Optimal design of N-Policy batch arrival queueing system with server's single vacation, setup time, second optional service and break down

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1 Ameria. Jr. of Mathematis ad Siees Vol., o.,(jauary Copyright Mid Reader Publiatios Optimal desig of -Poliy bath arrival queueig system with server's sigle vaatio, setup time, seod optioal servie ad break dow T. Poogodi & S. Aatha Lakshmi Departmet of Siee ad Humaities, Faulty of Egieerig, Aviashiligam Deemed Uiversity, Coimbatore M.I. Afthab egum Departmet of Mathematis, Aviashiligam Deemed Uiversity, Coimbatore ASTRACT This paper deals with a M [] /(G,G / queueig system i whih all the ustomers udergo first essetial servie (FES ad oly some of them reeive seod optioal servie(sos by the same server. I additio to this the server is ureliable ad hee subjeted to radom breakdows while i servie ad the server leaves for sigle vaatio whe the system is empty. After returig from vaatio if there are or more ustomers i the system the the server does setup work before it starts the servie. Expliit aalytial expressios for various performae measures are derived. A ost model for the optimal operatig - Poliy that miimizes the total expeted ost per uit time is determied ad umerial aalysis is arried out. Mathematis subjet lassifiatio: 6K5 Keywords: M [] /(G,G / queueig system, sigle vaatio, -Poliy, setup time, breakdow, optimal ost model. Itrodutio I day-to-day life, oe eouters umerous examples of queueig situatios, where all arrivig ustomers require the mai servie ad oly some may require the subsidiary servie provided by the server. K.C.Madha [] has doe some iitial work o the stea state behaviour of M/G/ queue with seod optioal servie ad later Choudhry ad Paul [] exteded the results of Madha [] to a bath arrival queue uder - poliy. The authors metioed above have foused o reliable servers. Queueig models with seod optioal servie ad breakdows aommodate real world situatios more losely. Therefore, it would be pratial to osider the -poliy for the bath arrival M [] /G/ queueig system i whih the servie is ureliable ad all the arrivig ustomers demad the first essetial servie (FES where as oly some of them demad the seod optioal servie (SOS. There are several vaatio poliies ad this paper deals with sigle vaatio poliy. Also the server eeds a radom amout of time for preparatory work whih is termed as server s setup time (or startup time. I this model, it is assumed that after returig from vaatio if the server fids (or more ustomers i the system the the server is tured o for the start up work of radom legth D ad as soo as the server fiishes the setup work, the busy period iitiates. 5

2 T. Poogodi & S. Aatha Lakshmi & M.I. Afthab egum The stea state behavior of queue size distributio is aalyzed for this model, usig the supplemetary variable tehique. Various performae measures suh as expeted system size ad expeted legth of the yle are also alulated. The PGF of the system size distributio at a arbitrary epoh is derived. The total expeted ost futio per uit time is derived ad the optimal value s* whih miimize the log ru average ost uder liear ost struture is developed for this model. umerial aalysis is also preseted to justify the measures alulated for the model. Model Desriptio The M [] /(G, G / Queueig System uder osideratio has the followig speifiatio. Compoud Arrival Proess The ustomers are assumed to arrive i bathes aordig to ompoud Poisso proess with arrival rate. The umber of uits arrive at a arbitrary istat is a radom variable whose probability distributio is give by Pr( k g k, k,, 3 -Poliy Setup Time ad Sigle Vaatio A yle begis right after the system beomes empty ad the server leaves the system for vaatio. After returig from vaatio, if the server fids (or more ustomers preset i the system the the server takes radom amout of setup time for preparatory work before startig the servie. The setup time is a radom variable with fiite momets whih has the geeral distributio D(t ad desity futio d(t. The ustomers arrivig durig the vaatio period ad setup period will joi the queue ad wait for their turs. If the server returig from vaatio fids less tha ustomers i the system the he stays idle i the system (i.e. the server takes oly sigle vaatio. The period durig whih the server remais idle i the system to start the preparatory work after returig from vaatio is alled build up period. The vaatio time follows the geeral distributio V(t with fiite momets ad desity futio v(t. usy Period ad Server s reakdow Immediately after the setup time the busy period starts ad ustomers are served oe by oe aordig to FCFS queue disiplie. Durig busy period the server provides eah uit two types of heterogeeous servies of whih, oe is optioal. (i.e. the server provides first essetial servie (FES to all the arrivig ustomers ad after the ompletio of FES the ustomers may leave the system with probability ( r (or may opt for the SOS with probability r ( r. Durig the servies (FES (or SOS the server may udergo breakdows aordig to the Poisso proess with rates a i, i,. Wheever the breakdows our the server is set immediately for repair ad the repair times follow the geeral distributios U i (t, i, with fiite mea ad variae. Oe the server gets repaired, he is set bak to the servie faility to resume the servie. Thus the vaatio period, setup period, busy period ad break dow periods ostitute a yle. It is also assumed that the arrival proesses, vaatio time, servie time ad setup time are idepedet of eah other. The model desribed above is depited i figure I. Stea State System Size Equatios To obtai the stea state system size equatios of the model usig supplemetary variable tehique, we employ the remaiig servie time, the remaiig setup time ad the remaiig vaatio time of the server as the supplemetary variables. The followig otatios ad probabilities are used to derive the stea state equatios of the model. - threshold - group arrival rate - arrival size radom variable g k - Pr( k, k,,3 5

3 Optimal desig of -Poliy bath arrival queueig system α k (h k - probability that k ustomers arrive durig a vaatio (setup time, k,, (z - the probability geeratig futio (PGF of (i ( g k - i-fold ovolutio of {g k } with itself ad g a i - Poisso breakdow rate orrespodig to the i th phase of servie, i, U i - repair time orrespodig to the breakdow a i. (t - the system size iludig oe i servie at time t. Let the abbreviatios C.D.F, p.d.f, LST respetively deote umulative desity futio, probability desity futio, Laplae Stieltjes trasform of the radom variables (R.V.. R.V C.D.F p.d.f LST FES S S (t s (t S (θ SOS S S (t s (t S (θ Vaatio time V V(t v(t Setup time D D(t d(t V D Repair time U i U i (t u i (t U i (θ Remaiig time of the R.V. at t S (t S (t V (t D (t U i (t, i, The server s states are deoted by the radom variable Y(t -,,,, 3 4, 5, if the server is idle durig buildup period if the server is o vaatio if the server is busy with FES if the server is dow with a ustomer i FES,if the server is busy with SOS if the server is dow with the ustomer i SOS if the server is i the setup state at time t FIGURE I 53

4 T. Poogodi & S. Aatha Lakshmi & M.I. Afthab egum The state of the system at time t a be desribed by the Markov proess. K(t {(Y(t, (t, S i (t, U ( t i, V (t, D (t, t, i, } The trasiet state system size probabilities are defied by : R (t Pr(Y(t, (t, to Q (x,t dt Pr{Y(t ; (t, x V (t x + dt}, P (x, t dt Pr((t, x S (t x + dt, Y(t, P (x, t dt Pr((t, x S (t x + dt, Y(t 3, (x, y, t dt Pr((t, S (t x, y U (t y + dt, Y(t, (x, y, t dt Pr((t, S (t x, y U (t y + dt, Y(t 4, D (x, t dt Pr((t, x D (t x + dt, Y(t 5, Stea State Equatios Uder the stea state, the system size probabilities are assumed to be idepedet of time ad the stea state equatios are give by, R Q ( ( R Q ( + Rk g k, k ( P (x ( + a P (x + ( r P ( s (x + (x, + P ( s (x (3 P (x ( + a P (x + ( r P + ( s (x + (x, + P + ( s (x + P (x g k, P (x ( + a P (x + ( r P + ( s (x + (x, k k + D ( s (x+ P + ( s (x + P k(x g k, (5 k P (x ( + a P (x + r P ( s (x + (x, (6 k P (x ( + a P (x + r P ( s (x + P (x k + (x,, (7 Q (x Q (x + (P ( ( r + P ( v(x (8 Q (x Q (x + Qk (x g k, (9 k g k (4 54

5 Optimal desig of -Poliy bath arrival queueig system D (x D (x + Q ( d(x+ Rk g k d(x, ( k D (x D (x + Q ( d(x + Rk g k d(x + Dk g k, k + k + ( (x, y (x, y + a P (x u (y ( k (x, y (x, y + a P (x u (y + k(x, y g k, (x, y (x, y + a P (x u (y (4 (x, y (x, y + a P (x u (y + (x, y g k, (5 k k (3 have, LST defiitio The followig Laplae Stieltjes Trasform (LST are defied to derive the PGF of the system size P i i ( θ, y D Q θx S i e P i (x ; θx e i (x, y, i, θx e D (x ; D (θ θx e Q (x ; V (θ e e θx e d S i (x, i, θx θx d D(x d V(x Takig the LST o both sides of equatios (3 to (5 ad usig the properties of LST of differetiatio, we θ P P ( ( + a P ( r P ( S ( θ, P ( S θ P P ( ( + a P ( r P + ( S ( θ, k P + ( S P g k, (7 k θ P P ( ( + a P ( r P + ( S ( θ, k D ( S P + ( S P g k, θ P P ( ( + a P r P ( S ( θ, k (6 (9 (8 55

6 T. Poogodi & S. Aatha Lakshmi & M.I. Afthab egum θ P ( θ P ( ( + a P r P ( ( S θ k P g k ( θ,, k θ Q Q ( Q (P ( ( r + P ( V (θ ( θ Q Q ( Q Qk g k, ( k θ D D ( D Q ( D (θ Rk g k D (θ (3 k θ D D ( D Q ( D (θ Rk g k D (θ D k g k, k + k + (4 (θ, y (θ, y (θ, y (θ, y (θ, y + a P (θ u (y (5 (θ, y + a P (θ u (y + ( θ, y k k g k, (θ, y+ a P (θ u (y (7 (θ, y + a P (θ u (y + ( θ, y The LST with respet to repair time are defied by, i ( θ, θ Ui θ y e i ( θ, y e θ y u i (y k Takig the LST o both sides of equatios (5 to (8 we have, θ θ θ θ (θ, θ (θ, (θ, θ (θ, (θ, θ (θ, (θ, θ (θ, (θ, θ a (θ, θ a k( θ, θ k (θ, θ a (θ, θ a k( θ, θ k k g k, (6 (8 ( P (θ U (θ (9 P (θ U (θ g k, (3 P (θ U (θ (3 P (θ U (θ g k, (3 56

7 Optimal desig of -Poliy bath arrival queueig system PGF of the system size probabilities The followig partial PGFs for z are defied to determie the system size distributio. R(z R z P i (z, θ D (z, θ Q (z, θ i P z, P i (z, D z, D(z, Q z, Q(z, Pi ( z, i, D ( z Q ( z i (z, θ, θ i ( θ, θ z i * * * * * * (z, θ, Idetities d dz d dz i ( θ, z, i, Here some importat idetities used i this paper are listed out. z z z z P k( θ gk k P k( θ gk k k( θ, θ gk k k + R k gk + D V (z P P 57 z (z, θ (z P P z (z, θ (z ( θ, θ θ k z (z, θ, (z z R k k g R(z (z z D ( V W(z z k R z E( [E(D E( k E(V k k g k z k g k z k k g k z k g k z + E(D E(V + E(V + E(V E(D ]

8 T. Poogodi & S. Aatha Lakshmi & M.I. Afthab egum * d dz D W(z z E( E(D Stea state solutios The losed form expressios after extesive simplifiatio for D (z, θ ad Q (z, θ are as follows. P ( (V Q V (z, θ θ W (z (33 D (z, θ (D D θ W (z (P ( V R(z W (34 Usig the equatios (9 ad ( ad (9 to (3, we get i (z, θ, i (z, θ, θ P i a i (z, θ U a i i i P i (z, θ (U U i (θ, i, (36 θ W (z (35 P (z, θ z(p ( (D V (H a D (z H ( θ h S a R(z W (37 P (z, θ r z H a (P ( (D D R(z W (z H ( θ h V a (H a S where P ( P ( ( r + P (, W (z (, H (z H a (( r + r H a, H a i S i ( h a i ad (38 h a i W (z + a i ( i U, i, The equatios (33, (34,(36, (37 ad (38 at θ θ respetively give, Q (z, D (z, ( V W ( D P ( (z (P ( V W (z R(z W 58

9 Optimal desig of -Poliy bath arrival queueig system i (z,, ai Pi (z, (Ui W (z, i, P (z, z(p ( (D V (z H D (H (h a a R(z W P (z, r z H Let P (z i a (P (P ( (D D i (, h a R(z W V (z H z + i ( z,, (H a zφ(z(h The P (z W(z (zh where φ(z P ( (( D V + D R(z W Let P Ι (z gives the PGF of the system size probabilities whe the server is idle. The P Ι (z Q (z, + D (z, + R(z P Ι (z ( V W P ( (z + ( D (P ( V W (z R(z W + R(z To alulate R(z,for, let Π ad Π gi Π -i ; Ψ α ad Ψ α i Π i i i Usig equatios( ad ( we get P R(z ( z P ( Ψ(z, where Ψ(z φ(z The P Ι (z W(z If P(z deotes the total probability geeratig futio of the umber of ustomers i the system i stea state, the, P(z P (z + P Ι (z. φ(z (zh (z, where φ(z ivolves the ukow P ( ad this a be alulated W(z(zH usig the ormalizig oditio P( Ad P ( ρ E( D + E( V + where ρ E( E(H, E(H E(S ( + a E(U + r E(S ( + a E(U Substitutig for P ( i P(z we have, 59

10 P(z ( ρ (z H (z zh (z T. Poogodi & S. Aatha Lakshmi & M.I. Afthab egum D ( W ( z V ( W W ( z ( z + D ( W E( D + E( V + ( z z Performae Measures I this setio, the probability that the server is o vaatio (P V, i busy period (P usy, i breakdow state (P r ad i setup state (P D are alulated. i. P V the probability that the server is o vaatio E(V P (, where P ( P ( ( r + P ( ii. P usy the probability that the server is busy E( (E(S + r E(S ρ usy, iii. P r the probability that the server is i break dow state lt (z,, + (z,, ( z P r E( (E(S a E(U + r E(S a E(U ρ r ote that P usy + P r E( E(H ρ iv. P D the probability that the server is doig his setup work E(D P ( Mea system size Let L deote the expeted system size of the ureliable M [] /G/ queue with -poliy, sigle vaatio ad setup time. The L d dz y alulatio, L + E( E(D Expeted Cyle Legth P ( z z ( E( E(H + + E(( E(H (ρ (E(D + E(V + + E( (E(D + ρ + E(DE(V + E(V Let E(T, E(, E(T, E(r, E(D ad E(C represet the expeted idle period, expeted busy period, expeted yle, expeted break dow period, expeted setup period ad the expeted ompletio period respetively.the the log-ru fratio of time the server is idle ad busy are give by,. E(T P Ι E(V P ( E(Ty E( E(T y P usy ρ usy 6

11 Optimal desig of -Poliy bath arrival queueig system E(r E(T y E(D E(T y P r ρ r P D E(D P ( From the above alulatios E(T y P ( E D + + ( E(V ρ The E(C Expeted ompletio period E( + E(r (ρ usy + ρ r E(T y ρ ρ + + E( D E(V Optimal Desig of -poliy with Sigle Vaatio I this setio, the total expeted ost futio per uit time for the model to alulate the optimal value S whih miimizes the liear ost futio is developed. A ost struture that has bee widely used i literature is employed. Let C y - tur o ost per yle C h - holdig ost per uit time C D - setup ost per uit time C V - reward per uit time due to vaatio C usy - operatig ost per uit time C r - breakdow ost per uit time T ( - total average ost per uit time C Ι - idle ost per uit t The T ( [ C y + CD E(D + CΙ E(T CV E(T y E(V] + C h L + C usy P usy + C r P r C h L ρ [E(D + E(V] + + E( E(D C + (E(D + E(V + y + + CD E(D CΙ CV E( + (E(D + E( D E(V + E(V E(V + + C usy ρ usy + C r ρ r 6

12 The T ( T. Poogodi & S. Aatha Lakshmi & M.I. Afthab egum A+ ( ((ρ C where A C usy ρ usy + C r ρ r + C h L A Ι + E(E(D C ( ρ (C y + C D E(D C V E(V + C h C (E(D + E(V + C h E( + C h + A (E(D + E(D E(V + E(V ad L ρ + ( E( E(H + E(( E(H ( ρ Theorem Let S be the optimal threshold value of that miimizes the average ost per uit time T ( uder the ost struture metioed. The S of the model is give by, S mi k / ( (E(D + E(V (C usy (ρ + E( E(D Ch + k Ch k + Ch (k > A Proof y alulatio, T S ( k+ T S (k k (h(k, C k Ck+ where h(k A + (E(D + E(V (C usy ( ρ + E( E(D C h + k C h + C h The sig of h(k determies whether T (k ireases (or dereases. k (k If k be the first iteger suh that h(k >, the h(k + h(k + C h ( (E(D + E(V + C h Ψ > This implies h(k + > wheever h(k > Therefore S first k,for whih h(k > (i.e. S mi {k>/h(k>} 6

13 Optimal desig of -Poliy bath arrival queueig system umerial results are provided for, UMERICAL AALYSIS (i expeted umber of ustomers waitig i the system ( L (ii the optimal value of (iii the total expeted ost per uit time (T ( (iv expeted busy period E( (v expeted legth of the yle E(T y for the model. For the omputatio work of the model, we make the followig assumptios: The bath size follows the geometri distributios (i.e. g k Pr ( k ( p p k, k, with mea E( p The servie time S of eah stage follows two-stage hyper expoetial distributios whose measures are the followig : The mea of S i, i, are E(S a µ + a The seod order momet of S i, i, are E (S a a + µ µ ad E (S µ ad E(S b µ b b + µ µ + b µ The set-up time D ad vaatio V follow Erlag 3-type distributios with mea E(D, E(V ad the γ η 4 4 seod order momets E(D ad E(V 3γ 3η. The repair time follows expoetial distributio with parameters β i, i, C h, C D 5, C y, C V, C usy, C r 5. The parameters a i, b i, i, are the same for all the tables. a., a.5, b, b 3. I Table (, it is verified that as the arrival rate ireases, the optimal expeted system size ad the total expeted ost also irease. The expeted legth of the yle ad the orrespodig utilizatio fator ρ ad ρ h are also give i the table for partiular values of p, γ ad η. 63

14 T. Poogodi & S. Aatha Lakshmi & M.I. Afthab egum Table p., γ.3, η. s E( T Cy L T C ( ρ ρ h I queueig models, it is obvious that, if the mea servie time is redued, the expeted queue size will be redued. E(S E(S + r E(S a a + µ + r µ b b + µ a be redued by ireasig ay oe of µ the µ ij values. The to justify the statemet, that queue legth is a ireasig futio of the mea servie time E(S, for ireasig values of µ ij the measures are alulated. The optimal total ost of the system ad the expeted busy period are also preseted i eah table. The value of r is fixed i tables ( to (5 as r.3. E(S 7, p. Table µ.5, µ, µ.5 µ E(S S E( L T C ( ρ ρ h E(S.866 Table 3 µ.5, µ, µ.5 µ E(S E( S L T C ( ρ ρ h

15 Optimal desig of -Poliy bath arrival queueig system It is show i Tables (4 ad (5 that the expeted queue legth a be dereased by reduig the setup time E(D ad reduig the vaatio time E(V. The fixed values of.5 ad p. are osidered to justify the result. Table 4 η. γ E(D S L ( T C Table 5 γ.3 η E(V S L ( The values of Table(6 give the optimum value of ad the miimum optimal expeted ost for the - poliy queueig model orrespodig to differet ost struture. Table 6 (γ, a, a, b, b, µ, µ, µ, µ, η (.3,.,.5,, 3,.5,., 6,.5,. Case : C h 5, C D 5, C usy (, p (.,. (.5,.3 (.,.6 (.5,.9 (.3,.3 T C S ( T C Case : C h 5, C D 5, C usy 5 (, p (.,. (.5,.3 (.,.6 (.5,.9 (.3,.3 T C S ( Case : 3 C h 5, C D 75, C usy (, p (.,. (.5,.3 (.,.6 (.5,.9 (.3,.3 T C S (

16 T. Poogodi & S. Aatha Lakshmi & M.I. Afthab egum Case : 4 C h 35, C D, C usy 5 (, p (.,. (.5,.3 (.,.6 (.5,.9 (.3,.3 T C S ( The followig table values justify the proedure (give i the theory of fidig the optimal values of for this model. The parametri values assumed to obtai the optimal values are metioed i the followig table. Table 7 p.3, a.5, a, b.5, b 3.5, µ, µ.5, µ 3, µ 4,.5, C h 5, C D 4, C usy, C y 5, C V 5, C r 8 L ( T C Colusio This is a extesio of the work o o- Markovia queueig system ombiig - Poliy with setup time ad vaatio,arried out by several researhers iludig Medhi ad Templeto [5], Mih [6], Lee ad Park [4], Lee et al. [7], Hur ad Paik [8]. ut these authors have foused oly o reliable servers. I this paper, for the o- Markovia ureliable queueig system with - Poliy,seod optioal servie,setup time ad vaatio,the PGF of the system size is preseted i losed form.further, various performae measures are derived ad the umerial aalysis is arred out to verify that the mea system size is a ireasig futio of arrival rate, dereasig futio of servie rate µ, setup rate γ ad vaatio parameter η. Further the optimal value of that miimizes the total ost is alulated umerially for various parametri values ad the proedures give above are justified. ibliography:. Mada, K.C., (, A M/G/ queue with seod optioal servie, Queueig System 34 : Choudhury, G. ad Paul, M., (6, A bath arrival queue with a seod optioal servie hael uder -poliy, Stohasti Aalysis ad Appliatios, 4 : Lee, H.S. ad M.M. Sriivasa, (987, Cotrol poliies for the M [x] /G queueig system, Teh Report, 5 (6,

17 Optimal desig of -Poliy bath arrival queueig system 4. Lee, H.W. ad J.O. Park, (997, Optimal strategy i -poliy produtio system with early setup, Joural of the Operatioal Res. Soiety, 48, Medhi, J. ad J.G.C. Templeto, (99, A poisso iput queue uder -poliy ad with a geeral startup time, Computers ad Oper. Res., 9, Mih, D.L., (988, Trasiet solutios for some exhaustive M/G/ queues with geeralized idepedet vaatios, Europea Joural of Oper. Res., 36, Lee, S.S., Lee, H.W. ad Chae, K.C., (995, O a bath arrival queue with -poliy ad sigle vaatio, Comput. Oper. Res.,, Hur, S. ad S.J. Paik, (999, The effet of differet arrival rates o the -poliy of M/G/ with server set up, Applied Mathe Modellig, 3,

Construction of Control Chart for Random Queue Length for (M / M / c): ( / FCFS) Queueing Model Using Skewness

Construction of Control Chart for Random Queue Length for (M / M / c): ( / FCFS) Queueing Model Using Skewness Iteratioal Joural of Sietifi ad Researh Publiatios, Volume, Issue, Deember ISSN 5-5 Costrutio of Cotrol Chart for Radom Queue Legth for (M / M / ): ( / FCFS) Queueig Model Usig Skewess Dr.(Mrs.) A.R. Sudamai