Solution of Non-homogeneous bulk arrival Two-node Tandem Queuing Model using Intervention Poisson distribution

Transcription

1 Volume-03 Issue-09 Sepembe-08 ISSN: (Onlne) RESEARCH REVIEW Inenaonal Jounal of Muldscplnay [UGC Lsed Jounal] Soluon of Non-homogeneous bulk aval Two-node Tandem Queung Model usng Inevenon Posson dsbuon * M. Hemanh Kuma; M. Venkaeswaan; 3 Tala Ganesh & 4 PRS Reddy * Depamen of Sascs, S Venkaeswaa Unvesy, Tupa, AP (Inda) Depamen of Humanes& Scences, S Venkaeswaa College of Engneeng, Tupa, AP (Inda) 3 Depamen of Mahemacs, NIT Hampu, HP (Inda) 4 Depamen of Sascs, S Venkaeswaa Unvesy, Tupa, AP (Inda) ARTICLE DETAILS Acle Hsoy Publshed Onlne: 07 Sepembe 08 Keywods Non-Homogenous Posson pocess, sae dependen sevce aes, Inevened Posson dsbuon, bulk avals. * Coespondng Auho Emal: hemanhkumasa[a]gmal.com ABSTRACT Tandem queung model s useful fo analyzng he suaons asng a banks, emples, hospals, cago handlng, ec. Howeve, n seveal communcaon newoks, he oupu of one sevce saon s an npu o he ohe sevce saon. Tha s, he sevce saons (ansmes) ae conneced n andem. Tandem queung models povde he basc famewok fo desgnng and monong of seveal communcaon sysems. In hs pape, s consdeed and suded he wo-node andem queung model (TNTQM) wh nonhomogeneous compound Posson bulk avals has suded usng Inevenon Posson dsbuon (IPD). By applyng sensvy analyss, he sysem pefomance measues ae hghly nfluenced by non-homogenous bulk avals ae and bach sze dsbuon paamees. Smulaon sudes showed ha he nevened paamee n nceasng he sevce ae n ems of uly, mean numbe of cusome who undeakes sevce and empness paen n he queue.. Inoducon The qualy of sevce (QoS) oung n weless newok whch has he mulple hops deal wh wo-node andem queung model (TNTQM) [5, 8]. An appoxmae analyss of open sysems of andem queues wh blockng caused by fne buffes beween seves poposed by [3]. Fo blockng and falues, [9] has consdeed he TNTQ sysem wh Benoull avals and phase ype seves, and[] developed he same havng load dependen sevce aes communcaon newoks, and fo bulk avals suded by [0]. In communcaon newoks such as Ehene LAN, MAN affc, WAN, and VBR affc exhb me-dependen aval aes and canno be modeled wh homogeneous Posson pocess o compound Posson pocess [, 4, 6]. The aval ae of TNTQ of affc suded by [6]wh me vayng. Two-node non-homogeneous compound Posson pocess depends on he avals of he cusome n bulky [].Usng andem queue, lmed wok has epoed n he leaue, hence n hs pape, we developed and analyzed he TNTQM usng IPD and also assumed ha he avals ae me dependen. The jon pobably geneang funcon, mean avals ae, sevce ae, mean wang me, houghpu has been deved usng dffeence dffeenal equaons fo queue and sevce as well. One specal case namely wo-node andem queung model wh non-homogeneous Inevened Posson bulk avals havng sae dependen sevce aes ae also suded explcly. The noaons used n hs pape as : Tme of avals; θ: Incdence paamee; ρ: Inevened paamee; λ: Aval ae; α: Avals of nfne populaon; μ: Fs Sevce ae; μ: Second Sevce ae. he numbe cusomes aved no fs queue n baches (he acual numbe of cusomes n any avng module s a andom vaable X wh pobably C k). he avals follow compound Posson pocess wh mean aval ae λ() wh bulk sze dsbuons {C k} and he sevce saons follow he same pocesses wh paamees μ and μ fo he fs and second sevce saons whch ae lnealy dependen on he conen of queue has he FIFO dscplne. The dagam of andem queue shown n Fg. Fg.. Schemac dagam of he poposed model Le n and n be he numbe of cusomes of fs and second queues and le P n n be he pobably ha hee ae, n cusomes n he fs queue and n cusomes n he second queue a me. In he dffeence-dffeenal equaons govenng he model, a, b and c ae abay consans deved usng he nal condons P, P 0fo 0 0,0 0 0, 0. Inevened Posson dsbuon (IPD) The nevened Posson Dsbuon (IPD) s an alenae fom of zeo-uncaed Posson dsbuon wh some nevenons [4, 5] and has wde applcaons ncludng elably, medcal and ec. The pobably mass funcon of IPD as e x x x e P( X x), 0 x! The mean and vaance of IPD ae. Queung Model The consdeed queung model has wo queues Q, Q, sevce saons S, S ae conneced n sees, and he cusomes wll ge sevce hough fs seve hen wll jon no he second queue whch s conneced o S n sees. Fuhe e E X V ( X ) ( e ) e and RRIJM 05, All Rghs Reseved 4 P a g e

2 Volume-03, Issue-09, Sepembe-08 RESEARCH REVIEW Inenaonal Jounal of Muldscplnay. Pefomance Measues wh IPD The numbe of cusomes n boh queues, he jon pobably geneang funcon s gven as K k e e k z z e PZ, Z, exp z k 0 0 k! m k e e k z z exp z k 0 0 k! e Expandng,, P 00 P Z Z we oban he pobably of zeo cusomes n he sysem as exp K k e e k e k 0 k! m k e e k e exp k 0 k! The geneang funcon of fs queue as P Z m k e e k e, exp z k 0 k! m k e e k e z k k! By expandng P(Z,) we oban he pobably of zeo cusomes n he fs queue sysem as 0 P m e e k e exp k 0 k! m K e e k e K k! The aveage numbe of cusomes n he fs queue s L e e e The ulzaon faco of he fs sevce saon su P 0. m k e e k e m k k exp e k e k 0 k! k k! The houghpu of he fs sevce saon s Thp U m k e e k e exp k 0 k! RRIJM 05, All Rghs Reseved 43 Page

3 Volume-03, Issue-09, Sepembe-08 RESEARCH REVIEW Inenaonal Jounal of Muldscplnay m k e e k e The mean wang me, vaance and k k! coeffcen of vaaon of a cusome n he fs queue as W L Thp e m e k V k e e k k! k e e CV V L 00 Agan expandng he,, P Z Z, we oban he pobably of zeo cusomes n he sysem as m k e e k e P0. exp k 0 k! m k k e exp k e k 0 k! of cusomes n he second queue s L e e e second sevce saon su P.0 m k e e k e exp k 0 k! The aveage numbe The ulzaon faco of he m k k e k e The houghpu of he k 0 k! second sevce saon sthp U m k e e k e exp k 0 k! m k e e k e k 0 k! vaance and coeffcen of vaaon of a cusome n he fs queue as W L Thp e The mean wang me, m e k e e V k. e.. 4 k k! e e 4 RRIJM 05, All Rghs Reseved 44 Page

4 Volume-03, Issue-09, Sepembe-08 RESEARCH REVIEW Inenaonal Jounal of Muldscplnay m e e k. e e k k! CV V L 00 The mean numbe of cusomes n ene sysem a me s L L L 3. An llusaon Though he numecal example, we dscussed he pefomance measues fo he poposed model. As dscussed n he second secon, he numbe of cusomes enes no he queue sysem n bulky way, once he sevce aken fom fs queue, hen wll jon no he second whch ae seally conneced. The aval ae and module of he cusomes follows compound Posson pocess and Unfom dsbuon especvely. Usng hs llusaon, he pefomance measues wee compued n ansen fom, and he paamees consdeed as: : 0.05, 0.06, 0.07, 0.08, 0.09; θ :.50,.0,.5,3, 3.5; ρ:.50,.0,.5, 3, 3.5; λ: 0.5, 0.6, 0.7, 0.8, 0.9; α: 0.0, 0.03, 0.04, 0.05, 0.06; μ : 9, 0,,, 3; μ : 6,7,8,9,30,3; The pobably of zeo cusomes n he queung sysem and magnal queues, he aveage numbe of cusomes, ulzaon faco wee compued fo dffeen values of he paamees, θ, ρ, λ, α, μ, μ, and wee pesened n Table. and houghpu of he sevce saons, mean wang me, vaance and coeffcen of vaaon of cusomes n each queue wee gven n Table.. The elaonshps beween he paamees and pefomance measues ae shown n Fg.. The smla gaphcal epesenaon s also made fo he pefomance measues wh espec o λ, μ₁ and μ especvely. Fom all he compasons, s obseved n he Table Concluson In hs pape he applcably of he nevened Posson dsbuon s made o demonsae he behavo of nonhomogeneous bulk queues unde wo node andem seve seup upon he smulaon sudes s obseved ha he nevened paamee n nceasng he sevce ae n ems of uly, mean numbe of cusome who unde ake sevce and empness paen n he queue. Howeve due o he pesence of compound paamee he wang me s obseved o have a gadual nceasng paen bu hs can be mnmzed f he sevce ae s nceased on he whole he mpang IPD n o pme wok of Non-homogeneous bulk avals as shown s mpac n explaned behavo of he queung sysem fuhe conollng hey paamees lke vaably, ncease he sevce ae he model povdes bee esul n ems of houghpu, uly and wang me. Refeences. A.V.S. Suhasn, K. Snvasa Rao, PRS Reddy (0), Tansen Analyss of Tandem Queung Model Wh Nonhomogeneous Posson Bulk Avals Havng Sae dependen sevce Raes Inenaonal Jounal of Advanced Compue and Mahemacal Scences, 3(3), pp. 7-89, hps://do.og/0.504/ijor Abyen P,, Baanuk R, Fland P, Red R, Vech D (00), Mul-scale naue of newok affc, IEEE Sgnal Pocessng Magazne, 9 (3), pp. 8-46, DOI: 0.09/ AlexandeBandwajn and Yung-L Lly Jow (988), An Appoxmaon Mehod fo Tandem Queues wh Blockng, Opeaons Reseach, 36 (), pp , hp:// 4. Cappe O, Moulnes E, Pesque J C, Peopulu and Y and X (00), Long Range Dependence and Heavy- al Modelng fo Tele affc Daa, IEEE Sgnal Pocessng Magazne, pp Ch. V. Raghavendan, G. Naga Sash, M. V. Rama Sunda,P.SueshVama(04),A Two Node Tandem CommuncaonNewok wh Feedback Havng DBA and NHP Avals, Inenaonal JounalofCompue and EleccalEngneeng,6(5),pp.4-435,do:0.7706/jcee. 04.v Dnda P.A., (006), Desgn, Implemenaon and pefomance of an Exensble oolk n esouce pedcon n Dsbued sysems, IEEE Tansacons on Paallel and Dsbued sysems, 7 (), pp J. DugaApaajha and K. Snvasa Rao (07), Two Node Tandem Queung Model wh Dec Avals o Boh he Sevce Saons Havng Sae and Tme Dependen Phase Type Sevce, Inenaonal Jounal of Compue Applcaons, 3 (), pp L. Le and E. Hossan (08), Tandem Queue Models wh Applcaons o QoS Roung n Mulhop Weless Newoks, IEEE Tansacons on Moble Compung, 7 (8), pp , DOI: 0.09/TMC LevenGün and Amand M. Makowsk (007), Maxgeomec soluon fo wo node andem queueng sysems wh phase-ype seves subjec o blockng and falues, Communcaons n Sascs. Sochasc Models, 5(3), pp , DOI: 0.080/ Nageswaaao, K., Snvasaao, K and Snvasaao, P (00), A andem communcaon newok wh dynamc bandwdh allocaon and modfed phase ype ansmsson havng bulk avals, Inenaonal Jounal of Compue Scence, Issues, Vol. 7, No., pp Padmavah, G., Snvasaao, K and Reddy, K.V.V.S (009), Pefomance Evaluaon of Paallel and Sees Communcaon newok wh dynamc bandwdh allocaon CIIT Inenaonal Jounal of Newokng and Communcaon, Vol., No.7, pp Pahasaahy, P.R and Selvaju, N. (00), Tansen analyss of a queue whee poenal cusomes ae RRIJM 05, All Rghs Reseved 45 Page

5 Volume-03, Issue-09, Sepembe-08 dscouaged by queue lengh, Mahemacal poblems n Engneeng, Vol.7, pp Paul J. Buke, (956), The oupu of a queueng sysem, Opeaons Reseach, 4 (6), pp , hps://do.og/0.87/ope R. Shanmugam (985), An nevened Posson dsbuon and s medcal applcaon, Bomecs, 4, pp , hp:// 5. R. Shanmugam (99), An nfeenal pocedue fo he Posson nevenon paamee, Bomecs, 48, pp , hp:// RESEARCH REVIEW Inenaonal Jounal of Muldscplnay 6. RakeshSngha, Shv Du Josh and Rajenda K P Bha (007), A Novel Dscee dsbuon and pocess o model self-smla affc, 9h IEEE Inenaonal Confeence on Telecommuncaon ConTel, pp W.E. Leland, M.S. Taqqu, W. Wllnge (994), On he selfsmla naue of Ehene affc (Exended veson), IEEE/ACM Tansacons on Newokng, (), pp. -5, DOI: 0.09/ Appendx Table.. Values of P 0.(), P.0(), P 00(), L (), L (), L(), U (), U () fo dffeen values of paamees θ ρ λ α μ μ P 0.() P 00() P. 0() L () L () L () U () U () RRIJM 05, All Rghs Reseved 46 Page

6 Volume-03, Issue-09, Sepembe-08 RESEARCH REVIEW Inenaonal Jounal of Muldscplnay Table.. Values of Thp (), Thp (), W (), W (), V (), V (), CV (), CV () fo dffeen values of paamee θ ρ λ α μ μ Thp () Thp () W () W () V () V () CV () CV () Table.3. Compasons RRIJM 05, All Rghs Reseved 47 Page

7 Volume-03, Issue-09, Sepembe-08 RESEARCH REVIEW Inenaonal Jounal of Muldscplnay Fg.. Pefomance Measues wh espec o Tme RRIJM 05, All Rghs Reseved 48 Page