A TASK OF THE STORAGE CONTROL THEORY IN TRANSPORT SYSTEMS USING RESAMPLING-METHOD

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1 A TASK OF THE STORAGE ONTROL THEOR IN TRANSPORT SSTEMS USING RESAMPLING-METHO Helen Afanaseva Fault of outer sene and Eletrons Transort and Teleounatons Insttute Loonosov Str. LV-9 Rga Latva ABSTRAT: The robablt of shortage absene s estated for the storage of soe transort sste. The ntensve outer ethods of statsts are used n orresondng roesses sulaton. The effen of suggested aroah takng the ean square error of the estator as the rteron s llustrated wth orresondng nueral exales. KEWORS: Transort sstes ontrol Storage ontrol theor Resalng ethods estators roertes INTROUTION The ter storage n onteorar busness fnshes to be sl alulatng ndator of atvt but beoes one of the ost ortant objets of anageent that nfluene the suess of all transort nsttuton. Takng nto aount growng oblt of nowadas transort t s ver ortant to organze onsderng logsts roesses of storage sul ore effentl. The sealsts suggest to analze and to otze suh olex stohast sstes on base of orresondng atheatal odels and sulaton. Let us onsder soe transort oan that has a nuber of robuses. There s storage of ver rare sares that are not used ver often but ther state of shortage s ver exensve. What s the otal level on ths storage? One of the was to solve suh roble s to buld the orresondng sulaton odel. If we have suffent data of te oents of sul and deand we an estate the orresondng robablt laws and ther araeters. Then we an use ths nforaton as the nut flow nto our sulaton odel of storage ontrol. The an roble that an arse here s that the nut data szes are not suffentl bg to redt the rght robablt law and estate araeters orretl. The suggested nonaraetr resalng-aroah hels to avod those dsadvantages of tradtonal ethods. In onteorar world wth owerful odern outers were oened new ersetves before the statstans. Intensve outer ethods of statsts [5] or resalng ethods allows to use the nut data for sulaton wthout revous estaton of dstrbuton lows or ther araeter sl n dfferent obnatons. Ths akes ossble to turn the data wth the dfferent sdes and to get the target araeters of sulaton wthout bas and wth saller value of varane. Resalng gves robust estators of araeter of nterest takng as effen rteron the ean square error. In the resent aer the resalng-aroah leentaton s onsdered to a task of storage ontrol theor [] n transort sstes.

2 Suose we have two sle ndeendent renewal roesses: deand { =...} and sul { =...} where { } and { } are the sequenes of unnegatve ndeendent rando varables eah sequene wth ts own oon dstrbuton [6][3]. The dstrbuton funtons of sequenes { } and { } are unknown but orresondng ntal sales of szes n and n are avalable. The renewals of the roesses orresond to the sul or to the deand of storage unts. The ntal storage level K s known. Then the robablt of nterest P{ >S -K } of the shortage absene s the robablt that the -th deand K oes later than the K -th sul S. K Then the robablt of nterest of the shortage absene s the robablt that the -th deand oes later that the K -th sul >S -K. It s also assued that the ntal storage level K s known. We wsh to nvestgate soe roertes of the dfferent estators of the shortage absene robablt. In the next seton soe ortant relatonshs for future uroses are gven. Then follows the seton where the resalng-estators of robablt of nterest are resented. The thrd seton resents the sef ases for soe dstrbutons on the base of resalngaroah. The fourth seton onsders the lassal estators of the robablt of nterest. Further n the ffth seton nueral exales llustrate the suggested aroah effen. The last seton onludes the aer.. SOME RELATIONSHIPS Now we desrbe our roble ore forall. Let us defne the dstrbuton funtons of sequenes { } and { } as F and F and dstrbuton denst funtons as f and f. The funtons F and F are unknown but orresondng sales H } and H } are avalable for sequenes { } and { n { n { } where H n and H n. Let us defne the dstrbuton and denst funton of the su of varables: F and. The sub ndex n ths notaton eans the nuber of addends n the su. The f entoned above funton F aordng to ths notaton s the dstrbuton funton of the su resented wth onl one eleent. The uer ndex eans the r.v. nae whose dstrbuton funton s onsdered. We are nterested n the te of the -th and K -th renewals: S K K. Our task s to estate the shortage absene robablt P{ S } that the -th K renewal of the deand roess { } oes later than the K -th renewal of the sul roess { }. Let s onsder the ndator funton Ψ x where x x x and are vetors of real nubers: x

3 f x x otherwse. Ψ Suose we have two vetors of r.v. and K. Our urose s to estate the shortage absene robablt Θ E Ψ. We wll estate Θ usng two dfferent aroahes: lassal and resalng. The lassal araetral aroah s wdel known. So we onsder the alternatve nonaraetral resalng-aroah leentaton n ths aer.. RESAMPLING-APPROAH At frst we onsder the resalng-aroah for the estaton of the shortage absene robablt. Ths ethod does not suoses the estaton of the dstrbuton araeters or the onstruton of the eral dstrbuton funtons to fnd haratersts of nterest as t s suosed b tradtonal ethods. Alternatvel we use rar data n dfferent obnatons and ths fat akes ossble to obtan unbased estators and derease ther varane. There are a lot of exales of resalng-aroah suessve leentaton to varous tasks for relablt sstes regresson odels for order statsts estaton [] [7]. Resalng-aroah suoses the followng stes. We hoose randol eleents fro the sale H and eleents fro the sale H. The eleents are taken wthout relaeent we rend that n n. Then we alulate the orresondng value of funton Ψ x usng forula. After that we return hosen eleents nto the orresondng sales. We reeat ths roedure durng r realzatons. Let j l d=.. d be the ndes of eleents fro the sale H { } that are hosen at the l-th realzaton. Then for the l- th realzaton we obtan the followng vetors: l l. j l j l j l x The resalng-estator j l j l j l Θ s the arthetal ean b r realzatons: Θ Ψ l l. 3 r r l Obvousl ths estator s unbased: We are nterested n the varane of ths estator. Let s denote the followng notatons: E Θ Θ. 4 μ E Ψ μ E 5 Ψ Ψ l l Ψ l μ E. 6 Then the varane of nterest an be alulated usng the followng forula:

4 where V Θ E Θ μ 7 E Θ * μ r r μ r. 8 In order to estate the varane of the estator we have frstl to fnd the exresson of the xed oent μ fro the forula 6. To alulate the oent μ the notaton of -ars an be used [4]. Let us denote W l l=...r { } a subset of the sale H whh was used for rodung the values of vetors l and l orresondngl W l H. Let us denote M ={... } M=M M. Let = be an eleent of M M. We sa that W l and W l rodue the -ar f and onl f W l and W l have oon eleents: W l W l. Let A ll denote the event subsales ll and l l rodue -ar but P ll be the robablt of ths event: P ll =P{A ll }. Beause of the fat realzatons l=..r are statstall equvalent we an ot the lower ndes ll and wrte P. Let then μ E Ψ l l Ψ A ' 9 μ ll P. μ M Therefore we need to alulate P and μ for all M. The robablt P an be alulated usng hergeoetral dstrbuton: n n P { } n where s bnoal oeffent. Now our task s to alulate μ M. Let s ntrodue soe new notatons for two dfferent realzatons l and l of the resalng-roedure. Usng sus entoned earler n forula let s nlude the uer ndex orresondng to the realzaton nuber. Then let s dvde eah su nto two arts n the followng wa. We onsder searatel the oon and the dfferent eleents of these sus for realzatons l and l : ξw j S l j where or S ξ l j df o df o S S ξw j S df l ' l l ' df l ' l S S o o s the su of sequenes { ξ } or { } eleents for the j-th realzaton j { l } df jk df jk or S s the su of n eleents fro j or j whh are absent n k or n n k k j { l } k j o o or S s the su of n oon eleents of l and l or l and l. n n

5 Therefore we an wrte: μ P { Ψ l l Ψ } df o df o df l ' l o df l ' l P{ S S S S o } df df df l ' l df l ' l P{ S S } P{ df z S df df l ' l z S df l ' l } f z dz o o where S f x the dstrbuton denst funton of r.v. С. Note that df for fxed value of r.v. = z the events ll ' z S df df and l ' l z S df l ' l are ndeendent. Then μ has the followng for: df df df μ P{ z S } P l ' l df l z S l ' { } f z dz. Therefore df df df ' ' z S P l l df z S l l F x z f x dx. R z P 3 The r.v. has the followng uulatve dstrbuton and robablt denst funtons: F z P{ z} P{ f o S o z} P{ z f x z f x dx. o S o z} 4 Note that the events z S df ' df and l ' l z S df l ' l df ll are equal robable. Therefore we an wrte the forula for μ alulaton n the followng wa: μ R z f z dz SPEIFI ASES Ths seton nludes soe exales when we llustrate the suggested aroah on well known dstrbutons. Exale: Exonental dstrbuton Let s onsder now another exale when n our renewal roesses r.v. and of nterest have exonental dstrbuton wth araeters and orresondngl. Then the dstrbuton funton F of the su wth araeters and df j fro forula has Erlang dstrbuton. The dstrbuton funton F of the other su S fro forula has also Erlang dstrbuton wth araeters and df j. We also defne

6 wth letter G wth orresondng ndes an addtonal funton to orresondng dstrbuton funton: G= Fx. Let us onsder the ntegral fro 3 n the followng wa: ax ax T T f G R where T s the underntegral exresson. All nteredate roofs and alulus for those two ntegral arts are gven n the []. Fnall we have: ax!!! ax G e F R 7 where x G - addtonal funton for Erlang dstrbuton funton wth araeters. Now onsder the robablt denst funton fro 4 for the r.v. S j o j o where oj and oj S have Erlang dstrbuton wth araeters and orresondngl. Then ax!!! G e f 8 where x G orresonds to addtonal funton for Erlang dstrbuton wth araeters and. The roof for forula 8 are gven n the []. Then usng forula 5 we an fnd the neessar xed oent. 4. LASSIAL APPROAH The lassal aroah to the estaton of the robablt of nterest s a araetral one. It suoses the ont estaton of the araeters of the dstrbuton f we know the dstrbuton te of the ntal sales H ={ }. We ntend to estate the araeters of the known tes of dstrbutons. Exale: Exonental dstrbuton Let s onsder an exale when r.v. and have exonental dstrbuton wth araeters and orresondngl. The su of exonentall dstrbuted r.v. has Erlang dstrbuton. The robablt of nterest } { S P Θ notaton fro forula now s: Θ e e f G!!!. 9

7 The lassal aroah suoses usng the ont estators nstead of the values of and : * * n n n S n. It gves the estator: Θ * * * * *! where. * * * * Now the exresson for varane of Θ s: V Θ E Θ E Θ and for the * * * ean square error ER Θ V Θ Θ E Θ. 5. NUMERIAL RESULTS onsder the ase when r.v. and have exonental dstrbuton wth araeters =.3 =.7. The real estators of robabltes of the shortage absene are resented n the Table the frst row of eah seton. Table Exerental results for real robabltes of shortage absene Θ lassal Θ * and Resalng Θ estators of Θ n =4 n =3 =6 =.3 =.7 n =4 n =3 =6 =.5 =.7 n = n =9 =4 =.3 =.7 K= K= K= K=3 Θ * B Θ * V Θ * ER Θ V Θ Θ * B Θ * V Θ * ER Θ V Θ Θ * B Θ * V Θ * ER Θ V Θ Let our sale szes be equal n and n and we onsder the -th unt s deand and dfferent ntal storage levels K=..3. All alulatons have erfored for r = realzatons n the ase of the resalng-aroah. We ntend to oare the varane of the estators of the resalng-aroah wth the ean square error of the lassal aroah. It s so beause of the resalng-aroah estators are unbased but the lassal ones on the ontrar have bas.

8 In the Table we an see the resalng-estators varane V Θ oarng wth * * lassal aroah estators varane V Θ bas B Θ and ean square error * ER Θ. The table shows how hanges the results deendng on dfferent sale szes n unt s nuber and ntal storage level K. Analzng table s results we an draw the onluson that the varane and orresondng ean square error of both aroahes dereases wth the nreasng of sale szes n and ntal storage level K. The varane of the resalng-estators s alost alwas saller than tradtonal one. However the resalng-estators are unbased. Takng as the rteron the ean square error resalng gves even better results for bg values of K. ONLUSION The resalng-aroah an be suessfull used for obtanng the estators of araeters of nterest of the renewal roesses. Obtaned forulas allow alulatng the varane of the estators for the resalng and lassal aroahes. Nueral exales show the effen of suggested aroah takng estators ean square error as effen rteron. Ths aroah an be the good alternatve to tradtonal one. AKNOWLEGEMENTS I would lke to thank sentf suervsor rofessor Alexander Andronov for hs useful valuable deas and oents durng ths aer rearaton. REFERENES [] AFANASEVA H. Resalng-aroah to a task of oarson of two renewal roesses.//in Proeedngs of the th Internatonal onferene on Analtal and Stohast Modellng Tehnques and Alatons RTU Rga [] ANRONOV A. MERKUREV u. Otzaton of statstal sale szes n sulaton. Journal of Statstal Plannng and Inferene. Vol [3] O.R. Renewal Theor. John Wle and Sons New ork 96.. [4] FIOSHIN M. Effen of Resalng Estators of Sequental Parallel Sstes Relablt. In Proeedngs of The Seond Internatonal onferene Sulaton Gang Tranng and Busness Proess Reengneerng n Oeratons. RTU Rga -7. [5] GENTLE E. Jaes. Eleents of outatonal statsts. Srnger-Verlag New ork. [6] ROSS Sheldon M. Aled Probablt Models wth Otzaton Alatons. over Publatons In. New ork 99. [7] WU.F.J. Jakknfe Bootstra and other resalng ethods n regresson analss. The Annals of Statsts Vol. 4 No