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 www.ijirset.com 9977
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 www.ijirset.com 9978
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 www.ijirset.com 9979
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 0. 0.47.046 5.797 0 0.6 0.476.082 5.980 0.7 0.482.2 6.72 0. 0.34 0.538 3.563 0.6 0.37 0.554 3.659 0.7 0.32 0.570 3.758 0.8 0.325 0.588 3.862 4 0. 0.235 0.362 2.726 0.6 0.238 0.372 2.795 0.7 0.24 0.382 2.866 0.8 0.244 0.393 2.940 0. 0.706 2.824 3.26 0 0.6 0.74 2.976 3.779 0.7 0.723 3.43 4.49 6 0.8 0.732 3.326.269 0. 0.47.046 5.797 0.6 0.476.082 5.980 0.7 0.482.2 6.73 0. 0.353 0.642 4.0358 0.6 0.357 0.66 4.48 0.7 0.36 0.682 4.265 0.8 0.366 0.704 4.387 Copyright to IJIRSET www.ijirset.com 9980
Iteratioal Joural of Iovative Research i Sciece, Egieerig ad Techology (A ISO 3297: 07 Certified Orgaiatio) 0. 0.94 8.823 77.288 8 0 0.6 0.952 23.80 97.280 0.7 0.964 32.29 30.606 0.8 0.976 48.780 97.265 0. 0.627.98 9.693 0.6 0.635 2.070 0.089 0.7 0.643 2.66 0.5 0.8 0.650 2.269 0.970 0. 0.47.046 5.797 0.6 0.476.082 5.980 0.7 0.482.2 6.73 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 34,327-350, 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., 39-334, 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), 3340-3344, 2. Copyright to IJIRSET www.ijirset.com 998