The M/M/c/N/K Interdependent Queueing Model with Controllable Arrival Rates and Reverse Balking

Transcription

1 ISSOlie : ISS it : Iteatioal Joual of Iovative eseah i Siee Egieeig ad Tehology A ISO 397: 7 Cetified Ogaizatio Vol. 5 Issue 5 May 6 The M/M/// Itedeedet Queueig Model with Cotollable Aival ates ad evese Balkig Sasikala.S Thiagaaja.M Assistat ofesso Cauvey College fo WomeTiuhiaalli Idia Assoiate ofesso St. Joseh s College Autoomous Tiuhiaalli Idia ABSTACT: I this ae a M/M/// itedeedet queueig model with otollable aival ates ad evese balkig is osideed. The fiite soue of ustomes aive i a oisso oesses with two aival ates -a faste ad slowe ate of aivals. The sevie times follow exoetial distibutio with aamete. This model is muh useful i aalysig the atiula situatios aisig at the laes like a data voie tasmissio omute ommuiatio system et. The steady state solutio ad the system haateistis ae deived ad aalysed fo this model. umeially esults ae give fo bette udestadig. EWODS: Fiite Caaity Fiite Soue Itedeedet Cotollable Aival ad Sevie ates evese balkig - Seve Bivaiate oisso oess. I. ITODUCTIO I eal atie it is ofte likely that a aival beome disouaged whe queue is log ad may ot wish to ete the queue.this tye of aival is alled balkig. The otio of ustome balkig aeas i queuig theoy i the woks of Haight []. He has aalysed M/M/ queue with balkig i whih queue legth is ifiite. Jai ad akesh uma [] have studied M/M// queueig system with evese Balkiga queueig system that idiates the obability of balkig will be low whe the queue size is moe. Alog with seveal othe assumtios it is ustomay to oside that the aival ad sevie oesses ae ideedet. Howeve i may atiula situatios it is eessay to oside that the aival ad sevies oesses ae ite deedet. A queueig model i whih aivals ad sevies ae oelated is kow as itedeedet queuig Model. Muh wok has bee eoted i the liteatue egadig itedeedet stadad queuig model with otollable aival ates..siivasa aoshobha ad.siivasa ao [3] have disussed M/M// itedeedet queuig model with otollable aival ates. A.Siivasa ad M. Thiagaaja [4567] have aalysed M/M// itedeedet queuig model with otollable aival ates M/M// itedeedet queuig Model with otollable aival ates M/M/// itedeedet queuig Model with otollable aival ates balkig eegig ad saes ad have aalysed M/M// / loss ad delay queueig system with itedeedet queuig Model with otollable aival ates ad o assig. B.Atlie isha ad M.Thiagaaja [8] have disussed M/M/// itedeedet etial queuig Model with otollable aival ates. eetly S.Sasikala ad M.Thiagaaja [9] have studied the M/M// itedeedet queuig Model with otollable aival ates ad evese balkig. I this ae a M/M/// itedeedet queueig model with otollable aival ates is osideed with the assumtio that the aival ad sevie oesses of the system ae oelated ad follows a bivaiate oisso oess. Hee the aival ate is osideed as -a faste ate of aival ad a slowe ate of aival. Wheeve the queue size eahes a etai esibed umbe the aival ate edues fom to ad it otiues with that ate as log as the otet i the queue is geate tha some othe esibed itege & <.Whe the otet Coyight to IJISET DOI:.568/IJISET

2 ISSOlie : ISS it : Iteatioal Joual of Iovative eseah i Siee Egieeig ad Tehology A ISO 397: 7 Cetified Ogaizatio Vol. 5 Issue 5 May 6 eahes the aival ate hages bak to ad the same oess is eeated. I setio the desitio of the model is give statig the elevat ostulates. I setio 3 the steady state equatios ae obtaied. I setio 4 the haateistis of the model ae deived I setio5 the aalytial esults ae umeially illustated. II.DESCITIO OF THE MODE a. Coside a -seve fiite aaity fiite soue queuig system i whih the ustomes aive aodig to the oisso flow of ates ad ad the sevie times ae exoetially distibuted with ate µ. It is assumed that the aival oess [X t] ad the sevie oess [X t] of the system ae oelated ad follows a bivaiate oisso oess havig the joit obability mass futio of the fom X x X... with aametes x ; t e mi x x t i j x =...< j x j t [ i t] [ - t] j! x j! x j! x j whee x i ; mi i i ad as mea faste aival ate mea slowe aival ate mea sevie ate ad mea deedee ate ovaiae betwee the aival ad sevie oess esetively. ; ; b. The aaity of the system is fiite.. The soue of queueig system is fiite. d. The queue disilie is Fist ome Fist seve. e. Whe the system is emty a ustome may balk with obability = -q. q ad may ete with obability f. Whe thee is at least oe ustome i the system the ustomes balk with a obability system with obability.. ad joi the The ostulates of the model ae i The obability that thee is o aival with evese balkig ad o sevie omletio duig a small iteval of time h whe the system is i faste ate of aivals is ' h o h ii iii The obability that thee is oe aival with evese balkig ad o sevie omletio duig a small iteval of time h whe the system is i faste ate of aivals is h o h The obability that thee is o aival with evese balkig ad o sevie omletio duig a small iteval of time h at state whe the system is i faste ate of aival is ' h o h Coyight to IJISET DOI:.568/IJISET

3 ISSOlie : ISS it : Iteatioal Joual of Iovative eseah i Siee Egieeig ad Tehology A ISO 397: 7 Cetified Ogaizatio Vol. 5 Issue 5 May 6 iv v vi vii The obability that thee is o aival with evese balkig ad o sevie omletio duig a small iteval of time h at state whe the system is i faste ate of aival is ' h o h The obability that thee is o aival with evese balkig ad o sevie omletio duig a small iteval of time h at state whe the system is i slowe ate of aival is ' h o h The obability that thee is o aival with evese balkig ad oe sevie omletio duig a small iteval of time h state whe the system is eithe i faste o slowe ate of aivals is h o h The obability that thee is oe aival with evese balkig ad oe sevie omletio duig a small iteval of time h whe the system is eithe i faste o slowe ate of aivals is h oh III.THE STEAD STATE EQUATIOS et be the steady state obability that thee ae ustomes i the system whe the aival ate is ad be the steady state obability that thee ae ustomes i the system whe the aival ate is. We obseve that exists whe = ; both & exist whe = ad exists whe = Futhe = = if >+. With this deedee stutue the steady state equatios ae Coyight to IJISET DOI:.568/IJISET

4 ISSOlie : ISS it : Iteatioal Joual of Iovative eseah i Siee Egieeig ad Tehology A ISO 397: 7 Cetified Ogaizatio Vol. 5 Issue 5 May 6 Coyight to IJISET DOI:.568/IJISET

5 ISSOlie : ISS it : Iteatioal Joual of Iovative eseah i Siee Egieeig ad Tehology A ISO 397: 7 Cetified Ogaizatio Vol. 5 Issue 5 May 6 Coyight to IJISET DOI:.568/IJISET Fom 3. to 3.8 we get = !...!...! A 3.4 Fom 3.9 to 3.3 we get = A

6 ISSOlie : ISS it : Iteatioal Joual of Iovative eseah i Siee Egieeig ad Tehology A ISO 397: 7 Cetified Ogaizatio Vol. 5 Issue 5 May 6 Coyight to IJISET DOI:.568/IJISET Whee = if > =!!...!!!!! A The obability that the system is emty a be alulated fom the omalizig oditio A......!!

7 ISSOlie : ISS it : Iteatioal Joual of Iovative eseah i Siee Egieeig ad Tehology A ISO 397: 7 Cetified Ogaizatio Vol. 5 Issue 5 May 6 IV. CHAACTESTICS OF THE MODE The obability that the system is i faste ate of aival is Sie exists oly whe we get o 4. The obability that the system is i slowe ate of aival is 4. exists oly whe = we get Sie 4.3 The exeted umbe of ustomes i the system is give by s so s 4.4 whee so 4.5 ad s 4.6 Theefoe s 4.7 Fom 3.4 ad 3.5we get Coyight to IJISET DOI:.568/IJISET

8 ISSOlie : ISS it : Iteatioal Joual of Iovative eseah i Siee Egieeig ad Tehology A ISO 397: 7 Cetified Ogaizatio Vol. 5 Issue 5 May 6 Coyight to IJISET DOI:.568/IJISET !! A S Usig ittle s fomula the exeted waitig time of the ustomes i the system is give by s s W Whee This model iludes the atiula ases as whe = this model edues to M/M/// itedeedet queueig model with otollable aival ates ad evese balkig. Whe teds to ad = this model edues to M/M/// queueig system with evese balkig. V. UMEICA IUSTATIOS Fo vaious values of q while ae fixed values omuted ad tabulated the values of S S W ad

9 ISSOlie : ISS it : Iteatioal Joual of Iovative eseah i Siee Egieeig ad Tehology A ISO 397: 7 Cetified Ogaizatio Vol. 5 Issue 5 May 6 Table q X X X X X X X VI.COCUSIO The obsevatios made fom the table 5. ae.whe ieases keeig othe aametes fixed ad deease but ad S iease. It is also obseved that the exeted system size is zeo whe q is..whe ieases keeig othe aametes fixed ad deease but ad S iease. 3.Whe the mea deedee ate ieases ad the othe aametes ae ket fixed ad iease ad S deease. 4. Whe the balkig ate deeases ad othe aametes ae ket fixed ad deease s iease egulaly ad attais maximum whe q is zeo. S W S EFEECES [] Haight F.A. Queuig with balkig Biometika Vol 44 o [] Jai.akesh uma Bhuede kuma Som A M/M// Queuig system with evese Balkig Ameia Joual of Oeatio eseah Vol.4 o [3] Siivasa ao. Shobha.T The M/M/ itedeedet queuig Model with otollable aival ates Oeatioal eseah soiety of Idia. Vol. 37 o. [4] Siivasa.A ad Thiagaaja.MThe M/M// itedeedet queuig model with otollable aival ates Iteatioal Joual of Maagemet ad Systems Vol o [5] Siivasa.A ad Thiagaaja.M The M/M/C/ ideedet queuig model with otollable aival ates Bulletia of Calutta Mathematial Soiety [6] Siivasa.A ad Thiagaaja.M The M/M/C// ideedet queuig model with otollable aival ates balkig eegig ad saes Joual of Statistis ad Aliatios os [7] Siivasa.A ad Thiagaaja.M The M/M// / loss ad delay queueig system with itedeedet queuig Model with otollable aival ates ad o assig Joual of Alied Statistial Siee [8] Atile isha B. Ad Thiagaaja M.The M/M/// itedeedet etial queuig Model with otollable aival ates Iteatioal Joual of Mathematial Siees ad Egieeig AliatiosVol.8o.VI7-44. [9] Sasikala.S ad Thiagaaja M.The M/M// itedeedet queuig model with otollable aival ates ad evese balkig Asia Joual of Cuet Egieeig ad MathsJa-Feb-46. Coyight to IJISET DOI:.568/IJISET