Matrix Geometric Method for M/M/1 Queueing Model With And Without Breakdown ATM Machines

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1 Reearch Paper America Joural of Egieerig Reearch (AJER) 28 America Joural of Egieerig Reearch (AJER) e-issn: p-issn : Volume-7 Iue- pp Ope Acce Matrix Geometric Method for M/M/ Queueig Model ith Ad ithout Breakdow ATM Machie Shoukry E.M.* 2 Salwa M. Aar* 3 Bohra A. Shehata*. * Departmet of Mathematical Statitic Ititute of Statitical Studie ad Reearch Cairo Uiverity. Correpodig Author: Shoukry E.M ABSTRACT: The matrix geometric method i ued to derive the tatioary ditributio of the M/M/ ueueig model with breakdow uig the traitio tructure of it Markov Chai. The M/M/ ueueig model for the ATM machie uig the matrix geometric method will be itroduced. e tudy the tability of the two ytem with ad without breakdow to obtai expreio of the performace meaure for the two model. Fially a comparative tudy wa itroduced betwee the M/M/ ueueig model with ad without breakdow. Keyword: Queue ATM Breakdow Matrix geometric method Markov Chai Date of Submiio: Date of acceptace: I. INTRODUCTION Queueig theory ad related model are ued to approximate real ueueig ituatio o the ueueig behavior ca be aalyzed mathematically; oe of thee ueueig ituatio i the automated teller machie (ATM) which i oe of the everal electroic bakig chael. It i amog the mot importat ervice facilitie i the bakig idutry. I claical ueueig model it i commo to aume that the erver ha ot failed or breakdow. However i real life ituatio a ueueig ytem might uddely breakdow ad hece the erver will ot be able to cotiue providig ervice ule the ytem i repaired.the ATM ytem i oe of the ytem which alway eed ome maiteace it eed to repair if it breakdow ad it eed to recharge with cah if it ru out of moey ad thi cae called alo breakdow.server breakdow i a avoidable pheomeo i the ervice facility of ueueig ytem. Sigle erver ueueig model with breakdow have may applicatio for itace i maufacturig ytem computer ytem commuicatio ytem etwork productio ytem ad may other of ytem which might uddely breakdow.i recet year ueueig model with erver failure or breakdow have emerged a oe of the importat area of ueueig theory. Avi- Itzhak ad Naor [2] preeted five iteretig model of ueueig problem with ervice tatio ubject to breakdow. Numerou reearcher icludig Federgrue ad So [4] Takie ad Segupta [] Núñez-Queija [8] were itereted i the breakdow model. ag et al. [2] elaborated o a iteretig approach to etimate the euilibrium ditributio for the umber of cutomer i the M [x] /M/ ueueig model with multiple vacatio ad erver breakdow. Kumar et al. [5] ued the probability geeratig fuctio to derive the tatioary ditributio of M/M/ ueueig model with multiple vacatio ad erver breakdow. Raj ad Chadraekar [9] ued the matrix geometric method ad uai-birth death proce for aalyzig the N-policy multiple vacatio ueueig model with erver breakdow ad repair; they alo provided a umerical example o their model. Ahour et al. [] derived the M/M/ ueueig model with multiple vacatio uig the matrix geometric method ad the traitio tructure of it Markov Chai. Some performace meaure were calculated ad a umerical illutratio wa give to compare the cae of M/M/ with ad without vacatio. The Matrix Geometric Method i a ueful tool for olvig the more complex ueueig problem. It i applied by may reearcher to olve variou ueueig problem i differet frame work. Neut [7] explaied variou matrix geometric olutio of tochatic model. The matrix geometric method i alo utilized to develop the computable explicit formula for the probability ditributio of the ueue legth ad other ytem meaure of performace. Thi paper i orgaized a follow. I Sectio II we itroduce ome of the ueueig ytem otatio I Sectio III we itroduce the M/M/ ueueig model for the ATM machie uig the matrix geometric method ad calculate ome meaure of performace with give umerical illutratio. Sectio IV. devoted to derive the M/M/ ueueig model with erver ubject to breakdow by uig the ame method. w w w. a j e r. o r g Page 246

2 America Joural of Egieerig Reearch (AJER) 28 Some meaure of performace are calculated ad a umerical example i itroduced. A comparative tudy betwee the two model with ad without breakdow will be doe i Sectio V. Fially the cocluio wa itroduced i Sectio VI. II. SOME QUEUEING SYSTEM NOTATIONS The M/M/ ueueig model will be derived uig the matrix- geometric method ad there are ome otatio will be ued i thi paper a followig:. : The arrival rate 2. : The ervice rate 3. : The breakdow rate 4. : The repair rate : The umber of cutomer i the ytem : The traffic iteity for the ytem 7. :The probability of idle ytem 8. U : The tate pace 9. Q : Ifiite geerate matrix. R : The rate matrix. III. THE M/M/ QUEUEING MODE ITH MATRIX GEOMETRIC METHOD The M/M/ ueueig model will be etablihed ad there are ome aumptio eed to be achieved i order to keep the model table which wa ummarized a followig: Figure : The ATM ueueig model.. Coiderig oe erver (oe ATM machie) a how i Figure. 2. Cutomer arrive idepedetly at the ATM machie accordig to Poio ditributio with rate. 3. The ervice time i expoetially ditributed with rate. 4. Firt-come-firt-erved (FCFS) i the ueue diciplie for the ytem. 5. Ifiite ueue legth for the ytem. 6. The tability coditio of the ytem i. (See Bakari et al.]3[(. 3. The traitio rate of the model I thi ubectio the ueueig model i formulated by a Quai Birth ad Death (QBD) Markov chai. The Markov chai ad traitio tructure were give by Neut [7] where B A A2 A A Q A2 A A A2 A A B A which i tri-diagoal matrix where A ada 2. w w w. a j e r. o r g Page 247

3 America Joural of Egieerig Reearch (AJER) Stability coditio Now we derive the coditio for the ytem to reach the teady tate where the tadard drift coditio (See atouche ad Ramawami [6]) which i eceary ad ufficiet coditio for the tability ad the QBD Markov chai which i give by: A A () 2 where the tability coditio i atified a; ( Statioary ditributio of the model From Neut [7] we get from the tatioary ditributio of the model ad A A ). (2) ( ) 3.4The meaure of performace Firtly we tart by derivig the expected umber of cutomer i the ytem ( ) a follow: Thu E ( ) (4) ( ) Secodly uig ittle' law to get:. The expected waitig time of cutomer i the ytem: ( ) 2. The expected umber of cutomer i the ueue: 3. The expected waitig time of cutomer i the ueue:. 3.5 Numerical illutratio for the M/M/ ueueig model A umerical illutratio will be give for the ATM machie uig the M/M/ ueueig model.e recorded the arrival ad the departure time for 8 cutomer from oe of the ATM machie durig two ruh hour. Ued the chi uare goode of fit tet to a pecific ditributio (Poio for arrival ad expoetial for ervice time) coeuetly we obtaied that the arrival rate i =4.2 cutomer/hour ad the ervice rate i =55.2 cutomer/hour for the ATM machie. To calculate the meaure of performace for the M/M/ ueueig model with the value of arrival rate ad ervice rate a followig; = =.7283=73% = =.27=27% = =2.7 ( ) cutomer = =.7 hour ( ) =.95 cutomer =.49 hour. The traffic iteity ( ) i directly proportioal with the arrival rate where it i 73%. The expected umber of cutomer i the ytem i 2.7 cutomer the ueue legth i.95 the expected waitig (3) (5) w w w. a j e r. o r g Page 248

4 America Joural of Egieerig Reearch (AJER) 28 time of cutomer i the ytem i.7 hour ad the expected waitig time of cutomer i the ueue i.49 hour. IV. M/M/ QUEUEING MODE ITH BREAKDON The M/M/: FCFS ueueig model with breakdow will be derived for the ATM machie uder ame aumptio from Sectio 2. with the additio of the followig aumptio: - The breakdow time i expoetially ditributed with rate. 2-The repair time i expoetially ditributed with rate. 3-Idle ad Buy period occur alteratively o ay time t belog to either oe of thee period where Idle period: a period at which erver i breakdow. Buy period: a period at which the erver i buy (active). 4. Markov Chai Ad Traitio Rate Of The Model I thi ubectio the ueueig model i alo formulated by a QBD Markov chai. The Markov chai ad traitio tructure were give with aumig that t we defie: X () t : The umber of cutomer i the ytem. Y () t : The tate of the erver. The coider the two-dimeioal tochatic proce Y ()= t where X () t i level proce ad Y () t i the backgroud proce. ( X ( t ) Y ( t )); t where Obviouly the proce ( X ( t ) Y ( t )); t i a cotiuou-time Markov chai with the followig partitioed tate pace: U where ad U (5) l l U () (6) U r ( r)( r) r (7) r where the elemet of the et are arraged i lexicographical order. (See Ahour et al. []). The the ifiiteimal geerator matrix i give by: B B B A A Q A2 A A A2 A A ( X ( t ) Y ( t )); t o which i tri-diagoal matrix. Hece B A B B ( ) A ( ) ad A2 4.2 Stability coditio i of the type (QBD) Markov chai. A A A A o: To derive the coditio for the ytem to reach the teady tate defie the matrix 2 w w w. a j e r. o r g Page 249

5 America Joural of Egieerig Reearch (AJER) 28 A Clearly The Cotiuou Time Markov Chai with geerator A i reducible with aborbig tate with tatioary vector x ad. The the tadard drift coditio (See atouche ad Ramawami [6]) which i eceary ad ufficiet coditio for the tability ad the QBD Markov chai i give by: xa e xa e (8) 2 where e i a colum vector with all elemet eual to oe. 4.3 Statioary ditributio of the model To derive the tatioary ditributio of the model we coider the balace euatio: where: Q (9) ( ) ad 2 Thi lead to the followig ytem of euatio: B B B A 2A2 A 2A 3A2 ( ) () () (2) (3) Give the geometric relatio: R 2 (4) For tability purpoe the pectral radiu of R mut be le tha oe.uig Euatio (4) to ubtitute i the get: Euatio (3) ad divided by A RA R A (5) 2 2 where the matrice 2 rate matrix R will be i the form: A A A are give from Subectio 3. ad by olvig Euatio (5) uig oftware the R ( ) ( ) ( ) The ormalizatio coditio i give by: Thi give: Uig Euatio (4) thi lead to: Sice Euatio (2) ca be writte a: e e ( ) I R e (6) e e (7) B ( A RA ) 2 The we will ue Euatio () (8) (9) to fid ad ad by uig Euatio (4) the tatioary vector foud. (8) (9) w w w. a j e r. o r g Page 25

6 America Joural of Egieerig Reearch (AJER) The meaure of performace Firtly we tart by derivig the expected umber of cutomer i the ytem ( ) a follow: E ( ) e (2) Uig Euatio (6) to get: Thi give; e 2 e 3 e 2 3 e 2 Re 3 R e 4 R e 2 3 ( ) 2 I R e Secodly uig ittle' law to get: ( ). (2) 4.5 Numerical applicatio of the breakdow model A umerical applicatio will be provided to tudy the performace of the M/M/ breakdow model which applied for ATM machie. we ue the ame data from Sectio 2. with the ame parameter of ad ad by aumig value of the parameter ad i order to calculate the meaure of performace for the ame ATM machie but with cae of breakdow. Calculatig the meaure of performace with the parameter of ( ad 54 ) a follow:.9 = cutomer. hour 3.58 cutomer.9 hour. The expected umber of cutomer i the ytem i 4.48 cutomer the ueue legth i 3.58 the expected waitig time of cutomer i the ytem i. hour ad the expected waitig time of cutomer i the ueue i.9 hour. V. A COMPARATIVE STUDY e will compare betwee the M/M/ ueueig model with ad without breakdow accordig to the meaure of performace where Table 4. illutrate the reult of the meaure of performace for the two model. Table I: Numerical Reult For The Model ith Ad ithout Breakdow Breakdow Model ithout-breakdow Model Arrival Rate P P = w w w. a j e r. o r g Page 25

7 America Joural of Egieerig Reearch (AJER) 28 By aalyzig the whole reult it i oberved that from Table 4. the expected umber of cutomer i the ytem ad i the ueue for breakdow model i greater tha the expected umber of cutomer i the ytem ad i the ueue for the model without breakdow. The expected waitig time of cutomer i the ytem ad i the ueue for the breakdow model i greater tha the expected waitig time of cutomer the ytem ad i the ueue for the model without breakdow. VI. CONCUSION The matrix geometric method i more coveiet to ue rather tha the claical method for ome ueueig model uch a M/M/ with erver breakdow. REFERENNES []. Ahour K.R. Aar S.M. ad Shehata B.A. (26). Statioary Ditributio of M/M/ Queue with Multiple Vacatio uig Matrix Geometric Method. The 5 th Aual Coferece o Statitic Computer Sciece ad Operatio Reearch Ititute of Statitical Studie ad Reearch Cairo Uiverity. Egypt. [2]. Avi-Itzhak B. ad Naor P. (963). Some ueueig problem with the ervice tatio ubject to breakdow. Operatio Reearch (3) [3]. Bakari H. R. Chamalwa H. A. ad Baba A. M. (24). Queueig Proce ad It Applicatio to Cutomer Service Delivery (A Cae tudy of Fidelity Bak PIC Maiduguri). Iteratioal Joural of Mathematic ad Statitic Ivetio (IJMSI) 2() 4-2. [4]. Federgrue A. ad So K.C. (99). Optimal maiteace policie for igle-erver ueueig ytem ubject to breakdow. Operatio Reearch 38(2) [5]. Kumar J. Shide V. ad Sigh B. (2). Aalyi of the M/M/ Queue Model with Multiple Vacatio ad Server Break- Dow. Joural of Iteratioal Academy of phyical Sciece 5(3) [6]. atouche G. ad Ramwami V. (999). Itroductio of Matrix Aalytic Method i Stochatic Modelig. Society for Idutrial ad Applied Mathematic. Philadelphia. [7]. Neut M.F. (98). Matrix geometric Solutio i Stochatic Model: a algorithmic approach. Joh Hopki Serie i Mathematical Sciece Joh Hopki Uiverity Pre Baltimore Md USA. [8]. Núñez-Queija R. (2). Sojour time i a proceor harig ueue with ervice iterruptio. Queueig ytem 34() [9]. Raj M. R. ad Chadraekar B. (25). Matrix geometric Method for Queueig Model with Subject to Breakdow ad N-Policy Vacatio. Mathematica Aetera 5(5) []. Taha H.A. (992).Operatio Reearch New York: Macmilla Publihig co. []. Takie T. ad Segupta B. (997). A igle erver ueue with ervice iterruptio. Queueig Sytem: Theory ad Applicatio 26(3) [2]. ag K-H. Cha M-C. ad Ke J-C. (27). Maximum etropy aalyi of the M [x] /M/ ueueig ytem with multiple vacatio ad erver breakdow. Computer & Idutrial Egieerig 52(2) Shoukry E.M.."Matrix Geometric Method For M/M/ Queueig Model ith Ad ithout Breakdow Atm Machie America Joural of Egieerig Reearch (AJER) vol. 7 o. 28 pp w w w. a j e r. o r g Page 252