M/M/1 queueing system with server start up, unreliable server and balking

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1 Research Joural of Mathematical ad Statistical Scieces ISS Vol. 6(2), -9, February (28) Res. J. Mathematical ad Statistical Sci. Aalysis of a -policy M/M/ queueig system with server start up, ureliable server ad balkig Abstract. Jayachitra Departmet of Mathematics, Hidustha Istitute of techology, Coimbatore, Tamil adu, Idia jayaasasi_akshu@rediffmail.com Available olie at: Received 24 th December 27, revised 3 th Jauary 28, accepted 5 th February 28 This paper is cocered with the optimal cotrol of a o reliable ad removable server i a -policy M/M/ queueig system with server start up ad balkig. Whe the queue legth reaches ( ), the server is immediately tured o but is temporarily uavailable to serve the waitig uits. The server eeds some startup time before providig the service. It is assumed that server breakdows accordig to a oisso process ad the repair time have a expoetial distributio. Arrivig customers may balk with a certai probability ad may depart without gettig service due to impatiece. ack of service occur where the server is busy or due to sudde breakdow. Explicit expressio for the umber of uits i the system are obtaied ad also derived various system measures. The total expected cost fuctio is developed to determie the optimal threshold at a miimum cost. The sesitivity aalysis is performed to examie the effect of various parameters i the system. Keywords: -policy, server breakdows, M/M/ queueig system, server startup, probability geeratig fuctio, balkig. Itroductio May practical queueig system especially those with balkig ad reegig have bee widely applied to may real life problems such as the situatios ivolvig impatiet telephoe switch board customers, the hospital emergig room hadlig critical patiets, the ivetory system with storage of perishable goods, etc. Queueig system with balkig, reegig or both have bee studied by may researchers. Haight first cosidered a M/M/ queue with balkig. A M/M/ queue with customers reegig was also proposed by Haight 2. The combied effects of balkig ad reegig i a M/M// queue have bee ivestigated by Acker ad Gafaria 3,4. Barrer 5 carried out oe of the early work o reegig where he cosidered determiistic reegig with sigle server Markovia arrival ad service rates. Abou-El-Atc ad Harri 6 cosidered multiple server queueig system M/M/C/ with balkig ad reegig. Wag ad Chag 7 exteded this work to study a M/M/R/ queueig system with balkig, reegig ad server breakdows. Multi-server Markovia queuig system with balkig ad reegig was discussed by Choudhury ad Medhi 8. M/M/ queueig system with the customers impatiece i the form of radom balkig was aalyzed by Maohara ad Jose 9. This paper provides a adequate service for its customers with tolerable waitig, wheever customer impatiece becomes sufficietly strog ad customers leave before beig served. The cocept of the policy was first itroduced by Yadi ad aor. Wag first proposed a maagemet policy for Markovia queueig systems uder the policy with server breakdows. Wag 2 ad Wag et al. 3 exteded the model proposed by Wag to M/Ek/ ad M/H2/ queueig systems, respectively. They developed the aalytic closed form solutios ad provided a sesitivity aalysis. Jayachitra ad James Albert 4 preseted a elaborated survey o various queueig models uder -policy. Jayachitra ad James Albert 5 discussed optimal cotrol of a -policy M/Ek/ queueig system with server startup ad breakdow. Also the authors 6,7 studied a Erlagia model uder server breakdows ad multiple vacatios ad provided a cost model to determie the optimal operatig policy at miimum cost. However to the best of our kowledge for M/M/ queueig system with server startup, - policy, ureliable server ad balkig, there is o literature which takes customers impatiece ito cosideratio. This motivates us to study a queueig system with -policy, server startup, breakdows ad balkig. Thus i the preset paper we cosider a M/M/ queueig system with server start up, -policy, ureliable server ad balkig. The model ca be realized i may real life situatios. The article is orgaized as follows. A full descriptio of the model is give i sectio 2. The steady state aalysis of the system state probabilities is performed through the geeratig fuctios i sectio 3. Steady state results o the expected umber of customers i differet states of the server are give i sectio 4. Special cases of this model are discussed i sectio 5. I sectio 6 the characteristic features of the system are ivestigated. Optimal cotrol policy is explaied i sectio 7. I sectio 8 sesitivity aalysis is carried out through umerical illustratio. Fially i sectio 9 coclusios ad future scope of the study are preseted. Iteratioal Sciece Commuity Associatio

2 Research Joural of Mathematical ad Statistical Scieces ISS Vol. 6(2), -9, February (28) Res. J. Mathematical ad Statistical Sci. The mai objectives of the aalysis carried out i this paper for the optimal cotrol policy are: i. To establish the steady state equatios ad obtai the steady state probability distributio of the umber of customers i the system i each state. ii. To derive expressios for the expected umber of customers i the system whe the server is i idle, startup, busy ad breakdow states respectively. iii. To formulate the total expected cost fuctios for the system, ad determie the optimal value of the cotrol parameter. iv. To carry out sesitivity aalysis o the optimal value of ad the miimum expected cost for various system parameters through umerical experimet. Model descriptio For the purpose of aalytical ivestigatio, we cosider the model with the followig assumptios: i. The arrival is poisso process with parameter λ. Arrivig cusomers at the server form a sigle waitig lie ad is served i the order of their arrivals. The server may serve oly oe uit at a time ad the service times are expoetially distributed with mea /µ. ii. The server is tured off whe the system becomes empty. As soo as the umber of customers i the system reaches, the server is immediately tured o but is temporarily uavailable to the waitig uits. It eeds to take expoetial startup time with parameter Ө. As soo as the startup period over, it serves the waitig uits immediately. iii. Whe the server is workig, it may breakdow at ay time with a oisso breakdow rate. iv. Whe the server fails, it is immediately repaired at a repair rate, where the repair times are expoetially distributed. v. A customer may balk from the queue with probability b whe the server is i idle or may balk with a probability b whe the server is i service made due to impatiece. Steady State Aalysis I steady state the followig otatios are used.,, = robability that there are customers i the system whe the server is idle. There are two situatios whe the server is idle. i. the server is tured off if ; ii. it is tured o ad performig startup if., = robability that there are customers i the system whe the server is tured o ad i operatio where =,2,3 2, = robability that there are customers i the system whe the server is i operatio ad foud to be broke dow where =,2,3 The steady state equatios are give as follows λ b µ = (),,, =, ( ) (2) ( + ), =, (3) ( + ), =, > (4) ( λ b + µ + ) = µ + (5),,2 2, ( λ b µ ), λ b, µ, + 2, + + = + + (2 ) (6) + + = ( ) ( λ b µ ), λ b, µ, + 2,, 2,, (7) ( + ) = (8) λ + = λ + ( 2 ) (9) ( b ) 2, b 2,, Solvig equatios (),(2),(3) ad (4) recursively, we fially get = R,, ( ) Where R =, +,, () robability geeratig fuctio: The techique of usig the probability geeratig fuctio may be applied i a recursive maer from equatio () to (9) to obtai the aalytic solutio of, i eat closed form expressio. Defie the probability geeratig fuctio of H (z), H (z), Q(z)ad R(z) respectively as follows: H ( z) = z (), = = (2) ( ), = H z z Q( z) R( z) = z (3) i=, = (4) z 2, = Applyig algebraic maipulatio techique to equatio (2) ad usig i equatio (), we get the followig z H ( z) = z, Usig equatio () i equatio (2) we get Rz H ( z) = Rz, (5) (6) Iteratioal Sciece Commuity Associatio 2

3 Research Joural of Mathematical ad Statistical Scieces ISS Vol. 6(2), -9, February (28) Res. J. Mathematical ad Statistical Sci. Multiplyig equatio (3) by z ad (4) by we get z H ( z) =, ( z) + Equatig (6) ad (7) we get Rz z = Rz ( z) + z ad summary over Multiplyig equatio () by z ad (5),(6)ad (7) by summary over we get + 2 Rz z ( λ b + µ + ) z + µ Q( z) + zr( z) = z Rz, (7) (7a) z ad (8) Applyig same procedure over the equatios (8) ad (9) to obtai the followig equatios R ( z ) = Q ( z ) (9) λ b + λ b z Substitutig equatio (9) i equatio (8) we get, ρb + where = + ρ + ( b b ) λ ρ = (26) µ et,, 2 ad 3 be the probability that the server is i idle, startup, busy ad break dow states respectively. The, = H () (i) = () (ii) 2 = Q () (iii) 3 = R () (iv) Expected umber of customers i the system We defie the expected umber of customers i the system as follows. = the expected umber of customers i the system whe the I server is idle. s = the expected umber of customers i the system whe the server is doig pre service (start up work). + z + Rz 2 λ, z + ( λ b + µ + ) z + µ Rz + z + z Q( z) = b z [ ] (2) w = the expected umber of customers i the system whe the server is workig. robability that the empty system: We evaluate the probability, usig ormalizig coditio. For this purpose H (), H (),Q() ad R() we evaluate (5),(6),(2) ad (9) respectively as H, from equatios () = (2) H () = (22), ρb + Q() = ρb + ρ b + R () = ρ b +,, et G(Z) represet the umber of customers i the system. (23) (24) G( Z) = H ( z) + H ( z) + Q( z) + R( z) (25) ow usig the ormalizig coditio G () value of probability =, we obtai the d = the expected umber of customers i the system whe the server is broke dow. = Expected umber of customer i the system. Usig the probability geeratig fuctios expected umber of uits i the system at differet states are preseted below. ( ) ' I = H() =, (27) 2 s = H () = + Q ' () w = ', 2 ρb 2 ( ) ρb ρb [ λ b µ ] + + = w ρb ρb + + ρb +, (28) ' λ b = R () = w + Q() d (3) The expected umber of customers i the system is give by ' ' ' ' = H () + H () + Q () + R ( z) (29) Iteratioal Sciece Commuity Associatio 3

4 Research Joural of Mathematical ad Statistical Scieces ISS Vol. 6(2), -9, February (28) Res. J. Mathematical ad Statistical Sci. = [ ] 2 ρb 2 + ( + ) + ρb + ρb + λ b µ ( ) ρb + ρb + ρb + ρb ρ + ( b b ) λ b, (3) Special Cases I this sectio we preset some existig results i the literature which are special cases of our model. Case (i): If Ө=,=, =, b =, b = ad =, our model represets the ordiary M/M/ queueig system with reliable case. Whe =, Ө =, =, b =, b = ad =, expressio (2 ) for Q(z) reduces to the special case stated i the expressios (2.4) of Gross ad Harris 8. Case (ii): If Ө=, b = ad b = our model represets the - policy M/M/ queueig system with ureliable case. Whe Ө=, b = ad b = expressio (2) for Q(z) ad (9) for R(z) reduces to a special case of expressio (8) ad (9) of Wag respectively. Case (iii): If Ө=, =, =, b =, b = ad =,equatio (26) reduces to the result of probability that the empty system i the ordiary M/M/ queueig model. Other System Characteristic et I, S, B, ad D deote the legth of the server off, startup, busy ad breakdow periods respectively. Applyig the memory less property of the oisso process, we fid that the mea legth of the idle period is E( I) = (32) The expected legth of the idle period, startup period, busy period ad breakdow period are deoted by E(I), E(S),E(B) ad E(D) respectively. The expected legth of the busy cycle is give by E(C) = E(I) + E(S)+E(B)+E(D) From equatios (2) to (24) we obtai the log ru fractio of time for the server is idle, start up, busy ad broke dow respectively. E( I ) H (), E( C ) = = (33) E(S) E( C) = H s () =, (34) E ( B) ρb = Q() = (35) E ( C ) ρ + (b b ) E( D) = R() = Q() (36) E( C) From equatio, (32) ad (33) we have the umber of busy cycles per uit time E( C) = (37), Optimal Cotrol policy We develop the total expected cost fuctio per uit time for the -policy M / M/ Queueig system with server startup, ureliable server ad balkig i which is a decisio variable. With the cost structure beig costructed, the objective is to determie the optimal operatig policy so as to miimize this fuctio. et C = holdig cost per uit time for each customer preset i the h system. C = Cost per uit time for keepig the server idle. o C =Set up cost per busy cycle. f C d =breakdow cost per uit time C = startup cost per uit time for preparatory work of the s server before startig the service. Iteratioal Sciece Commuity Associatio 4

5 Research Joural of Mathematical ad Statistical Scieces ISS Vol. 6(2), -9, February (28) Res. J. Mathematical ad Statistical Sci. C b =cost per uit time for the server keepig o ad i operatio. C k = Cost per uit time whe a customer balks. C f E ( I ) E( S) E( B) E ( D) Tcos t ( ) = Ch + + Co + Cs + Cb + Cd + E( C) E ( C) E ( C) E( C) E ( C) (38) E( I) E( S) E( B) E( D) Ck λ( b ) + λ( b ) E( C) + + E( C) E( C) E( C) Usig equatios (3), (33)-(37), the results of (38) ca be explicitly expressed which is a very log ad complex formula for T ( ) cos. We obtai the optimal value which miimizes t the cost fuctio T ( ) cos ad settig the result to be zero. i.e, T ( ) cos t = t, by differetiatig it with respect to (39) The solutio to (39) may ot be a iteger ad the optimal positive iteger value of is oe of the itegers surroudig * which gives a smaller cost T cost. Here it should be poited out explicitly that the solutio really gives the miimum value ad 2 T 2 cost ( ) = at =* is greater tha zero whe the values of system parameters satisfy suitable coditios. However it is quite tedious to preset the explicit expressio. Therefore we will perform the umerical illustratios to demostrate that the fuctio is really covex ad the solutio gives a miimum. Sesitivity Aalysis I the course of aalysis, sesitivity aalysis has bee carried out to fid the optimum value of (ie., *) ad the miimum cost based o chages i the system parameters usig MATAB. I order to arrive at the coclusios, the followig arbitrary values of the system parameters are cosidered. Case : C h =5, C =, C b =, C d =2, C s =25, C f =5, C k = Case 2: C h =5, C =2, C b =2, C d =25, C s =25, C f =5, C k =5 Case 3: C h =5, C =5, C b =4, C d =35, C s =5, C f =, C k =4 Case 4: C h =, C =, C b =5, C d =4, C s =6, C f =, C k =7 Case 5: C h =5, C =, C b =5, C d =5, C s =8, C f =5, C k = Table-: The optimal value of ad its miimum expected cost for λ=.3, µ =.2, =3., b =.4, b =.2 (Ө,) (.4,.2).6,.2) (.8,.2) (.5,.) (.5,.2) (.5,.3) Case(i) Case(ii) Case(iii) Case(iv) Case(v) * T cost (*) * T cost (*) * T cost (*) * T cost (*) * T cost (*) Iteratioal Sciece Commuity Associatio 5

6 Research Joural of Mathematical ad Statistical Scieces ISS Vol. 6(2), -9, February (28) Res. J. Mathematical ad Statistical Sci. Table-2: The optimal value of ad its miimum expected cost for λ=.3, Ө =.2, =.5, b =.4, b =.2 Case(i) Case(ii) (µ,) (.,3.).2,3.) (.4,3.) (.4,2.) (.4,4.) (.4,6.) * T cost (*) * T cost (*) Case(iii) Case(iv) Case(v) * 2 T cost (*) * T cost (*) * T cost (*) Table-3: The optimal value of ad its miimum expected cost for = 3, µ =.2, =.5, b =.4, b =.2. (λ,ө) (.2,4) (.4,.4) (.6,.4) (.4,.) (.4,.3) (.4,.5) * 8 8 Case(i) T cost (*) Case(ii) Case(iii) Case(iv) Case(v) * T cost (*) * T cost (*) * 9 2 T cost (*) * T cost (*) Table-4: The optimal value of ad its miimum expected cost for λ=.3, µ =.2, =3., =.5, Ө=.4 (b,b ) (.4,.3) (.45,.3) (.5,.3) (.4,.2) (.4,.3) (.4,.4) Case(i) * T cost (*) Case(ii) * T cost (*) Case(iii) Case(iv) Case(v) * T cost (*) * T cost (*) * T cost (*) Iteratioal Sciece Commuity Associatio 6

7 Research Joural of Mathematical ad Statistical Scieces ISS Vol. 6(2), -9, February (28) Res. J. Mathematical ad Statistical Sci. The optimal value of ad its miimum expected cost T coat (*) for the above five cases are show i Table- for various values of (Ө,). It is observed that: i. T cost (*) decreases as Ө icreases ad T cost (*) icreases as icreases. ii. * decreases as Ө icreases. Ituitively * is isesitive to the chages i. The optimal value of ad its miimum expected cost T cost (*) for the above five cases are show i Table-2 for various values of (µ,). From Table-2 we fid that i. * icreases whe µ chages from. to.4. ad * remais uchaged whe chages from 2. to 6.. ii) T cost (*) decreases as µ ad icreases respectively. The optimal value of ad its miimum expected cost T cost (*) for the above five cases are show i table (3)for various values of (λ,r). From Table-3 we fid that i. * icreases whe λ icreases ad *decreases whe Ө icreases. ii. T cost (*) icreases whe λ icreases ad T cost (*) decreases as Ө icreases. From Table-4 we fid that i. * slightly icreases whe b icreases from.4 to.5 ad * slightly decreases whe b chages from.2 to.4. ii. T cost (*) icreases whe b icreases ad T cost (*) decreases whe b icreases. Over all we coclude that: i., ad do ot affect *, ii. Ө, b ad b rarely affect *. iii. λ ad µ affect * sigificatly, iv. C h ad C f have much stroger effect o * Figure-: Arrival rate λ Vs Total cost Tc by varyig service rate µ. Figure-2: Service rate µ Vs Total cost T c by varyig repair rate. Iteratioal Sciece Commuity Associatio 7

8 Research Joural of Mathematical ad Statistical Scieces ISS Vol. 6(2), -9, February (28) Res. J. Mathematical ad Statistical Sci. Figure-3: Service rate Vs Total cost by varyig. Figure- displays the correlatio betwee the arrival rate λ ad the total cost (T c ) by varyig service rate µ. We ca observe that as the service rate goes o icreasig, the total cost goes o decreasig. I Figure-2 the total cost goes o decreasig as we go o icreasig the repair rate. I Figure-3 the total cost goes o icreasig as we go o icreasig the breakdow rate. Coclusio Optimal strategy aalysis of a -policy M/M/ queueig system with server startup, ureliable server ad balkig is studied. Some of the system performace measures have bee derived. A cost fuctio has bee formulated to determie the optimal value of. Sesitivity aalysis has bee carried out through umerical illustratios. These umerical values will be useful i aalyzig practical queueig system ad make decisio The preset study ca be exteded by cosiderig that the service time, breakdow time ad repair time follow geeral distributio. Refereces. Haight F.A. (957). Queuig with balkig. Biometrika, 44, Haight F.A. (959). Queueig with Reegig. Metrika, 2, Acker Jr. C.J. ad Gafaria A.V. (963). Some Queuig roblems with Balkig ad Reegig. I. Operatios Research, (), Acker Jr. C.J. ad Gafaria A.V. (963). Some Queuig roblems with Balkig ad Reegig. II. Operatios Research, (6), , 5. Barer D.Y. (957). Queueig with Impatiet Customers ad Ordered Service. Operatios Research, 5(5), Abou El-Ata M.O. ad Hariri A.M.A. (992). The M/M/C/ queue with balkig ad reegig. Computers ad Operatios Research, 9(8), Wag K.H. ad Chag Y.C. (22). Cost aalysis of a fiite M/M/R queueig system with balkig, reegig ad server breakdows. Mathematical Methods of Operatios Research, 56(2), Choudhury A. ad Medhi. (2). Balkig ad reegig i multiserver Markovia qeueig systems. Iteratioal Joural of Mathematics i Operatioal Research, 3(4), Maohara M. ad Jose K.M. (2). Markovia Queueig system with radom balkig. Opsearch, 48 (3), Yadi M. ad aor. (963). Queueig systems with a removable service statio. Joural of the Operatioal Research Society, 4(4), Wag K.H. (995). Optimal operatio of a Markovia queueig system with removable ad o reliable server. Microelectroics Reliability, 35(8), Wag K.H. (997). Optimal cotrol of a M/Ek/ queueig system with removable service statio subject to breakdows. Joural of the Operatioal Research Society, 48(9), Wag K.H., Chag K.W. ad Sivazlia B.D. (999). Optimal cotrol of a removable ad o-reliable server i a ifiite ad a fiite M/H2/ queueig system. Applied Mathematical Modellig, 23(8), Jayachitra. ad Albert A.J. (24). Recet developmets i queueig models uder -policy: A short survey. It. J. Math. Arch., 5(3), Jayachitra. ad James Albert A. (24). Optimal cotrol of a -policy M/Ek/ Queueig System with Server Start up ad Breakdows. Far East joural of Mathematical Scieces, 94(), Jayachitra. ad Albert James A. (24). Optimal Maagemet policy for Heterogeeous Arrival M/Ek/ Queueig System with server Breakdows ad Multiple Iteratioal Sciece Commuity Associatio 8

9 Research Joural of Mathematical ad Statistical Scieces ISS Vol. 6(2), -9, February (28) Res. J. Mathematical ad Statistical Sci. vacatios. Advaces i theoretical ad Applied Mathematics, 9(), Jayachitra. ad Albert James A. (25). erformace Aalysis of a -policy M/Ek/ Queueig system with Server startup, Ureliable server ad Balkig. Iteratioal Joural of Mathematical Archieve -IJMA, 6(4), Gross D. ad Harris C.M. (985). Fudametals of Queueig Theory. 2 d Editio. Wiley, ew York. Iteratioal Sciece Commuity Associatio 9