FEATURE ANALYSIS ON QUEUE LENGTH OF ASYMMETRIC TWO-QUEUE POLLING SYSTEM WITH GATED SERVICES *

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1 Journal of Theoretcal and Appled Informaton Technoloy 0 th January 03. Vol. 4 No JATIT & LLS. All rhts reserved. ISSN: E-ISSN: FEATURE ANALYSIS ON QUEUE LENGTH OF ASYMMETRIC TWO-QUEUE POLLING SYSTEM WITH GATED SERVICES * WANG XINCHUN, LIU YUMING, CHENG MAN, AND YE QING Department of Physcal and Electroncs, Chuxon Normal Unversty, Chuxon, Chna. Emal: xxxywxch@sna.com, luyum@cxtc.edu.cn, chenm@cxtc.edu.cn, yq@cxtc.edu.cn ABSTRACT On the bass of establshn mathematcal model and defnn the parameters and treatn embedded Marov chans and probablty eneratn functon as math tools, the paper analyzed dscrete-tme, nonsymmetrc dual-threshold servce queue system precsely, and t derved frst order and second order characterstcs of the system. The paper calculates the averae queue lenth and averae watn delay of nformaton roup. Smulaton experment based on system runnn mechansm and theoretcal calculatons have ood consstency. The analyss lad the sold foundaton for research of ated polln system based on asymmetrc mult queue; t further deepens the percepton of asymmetrc ated polln system; t s meannful to domnate polln system flexbly. Keywords: Asymmetrc; Two-Queue; Gated Polln System; Feature Of Averae Queue Lenth; Smulaton And Theoretcal Calculaton. INTRODUCTION As for research dffculty on asymmetrc polln system s too reat, t s enerally that people usually treat analyss and dscusson of performance of the asymmetrc two polln system as the startn-pont so as to lad sold foundaton for the research of asymmetrc mult queues ated polln system. It s enerally beleved that queue servce system based on perodc nqury has many applcatons n communcaton system and computer networ. As for queue servce system based on perodc nqury, the number of nformaton roup, servce tme for nformaton roup and shftn tme at any tme are varable. So, t s qute dffcult to analyze related performance. In the study of asymmetrc polln system Ref - ve some precse resoluton; however, as there are some problems n analyss method the obtaned results are partal under certan qualfyn condtons. On the bass of Ref -9 the paper adopts the method of embedded Marov chans and probablty eneratn functon to resolve dscretetme and polln asymmetrc double ated servce system, deduce one-order and two-order characterstcs of the system, calculates the averae watn delay of the system. The paper draws some useful conclusons by comparn smulaton results on the bass of the runnn mechansm and theoretcal calculatons.. MATHEMATICAL MODEL OF THE SYSTEM Fure. System Structure Model Of Gated Polln System Of Asymmetrc Two Queues.Structure model of the system Fure. System structure model of ated polln system of asymmetrc two queues Cyclc polln asymmetrc and two-queue ated *Funded Proect: The Scence Study Foundaton of Yunnan Provncal Department of Educaton n Chna under Grant NO. Y040 8

2 Journal of Theoretcal and Appled Informaton Technoloy 0 th January 03. Vol. 4 No JATIT & LLS. All rhts reserved. ISSN: E-ISSN: servce system accepts servces accordn the way of FIFO for nformaton roups belonn to the same queue. Both Termnal and Termnal wor n the means of ated servce. In Queue, when the servce for the former ated nformaton roup s expred and new-comn nformaton durn the course of the servce wll be servced at the next perod and swtch to the Queue after one converson perod. Queue fnshes the servce of current ated nformaton roup, then, the system swtches to Queue and performs the next round servce after a shftn tme. The system structure model s llustrated n Fure. System Parameters and Worn Condton For any queue, nformaton roups arrved at any unt tme s ndependent dentcally dstrbuted. Dstrbuton probablty eneral functon, mean and varance of nformaton roup n the queue are ' '' A ( z ) λ = A () σ λ = A ( ) + λ λ respectvely. In the same way, the probablty functon of the shftn tme, mean and varance ' '' are B ( z ), β = B (), σ β = B ( ) + β β respectvely, where, =,. As lon as the buffer storae capacty s lare enouh for each queue, there wll not exsts the loss of roup nformaton. Each customer n the queue s served by the rule of Frst-come, Frst-serve. For the convenence of analyss, the paper defnes stochastc varable u as the shftn tme that water shfts from Queue to Queue + at the tme t n. v s the servce tme for Queue at the tme t n by the water. µ ( u ) s the number of nformaton roups when the water oes nto Queue at the tme u. η ( v ) s the number of nformaton roups when the water oes nto Queue at the tme v. ξ s the number of nformaton roups of Queue at the tme of t n. =,; =, as for non-symmetrcal two queues..3 Status equaton of the system For the system of asymmetrc two-queue ated servce there should be ( ) r λ + () + () βλ ξ n + ) = 0 + η ( v ) + µ ( ) () ( u Resolve (), (8), (9) and (0) ontly, the paper obtans + () ξ ( n ) = ξ( n) + η( v) + µ ( u) At the tme t n+ when queryn ated servce queue..4 Probablty eneratn functon of the system Probablty eneratn functon of the system of asymmetrc two-queue ated servce s G + ( z, z ( n ξ ) = lm E z n = + ) (3) Combned (), () and (3) the paper obtans probablty eneratn functon of the system of asymmetrc two-queue ated servce s G( z, z) = R A z A z G z B A z A z ( ( ) ( )) (, ( ( ) ( ))) G( z, z) = R A z A z G B A z A z z ( ( ) ( )) ( ( ( ) ( )), ) G z z (4) (5) The (, ) n (4) s the probablty eneratn functon of the status dstrbuton of the Queue. The G ( z, z ) n (5) s the probablty eneratn functon of the status dstrbuton of the Queue. 3. ANALYSIS OF THE SYSTEM PERFORMANCE 3. Resolve one-order characterstcs Defne G ( z, z ) ( ) = lm z, z z =,; =, () Solve one-order partal dervatve of (4) and (5) by (), and obtan the follown by reducton ( + β λ ) rλ + () () = () ( + β λ ) rλ () = (8) ( r λ + β λ ) () = (9) = (0) 9

3 Journal of Theoretcal and Appled Informaton Technoloy 0 th January 03. Vol. 4 No JATIT & LLS. All rhts reserved. ISSN: E-ISSN: ( r r + ρ r ) λ () = () ρ ( r r + ρ r ) λ () = () ρ Averae lenth of queue () = ( r + r ) λ (3) ρ ( r + r ) λ () = (4) ρ Averae cycle tme r = r r θ = T θ = = ρ ρ = T + (5) 4. COMPARISON BETWEEN SIMULATION AND THEORETICAL CALCULATION RESULTS Accordn to the mechansm of the system the paper compares computer smulaton wth theoretcal calculaton results. Assume arrval of each queue at any tme slot obeys Posson dstrbuton and smulaton and theoretcal calculaton adopt the same parameters. M = 0 Set polln number. When r = r =, λ = 0. 0, λ = smulaton and theoretcal calculaton results of averae lenth of Queue and varyn wth servce rate s llustrated n Fure. Set polln number M = 3 0. When = r = r, β, β 3 = = smulaton and theoretcal calculaton results of averae lenth of Queue and varyn wth arrval rate s llustrated n Fure 3. Set polln number M = 5 0. When β = β = 0, λ = 0. 0, λ = smulaton and theoretcal calculaton results of averae lenth of Queue and varyn wth shftn tme s llustrated n Fure 4. Assume queues are = 5 0 symmetrcal, set polln number smulaton and theoretcal calculaton results of averae lenth of Queue and varyn wth the varaton of load are llustrated n Fure 5. M 5. ANALYSIS AND DISCUSSION We can obtan t from Fure that the averae watn delay ncreases wth the ncrease of β when λ and γ are fxed. Compare two sets of curves n Fure and you wll fnd that he averae watn delay ncreases remarably f λ s larer. Averae watn delay s larer f λ s larer and when β s fxed. Smulaton and theoretcal resolutons (3) and (4) have ood consstency. Althouh smulaton and theoretcal resolutons approxmately equal, there s a devaton between them. Ths s manly due to fewer statstcs polln, only M = 0. If ncreasn polln number, provn data qualty and decreasn devaton. Averae lenth of watn queue Servce rate Fure. Comparson Between Smulaton And Theoretcal Calculaton Results Of Averae Lenth Of Queue And Varyn Wth Servce Rate Averae lenth of watn queue Arrval rate Fure 3. Comparson Between Smulaton And Theoretcal Calculaton Results Of Averae Lenth Of Queue And Varyn Wth Arrval Rate 30

4 Journal of Theoretcal and Appled Informaton Technoloy 0 th January 03. Vol. 4 No JATIT & LLS. All rhts reserved. ISSN: E-ISSN: We can also obtan t from Fure 3 that the averae watn delay ncreases wth the ncrease of λ when β and γ are fxed. Compare two sets of curves n Fure 3 and you wll fnd that the averae watn delay ncreases remarably f β s larer. Averae watn delay s larer f β s larer and when λ s fxed. Smulaton and theoretcal resolutons (3) and (4) have ood consstency. Smulaton and theoretcal resolutons approxmately equal and there s only lttle devaton between them. Ths s manly due that statstcal polln number has reached M = 3 0. If ncreasn polln number M we can mprove the qualty of data, and further reduce the error. Averae lenth of watn queue Shftn tme Fure 4. Comparson Between Smulaton And Theoretcal Calculaton Results Of Averae Lenth Of Queue And Varyn Wth Shftn Tme Averae lenth of watn queue load Fure 5. Comparson Between Smulaton And Theoretcal Calculaton Results Of Averae Lenth Of Queue And Varyn Wth The Varaton Of Load We can obtan t from Fure 4 that the averae watn delay ncreases wth the ncrease of γ when λ and β are fxed. Compare two sets of curves n Fure 4 and you wll fnd that the averae watn delay ncreases remarably f ρ s larer. Averae watn delay of the ones wth heaver load s larer when shftn tme s fxed. Expermental results and theoretcal expressons (3) and (4) have ood consstency. Smulaton and theoretcal resolutons approxmately equal and there s only lttle devaton between them. Ths s manly due that statstcal polln number has reached M = 3 0. If ncreasn polln number M we can mprove the qualty of data, and further reduce the error. We can also obtan t from Fure 5 that the averae watn delay ncreases wth the ncrease of ρ when γ s fxed. Smulaton and theoretcal resolutons (3) and (4) have ood consstency. The devaton s least. Ths s manly due that statstcal polln number has reached M = 5 0. If ncreasn polln number M we can mprove the qualty of data, and further reduce the error. Compare Fure, 3, 4, 5 we can fure out that bas between smulaton results and theoretcal value further reduce wth the ncrease of polln number M.The experments show the error can be manpulated wthn % f the load of asymmetrc two-queue satsfes ρ and polln number = 8 reaches M 0. Expresson (3) and (4) transform to the Expresson () n Reference [] when the networ s runnn under the crcumstance of symmetry parameters and N =. REFERENCE [] Feruson M J, Amnetzah Y J. Exact results for nonsymmetrc toen rn systems. IEEE Trans Commun, 985, 33 (3): 3-3. [] Evertt D. Smple approxmatons for toen rns. IEEE Trans Commun, 98, 34: 9-. [3] Tanc T, Taahash Y. Exact analyss of asymmetrc polln systems wth snle buffers. IEEE Trans Commun, 988, 3 (0): 9-. [4] Ibe O C, Chen X. Performance analyss of asymmetrc snle-buffer polln systems. Perform Eval, 989, 0: -4. 3

5 Journal of Theoretcal and Appled Informaton Technoloy 0 th January 03. Vol. 4 No JATIT & LLS. All rhts reserved. ISSN: E-ISSN: [5] Ibe O C, Chen X. Approxmate analyss of asymmetrc snle-server toen-passn systems. IEEE Trans Commun, 989, 3 (): 5-5. [] Muheree B, Kwo C K, Lantz A C, Moh W L M. Comments on exact analyss of asymmetrc polln systems wth snle buffers. IEEE Trans Commun, 990, 38 () : [] Zhao Donfen, Zhen Sumn. Messae Watn Tme Analyss for a Polln System wth Gated Servce. Journal of Chna of communcaton, 994, 5(): 8-3. [8] Zhao Donfen. Study on scheduln rules of producton flows wth two prorty control. Informaton and Control, 998, (5): [9] Yu Yn, Zhao Donfen. Performance analyss of exhaustn and ated polln system. Journal of Yunnan Normal Unversty, 00, (4):