Control chart for number of customers in the system of M [X] / M / 1 Queueing system
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1 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. (Mrs.) S.Muthulakshmi 2 Assistat Professor, Departmet of Sciece ad Humaities,Aviashiligam Uiversity, Coimbatore, Tamil Nadu, Idia Professor, Departmet of Mathematics, Aviashiligam Uiversity, Coimbatore, Tamil Nadu, Idia 2 Abstract:Queueig models characteried by bulk arrival or bulk service of either fixed or variable sies are commoly foud i modellig of traffic ad trasportatio systems, complex computer ad telecommuicatio systems, ivetory repleishmet system ad other real-life applicatios. Cotrol chart techique eables to moitor the performace of these queues. Average queue legth ad average waitig time are the mai observable performace characteristics of ay queueig system. I this paper cotrol limits are established for M [X] /M / queueig model whe the batch sie follows geometric distributio. Numerical results are added to highlight its applicatios. Keywords:bulk arrival, queue legth, geometric distributio, Poisso arrival, expoetial service, cotrol limits. I. INTRODUCTION I geeral, i queueig models, it is assumed that the customers arrive sigly at service facility. But this assumptio is violated i may real life queueig situatios. Letters arrivig at a post office, ships arrivig at a port i a covoy, people attedig a weddig receptio etc., are some of the examples of queueig situatios i which customers arrive i groups. Queueig model cosistig of bulk arrival has bee discussed by Gross ad Harris (998), Carme Armero ad David Coesa (00) ad several others. Cotrol chart is a quality cotrol techique evolved iitially to moitor productio processes. Motgomery (05) proposed a umber of applicatios of Shewhart cotrol charts i assurig quality i maufacturig idustries. Shore (00) developed cotrol chart for radom queue legth of M /M /S queueig model by cosiderig the first three momets ad also Shore (06) developed Shewhart-like geeral cotrol charts for G/G/S queueig system usig skewess. Khaparde ad Dhabe (0) costructed the cotrol chart for radom queue legth of M/M/ queueig model usig method of weighted variace. Poogodi ad Muthulakshmi (2) aalysed umber of customers i system of M/E k / queueig model usig cotrol chart techique. I this paper Shewhart cotrol chart for umber of customers i the system of bulk arrival queueig model M [X] /M / is proposed. Sectio 2 relates to the queueig model descriptio. Mea ad variace of umber of customers i the queueig system are derived i sectio 3. Parameters of cotrol chart for umber of customers are established i sectio 4. Numerical results are obtaied i sectio 5 to aalyse the effect of parameters. II. M [X] /M/ MODEL DESCRIPTION Cosider a sigle server queueig model i which the arrivals occur i batches accordig to Poisso process with rate 0. The batch sie X is a radom variable with P(X = k) = C k, k =, 2, 3,. Customers are served oe by oe ad the service time distributio is expoetial with rate µ. Let P be the probability that there are customers i the system. Copyright to IJIRSET
2 Iteratioal Joural of Iovative Research i Sciece, Egieerig ad Techology The steady state equatios goverig this model are 0 = - ( + µ) P + µ P + + P kc k ( ) k (A ISO 3297: 07 Certified Orgaiatio) 0 = -P 0 + µp () The system of equatios () may be solved usig geeratig fuctio approach. Defie the geeratig fuctios of the steady state probabilities {P }ad the batch sie distributio {c } respectively as P() 0 P, C() c, Multiplyig equatio () by appropriate powers of ad summig, we get μ 0 λ P μ P P λ P kc k (2) 0 k k k Cosider Pk c k c k Pk C()P(). (3) k k k Usig equatio (3), equatio (2) becomes μ 0 = λp() μ(p() P0 ) (P() P0 ) λc()p() Solvig for P(), we get μp P() = 0 ( ), (4) μ( ) λ( C()) The geeratig fuctio of the complemetary batch sie probabilities Pr(X > x) = -C x = C ~ is give by C ~ () ~ c C() Takig r = /µ, equatio (4) yields P 0 P() = rc ~ () E(X(X )) Clearly, C ~ () E(X) ad C ~ () 2 Usig the ormaliig coditios, we obtai P 0 = - ρ, where ρ = r E(X). If N s ad N q are umber of customers i the system ad umber of customers i the queue respectively the ρ re(x ) N s = 2( adn q = N s ρ. 2 x Copyright to IJIRSET
3 Iteratioal Joural of Iovative Research i Sciece, Egieerig ad Techology (A ISO 3297: 07 Certified Orgaiatio) Assume that the umber of customers i ay arrivig batch follows geometric distributio with parameter. The the probability mass fuctio of the batch sie is c x = (-) x-, 0<< ( ) The C () = ad E(X) with α r ρ α (5) From equatios (4) ad (5), we obtai ( (α ( 0 Compariso of like powers of o both sides gives P () = 0 α α ( ρ P = ρ( ( (α (, 0. III. MEAN AND VARIANCE Let N s deote the umber of customers i the system (both i queue ad i service).the the expected umber of customers i the system is ρ E(N s ) = (6) ( ( ad the variace of the umber of customers i the systemis ρ( α( ) Var (N s ) = (7) 2 2 ( ( IV. PARAMETERS OF THE CONTROL CHART The parameters of Shewhart cotrol chart, uder the assumptio that the umber of customers i the system follows ormal distributio, are give by UCL = E (N s ) + 3 Var(N s ) (8) CL = E (N s ) LCL = E (N s ) - 3 Var(N s ) The parameters of the cotrol chart for M [X] /M / queueig model, usig (6) ad (7) i (8), are obtaied as ρ CL = ( ( UCL = ρ 3 ρ( α( ) ( ( Copyright to IJIRSET
4 Iteratioal Joural of Iovative Research i Sciece, Egieerig ad Techology (A ISO 3297: 07 Certified Orgaiatio) LCL = ρ 3 ρ( α( ) ( ( V. NUMERICAL ANALYSIS Numerical aalysis is carried out to aalye the performace of queueig system with referece to the parameters λ, µ ad α. As LCL values are egative for the selected values of the parameters, they are cosidered as ero ad therefore ot show i the table as a separate colum. Table gives the traffic itesity ad the cotrol chart parameters for umber of customers i the queueig system for various values of λ, µ ad α. TABLE CONTROL CHART PARAMETERS FOR THE SYSTEM SIZE OF M [X] /M / MODEL λ µ α ρ CL UCL Copyright to IJIRSET
5 Iteratioal Joural of Iovative Research i Sciece, Egieerig ad Techology (A ISO 3297: 07 Certified Orgaiatio) Numerical values i the table reveal the followig features:. (a) icrease i α icreases the parameters ρ, CL ad UCL for fixed values of λ ad µ. (b) icrease i µ decreases the parameters ρ, CL ad UCL for fixed values of λ ad α. (c) icrease i λ icreases the parameters ρ, CL ad UCL for fixed values of µ ad α. VI. CONCLUSION The preseted model has potetial applicatios i practical systems such as maufacturig systems, telecommuicatio systems ad computer etworks. REFERENCES [] Carme Armero ad David Coesa, Predictio i Markovia Bulk arrival queues, Spriger, Queueig systems 34, , 00. [2] Gross. D. ad Harris. M., Fudametals of queueig theory, 5th editio, Joh Wiley & Sos, Ic., 998. [3] Khaparde.M.V. ad Dhabe.S. D., Cotrol chart for radom queue legth N for (M/M/): ( /FCFS) Queueig model, Iteratioal Joural of Agricultural ad Statistical scieces, Vol., , 0. [4] Motgomery D.C., Itroductio to statistical quality cotrol, 5th editio, Joh Wiley & Sos, Ic., 05. [5] Shore. H., Cotrol charts for the queue legth i a G/G/S System, IIE Trasactios, 38, 7-3, 06. [6] Shore. H., Geeral cotrol charts for attributes, IIE trasactios, 32, 49-60, 00. [7] Poogodi.T ad Muthulakshmi. S., Radom queue legth cotrol chart for (M/Ek/): ( /FCFS) queueig model, Iteratioal Joural of Mathematical Archive- 3(9), , 2. Copyright to IJIRSET 998
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