Optimal Strategy Analysis of an N-policy M/E k /1 Queueing System with Server Breakdowns and Multiple Vacations

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1 Iteratoal Joural of Scetfc ad Research ublcatos, Volume 3, Issue, ovember 3 ISS Optmal Strategy Aalyss of a -polcy M/E / Queueg System wth Server Breadows ad Multple Vacatos.Jayachtra*, Dr.A.James Albert** *.hd Research Scholar,Mathematcs, arpagam Uversty ** Dea, Mathematcs, arpagam Uversty Abstract- Ths paper studes the optmal cotrol of a -polcy M/E / queueg system wth server breadows ad multple vacatos. The server s tured o whe uts are accumulated the system. The server s tured off ad taes a vacato wth expoetal radom legth wheever the system s empty. If the umber of uts watg the system at ay vacato completo s less tha, the server wll tae aother vacato. If the server returs from a vacato ad fds at least uts the system, t mmedately starts to serve the watg uts. It s assumed that the server breadow s accordg to osso process ad the repar tme has a expoetal dstrbuto. We derve the dstrbuto of the system sze ad employ the probablty geeratg fucto to obta the mea queue legth. It s proved that the servce stato s busy the steady state s equal to the traffc testy. The total expected cost fucto per ut tme s developed to determe the optmal operatg polcy at mmum cost. Ths paper provdes the mmum expected cost ad the optmal operatg polcy based o umercal values of the system parameters. Sestvty aalyss s also provded. Idex Terms- : M/E / queueg system, multple vacatos, -polcy, probablty geeratg fucto, Server breadows. T I. ITRODUCTIO hs paper cosders the modelg of a producto system at whch arrvals of producto order follows a osso process at a rate λ. The producto tmes of the orders are made up of depedet ad detcally dstrbuted expoetal radom varables wth mea /µ whch yelds a Erlag type dstrbuto. The system operato starts (tured o) oly whe orders have accumulated ad s shut dow (tured off) whe o orders are preset. Whe the server s worg he may meet upredctable breadow but t s mmedately repared. Whe the system s tured off, the server leaves the system for a radom perod of tme called vacato. O returg to the system f the server fds less tha uts the system mmedately he taes aother vacato. Ths producto system ca be modeled by a M/E / queueg system wth server breadows ad multple vacatos uder - polcy. The cocept of polcy was frst troduced by Yad ad aor [6]. ast wor regardg queueg systems uder the polcy may be dvded to two categores: () cases wth server s vacatos ad () cases wth server s breadows. For cases wth server breadows, Wag [] frst proposed a maagemet polcy for Marova queueg systems uder the polcy wth server breadows. Wag [4] ad Wag et al. [5] exteded the model proposed by Wag [] to M/E/ ad M/H/ queueg systems respectvely. Vasatha umar ad Chada [7] preseted the optmal strategy aalyss of two phase M/E/ queueg system wth server breadows ad gatg. Also they obtaed the total expected cost fucto for the system ad determe the optmal value of the cotrol parameter. Exstg research wors, cludg those metoed above, have ever covered cases volvg both server breadows ad vacatos. Queueg models wth server breadows ad vacatos accommodate the real-world stuatos more closely. The purpose of ths paper s threefold.. The steady state equatos are establshed to get the steady state probablty dstrbuto ad to show that t geeralzes the prevous results.. We formulate the system total expected cost order to determe the optmal operatg -polcy umercally at the mmum cost for varous values of system parameters whle matag the mmal servce quatty. 3. We perform a sestvty aalyss II. MODEL DESCRITIO For the purpose of aalytcal vestgato, we cosder the model wth the followg assumptos:. The arrval s posso process wth parameter λ ad wth servce tmes accordg to a Erlag dstrbuto wth mea /µ ad stage parameter. The Erlag type dstrbuto s made up of depedet ad detcal expoetal stages, each wth mea /µ. A customer goes to the frst stage of the servce (say stage ) the progresses through the remag stages ad must

2 Iteratoal Joural of Scetfc ad Research ublcatos, Volume 3, Issue, ovember 3 ISS complete the last stage (say stage ) before the ext customer eters the last stage. We assume that customers arrvg at the servce stato form a sgle watg le ad are served the order of ther arrvals.. Whe the system s tured off, the server leaves the system for a radom perod of tme called vacato whch s expoetally dstrbuted wth parameter θ. 3. Whe the server s worg, the server may breadow at ay tme wth a osso breadow rate α. 4. Whe the server fals, t s mmedately repared at a repar rate β, where the repar tmes are expoetally dstrbuted. III. STEADY STATE RESULTS I steady state the followg otatos are used.,, = robablty that there are o customers the system whe the server s o vacato,, = robablty that there are customers the system ad the customers servce s stage whle the server s o vacato.,, = robablty that there are customers the system ad the customers s the stage whle the server s busy.,, = robablty that there are customers the system ad the customer servce s stage whe the server s operato but foud to be broe dow. The steady state equatos are gve as follows (),,,, ( ) (),,,, (3),,,, ( ),, o,, ( ) (4) ( ),,,,,, ( ) (5) ( ) (6),,,,,, ( ),,,,,,,, ( ) (7) ( ),,,,,,,, ( ) (8) ( ),,,,,,,,,, ( ) (9) ( ),,,,,,,, (, ) () ( ) (),,,, ( ),,,,,, ( ) ( ) () ( ),,,, ( ) (3) ( ),,,,,, ( ) (4) Solvg equatos (),() ad (4) recursvely, we fally get R,,,, ( ),,,, (5)

3 Iteratoal Joural of Scetfc ad Research ublcatos, Volume 3, Issue, ovember 3 3 ISS where R IV. ROBABILITY GEERATIG FUCTIO The techque of usg the probablty geeratg fucto may be appled a recursve maer from equatos () to (4) to obta the aalytcal soluto of,, a eat closed form of expresso. Defe the probablty geeratg fucto of G (z), G (z) ad G (z)respectvely as follows: (),,,, G z z (6) H () z z (7),, G ( z) H ( z) (8) G () z z (9),, G ( z) G ( z) () where z Applyg algebrac mapulato techque to equatos () ad (3), we get the followg z Rz G () z z Rz,, Multplyg equato (5) by z ad (7)ad () by z ad summary over we get () H ( ) ( ) ( ) z r rz s H z tg z () s t Where r Multplyg equato (6) by z (8) ad (9) by z ad summg over we get Rz r s H ( z) rzh ( z) H ( z) tg ( z) z Rz,, (3) Aga multplyg () ad (3),() ad (4) respectvely by approprate powers of z ad summary over we fd G( z) H( z) z G( z) H ( z) z G z equato (3) we get Substtutg () (4) (5)

4 Iteratoal Joural of Scetfc ad Research ublcatos, Volume 3, Issue, ovember 3 4 ISS ( ) H ( ) z r rz s t H z z Therefore H( z) r rz s t H( z) z Solvg () ad () equato (9) we obta Rz rz Rz H ( z) z r rz s t z,, (6) (7) We solve equatos (6) ad (7) for () G z ad G () z to obta the followg Rz z Rz G ( z) z s t z r rz s t z z r rz s t z,, (8) G( z) G( z) z (9) We evaluate the probablty,, usg ormalzg codto. For ths purpose we evaluate G(), G () ad G () from equatos (),(8) ad (9) respectvely as G (),, (3) G (),, (3) G () G () (3) ow usg the ormalzg codto gve by G () G () G () G (), we obta the value of probablty that the system empty s,, (33)

5 Iteratoal Joural of Scetfc ad Research ublcatos, Volume 3, Issue, ovember 3 5 ISS Thus V. SOME OF THE ERFORMACE MEASURES Deote the log ru fracto tme for whch the server s o vacato, busy ad broe dow by, ad respectvely. G () G() (35) G () (36) We defe the expected umber of customers the system as follows. L = the expected umber of uts the system whe the server s o vacato. v L = the expected umber of uts the system whe the server s worg. b L = the expected umber of uts the system whe the server s broe dow. d L = Expected umber of customers the system. s To fd L v L v, we compute G ' () G ' () ad G ' () ' G () Smlarly, we compute Lb equato () by applyg L Hosptal rule twce to obta. (34) (37) equatos (8) ad (9) respectvely, by applyg L hosptal rule twce to obta s r Lb (38) Ld G ' () ' Ld G() G () (39) L G () G () G () s ' ' ' s r ( ) ( ) L s (4)

6 Iteratoal Joural of Scetfc ad Research ublcatos, Volume 3, Issue, ovember 3 6 ISS VI. SECIAL CASES I ths secto we preset some exstg results the lterature whch are specal cases of our model. Case (): If Ө=,α= ad β=, expressos (37),(38) ad (4) reduces to a specal cases of L off, L o ad L respectvely of Wag ad Huag[3](p.9) Case (): If =, Ө=,α= ad β=, the expresso (33) reduces to a specal case of expresso (.8) of Gross ad Harrs[] VII. OTHER SYSTEM CHARACTERISTICS A grad vacato ad the grad vacato process are defed to vestgate the operatg characterstcs of our model. The frst grad vacato (G ) starts from the pot the system becomes empty ad the server leaves for the frst vacato (V ) ad lasts utl the server fds oe (or) more customers after returg from a vacato. At the ed of the frst grad vacato, f the umber of customers the queue s less tha, the server leaves for aother vacato ad t becomes the ew startg pt of the secod grad vacato (G ). Ths secod grad vacato cotues utl a dfferet system state s observed after a vacato. Grad vacatos (G,G.) cotue ths maer utl the umber of uts observed after a grad vacato s foud to be greater tha or equal to. The grad vacato process s a process mbedded the dle perod whch the mbedded states are the umber of uts the queue just after the server leavg for grad vacatos. Oe cycle begs whe the system s empty ad the server taes a vacato. The server remas o vacato utl there are at least uts the system whe t returs from a vacato. We call ths, the dle perod. The busy perod s tated whe the server starts servg the watg uts ad termates whe there are o uts the system. Whle provdg servce the server may breadow ad set for repar mmedately. Ths s called breadow perod. The dle perod I s developed by meas of the grad vacato process. It s defed that = the probablty that the grad vacato process passes through state. = the probablty that customers arrve durg a vacatos The usg the cocept of grad vacato process, t s determed =,.. Calculatg recursvely t s determed Sce the expected legth of a grad vacato s EV ( ) = the s the expected legth of the grad vacato ( ) ( ) whch starts wth customers. Hece we have the expected legth of the dle perod s gve by EI () (4) ( ) Substtutg for ad we have EI ( ) (4) If E(B),E(D) ad E(C) deote the expected busy perod, broe dow perod ad busy cycle respectvely, we have E(C) = E(I) + E(B)+E(D) From equatos (34) to (36) we obta the log ru fracto of tme for the server s dle, busy ad broe dow respectvely: EI () (43) EC ( ) EB ( ) (44) EC ( )

7 Iteratoal Joural of Scetfc ad Research ublcatos, Volume 3, Issue, ovember 3 7 ISS ED ( ) (45) EC ( ) Thus we have the umber of cycles per ut tme EC ( ) VIII. OTIMAL -OLICY (46) We develop a steady state total expected cost fucto per ut tme for the -polcy M / E /Queueg system wth server vacatos ad breadows, whch s a decso varable. Followg cost structure s costructed, our objectve s to determe the optmal operatg polcy so as to mmze ths cost fucto. Let C = holdg cost per ut tme for each customer preset the system. h C = Cost per ut tme for the operatg servce stato. o C =Set up cost per cycle. s C =breadow cost per ut tme d C =removable cost per ut tme for removg the servce stato. r C =removable per ut tme for the server beg o vacato. Usg the defto of each cost elemets ad ts correspodg characterstcs, the total expected cost fucto per ut tme s gve by E B E D EI ( ) T( ) ChLs Co Cs Cr Cd C (47) E C E C E C E( C) We obta the optmal value *,whch mmzes cost fucto by dfferetatg t wth respect to ad settg the result to be zero..e., ( T( )).The soluto to (47) may ot be a teger ad the optmal postve teger value of s oe of the tegers surroudg * whch gves a smaller cost T. Here we should be poted out explctly that the soluto really gves the mmum value ad ( T( )) at =* s greater tha zero whe the values of system parameters satsfy sutable codtos. However, t s qute tedous to preset the explct expresso. Therefore we wll perform the umercal expermets to demostrate that the fucto s really covex ad the soluto gves a mmum. IX. SESITIVITY AALYSIS I the course of aalyss, sestvty aalyss has bee carred out to fd the optmum value of (e.,*), expected system legth ad mmum cost based o chages the system parameters by usg MATLAB. I order to arrve at the coclusos, the followg arbtrary values of the system parameters are cosdered. λ=, α = 3, µ = 5, β = 5, C h =5, C =, C s =5, C d =, Cˠ=5, C r =5, Ө=3, = It s otced from the results table that as λ creases, the value of *, L (*) ad expected cost T(*) creases. Table : λ * 9 3 L(*) T(*) Computed values table shows that as µ creases * creases, the expected cost ad queue legth decreases as µ creases. Table : µ * L(*) T(*)

8 Iteratoal Joural of Scetfc ad Research ublcatos, Volume 3, Issue, ovember 3 8 ISS It ca be observed from table3 that * creases for smaller values of β ad does ot chage for larger values β, the expected cost ad queue legth creases wth crease β. Table 3: β * L(*) T(*) From table 4, t s observed that * remaed uchaged whe α creased from. to.9. Thus * s sestve to the chages α. L(*) ad T(*) decreased as α creased. Table 4: α * L(*) T(*) Computed values table 5 shows that * s sestve to the chages Ө. L (*) ad T (*) decreases as Ө creases. Table 5: Ө * L(*) T(*) X. COCLUSIO Optmal strategy aalyss of -polcy M/E / queueg system wth server breadows ad vacatos has bee studed. Some of the system performace measures have bee derved. A cost fucto s formulated to determe the optmal value of. Sestvty aalyss s carred out through umercal llustratos. These umercal values wll be useful aalyzg practcal queueg system ad mae decso to mprove the grade of servce by selectg approprate system descrptors. The preset study ca be exteded by worg vacato. The future scope of the study s the cost ad proft aalyss of ths model. REFERECES [] D.Gross ad C.M.Harres, Fudametal of Queueg Theory. d Edto.Wley, ew Yor. (985). [].H.Wag,(995). Optmal operato of a Marovaqueueg system wth removable ado relable server, Mcroelectrocs Relablty,35, (995).pp [3].H.Wag ad H.M.Huag,(995), Optmal cotrol of a M/E / queueg system wth a removable servce stato, Joural of Operatoal Research Socety, 46,,(995),pp.4-. [4].H.Wag,, Optmal cotrol of a M/E/ queueg system wth removable servce stato subject to breadows. Joural of the Operatoal Research Socety,48,,(997) [5].H. Wag.,.W.Chag,adB.D.Svazla,999). Optmal cotrol of a removable ad o-relable server a fte ad a fte M/H / queueg system. Appled Mathematcal Modelg,3,(999),pp [6] M.Yad ad.aor, Queueg system wth a removable servce stato. Opl Res. Quarterly.4,(963)pp [7]V.Vasata umar,b.v.s..har rasad,.chada,..r.rao. Optmal Strategy Aalyss of a -polcy Two-hase M/E / Queueg System wth Server Breadows ad Gatg.Appled Mathematcal Sceces,4(66),(),pp AUTHORS Frst Author.Jayachtra, h.d. Research Scholar, arpagam Uversty, Combatore.E.mal: jayaasas_ashu@redffmal.com Secod Author Dr.A.James Albert, Dea, Arts, Scece ad Humates,arpagam Uversty, Combatore, Taml adu E.mal:asralbert@gmal.com. Correspodece Author.Jayachtra, h.d. Research Scholar, arpagam Uversty, E.mal: jayaasas_ashu@redffmal.com