Reliability Model of Tracking, Telemetry, Command and Communication System using Markov Approach

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1 Internatona Journa of Perforabty Engneerng, Vo. 8, No., Juy,, pp. -6. RAMS Consutants Prnted n Inda Reabty Mode of Trackng, Teeetry, Coand and Councaton Syste usng Markov Approach HUA YAN and XIAO-YUE WU Coege of Inforaton Systes and Manageent, Natona Unversty of Defense Technoogy, Changsha, Chna (Receved on January, ; Revsed on May, ) Abstract: For the reabty anayss of trackng, teeetry and coand (TT&C) and councaton systes, ost exstng odeng ethods can ony dea wth genera TT&C and councaton s. In ths paper, a fora descrpton of TT&C and councaton s gven to factate the reabty odeng of such systes. A contnuous-te Markov chan (CTMC) ode s but for an de. A ode for TT&C and councaton s n consecutve fght cyces s proposed, n whch the s are cobned to a new copcated one. Exapes wth nuerca resuts show the effectveness of the proposed approach.. Introducton Keywords: TT&C and Councaton syste, reabty ode, Markov process. The TT&C and councaton syste pays very portant roes n space fght actvtes. The an s of t are trackng, surveyng, ontorng and controng the spacecraft [], and councaton, such as voce, teevson and data transsson []. Therefore, ts syste reabty s essenta to the success of aunchng, otonng and andng of spacecraft. Varous technques have been deveoped for the reabty anayss of TT&C and councaton systes: Woan [3] constructed a reabty ode of space systes whch s defned n a set-theoretc fraework; Bondava et a. [] proposed a odeng approach based on a herhca ethodoogy to sove the Rosetta space appcaton provded by Matra Mon Space. Essentay, the TT&C and councaton can be regarded as a phased-sson syste (PMS). For exape, the has dverse types and ut-phase. PMS has been wdey studed n terature [-8]. The odeng and anayzng ethods n terature ay be roughy cassfed nto two ajor groups: cobnatora ethods and Markov ethods. Cobnatora ethods ncude reabty bock dagra (RBD) [, 9] and faut tree anayss [, ]. These ethods are straghtforward, effcent, but ted to hande non-reparabe syste. However, the ethods based on Markov process can correcty treat dependency between dfferent coponents and two successve phases. Generay, the TT&C and councaton s not an ordnary knd of PMS. For exape, t ncudes phases referred to as de s, n whch no s peented. In ths paper, by defnng concepts and notatons of the TT&C and councaton syste, odes are but for de s s n consecutve fght cyces.. Fora Descrptons of TT&C and Councaton Task Soe reevant concepts and notatons of the TT&C and councaton syste are gven as foows. * Correspondng author s ea: yanhua_83@63.co

2 H. Yan and X.-Y. Wu Fght Phase: the spacecraft fght process can be dvded nto severa phases. Let fp represent the th fght phase. Fght Cyce: the perod of spacecraft orbtng around the earth s caed a fght cyce. The TT&C and councaton syste shoud perfor varous s durng each fght j,* cyce to support space sson. Let fc represent the j th fght cyce, n whch * = key s the crtca fght cyce, and * = Nonkey s the non-crtca one. Resources Set of a Task: a needs a nuber of TT&C and councaton resources to peent t. Resources ncude statons, centers, shps, obe statons, and factes or sub-systes. If a resource has the abty to peent TT&C and councaton, t ust see the spacecraft. Due to the restrcton of orbt vsbty, a resource can see a spacecraft ony wthn soe speca te perods, and dfferent resources have dfferent vsbe te perods. It eans that for the sae, dfferent resources ay have dfferent peentng te perods. The resource set of a s denoted as Resources, eeent n Resources s defned as resk = { rnae, rst, ret}, where, rnae s the nae of resource, rst and ret are the start te and the end te respectvey for peentng. For exape, Resources = {( r, t, t ),( r, t3, t )} eans resource r and r peent a, and the start te and end te are t, t, and t 3, t respectvey. TT&C and Councaton Task (TTCCT): s peented by the TT&C and councaton syste have utpe types, and they are perfored durng a certan fght cyce of a fght phase. The TTCCT can be defned as j,* = { type, fp, fc, tst, tet, Resources}, where, type s the type of, fp j,* represents that s perfored durng the th fght phase, fc represents that s perfored durng the j th fght cyce, tst and tet are the start te and the end te of respectvey, and Resources s the resource set of. Task Arc: as dscussed above, resources n a ay have dfferent peentng te perods. Therefore, the syste confguraton and success crteron of TTCCT ay change over te. In order to anayze the reabty of TTCCT, t shoud be transfored nto a sequence of snge phase systes. Each snge phase syste s caed a, and the th denoted as. Let Arcs( ) be the set of a s n. The syste topoogy n each can be expressed by a RBD or faut tree. Ide Task Arc: a specfc, n whch a resources are de (runnng but not peentng any ). The technque of handng de s dfferent fro that of genera, as w be dscussed ater. 3. Reabty Mode of TT&C and Councaton Task Based on Markov Process The an assuptons for the ode n ths paper are ade as foows: () The TT&C and councaton resources have two states: good or faure, ther faure and repar tes are ndependenty exponentay dstrbuted. () Repar s perfect, and aways restores the resource to as-good-as-new. (3) The transton te between any two successve s s nstantaneous. () If a TTCCT fas n one, the entre s faed.

3 Reabty Mode of TT&C and Councaton Syste usng Markov Approach 3 3. Reabty Mode of TTCCT wth Ide Task Arc Assue there s a TTCCT, ts fght phase and fght cyce are fp and,key fc respectvey. Then, the reabty ode of can be represented as foows:, = { type, fp, fc key, t, t, Resources} () Resources( ) = { res, res, res, res, res } () 3 res = { r, t, t }, res = { r, t, t }, res = { r, t, t }, res = { r, t, t }, res = { r, t, t } (3) Fgure : Te sequence of resources n Fgure 3: Markov ode of n Fgure : Reabty bock dagra of non-de n Assue that the resources r r are factes of TT&C center. Te sequence of resources n s shown n Fgure. can be dvded nto three s, that s Arcs( ) = {,, }, where s an de. Fgure shows the RBD of each non-de. The RBD represents the syste ogc structure. Fro Fgure we can see that the structura confguraton changes wth te. Markov ode for ordnary has been consdered n severa terature [6]. For ordnary, the faure states of syste are regarded as absorbng states as the w fa once the syste enterng nto these states. However, n the de nterva, the syste s not requred to accopsh any, so there s no absorbng state. At the sae te, the coponent faure and repar actvtes are perfored noray as coponents are st runnng durng the de te nterva. In Fgure, s de, assue the faure rate and repar rate of r are λ and Fgure 3, where, the states are denoted as a bnary strng resource r k, t take vaue f r k s good and otherwse. 3. Reabty Mode of TTCCTs n Mut-Fght Cyces µ, the Markov ode of s gven n c c... c n, c k s the state of It s necessary to cobne the TTCCTs, of the sae type, n ut-fght cyces, to a new copcated one to cacuate ts reabty. The rues of cobnng TTCCTs of sae

4 H. Yan and X.-Y. Wu type are as foows: () the type and fght phase keep consstent wth orgna TTCCTs; () the fght cyce contans a cyces of orgna ones; () the start te and the end te are the earest start te and the atest end te of orgna ones respectvey; (v) the resources set s the unon of orgna TTCCTs resources set. For exape, for two TTCCTs 3,, = { typea, fp, fc key, t, t, Resources} () 3 Resouces( ) = {( r, t, t ),( r, t, t ),( r, t, t )} () = { typea, fp, fc, t, t, Resources} (6), key 3 6 Resouces( ) = {( r, t, t ),( r, t, t ),( r, t, t )} (7) Let t < t < t 3 < t < t < t 6, the cobned new can be represented as: (, key ),(, key ) = { typea, fp, fc, t, t, Resources} (8) new Resouces( new ) = {( r, t, t3 ),( r, t, t ),( r3, t, t3),( r, t, t6 ),( r, t, t),( r3, t, t6 )} (9) As shown above, the resources set of new has eeents wth sae resources nae. For exape, naes of resources n ( r, t, t 3) and ( r, t, t 6 ) are sae. However, they are consdered as dfferent n the ode, for ther start te and end te are dstnct. Accordng to the TTCCT reabty ode, the s for are Arcs( ) = {,...,,...}. Our approach adopted s to construct a contnuous-te dscrete-state Markov ode for every, then sove each snge Markov ode by gettng ts nta condtons n such a way: the nta states probabty for + s set equa to the fna states probabty of. Then, the reabty of ast s the reabty of. Exapes. In the foowng exapes, assue that the ean te between faures (MTBF) and the ean te to repar (MTTR) of each resource are 3n and n respectvey. We use the nuerca ethod ntroduced n [] to sove the Markov odes of each.. Ide Task Arc For, the RBD of each s shown n Fgure. Three s are, and. Assue the duraton of each s n, 3n and n respectvey. s de. If we deete fro, we get a new TTCCT. can be descrbed as foows: 3, key = {orbt easureent, fp, fc,,, Resources} () Resources( ) = {( r,,),( r,,), ( r,,),( r,,), () ( r,,), ( r,,),( r,,)} 6 3 Arc( ) = {,, } (), key {orbt easureent,, } = fp, fc,, Resources (3) Resources( ) = {( r,,),( r,,),( r,,),( r,, ),( r,,)} () 3 Arc( ) = {, } ()

5 Reabty Mode of TT&C and Councaton Syste usng Markov Approach The resutng reabty of each s shown n Tabe. We can see fro Tabe that the reabty of s ower than that of, because of s de, and there are st runnng resources durng the de. Tabe : Reabty of Task s of, reabty reabty reabty TTCCTs n Mut-Fght Cyce For, they beong to two consecutve fght cyces. Let the aount of te between be n. If we focus on the reabty of consderng as a snge TTCCT, then we can cobne the and get a new. Accordng to the cobnaton rue gven n secton 3., can be descrbed as foows: (, key ),(3, key ) = {orbt easureent, fp, fc,,3, Resources} (6) Resources( ) = {( r,, ),( r,, ),( r,,),( r,,),( r,,),( r,7,8), 3 ( r,7,8),( r,,3), ( r,,3),( r,7,8),( r,,3),( r,,3)} 3 3 (7) Arc( ) = {,,,,, } (8),key of s an de because t s the nterva process between fc and 3,key fc, and durng ths te there s no requred to be perfored. and of are actuay the two s n, and 3, and of are correspondng to the three s of. The resutng reabty of are shown n Tabe I. Coparng the reabty of s of, n Tabe I, We can see that after the de, the resutng reabty of s decned evdenty. As the nterva between two s n consecutve fght cyces s generay a ong te, restartng the resources durng ths te to restore the to as-good-as-new s portant for provng the reabty.. Concusons In ths paper, we study the reabty odeng of the TT&C and councaton. Accordng to the exape resuts, athough an de contans no absorbng state, t affects the reabty especay when ts astng te s ong enough. If the resources can be restarted durng the de, the reabty w be proved. But n ths stuaton, the preparng te after resource openng and the resource cose te ust be consdered n odes. Therefore, future reseh work s needed to dea wth resource restartng stuaton. Acknowedgent The authors woud ke to thank the revewers for ther coents and suggestons that greaty heped the quaty of ths paper. Ths work has been supported by NSFC (Grant No.

6 6 H. Yan and X.-Y. Wu 779). References [] Yu, Z.J. Status Quo and Deveopent of Spacefght TT&C Systes. Engneerng Scence 6; : -6. [] Xa, N.Y. Space TT&C Syste, Bejng: Natona Defense Industry Press,. [3] Woan, W. A Mode for the Reabty Estaton of Space Systes. IEEE Transactons on Reabty 963; R-(): [] Bondava, A., I. Mura and M. Ne. Anaytca Modeng and Evauaton of GUARDS nstances: exape for Space Appcatons. LAAS-CNRS ESPRIT Project 76:GUARDS Report D3A/AO/6/B 997. [] Esary, J.D. and H. Zehs. Reabty Anayss of Phased Mssons. Proceedngs of the Conference on Reabty and Faut Tree Anayss: SIAM 97; Phadepha, USA: [6] Aa, M. and U.M. A-Saggaf. Quanttatve Reabty Evauaton of Reparabe Phased-Msson Systes Usng Markov Approach. IEEE Transactons on Reabty 986; R-3(): [7] K, K. and K.S. Park. Phased-Msson Syste Reabty under Markov Envronent. IEEE Transactons on Reabty 99; 3(): [8] Wang, D.Z. and K.S. Trved. Reabty Anayss of Phased-Msson Syste wth Independent Coponent Repars. IEEE Transactons on Reabty 7; 6(3): -. [9] Dugan, J. B. Autoated Anayss of Phased-Msson Reabty. IEEE Transactons on Reabty 99; (): -. [] Zang, X.Y., H.R. Sun and K.S. Trved. A BDD-Based Agorth for Reabty Anayss of Phased-Msson Systes. IEEE Transactons on Reabty 999; 8(): -6. [] Xng, L.D. and J.B. Dugan. Anayss of Generazed Phased-Msson Syste Reabty, Perforance, and Senstvty. IEEE Transactons on Reabty ; (): 99-. [] Wu, X.Y. Nuerca Method for Reabty Anayss of Phased-Msson Syste Usng Markov Chans. Proceedngs of the 7th Internatona Conference on Matheatca Methods n Reabty: theory, Methods, Appcatons, Bejng, Chna: Hua Yan s a Ph.D. student of the Coege of Inforaton Syste and Manageent, Natona Unversty of Defense Technoogy, Changsha, Chna. Hs reseh nterests focus on syste reabty odeng and anayss. Xao-Yue Wu receved hs Ph.D. degree n Manageent Scence and Engneerng fro the Natona Unversty of Defense Technoogy n. He s a professor of the Coege of Inforaton Syste and Manageent, Natona Unversty of Defense Technoogy, Changsha, Chna. Hs reseh nterests ncude syste reabty odeng and anayss, test and evauaton, and decson anayss.