Fuzzy Erlangian Queuing System with State Dependent Service Rate, Balking, Reneging and Retention of Reneged customers

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1 Iteatioa Joa of Basic & Aied Scieces IJBAS-IJENS Vo:4 No: 6 Fzzy Eagia Qeig System with State Deedet Sevice Rate Bakig Reegig ad Retetio of Reeged cstomes MS E Paomy Deatmet of Statistics Facty of Commece A-Azha Uivesity Gis' Bach Dkhia Egyt Deatmet of Mathematics Facty of Sciece Sama bi Abdaziz Uivesity A-Khaj Sadi Aabia E-mai: dmahdy_eaomy@yahoocom Abstact-- The aim of this ae is to deive the aaytica sotio of fzzy tcated Eagia sevice ee with state-deedet ate bakig eegig ad etetio of eeged cstomes FM / FE // N ( ) We obtai the obabiities that thee ae "" its i the system ad the it i the sevice occies stage "s " (s = s ) We teat this ee fo geea vaes of k ad N Idex Tem-- Fzzy ee; Membeshi fctio; Eagia sevice ee; Retetio of eeged cstomes - INTRODUCTION This ae cosides the eig system FM / FE // N with state deedet sevice ate bakig eegig ad etetio of eeged cstomes cocets The Eag distibtio deoted by E is a secia case of the gamma distibtio is amed afte AK Eag who ioeeed eeig system theoy fo its aicatio to cogestio i teehoe etwoks The o-tcated ee: / / M E was soved by Mose [4] at = ad white et a [6] Who obtaied the sotio i the fom of a geeatig fctio ad the obabiities cod be obtaied by a owe seies exasio Ritha ad Seeekha[5] teated Fzzy N- Poicy ees with ifiite caacity A Seedy [] gave a aaytica sotio of the ee: / // M E N with bakig oy This wok had bee foowed by Kotb [] who stdied the aaytica sotio of the state-deedet Eagia ee: M / E // N with bakig by sig a vey sef emma E- aomy[ ] stdied the same system withot etetio eeged ad fzzy cocets I this ae we teat the aaytica sotio of the ee: / // FM FE N fo fiite caacity cosideig by sig a ecece eatios We obtai s the obabiities that thee ae " " its i the system ad the it i sevice occies stage " s " ( s ) i tems of P We coside etetio of eeged cstomes that is the eeged cstome may eave the ee withot gettig sevice with obabiity ad may emai i the ee fo his sevice with obabiity ( ) (FIFO) The obabiity of a emty system P is aso obtaied The disciie cosideed is fist i fist ot - THE PROBEM ANAYSIS Coside the sige chae sevice time Eagia ee havig sevice stages each with ate with the state deedet The mea sevice ate is give by: ( ) k ( ) k N whee is the ate of time t havig the ( d f ) give by: Ai 4 IJENS

2 Iteatioa Joa of Basic & Aied Scieces IJBAS-IJENS Vo:4 No: 7 t f t e t This meas that the its ae seved with two diffeet ates o deedig o the mbe of its i the system whethe k o k N esectivey Aso coside a exoetia ite-aiva atte with ate baks (does ot ete the ee) whee : {a it jois the ee} N; Fo it is cea that: Assme be the obabiity that a it N N Assme the obabiities: P s = { its i the system ad the it i sevice beig i stage s } whee: N s P = obabiity of a emty system ie the day obabiity The steady state diffeece eatios ae: P P () P s P s s s P s P s s s s s () P P s k () P P P s k Pk s Pk s k Pk s s k Pk Pk k Pk s P s P s P s s P P P s k (4) k N (5) N P s P s N P s s N (6) N PN PN s Smmig () ove s ad sig () gives P P (7) s s Ai 4 IJENS

3 Iteatioa Joa of Basic & Aied Scieces IJBAS-IJENS Vo:4 No: 8 Smmig () ove s sig (7) ad addig the ests obtaiig fo k eads to: P P k (8) s s Simiay smmig (4) ove s ad sig (8) at =k yieds P P k (9) k k s k s Smmig (5) ove s ad sig (9): s s P P k N () Fom eatio oe ca easiy show that P P Makig se of eatio () yieds s s P P s () Uo sig the fist eatio of () ad (8) we get the ecece eatio i s s P s P i P i k i i Aso fom the fist Eatio of (5) ad () we obtai () i s s P s P i P i k N i i Fiay sig eatio (6) ad eatio () at =N gives: () PN s N PN i s (4) whee: is k k N Eatios () (4) ae the eied ecece eatios that give a obabiities i tems of P which it-sef may ow be detemied by sig the omaizig coditio: s s P P (5) Hece a the obabiities ae cometey kow i tems of the ee aametes Ai 4 IJENS

4 Iteatioa Joa of Basic & Aied Scieces IJBAS-IJENS Vo:4 No: 9 - EXAMPE The foowig exame istates the theoetica ests I the system: M / E // N with state-deedet M / E // 4 ) bakig eegig ad etetio eeged et k = = ad N = 4 ( ie the ee i the eatios () (5) the ests ae : whee: P a P P a P P a P P b P P b P P b P P c P P c P P c P P d P P d P P d P a a a b a a a b a a a a b a a a a a c b b b c b b b b c b b b b b d c c c d c c d c Fom the omaizig coditio: P P P P P s s s 4 s s s s s we have: P a a a b b b c c c c d d Theefoe the exected mbes i the system ad i the ee ae esectivey Ai 4 IJENS

5 Iteatioa Joa of Basic & Aied Scieces IJBAS-IJENS Vo:4 No: 4 Ps a a a b b b c c c 4d d d P s 4 s P s b b b c c c d d d P Aso the exected waitig time i Kotb the system ad the ee ae obtaied as foows: W W ad whee is the mea ate of its actay eteig the system 4 FUZZY SYSTEM We coside a etetio of eegig of cstomes eig mode with fiite caacity i which aivig cstomes foow a Poisso ocess with a fzzy aiva ate ad sevice times ae Eagia with a fzzy sevice ate ad ae imecise ad cetai If ad ae defied by tiaga fzzy mbes sch that : [ ]; [ ] [ ] ad [ ] Whee: ad The membeshi fctio of ( ) ad ( ) ae defied as foows if if ( ) if if ; if if ( ) if if Usig the cocet of -ct method ad the oeatios o tiaga fzzy mbes to fid fzzy obabiities ad fzzy efomace meases [ ( ) ( )]; [ ( ) ( )] ( ) [ ] ( ( )) ( ( ) ) ( ) [ ] ( ( )) ( ( ) ) [ ] ( ) ( ) ( ) ( ) ( ( )) ( ( ) ) ( ) [ 4] ( ( )) ( ( ) ) ( ) Ai 4 IJENS

6 Iteatioa Joa of Basic & Aied Scieces IJBAS-IJENS Vo:4 No: P a P P a P P a P P b P P b P P b P P c P P c P P c P P d P P d P P d P P a a a b b b c c c c d d a a a b b b c c c 4 d d d [ ] a a a b b b c c c d d d b b b c c c d d d [ ] a a a b b b c c c d d d Whee: Ad a a a b b b c c c 4 d d d a a a b b b c c c d d d a a a b b b c c c 4 d d d a a a b b b c c c d d d b b b c c c d d d a a a b b b c c c d d d b b b c c c d d d a a a b b b c c c d d d Aso as befoe we fid : Whee: ad b b b c c c d d d a a a b b b c c c d d d b b b c c c d d d a a a b b b c c c d d d Ai 4 IJENS

7 Iteatioa Joa of Basic & Aied Scieces IJBAS-IJENS Vo:4 No: Fom Fzzy itte`s foma we get: W W ad 5- SPECIA CASES Case : et ad k k N withot fzzy cocets O ests agee with the ests of A-Seedy [] Case : Rests of both [] ad [] has bee obtaied by ettig i the eatios () (4) i o ests Case : Rests of [6] ca be obtaied by ettig k N ad i o ests ad Theefoe the eig mode withot fzzy cocet stdied i this ae edces to M / E // N with state-deedet sevice ate bakig ad eegig as i [] Case 4 Whe thee is o cstome etetio 6- CONCUSION I this ae the tcated Eagia sevice ee is stdied with state-deedet bakig eegig etetio of eeged cstomes ad fzzy aamete The ecece eatios that gave a the obabiities i tems of P ae deived We istate the method by a exame is give to obtai some efomace meases sch as ad REFERENCES [] A- Seedy RO (988): " The Geea Sotios of the Tcated No Makovia Qees "PhD Thesis Facty of Sciece Tata ivesity Egyt [] E-aomy M S (8):" O a tcated Eagia eig system with State-Deedet Sevice Rate Bakig ad Reegig" Aied Mathematica Sciece vo o [] Kotb KAM (994) : " State-Deedet Qees " Ph D Thesis Facty of Sciece Tata ivesity Egyt [4] Mose PM (958) " Qees Ivetoies ad Maiteace " Joh Wiey New yok [5] Ritha W ad Seeekha B ():" Fzzy N Poicy Qees with Ifiite Caacity" Joa of hysica scices vo [6] White JA ; Schmidt JW ad Beett G K (975) " Aaysis of Qeeig System " Academic Pess New Yok Ai 4 IJENS