A Two Stage Batch Arrival Queue with Reneging during Vacation and Breakdown Periods

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1 America Joural of Operatios Research, 3, 3, ublishe Olie November 3 ( A Two Stage Batch Arrival ueue with Reegig urig Vacatio a Breakow erios Moita Baruah, Kailash C. Maa, Tillal Elabi Bruel Busiess School, Bruel Uiversity, Loo, UK College of Iformatio Techology, Ahlia Uiversity, Maama, Bahrai Moita.Baruah@bruel.ac.uk, moitabrh@gmail.com, kmaa@ahlia.eu.bh, Tillal.Elabi@bruel.ac.uk Receive September 3, 3; revise October 3, 3; accepte November 7, 3 Copyright 3 Moita Baruah et al. This is a ope access article istribute uer the Creative Commos Attributio Licese, which permits urestricte use, istributio, a reprouctio i ay meium, provie the origial work is properly cite. ABSTRACT We stuy a two stage ueuig moel where the server provies two stages of service oe by oe i successio. We cosier reegig to occur whe the server is uavailable urig the system breakow or vacatio perios. We cocetrate o erivig the steay state solutios by usig supplemetary variable techiue a calculate the mea ueue legth a mea waitig time. Further some special cases are also iscusse a umerical examples are presete. Keywors: Two-Stage Service; Batch Arrivals; Breakows; Reegig; Steay State ueue Sie. Itrouctio ueues with impatiet customers have attracte the attetio of may researchers a we see sigificat cotributio by umerous researchers i this area. Oe of the earliest works o balkig a reegig was by Haight [,], Barrer [3], which was the first to itrouce reegig i which he stuie etermiistic reegig with sigle server markovia arrival a service rates. Aother early work o markovia reegig with markovia a arrival a service patter was by Acker a Gafaria [4], Haghighi et al. [5] stuie a markovia multiserver ueuig system with balkig a reegig. Coseuetly, we see a lot of evelopmets i the stuy of ueues with impatiet customers i recet years. A M/G/ ueue with etermiistic reegig was stuie by Bae et al. [6]. We refer to some authors like Zhag et al. [7], El-auomy [8], Altma a Yechiali [9,], Kumar a Sharma [] who stuie ueues with impatiet customers i ifferet cotexts. Vacatio ueues are a importat area i the literature of ueuig theory. Sice the past two ecaes it has emerge as a importat area of stuy ue to its various applicability i real life problems such as telecommuicatio egieerig, maufacturig a prouctio iustries, computer a commuicatio etworks etc. A few Correspoig author. of early works o ueues with vacatios are see by authors like Levy a Yechailai [], Doshi [3], Keilso a Servi [4]. A two stage batch arrival ueuig system where customers receive a batch service i the first a iiviual service i the seco stage was stuie by Doshi [5] i the past. I recet years, extesive amout of work has bee oe o batch arrival ueues with vacatios a breakows. We metio a few recet papers by Kumar a Arumugaatha [6], Chouhury, Taj a aul [7], Maraghi, Maa a Darby-Dowma [8], Khalaf, Maa a Lukas [9]. I this paper we cosier a batch arrival ueue where service is offere i two stages of service, oe by oe i successio. We exte a evelop this moel by aig ew assumptios reegig a system breakows. Customers may reege (leave the ueue after joiig) urig server breakows or urig the time whe the server takes vacatio ue to impatiece. This is a very realistic assumptio a ofte we come across such ueuig situatios i the real worl.. The Mathematical Moel a) Customers arrive i batches followig a compou oisso rocess with rate of arrival λ. Let λ c i t ( i,,3, ) be the first orer probability that customers arrive at the system i batches of sie i, at x, x t, where the system at a short iterval of time ( ] Ope Access

2 M. BARUAH ET AL. 57 a c i i i c. b) The server provies two stages of heterogeeous services oe after the other i successio. A arrival batch shall receive the service offere at two stages oe by oe i successio, efie as the first stage (FS) a seco stage (SS) service respectively. The service isciplie is assume to be o a first come first serve basis (FCFS). We assume that the service time S j (j, ) of the j th stage service follows a geeral probability istributio with istributio fuctio Bj ( s j ), bj ( s j ) beig the probability esity fuctio a E( S j ) as the th momet of the service time, j,. Let μ j ( x) be the coitioal probability of stage j service urig the perio ( x, x t] give elapse time is x such that μ bj x j ( x) j, B x () a j s bj ( sj) μj ( x) exp μj ( x) x, j,. () c) Oce the seco stage service (SS) of a uit is complete the server is assume to take vacatio with probability p or may cotiue to offer service with probability ( p). As soo as the vacatio perio of the server is over, he jois the system to cotiue service of the waitig customers. We assume the vacatio time to be a raom variable followig geeral probability law with istributio fuc- tio give by W ( v ) a esity fuctio by w( v ) a EV is the th momet (,, ) tio time. Here we assume that ( x) of vacaφ be the coitioal probability of a vacatio perio urig the iterval ( x, x x], give that elapse time is x, so that W ( x) φ ( x) (3) W x a thus v wv φ( v) exp φ( x) x (4) ) I aitio, customers arrivig for service may become impatiet a reege (leave the ueue) after joiig urig vacatios a breakow perios. Reegig is assume to follow expoetial istributio with parame- γ t ter γ. Thus f () t γe t, γ >. Thus γ t is the probability that a customer ca reege urig a short iterval of time ( tt, t]. The system may fail or be subjecte to breakow at raom. The customer receivig service urig breakow returs back to the hea of the ueue. We assume that time betwee breakows occur accorig to a oisso process with mea rate of breakow as α >. Further the repair times follows a geeral (arbitrary) istributio with istributio fuctio F(x) a esity fuctio f(x). Let the coitioal probability of completio of F ( x) the repair process is β(x)x such that β ( x) F x r F r r exp x x. a thus β β 3. Defiitios a Notatios We assume that steay state exists a efie, j x robability that there are ( ) customers i the system icluig oe customer i type j service, j, a elapse service time is x. Thus x x is the, j, j correspoig steay state probability irrespective of elapse time x. V ( x ) probability that there are ( ) customers i the ueue a server is o vacatio a elapse vacatio time is x. V V x x is the correspoig steay state probability irrespective of elapse vacatio time x. Steay state probability of the server is ile as the server takes vacatio. The robability Geeratig Fuctios are efie as: j (, ), j; j, j x x ; j, (5) (, ), ;, ; (6) R x R x R R (, ) ; ; (7) V x V x V V C i i c. (8) i 4. Euatios Goverig the System Uer this moel we costruct the ifferetial euatios as, x x, x x i { λ μ α} λ c x i i, (9) Ope Access

3 57 M. BARUAH ET AL., x x, x x i { λ μ α} λ c x i i, R, x x R x x { λ β γ} λ cr x γr i i i ( λ β) γ x R x x R x R x () () () V x x V x x { λ φ γ} γ λ cv x V i i i ( λ φ) γ x V x x V x V x (3) (4), μ λ p x x x φ β V x x x R x x x (5) The above ifferetial euatios ow have to be solve subject to the followig bouary coitios: λ ( ) μ c p x x x,, R, ( x) β( x) x V( x) φ( x) x (6) μ x x x (7),, μ, (8) V p x x x α α R x x x x,, α α,, (9) R ( ) () 5. ueue Sie Distributio at Raom Epoch Multiplyig Euatios (9) a () by a summig over from to, yiels (, ) {( ) } (, ) x λ λ x C μ x α x () { λ λ μ α} ( ),, () x x C x x Applyig the same process i Euatio () a usig () gives γ R( x, ) λ λc( ) β( x) γ R( x, ) x (3) Similarly from (3) a (4), we get (, ) V x x C x γ λ λ φ γ (4) Further itegratig Euatios ()-(4) over limits to x gives us the followig ( x, ) (5) (, ) exp ( λ λc( ) x μ() t t ( x, ) (6) (, ) exp ( λ λa( ) x μ() t t R( x, ) γ (7) R(, ) exp λ λc( ) γ x β() t t V ( x, ) γ (8) V (, ) exp λ λc( ) γ x φ() t t Next multiplyig the bouary coitios by suitable powers of a takig summatio over all possible values of a usig the GF s we get after simplificatio (, ) ( λ λ) C μ p x, x x β φ R x, x x V x, x x μ (9), x, x x (3) μ V, p x, x x (3) Now multiplyig Euatio (9) by, summig over from to, a usig () a GF s we have (, ) α R (3) Ope Access

4 M. BARUAH ET AL. 573 Agai itegratig Euatios (5)-(8) with respect to x, gives us where a ( λ λc( ) α) C( ) B ( ) (, ) λ λ α ( λ λc( ) α) C( ) B ( ) (, ) λ λ α V ( ) V (, ) R( ) R(, ) W λ λc( ) γ γ λ λc( ) γ γ γ F λ λc( ) γ γ λ λc( ) γ ( ) ( ) λ λc α x λ λ α ( λ λc( ) λ λ α e B C e B x, B C B x γ λ λc( ) γ x γ F λ λc( ) γ e F x γ λ λc( ) γ x γ W C W x λ λ γ e (33) (34) (35) (36) are the Laplace-Steiltjes trasform of the first stage service time, seco stage service time, repair time a vacatio time respectively. Now we etermie the itegrals a μ x, x x, μ x, x, x, β R x, x x φ V x, x x by multiplyig the RHS of Euatios (5)-(8) by μ( x), μ( x), β ( x) a φ ( x) respectively a itegrate with respect to x a obtai ( x, ) μ( x) x (, ) B ( λ λc( ) (37) ( x, ) μ( x) x (, ) B( λ λc( ) (38) γ R( x, ) β( x) x R(, ) F λ λc( ) γ (39) γ V ( x, ) φ( x) x V (, ) W λ λc( ) γ (4) Let us take γ λ λc( ) α m; λ λc( ) γ k Utiliig (37)-(4) i Euatios (9)-(3) we obtai (, ) ( λ λ) ( ) (, ) R(, ) F ( k) V(, ) W ( k) C p B m (4) (, ) (, ) B m (4) (, ) (, ) V p B m (43) Usig (4) i (43) we get, (, ) (, ) V p B m B m (44) Agai from (3) usig (33) a (34) we get R (, ) α, B m, B ( m ) m (45) Now usig (4), (44) a (45) i Euatio (4), we, solve for ( ) ( m λc λ ) a (46) ( ) D m p B m B m pb m B m W k ( ), V (, ) { } α F k B m B m ( λ λ) (47) m C B m (48) ( λ λ) pm C B m B m (49) Substitutig the values from (46), (48) a (49) i (33)-(36), we obtai ( ) ( ) λc λ B m (5) ( ) λc λ B m B m (5) Ope Access

5 574 M. BARUAH ET AL. ( ) F ( k) α λc λ B m B m R V k ( λ λ) W ( k) pm C B m B m k (5) (53) Let eote the probability geeratig fuctio of the ueue sie irrespective of the state of the system. R V N( ) (54) I orer to etermie the probability of ile time, we use the ormaliig coitio (). Agai sice (5) is ietermiate of the form / at, we use L Hopital s Rule o Euatio (5) to obtai () λ ( ( ) ) ( α) ( ) E I B λe I α λe I γ E R α λe I α λe I γ E R pα λe I γ E V B α B α (55) (55) is the steay state probability that server is proviig service i stage. () λ α ( α ) ( ( ) ) The steay state probability that server is proviig service i stage two is ( ) E I B B λe I α λe I γ E R α λe I α λe I γ E R pα λe I γ E V B α B α (56) R () αλe( I) E( R) B ( α ) B( α ) ( λe( I) α( λe( I) γ ) E( R) ) α λe( I) α( λe( I) γ ) E( R) pα( λe( I) γ ) E( V) B ( α) B( α) (57) V () pαλe( I) E( V) B ( α ) B( α ) ( λe( I) α( λe( I) γ ) E( R) ) α λe( I) α( λe( I) γ ) E( R) pα( λe( I) γ ) E( V) B ( α) B( α) where C(), C () E( I) batch of customers, F ( ) E( R) is the mea of arrivig is the mea repair time a W ( ) E( V) is the mea vacatio time. Thus the ormaliatio coitio yiels (58) { } { } αγ E( R) B ( α ) B ( α ) pαγ B α B α λe I αe R αe R pαe V B α B α { } (59) A therefore the traffic itesity (utiliatio factor) ρ is { } { } αγe( R) B ( α) B ( α) pαγb ( α) B ( α) λe I αe R αe R pαe V B α B α ρ < (6) 6. Average ueue Sie a Average Waitig Time, the mea ueue sie is of the Sice L ( ) / form, we apply L Hopital s Rule twice a obtai D ( ) N ( ) N ( ) D ( ) L lim (6) D ( ) where primes a ouble primes eote the first a Ope Access

6 M. BARUAH ET AL. 575 seco erivatives respectively. () λ { α } λ { α α } ( α) ( α) N E I E R E I E R p E V B B { } () λ ( α) ( α) α ( α) ( α) α ( α) ( α) N E I I B B B B E R p B B E V {{ } λe I αe R pαe V B α B α B α B α ( ) } α λe I γ E R B α B α λe I γ E V B α B α { } { λe I α λe I γ E R pα λe I γ E V α} B α B( α) λ α( λ γ) ( ) ( ) D E I E I E R () λ ( α ) ( α) ( α) { α α } D E I I E R B B E R p E V { } ( ) { } αγ E( R) B ( α ) B ( α ) α λe I γ E R B α B α pα{ γe( V) λe( I) γ E V } B ( α) B ( α) { λe( I) α( λe( I) γ ) E( R) pα( λe( I) γ ) E( V) } B ( α) B ( α) B ( α) B ( α) { } (6) (63) (64) (65) where is give by (59), F" E(R ), W" E(V ) are the seco momets of repair time a vacatio time respectively, a E( I I ) is the seco factorial momet of batch of arrivig customers. Hece utiliig (6)-(65) i (6), we ca obtai the mea legth of ueue sie at raom epoch L a the mea waitig time of the L ueue W ca be obtaie by usig W. Alteraλ tively, we ca fi L L ρ, the mea ueue sie of L the system a W, the mea waitig time i the sys- λ tem. 7. Special Cases 7.. Case. No Server Vacatios I this case, the server has o optio to take a vacatio. Hece V ( ). Thus lettig p i our results (5)-(5), we obtai the followig a our moel reuces to a two stage batch arrivals with reegig urig breakows. R ( λc( ) λ) B ( m) ( ) ( ) (66) λc λ B m B m (67) α λc λ B m B m F k (68) k a D m B m B m α F k B m B m The probability of ile time is { } { } ( ) ( ) αγ E( R) B ( α ) B ( α ) λe I αe R αe R B α B α A the traffic itesity ρ is { } { } ( ) ( ) αγ E( R) B ( α ) B ( α ) λe I αe R αe R B α B α (69) (7) (7) Further puttig p i the Euatios (6)-(65) we ca obtai the mea ueue sie L a mea waitig time W. 7.. Case. No Reegig I this case we cosier that customers o ot reege urig breakows a vacatios. So we let the parameter γ. The k λ λc( ). The our moel reuces to a two stage batch arrival ueuig system with vacatios a breakows. Thus Euatios (5)-(53) chages to λc λ B m (7) Ope Access

7 576 M. BARUAH ET AL. R λc λb m B m (73) ( ) α F λ λc B m B m (74) where W ( ) ( λ λ ) pm W C B m B m (75) { λ λ } ( ){ α λ λ } D m p pw C B m B m F C B m B m The probability of ile time E( R) λe( I) E ( R) pe( V) αb ( α) B( α) B α B α α (76) A the utiliatio factor is E( R) ρ λe( I) E ( R) pe( V) αb ( α) B( α) B α B α α (77) The result (7)-(75) agrees with the result obtaie by Margahi, Maa a Darby-Dowma for a Batch arrival ueue with two stage Heterogeeous service, Beroulli scheule vacatios a geeral repair times Case 3: No System Breakows a No Reegig I this case, the server oes ot face ay breakow a there is also reegig of customers. So we take α, γ i our Euatios (5)-(53). The m λ λc( ) a k λ λc( ).. Thus we have ( λ λ ) B C (78) ( λ λ ) ( λ λ ) B C B C (79) where ( λ λ ) ( λ λ ) ( λ λ ) pb C B C W C V (8) { λ λ } ( ) λ λ ( λ λ ) ( ) ( ) D p pw C B C B C The probability geeratig fuctio of the ueue sie takes the form ( ) {( p) pw ( λ λc( ) )} B ( λ λc( ) ) B( λ λc( ) ) (8) The probability of ile time is { ( ) } λe I E S E S pe V a the traffic itesity is { } ρ λe I E S E S pe V < Thus (8) gives the steay state ueue sie of a X M /( G, G) / vacatio ueue, i.e. a two stage batch arrival vacatio ueue. Ope Access

8 M. BARUAH ET AL Case 4: Expoetial Service Time a Expoetial Vacatio Time I this case we assume that the service time for the two stages of service with service rate μ >, a μ >, respectively are expoetially istribute. Further the repair time a vacatio are all expoetially istribute with repair rate β > a vacatio rate ϕ >. Thus μ μ B ( m) ; B( m) ; μ m μ m where β φ F ( k) ; W ( k) β k φ k E( R) ; E( R ) β β EV ; EV φ φ ; m λ λc α k λ λc γ γ (8) Utiliig the above relatios i Euatios (5)-(53), we obtai μ λc( ) λ μ m ( ) (83) D ( ) R μ μ λc( ) λ μ m μ m (84) D α λc( ) λ μμ V where ( μ m)( μ m) ( β k) ( μ m)( μ m)( φ k) (85) μμ pm λc ( ) λ (86) φ μμ μμ β ( λ λc( ) ( p) p α φ k( μ m)( μ m) ( μ m)( μ m) β k Therefore the probability that the server is proviig service i first stage at raom poit of time is () μ λe( I) μ α α( λe( I) γ) p μμ λe( I) α λe( I) α( λe( I) γ) β β φ ( μ α )( μ α ) (87) robability that server is proviig service i seco stage at raom poit of time is () μ μ λe( I) ( μ α ) μ α α( λe( I) γ) p μμ λe( I) α λe( I) α( λe( I) γ) β β φ ( μ α )( μ α ) (88) robability that the server is uer repairs at raom poit of time is R () αλe I μμ β ( μ ( μ α( λe( I) γ) p μμ λe( I) α λe( I) α( λe( I) γ) β β φ ( μ α )( μ α ) (89) A probability that server is o vacatio at raom poit of time is give by Ope Access

9 578 M. BARUAH ET AL. V () λe I μμ pα φ ( μ ( μ α( λe( I) γ) p μμ λe( I) α λe( I) α( λe( I) γ) β β φ ( μ α )( μ α ) (9) robability that the server is ile but available i the system is λe I α α pα μμ β β φ ( μ ( μ αγ μμ μμ pαγ β μ α μ α μ α μ α ( )( ) ( )( ) (9) a D () Similarly the mea ueue sie a mea waitig time ca be erive by fiig N (), N (), D () a utiliig i Euatio (6). α α pα μμ N () λe( I) λe( I) β β φ ( μ α )( μ α ) α μμ pα μμ N () λe( I I ) β ( μ α)( μ α) φ ( μ ( μ ( ) ( ) μμ φ α λe I γ μμ p λe I γ λe( I) β ( μ α)( μ α) μ α μ α ( E( I) γ) α λ D () λe( I) β ( E( I) ) p ( E( I) ) α λ γ α λ γ μμ α λe( I) β φ μ α μ α () λe( I I ) α μμ α pα β ( μ ( μ β φ D ( E( I) ) ( E( I) ) p ( E( I) ) ( )( ) ( E( I) ) αγ λ γ μμ γ λ γ μμ pα β ( )( ) φ β μ α μ α φ ( μ α )( μ α ) α λ γ α λ γ μμ μμ λe( I) α β φ μ α μ α μ α μ α ( ) ( ) ( )( ) 8. Numerical Illustratio I orer to see the effect of ifferet parameters especially the reegig a breakow parameter o the ifferet states of the server, the utiliatio factor a proportio of ile time, we compute some umerical results. We cosier the service time, vacatio time a repair time to be expoetially istribute to umerically illustrate the feasibility of our results. Further we assume that arrivals come oe by oe, i.e. E( I ) a E( I I ) with arrival rate λ, first stage service rate μ 4 a seco stage service rate μ 8 All the parameters are selecte such that the steay state coitio is satisfie. To moitor the effect of the reegig γ a breakow α o the behavior of the ueuig moel, we take λ, μ 4, μ 8, β, φ 7, p.5, while γ varies from 5, 8, 9 a α varies from to 4, i Table. Ope Access

10 M. BARUAH ET AL. 579 Table. Computig measures of ueue characteristics whe λ, μ 4, μ 8, β, φ 7, p.5. γ α ρ L L W W L W Figure. Effect of γ 5 a α o the mea ueue sie L a mea waitig time W. 3 4 L W Figure. Effect of γ 8 a α o the mea ueue sie L a mea waitig time W ρ Figure 3. Effect of reegig γ a breakow at α o the proportio of ile time a utiliatio factor ρ. Thus from Table, we observe that as we icrease the parameter of customer s impatiece reegig γ, for varyig values of breakow parameter α, the utiliatio factor ecreases, while the probability of server ile time icreases. Figures a show the effect of reegig a breakow o the mea ueue sie L, mea waitig time W. It is clear that as breakow occurs, it icreases the average legth of ueue a ue to customers reegig from the ueue, the average waitig time also ecreases. Agai from Figure 3, it is clear that ue to breakow of the system a reegig, the proportio of ile time of the server icreases a utiliatio factor or busy perio ecreases. The tres show by the above table are as expecte. Ope Access

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