Analysis Of Single Server Queueing System With Batch Service. Under Multiple Vacations With Loss And Feedback

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1 Aalysis Of Sigle Server Queueig System With Batch Service Abstract Uder Multiple Vacatios With Loss Ad Feedback G. Ayyappa 1 G. Devipriya 2* A. Muthu Gaapathi Subramaia 3 1. Associate rofessor, odicherry Egieerig College, odicherry, Idia 2. Assistat rofessor, Sri Gaesh College of Egieerig & Techology,odicherry, Idia 3. Associate rofessor, Kachi Mamuivar Cetre for ost Graduate Studies, odicherry, Idia * of the correspodig author: devimou@yahoo.com Cosider a sigle server queueig system with foxed batch service uder multiple vacatios with loss ad feedback i which the arrival rate λ follows a oisso process ad the service time follows a expoetial distributio with parameter μ. Assume that the system iitially cotai k customers whe the server eters the system ad starts the service i batch. The cocept of feedback is icorporated i this model (i.e) after completio of the service, if this batch of customers dissatisfied the this batch may joi the queue with probability q ad with probability (1-q) leaves the system. This q is called a feedback probability. After completio of the service if he fids more tha k customers i the queue the the first k customers will be take for service ad service will be give as a batch of size k ad if he fids less tha k customers i the queue the he leaves for a multiple vacatio of expoetial legth α. The impatiet behaviour of customer is also studied i this model (i.e) the arrivig customer may joi the queue with probability p whe the server is busy or i vacatio. This probability p is called loss probability. This model is completely solved by costructig the geeratig fuctio ad Rouche s theorem is applied ad we have derived the closed form solutios for probability of umber of customers i the queue durig the server busy ad i vacatio. Further we are providig the aalytical solutio for mea umber of customers ad variace of the system. Numerical studies have bee doe for aalysis of mea ad variace for various values of λ, µ, α, p, q ad k ad also various particular cases of this model have bee discussed. Keywords : Sigle Server, Batch Service, Loss ad Feedback, Multiple vacatios, Steady state distributio. 1. Itroductio Queueig systems where services are offered i batches istead of persoalized service of oe at a time. A service may be of a fixed size (say) k, such a system is called fixed batch size. Bailey (1954), was the first to cosider bulk service. Such models fid applicatios i several situatios, such as trasportatio. Mass trasit vehicles are atural batch servers. There are a great umber of umerical ad approximatios methods are available, i this paper we will place more emphasis o the solutios by robability Geeratig Fuctio. Formulatio of queues with feedback mechaism was first itroduced by Takacs. The cocepts loss ad feedback are itroduced for customers oly. After completio of the service, if the customer dissatisfied the he may joi the queue with probability q ad with probability (1-q) he leaves the system. This is called feedback i queueig theory. If the server is busy at the time of the arrival of customer, the due to impatiet behaviour of the customer, customer may or may ot joi the queue. This is called loss i queueig theory. We assume that p is the probability that the customer jois the queue ad (1-p) is the probability that he leaves the system without gettig service (due to impatiet). Feedback queues play a vital role i the areas of Computer etworks, roductio systems subject to rework, Hospital maagemet, Super markets ad Bakig busiess etc. Takacs (1963), itroduced the cocept of feedback queues. Disey, McNickle ad Simmo (1980), D Avigo ad Disey (1976), Krishakumar (2002),Thagaraj ad Vaitha. (2009,2010). Ayyappa et al., (2010), Farahmad.K ad T. Li(2009) are a few 79

2 to be metioed for their cotributio. I this paper we are goig to cocetrate o a very special batch service queue called the fixed size batch service queue uder multiple vacatios with loss ad feedback. The model uder cosideratio is described i Sectio 2. I Sectio 3 we aalyze the model by derivig the system steady state equatios ad probability geeratig fuctios. Usig these geeratig fuctios, steady state probabilities are obtaied i Sectio 4. The operatig characteristics are obtaied i Sectio 5. A umerical study is carried out i Sectio 6 to test the effect of the system performace measure discussed i Sectio 5. We are providig the aalytical solutio for mea umber of customers ad variace of the system. Numerical studies have bee doe for aalysis of mea ad variace for various values of λ, µ, α, p, q ad k ad also various particular cases of this model have bee discussed. 2. DESCRIBE OF THE MODEL Cosider a sigle server queueig system with foxed batch service uder multiple vacatios with loss ad feedback i which the arrival rate λ follows a oisso process ad the service time follows a expoetial distributio with parameter μ. Assume that the system iitially cotai k customers whe the server eters the system ad starts the service i batch. The cocept of feedback is icorporated i this model (i.e) after completio of the service, if this batch of customers dissatisfied the this batch may joi the queue with probability q ad with probability (1-q) leaves the system. This q is called a feedback probability. After completio of the service if he fids more tha k customers i the queue the the first k customers will be take for service ad service will be give as a batch of size k ad if he fids less tha k customers i the queue the he leaves for a multiple vacatio of expoetial legth α. The impatiet behaviour of customer is also studied i this model (i.e) the arrivig customer may joi the queue with probability p whe the server is busy or i vacatio. This probability p is called loss probability. Let < N(t),C(t) > be a radom process where N(t) be the radom variable which represets the umber of customers i queue at time t ad C(t) be the radom variable which represets the server status (busy/vacatio) at time t. We defie,1 (t) - probability that the server is i busy if there are customers i the queue at time t.,2 (t) - probability that the server is i vacatio if there are customers i the queue at time t The Chapma- Kolmogorov equatios are ( t) ( p (1 q)) ( t) (1 q) ( t) + (t) (1) ' 0,1 0,1 k,1 k,2 ( t) ( p (1 q)) ( t) p ( t) (1 q) ( t) + (t) for = 1,2,3,... (2) ',1,1 1,1 k,1 k,2 ( t) p ( t) (1 q) ( t) (3) ' 0,2 0,2 0,1 ( t) p ( t) p ( t) (1 q) ( t) for = 1,2,3,...,k-1 (4) ',2,2 1,2,1 ',2( t) ( p ),2( t) p 1,2 ( t) for k (5) 80

3 3. EVALUATION OF STEADY STATE ROBABILITIES I this sectio we are fidig the closed form solutios for umber of customers i the queue whe the server is busy or i vacatio usig Geeratig fuctio techique. Whe steady state prevails, the equatios (1) to (5) becomes ( p (1 q)) 0,1 (1 q) k,1 + k,2 (6) ( p (1 q)) p (1 q) + for = 1,2,3,... (7),1 1,1 k,1 k,2 p (1 q) (8) 0,2 0,1 p p (1 q) for = 1,2,3,...,k-1 (9),2 1,2,1 ( p ),2 p 1,2 for k (10) Geeratig fuctios for the umber of customers i the queue whe the server is busy or i defied as the vacatio are G() z,1z ad H() z,2 0 0 Multiply the equatio (6) with 1 ad (7) with z o both sides ad summig over = 0 to, we get z k1 k1 k1 k,1,2 0 0 G( z)[ pz ( p (1 q)) z ] H( z) (1 q) z z (11) Addig equatio (8), (9) ad (10) after multiply with 1, z ( = 1, 2, 3,..., k-1) ad z ( = k, k+1, k+2,... ) respectively, we get H( z)[ p(1 z)] (1 q) z z From the equatios (11) ad (12), we get k1 k1,1 (12),2 0 0 H( z) p(1 z) Gz () k1 k pz ( p (1 q)) z (1 q) (13) From equatio (12), we get k1 k1,1,2 0 0 (1 q) z z H( z) p(1 z) (14) Equatio (14) represets the probability geeratig fuctio for umber of customers i the queue whe the server is i vacatio. From the equatios (13) ad (14), we get 81

4 Gz () k1 k1 ( p(1 z)) (1 q),1 z,2 z 0 0 k1 k [ pz ( p (1 q)) z ][ p(1 z)] (15) Equatio (15) represets the probability geeratig for umber of customers i the queue whe the server is busy. ut z = 1 i equatio (13), we get p G(1) H(1) k(1 q) p (16) The ormalized coditio is G(1) H(1) 1 (17) Usig the equatio (1) i (17), we get k(1 q) p H(1) k(1 q) (18) From the equatios (16) ad (18), we get Steady state probability that the server is busy = G(1) p k(1 q) Steady state probability that the server i vacatio = k(1 q) p H(1) k(1 q) The geeratig fuctio G(z) has the property that it must coverge iside the uit circle. We otice that the deomiator of G(z), k 1 k pz ( p (1 q)) z (1 q) has k+1 zero's. Applyig Rouche's theorem, we otice that k zeros of this expressio lies iside the circle z 1 ad must coicide with k zeros of umerator of G(z) ad oe zero lies outside the circle z 1. Let z 0 be a zero which lies outside the circle z 1. As G(z) coverges, k zeros of umerator ad deomiator will be cacelled, we get A Gz () ( p(1 z)) ( z z ) 0 (19) ut z = 1 i the equatio (19), we get G(1) A (1 z ) 0 (20) Usig (16) ad (18) i (20), we obtai 82

5 A 2 p (1 z0 ) k(1 q) (21) From the equatio (19) ad (21), we get p (1 z0 ) Gz () ( k(1 q) p) ( z z )( p pz) 0 (22) By applyig partial fractios, we get (1 z0 ) p p z G() z ( k(1 q) p) ( p pz ) p z 1 z (23) Comparig the coefficiet of z o both sides of the equatio (23), we get,1 ( k(1 q) p) ( s r) (1 rs ) (s 1-1 r ) for = 0,1,2,... (24) where r 1 p ad s = z p 0 Usig equatio (24) i (8) ad (9), apply recursive for = 1,2,3,...,k-1, we get (1 q),2 t,1 p t0 for = 0,1,2,3,...,k-1 (25) p p k1,2 k1,2 for k (26) Equatios (24), (25) ad (26) represet the steady state probabilities for umber of customers i the queue whe the server is busy / i vacatio. 4. STABILITY CONDITION The ecessary ad sufficiet coditio for the system to the stable is p k(1 q) < ARTICULAR CASE If we take p =1 ad q = 0, the results coicides with the results of the model sigle server batch service uder multiple vacatio. 6. SYSTEM ERFORMANCE MEASURES I this sectio, we will list some importat performace measures alog with their formulas. These measures are used to brig out the qualitative behaviour of the queueig model uder study. Numerical study has bee dealt i very large scale to study the followig measures a. robability that there are customers i the queue whe the server is busy 83

6 ,1 ( k(1 q) p) ( s r) 1 p Where r1 ad s = z p 0 (1 rs ) (s 1-1 r ) for = 0,1,2,... b. robability that there are customers i the queue whe the server i vacatio (1 q),2 t,1 p t0 for = 0,1,2,...,k-1 p p k1,2 k1,2 c. Mea umber of customers i the queue for k L ( ) q,1,2 0 d.variace of the umber of customers i the queue V ( x) ( L ) 2 2 2,1,2 q NUMERICAL STUDIES The values of parameters λ, µ, α, p, q ad k are chose so that they satisfy the stability coditio discussed i sectio 4. The system performace measures of this model have bee doe ad expressed i the form of tables for various values λ, µ, α ad k. Tables 3, 6 ad 9 show the impact of arrival rate λ ad k over Mea umber of customers i the queue. Tables 10 show the impact of p ad k over Mea umber of customers i the queue. Further we ifer the followig Mea umber of customers i the queue icreases as arrival rate λ icreases. 1 ad 2 are probabilities of server i busy ad i vacatio Tables 1, 4ad 7 show the steady probabilities distributio whe server is busy for various values of λ ad differet batch size k. Tables 2, 5 ad 8 show the steady state probabilities distributio whe the server is i vacatio for various values of λ ad differet batch size k. 84

7 Table 1: Steady state robabilities distributio for various values of λ ad µ = 10, α = 5, p = 0.8, q = 0.2 whe the server is busy ad batch size is K = 2 Λ p 01 p 11 p 21 p 31 p 41 p 51 p 61 p 71 p 81 p 91 p Table 2: Steady state robabilities distributio for various values of λ ad µ = 10, α = 5, p = 0.8, q = 0.2 whe the server is i vacatio ad batch size is K = 2 Λ p 02 p 12 p 22 p 32 p 42 p 52 p 62 p 72 p 82 p 92 p

8 Table 3: Average umber of customers i the queue ad Variace for various values of λ, p = 0.8, q = 0.2 ad µ = 10, α = 5 ad batch size is K = 2 Λ µ α 1 2 Mea Variace Table 4: Steady state robabilities distributio for various values of λ ad µ = 10, α = 5, p = 0.8, q = 0.2 whe the server is busy ad batch size is K = 4 Λ p 01 p 11 p 21 p 31 p 41 p 51 p 61 p 71 p 81 p 91 p

9 Table 5: Steady state robabilities distributio for various values of λ ad µ = 10, α = 5, p = 0.8, q = 0.2 whe the server is i vacatio ad batch size is K = 4 Λ p 02 p 12 p 22 p 32 p 42 p 52 p 62 p 72 p 82 p 92 p Table 6: Average umber of customers i the queue ad Variace for various values of λ, p = 0.8, q = 0.2 ad µ = 10, α = 5 ad batch size is K = 4 Λ µ Α 1 2 Mea Variace

10 Table 7: Steady state robabilities distributio for various values of λ ad µ = 10, α = 5, p = 0.8, q = 0.2 whe the server is busy ad batch size is K = 6 Λ p 01 p 11 p 21 p 31 p 41 p 51 p 61 p 71 p 81 p 91 p Table 8: Steady state robabilities distributio for various values of λ ad µ = 10, α = 5, p = 0.8, q = 0.2 whe the server is i vacatio ad batch size is K = 6 Λ p 02 p 12 p 22 p 32 p 42 p 52 p 62 p 72 p 82 p 92 p

11 Table 9: Average umber of customers i the queue ad Variace for various values of λ, p = 0.8, q =0.2 ad µ = 10, α = 5 ad batch size is K = 6 Λ µ Α 1 2 Mea Variace Table 10: Average umber of customers i the queue ad Variace for various values of p, λ = 5, q = 0.2 ad µ = 10, α = 5 ad batch size is K = 2 p µ λ 1 2 Mea Variace Coclusio: The Numerical studies show the chages i the system due to impact of batch size, vacatio rate, arrival rate. The mea umber of customers i the system icreases as batch size ad arrival rate icrease. The mea umber of customers i the system decreases as vacatio rate icreases. Various special cases have bee discussed, which are particular cases of this research work. This research work ca be exteded further by itroducig various cocepts like bulk arrival, iterruptio etc. Refereces G. Ayyappa, A.Muthu Gaapathi Subramaia ad Gopal sekar (2010),M/M/1 Retrial Queueig System with 89

12 Loss ad Feedback uder No-pre-emptive riority Service by Matrix Geometric Method, Applied Mathematical Scieces, Vol. 4, 2010, o. 48, Bailey, N.T.J., (1954), O queueig process with bulk service, Joural of Royal Statistical Society series, B16, G.R. D Avigo, R.L. Disey, (1976), Sigle server queues with state depedet feedback, INFOR, 14, R.L. Disey, D.C. McNickle, B. Simo, (1980), The M/G/1 queue with istataeous beroulli feedback, Nav. Res. Log. Quat, 27, Farahmad.K ad T. Li (2009). Sigle server retrial queueig system with loss of customers. Advaces ad Applicatios, Vol 11 pp Systems, 6, No.2, Krishakumar. B, S. avai Maheswari,ad A.Vijayakumar (2002),The M/G/1 retrial queue with feedback ad startig failure, Applied Mathematical Modellig, L. Takacs, (1963), A sigle server queue with feedback, The Bell System Tech. Joural, 42, Thagaraj,V. ad Vaitha,S. (2009). A two phase M/G/1 feedback queue with multiple server vacatio. Stochastic Aalysis ad Applicatios, 27: V. Thagaraj, S. Vaitha, (2010), M/M/1 queue with feedback a cotiued factio approach, Iteratioal Joural of Computatioal ad Applied Mathematics, 5, V. Thagaraj, S. Vaitha,(2009), O the aalysis of M/M/1 feedback queue with catastrophes usig cotiued fractios, Iteratioal Joural of ure ad Applied Mathematics,

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Control chart for number of customers in the system of M [X] / M / 1 Queueing system

Control chart for number of customers in the system of M [X] / M / 1 Queueing system Iteratioal Joural of Iovative Research i Sciece, Egieerig ad Techology (A ISO 3297: 07 Certified Orgaiatio) Cotrol chart for umber of customers i the system of M [X] / M / Queueig system T.Poogodi, Dr.