Feedback Queue with Services in Different Stations under Reneging and Vacation Policies

Transcription

1 Internatonal Journal of Apple Engneerng Research ISSN Volume 1, Number (17) pp Research Ina Publcatons. Feebac Queue wth Servces n Dfferent Statons uner Renegng an Vacaton Polces Sunar Rajan B Research Scholar, Research & Development Centre, Bharathyar Unversty, Combatore, Taml Nau , Ina. Corresponng author Orc I: Ganesan V Assocate Professor (Ret.), Department of Statstcs, Peryar E.V.R. College, Truchrappall, Taml Nau - 63, Ina. Rta S Assocate Professor, School of Mathematcs, Department of Statstcs, Peryar Unversty, Salem, Taml Nau , Ina. Abstract Ths paper escrbes a sngle server queung system where the unts arrve n batches of varable sze uner Posson stream. The server proves servces n statons. The servce tmes n each staton follows general strbuton. The prncples of feebac, vacaton an renegng are employe n the system. The steay state probablty functons that the server s provng servce n any servce staton an that on vacaton are erve. The corresponng steay state probabltes are also obtane. The expecte number of unts n the queue has been obtane for some specal cases. Keywors: Sngle server, feebac queue, renegng, steay state probablty, vacaton INTRODUCTION An emergng area of queueng theory s the bul queue, n whch arrval an/or eparture can happen n batches ether fxe or n varable sze. Vtal applcatons of ths queueng moel can be seen mple n many areas le communcaton system, computer networs, proucton nustry, logstc sector etc. Queueng moels wth vacaton polces have been stue by many researchers nclung renegng. In real lfe, there are some queueng stuatons when some unt s scourage by long wats n the queue. The unts may be ece to balng or renegng. Balng an renegng have attracte the attenton of many researchers an stuy on queues wth behavor of such unts has evelope extensve amount of lterature. Concept of renegng n queueng systems was ntrouce by Ancer an Gafaran [1] an Daley []. Bae et al., [3] have stue the watng tme of M/G/1 queue wth mpatent customers. Meh [4] has explane a sngle server Posson arrval queue along wth a secon optonal channel. Altman an Yechal [5] have analyze customer mpatence n queues wth server vacaton. Chouhury an Meh [6] have stue balng an renegng n mult-server Marovan queueng systems. Vacaton polces n queues were extensvely stue prevously by Baba [7], Dosh [8], Kelson an Serv [9], Maan et al., [1, 11, 1], Borthaur an Chouhury [13, 14, 15]. Ganesan an Sunar Rajan [16] have stue bul arrval queue wth breaown analyss. Thangaraj an Vantha [17] have scusse two-stage heterogeneous servce, compulsory vacaton an ranom breaowns. Ganesan an Sunar Rajan [18] have analyze a queue wth heterogeneous servces an ranom breaowns. Ayyappan an Sathya [19] have stue three stage heterogeneous servce an server vacaton of M x /G/1 feebac queueng moel as well as two types of ranom breaowns an multple vacatons wth restrcte amssblty []. Monta Baruah et al.,[1] have stue bul nput queue whch have secon optonal servce along wth renegng urng vacaton. The present stuy eals M x /(G 1, G, G )/1 queueng moel, where the unts arrve n batches of varable sze efne by Baley [] an once the unts enter the ntal servce process, t must go through statons of servce on frst n frst out (FIFO) scplne. The servce tme of ths moel follows general strbuton. The unt follows Bernoull feebac after completon of statons of servce an f t has not receve a qualty servce, the unt wll rejon at the en of the queue wth probablty p or leaves forever from the system wth probablty (1-p). Whenever the system s empty, the server may go on vacaton for a ranom uraton. Here, the server aopts multple vacaton polcy untl at least one batch of arrval present n the system. The vacaton peros of the server are strbute accorng to general strbuton. It s assume that when the server s on vacaton the unt may renege from queue an t follows exponental strbuton wth parameter β

2 Internatonal Journal of Apple Engneerng Research ISSN Volume 1, Number (17) pp Research Ina Publcatons. ASSUMPTIONS AND DESCRIPTIONS OF THE QUEUEING MODEL Ths queueng moel s escrbe uner the assumptons are as follows: Unts arrve n batches of varable sze an follow Posson stream. Let a θ t (θ = 1,, n) be the frst orer probablty that a batch of unts arrve to the system urng the small nterval (t, t + t), where a θ 1, j =1 a θ = 1 an > s the mean arrval rate of batches. Each unt unergoes statons of heterogeneous servce prove by a sngle server on a frst n frst out (FIFO) scplne. The servce tmes of statons follow fferent general strbutons functon B (V) an ensty functon b (V) B (V), = 1,, assumes that they have fnte moments E (s m ) for m>1 an = 1,,.. After completon of statons of servce, f any unts requre repeatng ts servce for any reason or for unsuccessful servce, the unt rejons at the en of the queue. Servce tme for a feebac unt s nepenent of ts prevous servce tme. Let s x x be the contonal probablty of completon of the th staton of servce urng the tme nterval, (x, x + x) gven that the elapse tme s x, so that s x = an b x 1 B x, = 1,,... (1) b x = s γ e s x x, = 1,,... () The unts may renege when the server s on vacaton. Ths renegng follows exponental strbuton wth parameter β. Thus f t = βe βt t, β > (3) Let t be the probablty that a customer can renege urng a short nterval of tme (t, t + t) After completon of servce, f there s no unt watng n the system, then the server goes for vacaton wth ranom uraton. On returnng from vacaton, the server nstantly starts servcng epenng on the avalablty of the unts. Otherwse, f no unt s watng n the system, then t goes for vacaton agan. The server contnues to go for vacaton untl t fns at least one batch n the system. The vacaton tme follows a general strbuton wth strbuton functon V(x) an ensty functon v(x) Laplace-Steltjes Transform (LST) V(t) an fnte moments E(V m ), m>1 v x x = 1 V x, = 1,, (4) an V v = β v e β x x (5) Inter-arrval tmes, servce tmes of each staton of servce an the vacaton tmes are nepenent. The steay state probabltes are efne as, () (x): Steay state probablty that, the server s provng servce n the th staton an there are n (>) unts n the queue exclung the one beng serve. V n(x): Steay state probablty that, the server s on vacaton an there are n (>) unts n the queue. Formulaton of steay state equatons Base on the postulates gven as above, the lmtng behavor of ths queung process at the statonary pont of tme can be one by usng below mentone Kolmogorov forwar equatons x P n x + x + s x x = n θ=1 a θ θ x, n 1 x P o x + + s x P o x =, = 1,, (7) x V n x + ( + x + β ) V n (x) = x V o x + ( + x ) V o x = n x x a θ V n θ + βv n+1 (6) θ=1 (8) The bounary contons for solvng the above equatons (6) to (8) are state as 1 = = q Vn+1 x γ x x + p = 1 x s j 1 x x s x x + q = +1 (9) x s x x, n (1) x, n =,1,,3, =, 3 4, (11) V n o = (1) V o = V x γ x + q = P o x s x (x) (13) Let(x) x be the contonal probablty of a completon of a vacaton urng the nterval (x, x + x), gven that the elapse vacaton tme s x, so that 11966

3 Internatonal Journal of Apple Engneerng Research ISSN Volume 1, Number (17) pp Research Ina Publcatons. Probablty Generatng Functon Let us efne the probablty generatng functons as follows P (x, z) = n= x z n P () x, z s x x = P (), z B In the same manner, the equaton () s reformulate as (4) P () z = n= z n V(z) = V(, z) [ 1 V A z +β β z ] (14) A z +β (14) β z V x, z = n= V n x z n V z = n= V n z n, z 1, x > Soluton of steay state equatons Multply equatons (11) an (8) by z n an tang summaton overall possble values of n an on smplfcaton we get, x P() x, z + ( A(z) + s (x)) P () x, z = (15) x V x, z + + x + β β z V x, z = (16) We multply equatons (1),(11) an (1) wth approprate powers of z an summng over sutable values of n, an get, zp 1, z = P (), z = V x, z γ x x + q + pz P q = = P x s x x x, z s x x V x, z γ x x P ( 1) x, z s 1 x x, =,3, (17) (18) an (5) V x, z γ x x = V(, z) V ( + β β z ) (6) V ( + β β z ) = e ( A z +β β z ) Usng (18) an (4), we have P, z = 1 D z 1 j =1 D z = z q + pz { B j } V(x) {V ( + β β z ) 1}]V (), = 1,, = j =1 B j (7) By applyng (7) n (3) an get the probablty generatng functons of the th staton of servce. V(, z) = V () (19) Now, use the equatons (13) an (17) an get, zp (1), z = V x, z γ x x + q + pz P = x, z s x x V () () By usng the prncpal of lnear fferental equaton n the equatons (15) an (16) whch prove, P () x, z = P () ( A z )x, z e s t t (1) V(x, z) = V(, z) e ( A z +β β z )x γ t t () Now, ntegrate equaton (1) wth respect to x an get, P () z = P (), z [ 1 B A z A z B = e A z x ] B x (3) s the Laplace transform of the servce tme n th staton. Agan, apply smple mathematcs an usng the expresson () n the equaton (1), whch becomes P z = 1 D z j =1 B j V 1 B V + β β z 1, = 1, (8) Smlarly, substtute the equaton (19) n the expresson (5) an get, 1 V + β β z V z = [ + β β ]V z (9) The probablty generatng functon of the whole system s obtane by usng the relatons (8) an (9), P z = V z + z P () z =1 (3) The computaton of expecte queue sze s very ffcult snce the expresson (3) s the sum of proucts of Laplace Transform of the probablty strbuton functons. In ths juncture, the number of servce statons s reuce an some specal cases are further scusse

4 Internatonal Journal of Apple Engneerng Research ISSN Volume 1, Number (17) pp Research Ina Publcatons. The steay-state probabltes that the server s provng th staton of servce an that the server s on vacaton are respectvely enote as P () (1) an V(1) On applyng z=1 n the expressons (8) an (9), whch become netermnate form. So use L Hosptal s rule an get the requre steay-state probabltes after usng some mathematcal applcatons, P () E (s )E(V) 1 = [ 1 q p =1 E(s E(s ) ]V an V 1 = E(V)V (31) (3) In orer to fn the value of V o(), apply normalzng conton, =1 P () 1 + V 1 = 1 (33) Now, use the expressons (31) an (3) n the conton (33) an get q p V = E(s Specal Cases 1 =1 ) E(s ) 1 E V)[ q P =1 E(s ] (34) Case (): The number of servce statons are assume as =3 Suppose, the number of servce staton s taen as three an the equaton (3) reuces to P z = V A z +β β B z 1 A z B A z B 3 A z [ z q+pz B 1 A z B A z q+pz B 1 A z B A z B 3 A z V (35) Dfferentatng P(z) w.r.t. z an lettng z=1 an get n etermnant form. So L Hosptal s rule s apple an get the followng results whch lea to expecte queue sze. N 1 = q A 1 + β E(V) (36) N 1 = q A 1 + β E V + E V A 1 β + A 1 + β A 1 E s 1 + E s + 1 E s 3 (37) D 1 = q p A 1 E s 1 + E s + E s 3 (38) D 1 = pa 1 E s 1 + E s + E s 3 + A 1 E s 1 + E s + Then A 1 E s 1 E s + A 1 E s 1 E s + E s 1 E s 3 + A 1 E s 3 + A 1 E s 3 ) p A 1 E s E s 3 L q = D 1 N 1 N 1 D 1 D 1 ( V ) (39) = q A 1 + β E V Where, + qe V A 1 β + A 1 A 1 + β + A 1 + β pa A A V = q p A 1 E s 1 + E s + E s 3 = E s 1 + E s + 1 E s 3 3 = E s 1 + E s + E s 3 4 = E s 1 + E s 5 = E s 1 + E s + E s 1 E s + E s 1 E s 3 an 6 = A 1 E s 3 + A 1 E(s 3 ) P A (1) E s E s 3 (4) Case (): Suppose the vacaton tmes follow Erlang strbuton wth shape parameter a an scale parameter b. Its mean an varance are respectvely a/b an a/b.then, the requre expecte queue sze s gven by L q = q A 1 + β a b 1 + a + q a A 1 β + A 1 A 1 + β b + A 1 + β pa A A V 6 (41) Case (): The servce tmes of the three servce statons are strbute exponentally wth same parameter. The vacaton tmes follow general strbuton. Then the expecte queue sze gven n (4) s reuce as L q = q A 1 + β E V { q p 5A 1 } + qe V A 1 β + 5A 1 A 1 + β q p 5A 1 + A 1 + β 8pA 1 + 3A A 1 + A 1 + (3 + p A 1 ]]{(q p) 5A 1 } ( V ) (4) Case (v): The unts are arrvng one by one an renegng s not allowe. Then the system s reuce as M/(G 1, G, G 3)/1 moel an ts expecte queue sze s obtane by usng A(z)=z an β=, as follows L q = qe V + qe V [ + p ] 1 V (43) = q p {(E s 1 + E s ) + E s 3 } an = E s 3 + (1 + p)e s E s

5 Internatonal Journal of Apple Engneerng Research ISSN Volume 1, Number (17) pp Research Ina Publcatons. CONCLUSION In ths paper, the steay state results of a sngle server queue wth some concepts, namely, feebac, vacaton, renegng an batch arrval are mathematcally erve. The expecte queue szes n fferent stuatons are scusse. Suppose one may ntrouce Erlang arrval an/or bul servce, then the researcher may get fferent types of valuable results. Ths queung moel an erve results can be use n communcaton networs an nustres. REFERENCES [1] Ancer, Jr.C.J an Gafaran, A.V, Some queung problems wth bulng an renegng. Operatons Research,1,1(1963), [] Daley, D.J, General customer mpatence n the queue G 1/G/1, Journal of Apple Probablty (1965), [3] Bae, J,Km,S an Lee, E.Y, The vrtual watng tme of the M/G/1 queue wth mpatent customers, Queung systems: Theory an Applcaton, 38,(1), [4] Meh,J,A sngle server Posson nput wth a secon optonal channel, Queung systems, 4,(), [5] Altman, E an Yechal, U, Analyss of customers mpatence n queues wth server vacaton, Queung system, 5, (6), [6] Chouhury, A an Meh, P, Bulng an renegng n mult-server Marovan Queung systems, Internatonal Journal of Mathematcs n operatonal research, 3, (11), [7] Baba, Y, On the M x /G/1 Queue wth vacaton tme, Operatons Research Letters 5,(1986), [8] Dosh, B.T, Queung systems wth vacatons A survey. Queung Systems, 1, (1986), [9] Kelson, J an Serv, L.D, Dynamc of the M/G/1 vacaton moel. Operatons Research, Vol 35, 4, (1987), [1] Maan, K.C, An M/G/1 Queung system wth compulsory server vacatons, Trabajos e Investgaton, 7, (199), [11] Maan, K.C an Abu-Dayyeh, W, Restrcte Amssblty of batches nto an M x /G/1 type bul queue wth mofe Bernoull Scheule server vacatons, ESSAIM: Probablty an Statstcs, 6, (), [1] Maan, K.C an Chouhury, G, An M x /G/1 Queue wth a Bernoull Vacaton Scheule uner restrcte amssblty polcy, Sanhya: The Inan Journal of Statstcs. 66(1),(4), [13] Borthaur, A an Chouhury, G, On a batch arrval Posson queue wth generalze vacaton. Sanhya Ser.B, 59,(1997), [14] Chouhury, G an Borthaur, A, The stochastc ecomposton results of batch arrval Posson queue wth a gran vacaton process, Sanhya. Ser.B, 6(3), (), [15] Chouhury, G A, batch arrval queue wth a vacaton tme uner sngle vacaton polcy, Computers an Operatons Research, 9(14), (), [16] Ganesan, V an Sunar Rajan, B, Bul Arrval Queue wth Breaown Analyss, IAPQR Transactons. 33(), (8), [17] Thangaraj, V an Vantha, S, M/G/1 Queue wth two-stage heterogeneous servce compulsory server vacaton an ranom breaowns, Int. J. Contemp, Math. Scences, 5, (1), [18] Ganesan, V an Sunar Rajan, B, A queue wth heterogeneous servces an ranom breaowns, Internatonal Journal of Mathematcs an Apple Statstcs. (), (11), , [19] Ayyappan G an Sathya K, M x /G/1 feebac queue wth three stage heterogeneous servce an server vacatons havng restrcte amssblty, Journal of Computatons an moelngvol.3. No.,(13), 3-5. [] Ayyappan G, Sathya, K an Muthu Ganapathy Subramanan A, M X /G/1 queue wth two types of breaown subject to ranom breaowns, multple vacaton an restrcte amssblty, Apple Mathematcs Scences, Vol.7, No.53, (13), [1] Monta Baruah, Maan K.C an Tllal Elab. A batch arrval queue wth secon optonal servce an renegng urng vacaton peros, Revsta Investgatons Operatonal. Vol.34, No.3,(13), [] Baley, N.T.J., A contnuous tme treatment of a smple queue usng generatng functons, J. Roy. Statst. Soc. Vol. B16, 1954, pp