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1 IJS 4 () July Sharma & al ehavour of Subytem of ommuncaton Satellte SOHSI HVIOU O OMMUNIION SUSYSM O OMMUNIION SLLI SK Mttal eepankar Sharma & Neelam Sharma 3 S he author n th paper have dcued the tochatc behavor of communcaton atellte for evaluaton of ome mportant relablty parameter One Standby control unt ha been taken to mprove ytem performance h Standby unt can take place through a perfect wtchng devce Supplementary varable technque ha been ued to convert a Non-Markovan proce n to Markovan one Steady-State behavor of the ytem ha obtaned ll the tranton State probablte n cae repar follow exponental tme dtrbuton have alo computed to mprove practcal utlty of the model Keyword: Supplementary varable erfect Swtchng Standby redundancy valablty and cot functon INOUION he heart of a atellte the communcaton ubytem It cont of multple tranponder whch a explaned earler receve and amplfy up-lnk gnal tranlate them n frequency and amplfy agan for retranmon a down-lnk gnal In the ngle converon tranponder (fg) only ngle frequency tranlaton proce take place from the receved gnal to the tranmtted gnal he author dvded the ytem nto fve ubytem namely (ntenna) (ecever) t (frequency tranlator) (tranmtter) and (two control unt n tandby) Laplace tranform and upplementary varable technque have been ued to olve and formulate of mathematcal model ll falure follow exponental tme dtrbuton where a all repar follow general tme dtrbuton Laplace tranform of varou tate probablte have been obtaned numercal example together wth t graphcal llutraton ha appended n lat to hghlght mportant reult of th tudy SSUMIONS he followng aumpton have been aocated wth th model: Intally the ytem work wth t full effcency epar faclte are alway avalable and after repar unt work lke new one 3 Swtchng devce ued to onlne tandby control unt perfect 4 alure are tattcally ndependent Nothng can fal from a faled tate 5 ll the falure and repar follow exponental and general tme dtrbuton repectvely ommuncaton Subytem ecver requency ranlator ranponder ranmtter ntenna Subytem erfect Swtchng evce ommuncaton ntenna ontrol unt g-: General block dagram of a atellte 8
2 IJS 4 () July Sharma & al ehavour of Subytem of ommuncaton Satellte (x) (m) ( y (y) ( x ( m (x) ( ) ( ) t ( x t (y) c ( y (z) (r) (z) (r) ( z ( r (x) ( r ( z ( n g-: State-ranton agram State: Good aled NOMNLU ollowng notaton have been ued throughout th tudy: ( ) : he probablty that at tme t the ytem n good tate of full effcency t ( j : he probablty that at tme t the ytem n faled tate due to falure of th ubytem and elaped repar tme le n the nterval j j where and j xy zrm xy zrn repectvely ( j) : he frt order probablty that th ubytem wll be repared n the tme nterval j j condtoned that t wa not repared up to the tme j 83
3 IJS 4 () July Sharma & al ehavour of Subytem of ommuncaton Satellte / / / / : alure rate of ntenna recever frequency tranlator tranmtter and whole / ytem due to envronmental reaon : alure rate of control unt I and II : Laplace tranform of functon ( M : t d d S = mean tme to repar S : jexp ( j) dj for j μ th unt 3 OMULION O MHMIL MOL: Ung contnuty argument we obtan the followng et of dfference-dfferental equaton whch are dcrete n pace and contnuou n tme governng the behavour of the model under conderaton: t t x x t dx y y t dy z z tdz r r tdr m m t n n t dn t t t t t x y z r m dm x x t y y t z z t r r t m m t () () (3) (4) (5) (6) 84
4 IJS 4 () July Sharma & al ehavour of Subytem of ommuncaton Satellte t n n n t (7) t t x x t dx y y t dy z z t r r tdr t t x t y t z t r x x t y y t z z t oundary condton are: t t t t t t t t r r t (8) (9) () () () (3) (4) (5) (6) t t t (7) t t t t t t t t t t (8) (9) () () () 85
5 IJS 4 () July Sharma & al ehavour of Subytem of ommuncaton Satellte Intal condton: and all other tate probablte at t are zero (3) 4 SOLUION O H MOL: akng Laplace tranform of equaton () through () by makng ue of ntal condton (3) and then on olvng them one by one we obtan the followng Laplace tranform of dfferent tate probablte: (4) m (5) (6) (7) (8) (9) (3) (3) (3) (33) (34) 86
6 IJS 4 () July Sharma & al ehavour of Subytem of ommuncaton Satellte (35) where S S S S (36) S S S S S S S (37) and j S j j for all and j (38) 5 GOI HVIOU O H SYSM: y makng ue of bel Lemma Vz lm lm t ay o t have the followng tme- ndependent tate probablte from equaton (4) through (35): provded the lmt on rght ext we (39) M (4) M (4) M (4) M (43) (ay) (44) M (45) 87
7 IJS 4 () July Sharma & al ehavour of Subytem of ommuncaton Satellte M M M M M where (46) (47) (48) (49) (5) (5) M S for all (5) d d (53) 6 SOM IUL SS: (a) when repar follow exponental tme dtrbuton Settng S j for all and j n equaton (4) through (35) one may obtan the followng tranton tate j probablte n th cae: (54) (55) (56) 88
8 IJS 4 () July Sharma & al ehavour of Subytem of ommuncaton Satellte 89 (57) (58) (59) (6) (6) (6) (63) (64) where (65) (66) and (67)
9 IJS 4 () July Sharma & al ehavour of Subytem of ommuncaton Satellte (b) valablty and proft functon for the ytem: We have up or up a b where a b On takng nvere Laplace tranform we obtan up t at at e e a b e a b bt (68) It nteretng to note that up lo t t down (69) gan the proft functon for the ytem gven by up t upt dt t 3 G t Where revenue cot per unt tme repar cot per unt tme and 3 ytem etablhment cot for one perod o here e a a b t 3 G t at e b bt a b (7) where a and b have mentoned earler (c) Numercal omputaton or a numercal computaton let u conder the data / unt tme / unt tme 3 5 / etup and t= y puttng thee value n equaton (68) (69) and (7) we plot varou graph hown n fgure (3) (4) and (5) 9
10 up( > IJS 4 () July Sharma & al ehavour of Subytem of ommuncaton Satellte to oberve the change n value of avalablty and proft functon able and 3 how the value of t t up down and G( repectvely 7 SULS N ISUSSION ll the tranton tate probablte n cae repar follow exponental tme dtrbuton have been computed to mprove practcal utlty of the model Steady-tate behavour of the ytem ha alo been obtaned One numercal computaton wth t graphcal llutraton ha mentoned n the end to hghlght mportant reult of tudy n examnaton of table - and fg-3 reveal that the avalablty of the ytem decreae catatrophcally n the begnnng but after t =4 t decreae n a contant manner lo fg-4 the graph roft functon v t and the correpondng value are gven n the table - crtcal examnaton of fg-4 yeld that proft functon negatve n tartng becaue ntally we nvet money to etablh the new ytem but G( become potve and t value tart ncreang up to t=6 and thereafter t agan decreae Maxmum value of G ( and t for t=6 t up t able- up( 8 6 up( t > g- 3 9
11 G( > IJS 4 () July Sharma & al ehavour of Subytem of ommuncaton Satellte G t t able- G( t > G g- 4 8 NS [] Satellte ommuncaton Sytem ngneerng rentce Hall Inc New Jerey 993 G Gordon and WL Morgan: [] rncple of ommuncaton Satellte John Wley & Son Inc NY pp [3] Hyder ; Joorel J S: Stochatc ehavor of a wo Unt old Standby edundant Sytem Subject to andom alure Mcroelectronc elab Vol 36() pp [4] Satellte ommuncaton Sytem John Wlley and Son ourth dton 3 757pp Satellte and atellte ubytem relablty: Stattcal data analy and modelng [5] ape V; akten : n Introducton to ontnuou-tme Stochatc rocee: heory Model and pplcaton" rkhauer ublcaton 4 [6] Sharma eepankar Sharma Jyot; tmaton of relablty parameter for telecommuncaton ytem Journal of combnatorc nformaton and ytem cence Vol9 No-4 pp5-6(5) [7] Sharma eepankar Geol K; Sharma Vnt: elablty and M evaluaton of telecommuncaton ytem ulletn of pure and appled Scence Vol4 () No pp (5) 9
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