ied widely. The waiting time ditribution and queue ize for a queue with gamma ditributed arrival proce with a pecial Markovian tructure and i.i.d. exp

Transcription

1 Modeling of Correlated Arrival Procee in the Internet Muckai K Girih SBC Technology Reource, Inc Willow Road Pleaanton, CA mgirih@tri.bc.com Jian-Qiang Hu Λ Dept. of Manufacturing Engineering Boton Univerity 15 St. Mary' Street Boton, MA hu@enga.bu.edu Abtract The correlation in the arrival procee in the Internet are very ignificant ince voice, video and data are tranported imultaneouly over high peed link. Detailed knowledge of the behavior of the arrival procee i of paramount importance in traffic engineering and network optimization. In thi paper, we propoe to ue Markov-modulated arrival procee to approximate the correlated arrival procee in the Internet. Two approximate procedure are developed to evaluate the parameter of the underlying Markov chain and the conditional denity function given a few moment and the lag-1 autocorrelation of the arrival proce. The reulting queueing ytem can be analyzed uing the exiting technique for the G/G/1 queue with Markov-modulated arrival procee. We alo how how our model can be ued in connection admiion control (CAC). Keyword: Internet; ATM Network; Autocorrelation; Markov-Modulated Arrival Procee, CAC. 1 Introduction The field of communication ytem ha been witneing tremendou growth with the development of many complex device and mechanim that have helped the exploive growth of the Internet. Integrated communication ytem have been deigned to facilitate the tranport of video and voice along with data and meage. High-peed router, Frame Relay and Aynchronou Tranfer Mode (ATM) are ome example. The arrival ditribution in thee network are typically burty and highly correlated. The analyi of queue with correlation in the input ditribution have poed eriou difficultie to the reearcher and the practitioner alike. In pite of the remarkable progre achieved in thi area, thee ytem are till not undertood completely. Ignoring correlation often lead to high level of inaccuracy in the performance meaure (ee Girih and Hu 1996a and Girih Λ Hi reearch i upported in part by the National Science Foundation under grant EEC ). A imulation tudy reported in Livny et al. (1993) conidered a ingle erver queue with correlation. Their reult point to the fact that the average waiting time deviate coniderably from that of etimation by ignoring the correlation. Traffic management in the Internet backbone node i a complex tak given the large number of witched and permanent virtual circuit that demand differing Quality of Service requirement from the network. In order to have effective network deign, operation and control, one need to be able to haveaccurate performance evaluation mechanim in place. The typical performance meaure of interet are the ditribution of the congetion meaure uch athe waiting time, buffer occupancy and buffer overflow probabilitie. One of the few model ued to tudy correlated queue i the G/G/1 queue with Markov-modulated arrival (MAP). For a review on Markov-modulated queue, ee PrabhuandZhu (1989) and the reference therein. The MacLaurin erie for the moment of the waiting and ytem time for thi ytem were derived in Zhu and Li (1993) and for the moment and the lag-1 autocorrelation of the interdeparture time were derived in Girih and Hu (1996b,c). It wa hown that Padé approximation i very effective in etimating thee performance meaure from their MacLaurin erie. Refer to Baker (1975) and Petruhev and Popov (1987) for a comprehenive treatment of the theory of Padé approximation. Mot of thee model aume that the interarrival and ervice time ditribution are known and aid in performance evaluation of the reulting queue. But, in practice, thee input ditribution need to be approximated from ome meaurable quantitie. Moreover, very little work ha been done in the area of modeling correlated arrival procee. It i the intent of thi paper to develop imple cheme to approximately characterize the arrival procee given ome pecific information about them. The dependence propertie of the arrival procee in packet-witched network have been tud-

2 ied widely. The waiting time ditribution and queue ize for a queue with gamma ditributed arrival proce with a pecial Markovian tructure and i.i.d. exponentially ditributed ervice time were tudied by Tin (1985). Heffe and Lucantoni (1986) developed method to model packetized voice and data uing Markov-modulated Poion procee. Fendick et al. (1989) invetigated correlated ingle erver queue under heavy traffic condition uing the concept of index of diperion. Simple approximation uing the lag-1 autocorrelation for the ingle erver queue and tandem queueing network were propoed in Whitt (1984). In many communication network, including the Internet, often only a few parameter of the arrival procee are known. For example one can eaily etimate the firt few moment and the autocorrelation. But, it become increaingly difficult and computationally complex to meaure and tore or in ome cae even impoible to etimate the other characteritic. In order to analyze the queue which have thee parameter characterizing the arrival proce, we need to approximate thi by a uitable interarrival time ditribution. Thi can then facilitate the ue of everal available tool to analyze and etimate the performance meaure of thi ytem. Motivated with thi obervation, we develop two procedure to approximately model a Markov-modulated Arrival Proce given a few parameter of the arrival proce. In both cae, we ueatwo-tate Markov chain for the ake of implcity. We believe that thi can be generalized to accommodate and incorporate other feature. In the firt cae, given the firt two moment and the lag-1 autocorrelation we approximate the conditional ditribution a Poion procee. In the econd cae, given the firt three moment and the lag-1 autocorrelation we approximate the conditional ditribution by mixed Erlang ditribution. We have propoed the ue of mixed Erlang ditribution only for illutrative purpoe. It hould be borne in mind that any other uitable probability ditribution can be ued to match the moment. We are alo motivated through another problem. Mot popular cheme in the performance evaluation of general queueing network i baed on the concept of parametric decompoition. The main idea in thi approach i to decompoe the network into ingle queue by capturing the interaction between node by a few parameter. Then each queueing ytem can be eaily analyzed through the everal approximation cheme propoed in the literature. The intereted reader i referred to Whitt (1983) for detail on one of the mot popular analyzer, called QNA. In QNA, the interaction between node are captured by the firt two moment of the arrival procee. The parameterized node are then analyzed baed on thee two parameter and the firt two moment of the ervice time ditribution. Though QNA work very well under a variety ofnetwork topology and traffic load, it give rie to large error when the correlation in the interarrival time random variable are ignificant ince it doe not take into account the autocorrelation. Thi can alo be oberved in other approximation technique that conider higher order of the moment of the arrival and ervice time ditribution a well (ee Girih and Hu 1996a and Girih 1996). Therefore the lack of reult for taking correlation in the analyi ha neceitated the need for modified parametric decompoition method. The reult of thi paper, thu, can be ued to conider the autocorrelation along with the moment in characterizing the interaction between node in general queueing network. Many Internet backbone network conit of overlay network that include ATM network a well. ATM network have been deigned to guarantee Quality of Service (QoS) for the connection that are admitted. Different type of traffic require different ervice qualitie. Voice and real time video connection require very low delay, delay variation and cell lo; wherea regular data connection can tolerate higher delay. Preently, mot ATM network upport the following QoS clae: CBR (contant bit rate), VBR (real time and non-real time variable bit rate), ABR (available bit rate) and UBR (unpecified bit rate). A Connection Admiion Control (CAC) algorithm determine if a call i admiible by computing the equivalent bandwidth required by the connection and comparing it to the available bandwidth in the network. We how how the reult on modeling of the arrival procee developed in thi paper can be applied to a popular connection admiion control algorithm. One of the pionering work in connection admiion control can be found in Guerin et al. (1991). They conidered a two-tate fluid-flow model to approximate the arrival procee. Thi proce can be completely decribed by three parameter: the peak cell rate and the mean of the burt and idle period (which are aumed to be exponentially ditributed). The remainder of thi paper i organized a follow. The general problem of modeling a Markovmodulated arrival proce given the moment and lag- 1 autocorrelation i preented in Section 2. A pecial cae with modeling Poion conditional arrival ditribution i dicued in Section 3. The cae with general conditional arrival procee i tudied in Section 4. The application of thi approach in Connection Admiion Control i outlined in Section 5 and thi paper i concluded in Section 6.

3 2 The Problem Let A denote the generic random variable repreenting the correlated arrival proce in the Internet and let A k ; k 1 be the interarrival time between arrival k 1andk, where A 0 =0. In thi ection, we outline the problem where the firt n non-central moment of the arrival proce (M 1 ;:::M n ) are known a well a it lag-1 autocorrelation (χ). Recall that M n = E[A n ] 8n 1; χ = E[A 1 A 2 ]: The objective i to identify a Markov chain, it tranition and tationary probabilitie and the moment of the conditional ditribution. Let fj n ; n 0g be the underlying Markov chain, E be the tate pace of the Markov chain, P =(p ij );i;j 2E be the tranition probability matrix and let ß i ;i 2E be the tationary probability of being in tate i. Further, we denote the conditional non-central moment a: M ijk = E[A k n jj n = i; J n+1 = j] The problem i to determine E, ß i, p ij, M ij1, :::, M ijn from the following et of equation. XXX i j k ß i p ij p jk M ij1 M jk1 = χ XX i j XX i j XX i j ß i p ij M ij1 = M 1 ß i p ij M ij2 = M 2 ::: = ::: ::: = ::: ß i p ij M ijn = M n Since thi general problem i not eaily olvable, in the following two ection, we conider two pecial cae. 3 Matching Two Moment and the Lag-1 Autocorrelation We look at the cae when the firt two moment and the lag-1 autocorrelation of a ditribution are known and our objective i to find a Markovmodulated Poion proce (MMPP) that fit thee parameter. Note that a MMPP ha Poion conditional arrival procee. Recall that the known moment are denoted by M 1 ;M 2 and the lag-1 autocorrelation by χ. We now conider a Markovchain which i irreducible, aperiodic and recurrentinatatepace E. We aume that thi Markov chain ha only two tate (E = f1, 2g) and the doubly-tochatic tranition probability matrix i given by: 1 p1 p P = 1 p 2 1 p 2 Then the teady tate probabilitie can be immediately determined to be: ß 1 = p 2 =(p 1 + p 2 ), ß 2 = p 1 =(p 1 + p 2 ) by olving the equation, ßP P = ß and ß i2e i =1. We now add retriction on the conditional probability ditribution of the above Markov chain. The conditional probability ditribution of the interarrival time given J n = i, for i = 1; 2 are exponential with mean x i ; i.e, F i (t) =P fa n» tjj n = ig =1 exp( t x i ); (1) where F i (:) repreent the cumulative ditribution function of the conditional arrival ditribution. What remain to be determined are p 1 ;p 2 ;x 1 ;x 2 from the following et of equation which can be readily etablihed by applying the implified aumption in thi ection to the general problem formulation in Section 2: p 2 (1 p 1 )x p 1 p 2 x 1 x 2 + p 1 (1 p 2 )x 2 2 = 2χ p 2 x 1 + p 1 x 2 =(p 1 + p 2 )M 1 (2) p 2 x p 1 x 2 2 = 1 2 (p 1 + p 2 )M 2 Let u denote the quared coefficient of variation a, c 2 =(M 2 =M 2 1 ) 1. Solving (2), we get p 1 + p 2 = M 2 2χ (3) M 2 2M 2 1 p 1 (c x 1 = M 1 (1 2 1) ± ) (4) 2p 2 p 2 (c x 2 = M 1 (1 2 1) ) (5) 2p 1 We notethatwehave one degree of freedom in determining parameter p 1 and p 2. 4 Matching Three Moment and the Lag-1 Autocorrelation Given the firt three moment and the lag-1 autocorrelation of the interarrival time random variable A, we addre the problem of finding a Markovmodulated arrival proce (MAP) that fit thee parameter exactly in thi ection. The dependencie will be captured by the Markov chain. Recall that we denoted the moment of A by M 1 ;M 2 ;M 3 and the lag-1 autocorrelation by χ. We now look at a Markov chain which i irreducible, aperiodic and recurrent in a tate pace E. We again aume that thi Markov chain ha only two tate (E = f1, 2g) but with probability tranition matrix P = p 1 p 1 p p

4 For thi Markov chain, it teady tate probabilitie are given by ß 1 = ß 2 =0:5. The lag-1 correlation coefficient, ffi i defined a: ffi = χ M 2 1 M 2 M 2 1 (6) We now add retriction on the conditional probability ditribution of the above Markov chain. The conditional interarrival ditribution given J n = i, for i =1; 2have x i ;y i ;z i a their firt three non-central moment, repectvely; i.e, for i = 1; 2, E[A n jj n = i] = x i E[A 2 n jj n = i] = y i E[A 3 n jj n = i] = z i What remain to be determined are p; x 1, x 2 ;y 1, y 2 ;z 1, z 2 from the following et of equation: px (1 p)x 1 x 2 + px 2 2 = 2χ x 1 + x 2 = 2M 1 y 1 + y 2 = 2M 2 z 1 + z 2 = 2M 3 The above equation can now be olved. Solving the firt two equation for x 1 and x 2 we get: χ M1 x 1 = M (7) 2p 1 χ M1 2 x 2 = M 1 (8) 2p 1 If the correlation i poitive, then χ 2M 2 1» p» 1 Now back to olving for the higher conditional moment, we have the following fundamental relationhip that the moment of any probability ditribution have to atify: M 1 0 M 2 M M 1 M 3 M Since the conditional moment have to atify the above too,weget x 2 1» y 1» 2M 2 x 2 2 y 2 1 x 1» z 1» 2M 3 y2 2 x 2 Therefore, we define two more parameter denoted by fi and fl that atify the contraint: 0» fi» 1, 0» fl» 1, uch that we can expre the higher moment a: y 1 = fix 2 1 +(1 fi)(2m 2 x 2 2) (9) y 2 = 2M 2 y 1 (10) z 1 = fl y2 1 +(1 fl)(2m 3 y2 2 ) x 1 x 2 (11) z 2 = 2M 3 z 1 (12) Hence fi and fl can be choen uch that the econd and third conditional moment atify the above contraint. After determining thee, the firt three conditional moment can nowbematched to a probability ditribution uch a a mixture of Erlang ditribution (ee Johnon and Taaffe 1989). It i now eay to evaluate the performance of the reulting queueing ytem uing any of the known approximation technique (ee Prabhu and Zhu 1989,Zhu and Li 1993, Girih and Hu 1996b,c and the reference therein). We ummarize the procedure for obtaining the parameter of the Markov-modulated arrival proce below: Algorithm for Modeling MAP: 1. Chooe a value for p. 2. Determine x 1 and x 2 from Equation (7) and (8). 3. Chooe 0» fi» 1 and evaluate y 1 and y 2 from Equation (9) and (10). 4. Chooe 0» fl» 1 and evaluate z 1 and z 2 from Equation (11) and (12). 5. Match the conditional moment to a mixed Erlang ditribution uing the reult of Johnon and Taaffe (1989). 6. Ue thi Markov-modulated proce a the interarrival ditribution and analyze the queue uing any of the exiting technique. One of the main iue in the election of the parameter, ff; fi; fl i the order of the mixed Erlang ditribution (n) that are obtained by matching the conditional moment. The procedure of Johnon and Taaffe (1989) for matching the firt three moment to a mixed Erlang ditribution of common order involve the following two inequalitie that n ha to atify: n M 2 1 ; (13) M 2 M1 2 n 2(M 2 M1 2 ) 2 + M 2M 1 2 M 1 M 3 M 1 M 3 (M 2 M1 2)(M (14) 2 2M1 2 ):

5 In order to avoid numerical error in the analyi, we would like to keep n a mall a poible. Our numerical experience provide a few guideline a to the election of the parameter. In general, ff hould be a cloe to zero a poible. We alo found that fl = :5 and:3» fi» :7, in general, give good reult. More reearch i needed to optimize the parameter value for any ytem. One poible way i to expre n given in (13) and (14) in term of ff; fi; fl and ue claical optimization technique to minimize n, thereby, obtaining the optimal parameter value. 5 The Connection Admiion Control Algorithm An important feature of ATM network i the concept of guarantee for the Quality of Service (QoS). A call requeting connection et up requet a certain QoS which, if approved, can be guaranteed for the lifetime of the connection. Currently there are four type of QoS clae upported by mot ATM network: CBR (Contant Bit Rate), VBR (Real time and Non-Real time Variable Bit Rate), ABR (Available Bit Rate) and UBR (Unpecified Bit Rate). From the perpective of an ATM network, a connection i determined to be admiible by implementing acac algorithm which determine the equivalentcapacity or bandwidth required for the connection. Thi equivalent bandwidth i compared with the available bandwidth and if ufficient capacity remain, then the connection i admitted, otherwie it i rejected. Due to the nature of taitical multiplexing of ATM network and the varying requirement of the different QoS clae, determination of the equivalent capacity or bandwidth i a non-trivial problem and ha been tudied by everal reearcher and practitioner. A pioneering work in the determination of bandwidth allocation in packet witching network can be found in Guerin et al. (1991). They develop an approximate technique to etimate the equivalent capacity for high-peed packet witching network. In thi ection, we tudy the relationhip of their approach to the modeling of correlated arrival procee that wa developed in thi paper. The model conidered in Guerin et al. (1991) i a two-tate fluid-flow model in which the arrival occur at a peak rate of R per econd or zero. The buy and idle period are ditributed exponentially with mean B and I, repectively. The three parameter, (R; B; ρ) completely pecify the arrival proce, where ρ i the utilization. During the buy period, the interarrival time i 1=R and during the idle period, the interarrival time i I. Alo, the average number of arrival during one cycle of buy and idle period i BR. Incorporating thi, we get: M 1 = I +(RB=R) ; RB M 2 = I +(RB=R ) ; RB χ = (2I=R)+(2RB=R2 ) : RB Noting that ρ = B=(B + I), the above expreion can be implified a: M 1 = B + I RB = 1 ρr ; M 2 = I 2 RB + 1 R = (1 ρ)2 B ρ 2 R R ; 2 χ = 2(B + I) R 2 B = 2 ρr : 2 Another way of looking at thi problem i to tranform the Markov-modulated arrival proce characteritic to that of the fluid-flow model. Given a MAP, thefirt two moment and the lag-1 autocorrelation can be determined eaily and the three parameter of the fluid model can be evaluated from the following: ρ = χ 2M 2 1 ; R = 2M 1 χ ; (15) B = χ(4m 2 1 M 2 χ 2 ) 2M 1 (2M 2 1 χ)2 : The et of Equation (15) form a ueful way to ue MAP modeled from a variety of ituation in applying the CAC algorithm for ATM network. 6 Concluion The arrival procee in the Internet backbone node are typically highly correlated ince they are contituted by the uperpoition of voice, video and data traffic tream. In order to facilitate the modeling of uch ytem given a few parameter of the arrival proce, we conidered two model that approximate the arrival procee by Markov-modulated procee (MAP). Both thee model conider a twotate markov chain and one cae with Poion conditional arrival procee and the other cae with general conditional arrival ditribution. We derived method to calculate the parameter of thee ditribution. Thi modeling technique ha ignificance in the analyi of many communication ytem which encounter burty traffic. Thee model can alo be ued to approximate the interarrival time ditribution at the downtream node in tandem queueing network in which the interdeparture time from the uptream node have ignificant correlation. We alo howed how our reult can be applied in connection admiion control.

6 An intereting direction for future reearch i to conider the long range dependence in our model. Thi can then enure the capture of the long range dependence characteritic in the approximating arrival procee. It would alo be ueful to generalize the reult of thi paper by matching more moment and increaing the tate pace of the underlying Markov chain. Reference [1] G. A. Baker, Eential of Pade Approximant, Academic Pre, [2] K. W. Fendick, V. R. Sakena and W. Whitt, Dependence in Packet Queue," IEEE Tranaction on Communication, 37, , [3] M. K. Girih, Higher Order Approximation for General Open Queueing Network," Ph.D. Diertation, Department of Manufacturing Engineering, Boton Univerity, Boton, MA, [4] M. K. Girih and J. Q. Hu, Higher Order Approximation for Tandem Queueing Network," QUESTA, 22, , 1996a. [5] M. K. Girih and J. Q. Hu, The Departure Proce of the Markov-modulated G/G/1 Queue," Invited Paper, INFORMS Conference, Wahington D.C., 1996b. [6] M. K. Girih and J. Q. Hu, The Departure Proce of the MAP/G/1 Queue," Journal of DEDS, ubmitted for publication, 1996c. [7] M. K. Girih and J. Q. Hu, Modeling Correlated Arrival Procee in ATM Multiplexer," Invited Paper, 9-th INFORMS Applied Probability Conference, Boton, MA, [8] W. B. Gong and J. Q. Hu, The MacLaurin Serie for the GI/G/1 Queue," J. Applied Prob., 29, , [9] W. B. Gong, S. Nananukul and A. Yan, Pade Approximation for Stochatic Dicrete Event Sytem," IEEE Tranaction on Automatic Control, 40, No. 8, , [10] W. B. Gong, A. Yan and D. Tang, Rational Repreentation for Performance Function of Queueing Sytem," Princeton Conference on Sytem & Information Science, March [12] M. A. Johnon andm. R. Taaffe, Matching Moment to Phae Ditribution: Mixture of Erlang Ditribution of Common Order," Stochatic Model, 5:4, , [13] H. Heffe and D. M. Lucantoni, A Markov Modulated Characterization of Packetized Voice and Data Traffic and Related Statitical Performance," IEEE J. on Selected Area in Communication, SAC-4, , [14] J. Q. Hu, Analyticity of Single-Server Queue in Light Traffic," QUESTA, 19, 63-80, [15] J. Q. Hu, The Departure Proce of the GI/G/1 Queue and It MacLaurin Serie," Operation Reearch, 44, No. 5, , [16] M. Livny, B. Melamed and A. K. Tioli, The Impact of Autocorrelation on Queueing Sytem," Management Science, 39, No. 3, , [17] B. Melamed, TES: A Cla of Method for Genarating Autocorrelated Uniform Variate," ORSA Journal on Computing, 3, No. 4, , [18] P. P.Petruhev and V. A. Popov, Rational Approximation of Real Function, Cambridge Univerity Pre, [19] N. U. Prabhu and Y. Zhu, Markov-Modulated Queueing Sytem," QUESTA, 5, , [20] P. Tin, A Queueing Sytem with Markov Dependent Arrival," J. Appl. Probab., 22, , [21] W. Whitt, The Queueing Network Analyzer," The Bell Sytem Technical Journal, 62, No. 9, , [22] W. Whitt, Approximation for Departure Procee and Queue in Serie," Naval Reearch Logitic Quarterly, 31, , [23] Y. Zhu and H. Li, The MacLaurin Expanion for a G/G/1 Queue with Markov-modulated Arrival and Service," QUESTA, , [11] R. Guerin, H. Ahmadi and M. Naghhineh, Equivalent Capacity and It Application to Bandwidth Allocation in High-Speed Network," IEEE Journal on Selected Area in Communication, 9, No. 7, , 1991.