ANALYSIS OF VANET CLUSTER PERFORMANCE USING MARKOV-MODULATED PROCESSES

Transcription

1 eon. tattc Method and her Applcaton Proceedng of the th Internatonal Conference Relablty and tattc n ranportaton and Communcaton Reltat 7 October Rga Latva p IBN ranport and elecommuncaton Inttute Lomonoova LV-9 Rga Latva ANALYI OF VANE CLUER PERFORMANCE UING MARKOV-MODULAED PROCEE Jelena Revzna ranport and elecommuncaton Inttute Lomonoova tr. Rga LV-9 Latva Ph.: E-mal: lena_revzna@t.lv Vehcular communcaton ytem a promng technology hch can provde cutomer th varou ervce from afety alert to n-car entertanment. Due to t huge applcaton potental t attract attenton both from academa and ndutry []. In th paper e preent the tudy of Markov-Modulated proce applcaton to the reearch of cluter node behavour n the VANE netork. Keyord: Vehcular Ad hoc netork VANE cluter cluter head Markov-Modulated proce. Introducton he modellng of Vehcular Ad hoc netork ha attracted much attenton of reearcher durng the lat fe year. uch partcular cae of moble ad hoc netork characterzed by a trong moblty of the node a hgh dynamc and pecfc topology a gnfcant lo rate and a very hort duraton of communcaton. Among applcaton of the VANE e may quote automatc drvng enhancng afety by propagatng emergency alert and dfferent paenger ervce. he dynamc VANE topology caue routng dffculte a ell a congeton from floodng [ 3 4]. A clutered tructure can make the netork appear maller and more table n the ve of each node. By cluterng the vehcle nto group of mlar moblty the relatve moblty beteen communcatng neghbour node ll be reduced leadng to ntra-cluter tablty [5]. Further to tablty an effectve cluterng algorthm mut be robut to the harh channel condton preent n the VANE envronment. Another problem th VANE routng protocol [6 7] floodng lead to netork congeton hch can be allevated by a clutered topology [8 9]. In th paper e decrbe an operaton of Vehcular Ad hoc netork VANE ung the theory of Markov-Modulated Brth-Death Procee [] thout gong nto the techncal detal of the VANE protocol. he condered mathematcal model the follong. A Markov-modulated proce defned a to-dmenonal contnuou-tme Markov chan X J. o-called Markov component J on the fnte tate pace C { m} correpond to a homogeneou contnuou-tme Markov chan []. h chan characterzed by tranton rate λ k C. he tate J k not changed durng the exponentally k dtrbuted tme th the parameter Λ k λ k λk λk m. After th tme the gven tate replaced by another tate th a probablty qk λ k / Λk. he tate of the component J change regardle of the tate of component X. If the tate J fxed then the component X on the tate pace E {} behave a a homogeneou contnuou-tme brth-death proce. In th proce the tranton are poble among neghbourng tate only. he tranton rate are follo: for the tranton from tate to tate ; for the tranton from the tate to tate. Let. I aumed that the Markov chan X J ergodc. It of nteret to tudy the teady tate probablte for the proce X J: π P{ X J } E C. he ret of th paper organzed a follo. ecton contan the mathematcal model of VANE cluter ung Markov-Modulated Proce of arrval. ecton 3 decrbe the calculaton of tatonary probablte. Numercal reult are preented n ecton 4. ecton 5 ummarze the man reult of the paper. he reference and acknoledgement fnh the artcle. 64

2 he th Internatonal Conference RELIABILIY and AIIC n RANPORAION and COMMUNICAION -. Propoed Model Decrpton We have a cluter of vehcle located and movng freely on a certan terrtory. he flo of vehcle enterng the cluter a Poon flo th a parameter λ. he oourn tme of the vehcle thn the cluter are aumed to be mutually ndependent random varable exponentally dtrbuted th the ntenty β therefore the mean oourn tme / β. Among the vehcle a leadng vehcle Head ngled out ervng a a tool for provdng connecton beteen the vehcle beng both thn the cluter and outde t. Generally peakng each vehcle enterng the gven cluter requre a connecton to Head but e can neglect the tme needed for th nce t neglgble. Addtonally f Head leave the cluter t ll be changed to another at tme. Every vehcle can generate a connecton requet clam to the Head. A a reult addreng to Head take place. he ntenty of the one vehcle requet to the Head equalν hle the tme before the next clam aumed exponentally dtrbuted ndependent of other mlar tme the number of vehcle n the cluter and the Head tate. he Head operate a a ngle-erver queung ytem. It can erve only one vehcle at a tme. Let the ervce rate equal to τ. hould Head happen to be buy at the moment of t beng turned to then the requet drected to the queue from hch t ll be accepted for ervce later. Our tak to determne the charactertc uch a mean number of vehcle atng for communcaton pendng to the Head EX the probablty that the vehcle ll have to at the begnnng of a connecton to the Head P etc. More generally e are ntereted n the probablty π that vehcle are n the cluter hle the number of tranmttng vehcle or pendng connecton equal to. he decrbed model the Markov-type modulated brth-death proce MMBDP [ 3]. Here the Markovan component extraneou factor J a number of vehcle n the cluter. he tranton rate from tate J are a follo: λ the ntenty of the tranton to tate hle β the ntenty of the tranton to tate >. Obvouly th proce decrbed by the Markovan queung ytem th an nfnte number of ervce place. he tatonary dtrbuton th repect to uch ytem ell knon [4] and t Poon dtrbuton th the parameter ρ λ / β. hu the probablty of vehcle preence n the cluter ρ ρ e! he econd component Х of our to-dmenonal proce a number of vehcle that have communcaton uch vehcle can only be one or pendng connecton. If there a tate of J then the tranton from tate X to tate realzed th the ntenty ν ; ; hle the tranton to the tate > ha the ntenty that depend nether on nor on. No let make a fe comment to the gven decrpton. Frt the Markov-modulated proce mple the component J changng regardle of the tate of component X. econdly the change of both component can not occur multaneouly. In order to provde for thee condton e aume that the vehcle leavng the cluter nether communcate nor are pendng. he only excepton the cae here only one communcatng vehcle n the cluter. In th cae hould t leave the cluter then of coure t communcaton nterrupted. Note that the number of vehcle beng connected or pendng normally much le than the total number of vehcle n the cluter o the accepted aumpton practcally do not affect the accuracy of the calculaton. paper []. No e can apply the method to calculate the tatonary tate probablte { π } a decrbed n 3. tatonary tate Probablte Calculaton Let aume that the maxmum value of the component of J and X the number of ther tate equal to m and m < accordngly. herefore the range of value of thee component ll C.. m E... We ue the condtonal probablte of tate of the econd component be { } { } X to calculate the tatonary probablte of tate { π } on condton that the tate of the frt component J equal to : π /. P{ X J } 65

3 eon. tattc Method and her Applcaton 66 Let u recall that the probablte } { are calculated accordng to the formula. Let Λ δ β λ be the um of off-dagonal element of the ro of nfntemal matrce for the component J and X. Here δ Kronecker ymbol: one equal f and equal othere. For tatonary probablte π e have a ytem of equaton a follo:. C E Λ β λ 3 Further e ue the follong -dmenonal column vector: a ell a dagonal matrx of -th order. Furthermore e ll need -th order hftng matrx caung the vector ro element beng hfted donard hft rght by one poton:. No the ytem of equaton 3 can be rtten a: C dag dag dag I Λ β λ here I dentty matrx and dag dagonal matrx th the dagonal. Fnally e have:. C dag dag dag I Λ β λ 4 he lat formula allo u to mplement an teratve procedure of calculatng the condtonal dtrbuton } {. At that t neceary to et the ntal dtrbuton C a ell a normalze the receved trantonal probablte contnuouly o that the um of element ould be equal to. he uncondtonal dtrbuton } { π gven by formula: π. 5 It allo one to calculate the average number of vehcle n the queung ytem: m X E π. 6

4 he th Internatonal Conference RELIABILIY and AIIC n RANPORAION and COMMUNICAION - 4. Expermental Reult he follong ntal data a accepted for our example. Charactertc of a cluter of vehcle: the vehcle arrval ntenty λ 5 the mean oourn tme of a vehcle /β.5 o that β.4. Charactertc of the queung ytem: the ntenty of one vehcle requet ν. the ervce rate. Wth repect to th ntal data e fnd that the average number of vehcle n the cluter for the tatonary regme ρ λ/β.5. For the Poon dtrbuton th th parameter the probablty of havng more than vehcle n the cluter neglgble o t ha been aumed that the number of vehcle n the cluter doe not exceed m. mlar conderaton have made t poble to determne that the maxmum number of vehcle n a queung ytem can be accepted a 5. he calculaton ere performed by the program rtten n Mathcad 4. A the ntal dtrbuton for all value the equally probable dtrbuton agnng the ame probablty / /6 to all poble tate: / f < / f. he program realzng the above-decrbed method of calculaton compute tatonary dtrbuton 4 and 5 very quckly. he follong table ee able 3 ho the obtaned value for the dtrbuton { } and { π }. he peed of convergence of the ued teratve procedure examned for the mean number of vehcle n the cluter ee able 4. { } { } able. he dtrbuton of a number of the vehcle n the cluter able. he fragment of the condtonal dtrbuton { } { able 3. he fragment of the tatonary dtrbuton π } able 4. Convergence of the teratve procedure K EX

5 eon. tattc Method and her Applcaton No e conder an analytcal model that ha explct formula for the ndce of nteret. Let compare the obtaned reult EX. th the reult obtaned by the explct formula. From no on e ll ue the follong to properte for the to ndependent exponentally dtrbuted random varable th parameter a and b: ther mnmum value ha an exponental dtrbuton th the parameter a b; the frt value le than the econd one th the probablty a / a b. Further e argue n the ay a follo. In fact our model can be repreented a a queung ytem M / M / /. At frt e fnd the ntenty of the flo of clam arrvng at the queung ytem of our. On arrval of a vehcle to the cluter to random varable are aocated th the node: the oourn tme n the cluter and the tme paed untl the frt addre to the queung ytem. Both value have an exponental dtrbuton th parameter β and ν repectvely. herefore t ll addre a queung ytem th the probablty ν /β ν. Hence the flo of ntal clam enterng the queung ytem ll be λν /β ν. here may be repeated clam to the queung ytem. No e hall take nto conderaton the ntenty of uch repeated clam. here a cheme of ndependent repeated tral th to outcome: ucce mean a ne repeated clam falure the oppote event. ucce poble f: the clam doe not leave the cluter durng ervce tme the probablty / β and on fnhng the ervce a ne clam ll arrve before the moment of leavng the cluter the probablty ν / β ν. he correpondng event are ndependent o the probablty of ucce equal to p ν / β β ν. he number of tral untl the frt falure ha a geometrc dtrbuton th the mean ν / β β ν β β ν β β βν / p / /. herefore the ntenty of the Poon flo of clam to the queung ytem Λ λν / β ν / p λν β / β β βν. [ ] No let u conder the dtrbuton of the oourn tme of clam thn the queung ytem. It a mnmum contng of to ndependent and exponentally dtrbuted random varable: the actual ervce tme the dtrbuton parameter and the tme before leavng the cluter the dtrbuton parameter β. Conequently the oourn tme on the erver ha an exponental dtrbuton th parameter β. hu e have the queung ytem M / M / / a follo: the Poon flo th ntenty /. Λ / the ervce rate β.4 one erver and the nfnte queue. he load coeffcent of the erver ω Λ/ β.5. he mean number of clam n the ytem calculated by the Erlang formula: ω. 5 E X ω. 5. ω. 5 herefore the value of EX. obtaned by the method decrbed n th tudy almot dentcal to the true value. he extng dfference le thn the lmt of the permble computatonal error. It mportant to emphaze that uggeted model doe not loe t gnfcance becaue t can be generalzed allong one to take nto account the dfferent dependence of the queung ytem charactertc on the tate of Markov component.e. the number of vehcle n the cluter. 5. Concluon We propoed a ell-etablhed probabltc approach to the cluter decrpton n the VANE netork. One baed on the Markov-Modulated Brth-Death procee. he future nvetgaton ll be connected th the reearch of the cluter tablty and more detaled decrpton of VANE cluterng mechanm. Acknoledgement Frt of all I h to expre my grattude to the upervor Profeor Alexander Andronov for h nvaluable gudance and encouragement n producng th ork. he artcle rtten th the fnancal atance of European ocal Fund. Proect Nr. 9/59/DP/.../9/IPIA/VIAA/6 he upport n Realaton of the Doctoral Programme elematc and Logtc of the ranport and elecommuncaton Inttute. 68

6 he th Internatonal Conference RELIABILIY and AIIC n RANPORAION and COMMUNICAION - Reference. Jn X. uy W. We Y.. A tudy of the VANE Connectvty by Percolaton heory. In Proceedng of IEEE Conumer Communcaton and Netorkng Conference CCNC Intellgent Vehcular Communcaton ytem Workhop IVC January pp UA: IEEE.. orrent-moreno M. Kllat M. Hartenten H. 5. he challenge of robut nter- vehcle communcaton. In Proceedng of the Conference on Vehcular echnology vol. ept. 5 pp UA: IEEE. 3. McDonald A. Znat A moblty-baed frameork for adaptve cluterng n rele ad hoc netork: elected area n communcaton. IEEE Journal Yu J. Chong P. A urvey of cluterng cheme for moble ad hoc netork: communcaton urvey & tutoral IEEE Journal Y. Gunter B. Wegel and H. Gromann. 7. Cluter-baed medum acce cheme for Vanet. In Proceedng of the Conference on Intellgent ranportaton ytem 3 th eptember-3 rd October 7 pp UA: IEEE. 6. Ln Y. W. Chen Y.. Lee. L.. Routng Protocol n Vehcular Ad Hoc Netork: A urvey and Future Perpectve. Journal of Informaton cence and Engneerng DOI:.48/ ch8. 7. N.-Y. eng Y.-C. Chen Y.-. & heu J.-P he broadcat torm problem n a moble ad hoc netork. In Proceedng of the 5th Annual ACM/IEEE Internatonal Conference on Moble Computng and Netorkng 5 9 Augut 999 pp UA: ACM. 8. Chen W. & Ca. 5. Ad hoc peer-to-peer netork archtecture for vehcle afety communcaton. IEEE Communcaton Magazne DOI:.9/MCOM m H. C tochatc Model: An Algorthmc Approach. UK: John Wley & on.. Andronov A. M.. Markov-Modulated Brth-Death Procee. Automatc Control and Computer cence DOI:.33/ Pacheco A. ang L. C. & Prabhu U. N. 9. Markov-Modulated procee and emregeneratve Phenomena. Ne Jerey: World c.. Ihler A. Hutchn J. & myth P. 7. Learnng to Detect Event th Markov-Modulated Poon Procee. ACM ranacton on Knoledge Dcovery from Data. 3 3: 3:3. DOI:.45/ Bolch G. Grener. de Meer H. & rved K.. 6. Queueng Netork and Markov Chan: Modelng and Performance Evaluaton th Computer cence Applcaton. UK: A. John Wley & on. 4. Ibe O. C. 9. Markov Procee for tochatc Modelng. UA: Academc Pre. 69