin-Hua, 2007, “Research present analysis, we describe that how we can make a on the traffic monitoring of the distributed network queuing model on the basis of queuing theory and based on human immune algorithm,” Microcomputer subsequently we derive the estimation after analyzing Information, 2007-18. 10

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