Derivation of Three Queue Nodes Discrete-Time Analytical Model Based on DRED Algorithm

Transcription

1 Seventh Internatonal Conference on Informaton Technology Dervaton of Three Queue Nodes Dscrete-Tme Analytcal Model Based on DRED Algorthm Jafar Ababneh, Hussen Abdel-Jaber, 3 Fad Thabtah, 3 Wael Had, EmranBadarneh The Arab Academy for Bankng & Fnancal Scences, Amman, Jordan The World Islamc Scences & Educaton (W.I.S.E.), Amman, Jordan 3 Phladelpha Unversty, Amman, Jordan, Jafar_Ababneh@yahoo.com, habdela@wse.edu.o, ffayez@phladelpha.edu.o, Whad@phladelpha.edu.o, Bemran@yahoo.com Abstract Ths paper proposes a dervaton of dscrete-tme queung network analytcal model based on dynamc random early drop (DRED) algorthm to manage and control congeston n early stages before t occurs, whch s referred to as the 3QN model. The proposed model conssts of three nterconnected queue processng nodes. Expressons were derved to calculate performance measures, namely; throughput, and average queung delay. In order to nvestgate and analyze the effectveness and flexblty of the proposed model, two scenaros were performed. These scenaros nvestgate the varaton of packets arrval probablty aganst throughput), and average queung delay. Besde that we compare between the three queue nodes of the proposed model usng the derved performance measures to dentfy whch queue node provdes better performance. Results show that queue node one has better performance than queue node two and three wth regard to throughput. All the above performance measure results for the queue nodes are obtaned only based on the queung network settng parameters. Therefore the node whch offers better performance results than others s solely based on the gven values for the settng parameter. Key Words: Dscrete-Tme Analytcal Model, Congeston Control, Queung Network; Performance Measure, Network Management and Control, DRED algorthm.. Introducton Computer networks and data communcatons systems,.e. nternet, are the fastest growth technologes n recent years, whch lead to magnfcent development n many applcatons especally the swft advance n Internet, audo and vdeo data traffc, etc. It s obvous that congeston, whch s occurred when the requred resources by the sources at a router buffer s more than the avalable network resources and t s reducng a computer network performance [, ] such as degrade the throughput, obtan hgh packet queung delay and packet loss and unfar share of network connectons [, 3, 4]. For controllng congeston n networks, dfferent congeston control methods have been developed [, 3, 5-]. The earlest queue congeston control mechansm was the end-to-end Transport Control Protocol (TCP) whch s domnant n the Internet [4, 5, 3]. In ths protocol, the source only reduces ts rate when t detects congeston [5, 3]. Moreover, there are Tradtonal Queue Management (TQM) algorthms; Drop-Tal (DT), Random Drop on Full, and Drop Front on Full [3], whch operate at the desgnated node, and drop packets only when the queue s full, However, they have several drawbacks, ncludng, lockout phenomenon, full queues, bas versus burst traffc, and global synchronzaton, and consequently they contrbute n degradng the Internet performance [,, 6]. Dfferent congeston control algorthms were proposed to overcome some of DT drawbacks; these algorthms were referred to as Actve Queue Management (AQM) algorthms [, 3, 5- ]. The AQM algorthms respond to congeston before buffers overflow and have several obectves [, ]. Random Early Detecton (RED) [4, 7], Dynamc Random Early Drop (DRED) [5], Adaptve RED [6], BLUE [, 6, 8], Random Exponental Markng (REM) [9-], Stablzed Random Early Drop (SRED) [], Gentle RED [7], and GREEN [5] are consdered an examples for AQM algorthms. Mathematcal (or analytcal) modelng, practcal experments and smulatons are methodologes used for analyzng mult-queue system performance durng congeston. However, mathematcal models are cheaper, easer to use and can help n gvng approprate solutons [, 8]. Nevertheless, buldng analytcal models for queung network systems usng dscrete-tme queues s not easy, and ths explans the lack of research works whch have done n ths feld [9-]. In relaton wth tme, mathematcal models for AQM algorthms can be developed as contnuous or dscrete-tme models [9, ]. Analytcal models were developed n to evaluate the performance of a dscrete-tme AQM-based queung systems of one and two-queue nodes [4,, ]. In ths paper, a new dscrete-tme queung network analytcal model based on DRED algorthm s proposed. Ths analytcal model can be utlzed as a congeston control method n fxed and wreless networks. Expressons were / $6. IEEE DOI.9/ITNG

2 derved to calculate performance measures, namely; throughput (T ), and average queung delay (D ). In order to nvestgate and analyze the effectveness and flexblty of the proposed model, two scenaros were performed to nvestgate the varaton of packets arrval probablty arr ) aganst (T ), and (D ). Furthermore, we compare three proposed nodes n the developed model n order to determne the queue node that produces better performance. The rest of ths paper s outlned as follows: Secton ntroduces a DRED algorthm. The new dscrete-tme queung network analytcal model s presented n Secton 3. Secton 4 descrbes a comparson between the queues nodes ( =,, 3) of the proposed model wth reference to the performance metrcs mentoned above. Fnally, conclusons and suggestons for future work are gven n Secton 5.. Dynamc Random Early Drop (DRED) Algorthm DRED was proposed n by Aweya et al [5], to control the problem of congeston n networks. DRED stablzes the queue length (Q) at a predetermned level (Q tar )) ndependently from the number of TCP connectons n the network. DRED depends on a fxed tme unt (t) and for each t, the current Q and the error sgnal (E) are computed n order to obtan packet probablty droppng p d. The calculated E depends on both the current Q and Qtar, and t can be expressed as: E(t) = Q(t) Qtar () Then, based on E (t), the fltered error sgnal F(t) can be calculated as: F(t) = F(t-) ( - wq) + wq E(t) () It can be notced that DRED uses a low-pass flter n calculatng the F(t) smlar to RED [5]. In DRED, the value of p d s updated for every t usng the followng expresson: F t p d t m n m ax p d t,, (3) K Where K s the capacty of the DRED router buffer. In [5], the pd parameter s updated only when the current Q s equal to or greater than the mnmum (nodrop) threshold (H mn ) for the sake of mantanng hgh-lnk utlzaton. In other words, there wll be no packet droppng when Q(t)<H mn. We can nfer that DRED reles on Q(t) n order to decde whether or not to drop packets. Hence, the congeston metrc for the DRED method s the current Q. DRED algorthm marks the arrval packet ether by droppng t or by addng an explct congeston notfcaton (ECN) bt n ts header [3]. 3. Proposed 3QN Dscrete-tme Analytcal Model DRED Based. The proposed queung network system s shown n Fgure. The analyss of the queung network s gven as: Fg.. Three queue nodes queung network (=,, 3) model. The queue nodes shown n the above fgure have a fnte capacty of packets, where K (=,, 3) correspond to the capactes for queue nodes, and 3, respectvely, ncludng packets that are currently n servce. we assume that the arrval process used s the dentcal ndependently dstrbuted (..d) Bernoull process a n ε {,}, n=,,,3, where a n represents packet arrval n a slot n. The thresholds H s expressed n Eqn. (4) as follows [5]: H = [.9k /] where =,, and 3 (4) Moreover, n Fgure, the probablty of packet arrval for the router buffer of queue node,, and 3 s λ (λ s defned below), only when the nstantaneous Q avg, s below H and thus there wll be no packet droppng (P d =). Whle, f the nstantaneous Q avg, reaches H ndex, the probablty of packet arrval for queue node reduces from λ to λ +3 (λ +3 s defned below) n order to allevate the congeston at the router buffer, and therefore P d ncreases from to (λ - λ +3 )/λ. Ths s snce the packet arrval probabltes (λ ) are decreased from probablty value to another fxed ones (λ +3 ), and the values of (P d ) are ncreasng from to another fxed probablty. To explan the queung network system shown n Fg., we analyze each queue node separately n order to come up wth the traffc equatons for each one of them. We assume that λ s the probablty of the packet arrval of the external packets (packets arrvng from outsde the network) n a slot. λ and λ +3 are the probabltes of packets that arrve at node before reachng the H ndex and after reachng the H ndex at the router buffer, respectvely. Further, β represent the probabltes of packet departure n a slot from nodes. We also assume that the queung network s n equlbrum, and the Q avg, process of each queue node s a Markov chan wth fnte state spaces. The state spaces for queue node are:,,,3,..., H, H, H,..., K, K ar p = p = p 3 = Where =,, 3 Fnally, we consder that λ > λ +3, β > λ thus, β > λ +3. Snce we analyze each queue node n the queung network system separately, the packet arrval probablty for each queue node requres an ndependent evaluaton. Eqns. (5- H H H3 3 P 3 β P 3 β P33 p p β 3 P P P 3 P 3 p p p3 886

3 6-7) represent the arrval probablty of packets for queue nodes, and 3, respectvely. r r r r (5) 3 3 r r r r (6) 3 3 r r r r (7) After solvng Eqns. (5-6-7) recursvely, we obtan the fnal form for λ, λ and λ 3 as n Eqns. (8-9-): represent the probabltes of packets leavng the queung network from queue nodes, and 3, respectvely. Lastly, (r, r o ) represent the routng probabltes of packets that come from outsde the network to queue nodes, and 3, respectvely. The state transton dagrams of queue nodes (=, and 3) are shown n Fgures, 3 and 4 respectvely. The symbols In Eqns. (8, 9, ), r,,, and 3 represents the routng probabltes of packets between the three queue nodes n the queung network, and (r, r ) r r rr r r33 r3r3 r3 r rr3 r3 r rr 3 r r r r r r33 r3 r rr3 r r33 r3r3 r r rr r r33 r3 r3 r3 r r r3 r3 r rr 3 r r r r r r33 r3r 3 r3 r r3 r r3 r r33 r r3 r r r r r3 r rr3 r r rr r3 r r r3 3 r r r r r r33 r3r3 r3 r r r3 r3 r rr3 (8) (9) ) H H3 K K Fg.. The state transton dagram of queue nodes,, and 3 n the proposed analytcal model 887

4 Dervng the balance equatons for queue nodes, and 3, by usng the state transton dagram n Fgure, we wll generally obtan Eqn. () for as shown below: () Where =, 3, 4,, H - H H H () H 3 H H H 3 3 (3) H 3 H H H 3 3 (4) H 3 Generally we obtan, (5) 3 Where = H +, H +, H +3,, K -, K -. Then fnally we can obtan. K K 3 K 3 (6) Where K =H + I. After solvng Eqns. (-6) n terms of we obtan generally Eqn. (7). (7) Let where,,3,4,..., H (8) To derve the equlbrum network queung network probabltes n the network queung we substtute Equaton (8) n Eqns. (-6) to acheve: (9) Where =,, 3 and =,, 3, 4,,H - H H 3 H 3 H 3 3 H H () Where =,, 3 and I=,, 3, 4,,I We start calculatng the frst performance measure of the proposed model (T ) whch defned as the number of packets that have been passed through the queue node successfully. The equaton used n the calculaton of (T ) s gvng below. K Packets/slot () T So we must calculate (whch s the probabltes of that queue nodes, and 3 are dle), by applyng Eqns. (9, ) n Eqn. (7), we obtan the fnal result for as n Eqn. (). H I H () The second performance measure s the average queue delay (D ), t defned as the tme a data packet spent n the buffer of the queue before beng processed and calculated by Eqn. (3). D = [Q avg, / T ] slots (3) The frst term Q avg, of Eqn. (3) for the queue nodes,, and 3 can be calculated usng the generatng functon P(z) as shown n Eqn. (4). P z K z (4) We utlze the P z equaton to obtan the Q avg, by producng the frst dervatve for the generatng functon P z at z, as shown n Eqn. (5). H H H 3 Qavg P H 3 I I I3 ( Results and Dscusson The results have been generated accordng to dfferent performance measures (, D ). In addton, an ndependent performance evaluaton for each node n the three nodes system s conducted to fgure out the node that has better performance. In the experments, the followng parameters settngs and assumptons are used: queue node has hgher prorty than the queue node and node has hgher prorty than the node

5 Fg. 3 Varaton of λ arr aganst T Queue node parameters r s set to.5, (=.75, =.9, =), and the routng probabltes of queue node (r =.4, r =.3, r =. and r 3 =.).Whereas, queue node r s set to.3, (=.75, =.9, =), and the routng probabltes of queue node (r =.4, r =.3, r =. and r 3 =.). Fnally queue node 3 r 3 s set to., (=, 75, 3 =.9, 3 =), and the routng probabltes of queue node 3(r 3 =.4, r 3 =.3, r 3 =. and r 33 =.). arr parameter has been set to [.66-.7] n order to see the performance effect for dfferent probabltes of packet arrval (moderate and hgh traffc load) wth.5 steps., and 3 were tuned to to observe the queue node performances wth small buffer szes. The performance measure results are obtaned based on the gven settng values for the above parameters. Thus, the performance whch node s provde better performance result s only gven based on the above parameter settng values. For our proposed analytcal model, the performance measure results for the proposed three queue nodes (,, and 3) analytcal model are shown n Fg Partcularly, we llustrated the results of queue nodes, and 3 wth respect to λ arr aganst Q avg, and T. After analyzng results of the frst performance measure whch s the relatonshp between packet arrval probablty and throughput, we obtaned that queue node acheved hgher throughput than nodes and 3. Ths returns to that the probablty of packet arrval of queue node s hgher than the probablty of packet arrval for queue nodes and 3 (the prorty). Also, queue node has hgher T results than node 3. It s obvous from Fg. 3 that node 3 s the best node n usng ts buffer occupances and the node s the worst because t uses hgher throughput wth same traffc load wth respect to dfferent routng probablty. Ths result s due to that the queue node receves more packets than ether queue node or 3, and queue node receves packets hgher than queue node 3 (see the routng probabltes). Fg. 4 Varaton of λ arr aganst D Accordngly, node has hgher delay results than node 3 wth reference to the settng values for the packet arrval probablty as shown n Fgure 4 whch means that node 3 s the best node snce t has the least delay among the three nodes. The reason that node s the worst s because t has the longest delay among nodes usng the same traffc load wth reference to dfferent routng probablty and the dssmlar prortes of the nodes. 5. Conclusons A dervaton of new dscrete-tme queung network analytcal model that conssts of three queue nodes s proposed n ths paper. Two performance measures for every queue node were derved for each queue node: throughput (T ) and average queung delay (D ). Ths dervaton s gven n order to compare the three queue nodes wth reference to ther two performance measures n the proposed analytcal model. A comparson ams to decde the queue node whch provdes better performance measure results. The two performance measures of the three queue nodes are obtaned based on the settng values of the queue node parameters. Therefore the node that offers better performance results than others s solely based on the gven values of the queue node settng parameters. The results ndcated that queue node provded the best T results. Furthermore, queue node s slghtly outperformed queue node 3 wth respect to both T results. In addton queue node 3 gave better D results than ether queue node or, and queue node produced better D results than queue node. The queue nodes and 3 offered the best results n terms of T and D respectvely; due to queue node has a hgher prorty than queue node whch t has a hgher prorty than queue node 3 n servng packets comng from outsde the queung network. We planned n future to apply our new dscrete- tme queung analytcal model n wreless networks such as cellular networks to control the congeston ncdent at base statons. Moreover, we ntend to extend our model by applyng multple arrvals and departures as rather than sngle arrval and departure. 889

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