A new approach on the renewal process with Geometric interarrival times

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Arthur Jacobs
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1 Ierol Reserch Jourl of ppled d Bsc ceces 3 vlble ole wwwrbscom I 5-838X / Vol 4 (6): cece Eplorer Publcos ew pproch o he reewl process wh Geomerc errrvl mes H mm d Mohmmd Mohmmd -Deprme of scs Fculy of Mhemcl ceces Uversy of Gul Rsh Ir - Deprme of scs Fculy of Mhemcl ceces Uversy of Gul Rsh Ir Correspodg uhor eml: smm@gulcr BTRCT: Ths pper develops he probbly of reewl process whose errrvl re depede d declly dsrbued (d) rdom vrbles (RVs) wh geomerc dsrbuo The dsrbuo of he umber of reewls (] d he o dsrbuo of he umber of reewls (] d [b] < < b re obed The usg he o dsrbuo he dsrbuo of he umber of reewls [b] < < b s obed Flly we show h every reewl processes whose errrvls re d dscree RVs hs depede d sory creme f d oly f s errrvls be geomerc dsrbuo Keywords: Reewl process Idepede creme ory creme Geomerc dsrbuo reewl process { ITRODUCTIO : s oegve eger vlued sochsc process h regsers he successve occurrece of eve durg he me ervl ( ] where he me duro bewee cosecuve "eve" re posve d RVs Le he successve errrvl mes be T T d (Krl d Tylor 975) uppose s bouded Borel subse of ( ) of he elemes of he se { > : I he cse ( ] process re d epoel RVs he T for deoes he umber we deoe by I s well ow h f errrvl mes reewl hs posso dsrbuo wh prmeer λ m ( ) Lebesgue mesure I prculr for [ b] < < b ( b ) m deog he hs posso dsrbuo wh prmeer λ (Kgm 993) d (Vrse 8) I lerure dffere dsrbuos re cosdered s errrvl mes reewl processes (Cuffe d Fredm 6) obed he ec dsrbuo of whe errrvl mes re sums of wo depede epoel RVs wh lely uequl prmeers (R 9) suded reewl process wh Webul errrvl mes (Prze 999) d (Vrse d mm 9)ppled he Erlg dsrbuo s errrvl mes of reewl process However lmos ll of hem he dsrbuo of d s epeco ws rgued bu he dsrbuo of he cse Erlg errrvl me ws obed (Prze 999) I hs pper seco we ob he dsrbuo E [ b] b [ b] < < b ws o ob The dsrbuo of d ( ) < < by usg he o dsrbuo of ( ) d he we ob dsrbuo of seco 3 The seco 4 we dscusses produc mesure of dscree rdom vecor d flly seco 5 gves ecessry d eough codo for sory d depede creme

2 Il Res J ppl Bsc c Vol 4 (6) DITRIBUTIO OF We ssume h he errrvl mes T T reewl process re dscree RVs hvg P T pq geomerc dsrbuo wh probbly mss fuco (pmf) p + q < p < d cumulve dsrbuo fuco F ( ) q Lemm uppose Proof T for d F ( ) P ( ) he p F ( ) q + q () We shll he resul usg mhemcl duco d he -fold covoluo of F() for dscree RVs s gve by F ( ) F ( u ) P ( X ) I c be obed he pmf of I flows u ( ) ( ) ( + ) P P P F F + p q hs boml dsrbuo wh prmeers ( p ) 3 JOIT DITRIBUTIO OF ( ) for by () we hve I he reewl process le T T be d geomerc RVs wh prmeer p Le [ b] < < b To ob he dsrbuo of we frs clcule he o dsrbuo ( ) () Theorem I he reewl process he o dsrbuo of ( ) s equl o b b p + s P ( s ) q s + s b s q Proof (3) For dffere vlue of d s hve P P (4) ( ) ( b ) ( ) ( ) ( ) ( s s + ) ( ) ( + ) ( ) ( + + ) P P < < b < (5) P s P < < < b < s (6) P P < < b < (7) P P < < < b < (8) P s P < < < < b < s (9) + + s + s + The equly (4) s obed by equo () Hece he cse s prove he heorem he cse d s oher cse re obed smlrly Le he heorem holds ow we 43

3 Il Res J ppl Bsc c Vol 4 (6) X Y X Y + Y + X Y + Y + Y 3 + s 3 X Y + Y + Y + Y 4 + s where Y B ( p ) Y Geo ( p ) Y B ( s p ) d ~ depede ~ 3 ~ The vlue of equo (9) equls b 3 + p b s Y Geo p d Y Y Y 3 d Y 4 re 3 + s + 4 s q () 4 3 We c clcule he summo () d ob equo (3) for d s by usg he relos d b b + + s s Corollry The dsrbuo of 3 he reewl process s b p b P ( s ) q p q s q s E b p The epeco of + s b s ( b ) s () 4 ~ s obed by () we hve 4 CHRCTERIZTIO OF THE REEWL PROCE frs we revew some coceps bou produc mesure Le T d W re dscree rdom vecors o mesure spces ( X χ µ ) ( Y γ ϑ ) he (TW) hs dsrbuo π µ ϑ χ γ ( µ ϑ s produc mesure d χ γ s σ -feld h geered by he mesureble recgles) We hve π ( E ) π '( E ) π "( E ) π '( E ) ϑ y : ( y ) E µ { { : X ( E ) ( y ) E { y π " µ : ϑ { y : y Y Remr uppose B χ γ d χ he ( ) { : E χ γ where π Y B ϑ y : y B µ { () Remr If T d T re depede rdom vecors wh dsrbuo µ d ϑ χ d γ respecvely he ( ) { { : X (3) (( ) ) µ { χ γ χ (4) { : P (( T T ) B ) P T B µ B χ γ P ( T ( T T ) B )) P T B B ow le s reewl process chrcerzed by he fc h successve errrvl mes T T d oegve eger RVs Codo () re 433

4 Il Res J ppl Bsc c Vol 4 (6) For ech ω ω s oegve eger for ( ω) d lm furher for ech ω ( ω ) s fuco of s o-decresg d rgh-couous d he po of dscouy he sup ω s < s ω s ecly Codo () The T T re depede hvg geomerc dsrbuo wh prmeer p Codo () () For < < < < creme re depede (b) The cremes hve egve boml e s s P ( s ) p ( p) s < Theorem Codo () d () re equvle he presece of codo () Proof of () () F d cosder he eves h hppe fer me By m { : + < we hve ( ) ( ) ( ) T T T T T + + ( ) ( ) T T defe he wg mes followg By [ ] [ ] or s + m or + s + m f d oly f m ( ) ( ) + s m (5) d ( ) ( ) + + s whch s he sme hg s T + + T m s Thus m{ m : T + + T s (6) Hece [ m ] { T + + T s < T + + T (7) + s m m + uppose µ s mesure fuco of d y by usg (4) we hve P < > y P T > + y ( + + ) ( + ) y P ( T > + y ) µ { q P ( T > ) µ { + + { : { : y q P ( T + > ) By usg depede of he T T ( + > + > + > < + ) P y T y T y ( + + ) y y y + y y P > y < q q P < q q q If H ( y ) ( y ) Thus ( ) ( ) P ( T T ) H P P (( T T ) H ) (8) ow he eve [ s m < u ] c be pu he form [( T T ) H ] where m u + d H s he se of for whch + + s < + + u Thus equo (8) gves m m + ( + ) P m u P P ( m u ) s s 434

5 Il Res J ppl Bsc c Vol 4 (6) From hs follow by duco o h f < < < he P P ( ) (9) Thus codo () mples equo (9) Bu from equo (9) d equo () follow he wo prs of codo () Proof of ()() P T q If () holds he > so h T hs geomerc dsrbuo To fd he o dsrbuo of T d T suppose h s < < s < d perform he clcule ( < < ) P s s P ( ) s s s s s ( s) ( s ) ( s ) s q s pq q q p s ( q q ) p q {( y y ): s< y s < y y G [ y y : < y < y ] y P p q Thus for recgle coed he ope se µ ' (( ) ) (' s mesure fuco of ( ) ) { y y :( y y ) By cluso ecluso hs holds for fe uos of such recgle d hece by usg couy from below for couble oes Therefore holds for G G ' f G ' s ope ce he ope ses form s - p q o G By smlr rgume y sysem geerg he Borel ses ( ) hs mss y ( ) hs mss p q o [ : < < < ] If ler rsformo g ( y ) y y y s defed by y y he T T g ( ) hs mss y ( pq ) Ths proves codo () COCLUIO I hs pper we cosder he reewl process whose errrvl mes re d geomerc dsrbuo We show h he reewl processes hs depede d sory cremes hs boml dsrbuo by ew mehod lso we show f reewl process hs wh dscree eger errrvls hs depede d sory cremes The s errrvls me hs geomerc dsrbuo REFERECE Krl Tylor HM (975) Frs Course ochsc Processes cdemc Press Kgm JFC(993) Posso Processes Oford sudes Probbly Vrse (8) O he Posso process Fr Es Jourl of Theorecl scs Cuffe BP Fredm MF (6) O he ec dsrbuo of delyed reewl process wh epoel sum ervl mes Europe Jourl of operol reserch 7: Re H (9) The Webull Dsrbuo Tylor & Frcs Group LLC 9 Vrse mm H (9) ome resuls o reewl process wh Erlg errrvl mes pplcos d ppled Mhemcs: Ierol Jourl 4: Prze E (999) ochsc processes IM's clsscs ppled Mh 435

REVISTA INVESTIGACION OPERACIONAL Vol. 25, No. 1, 2004. k n ),

REVISTA INVESTIGACION OPERACIONAL Vol 25, No, 24 RECURRENCE AND DIRECT FORMULAS FOR TE AL & LA NUMBERS Eduardo Pza Volo Cero de Ivesgacó e Maemáca Pura y Aplcada (CIMPA), Uversdad de Cosa Rca ABSTRACT