Construction of Control Chart for Random Queue Length for (M / M / c): ( / FCFS) Queueing Model Using Skewness

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1 Iteratioal Joural of Sietifi ad Researh Publiatios, Volume, Issue, Deember ISSN 5-5 Costrutio of Cotrol Chart for Radom Queue Legth for (M / M / ): ( / FCFS) Queueig Model Usig Skewess Dr.(Mrs.) A.R. Sudamai Ramaswamy, Mrs. B.Veila *Departmet of Mathematis, Aviashiligam Istitute for HomeSiee ad Higher Eduatio for Wome, Coimbatore-4. **Departmet of Mathematis, Sri Ramakrisha Egieerig College, Coimbatore-. Dr. (Mrs.) A.R. Sudamai Ramaswamy, Ph.D, arsudamairamaswamy@hotmail.om Correspodee Author: Mrs. B.Veila, M.Phil, veila.bs@gmail.om ABSTRACT I this paper, first-ome, first served, multiple hael Poisso/epoetial queueig system M / M / with ifiite waitig apaity is osidered. For this model, we itrodue a ew proedure for ostrutio of i) Shewhart s otrol hart C, ii) Cotrol hart C for radom queue legth N usig the method o skewess suggested by Shore() ad the otrol harts C ad C are ompared. Ide Terms: Radom queue legth, Average ru legth, False alarm rate, Average queue legth.. INTRODUCTION Waitig i lie for servie is oe of the most upleasat eperiees of life o this world. Barrer (957) says, I ertai queueig proesses a potetial ustomer is osidered lost if the system is busy at the time servie is demaded. If ot served durig this time, the ustomer leaves the system ad is osidered lost. I queueig system the ustomer satisfatio a be ireased by ostrutig otrol harts for N ad providig otrol limits for N so as to make effetive utilizatio of time. If otrol limits are displayed so that ustomer a have prior idea about otrol limits. With this fator, a attempt is made to fid otrol limits for radom queue legth N for (M/M/) : ( /FCFS) queueig model. The pioeerig work i this diretio was made by Haim Shore i. Two otrol harts C ad C are ostruted for radom queue legth N for (M/M/) : ( /FCFS)

2 Iteratioal Joural of Sietifi ad Researh Publiatios, Volume, Issue, Deember ISSN 5-5 queueig model by Khaparde ad Dhabe () ad also C ad C 4 are ostruted for radom queue legth N for (M/M/) : ( /FCFS) queueig model by Khaparde ad Dhabe () where Cotrol hart C : is the Shewhart otrol hart, Cotrol hart C : is the otrol hart usig method of weighted variae, Cotrol hart C : is the otrol hart for radom queue legth N based o skewess ad Cotrol hart C 4 : is the otrol hart usig Nelso s power trasformatio. I this paper ostrutio of otrol hart C for radom queue legth N for (M/M/) : ( /FCFS) queueig model is doe usig method based o skewess. Performae of otrol hart C is ompared with otrol hart C.. (M / M / ) : ( / FCFS) QUEUEING MODEL I this model arrival follows Poisso distributio with parameter ad servie time follows idepedetly ad idetially distributed epoetial distributio with parameter. There are idetial servers preset i the system. Figure I represet the state-trasitio diagram for this model. + () FIGURE I : STATE-TRANSITION RATE DIAGRAM FOR M / M / MODEL Let radom variable N deote the umber of ustomers i the system both waitig ad i servie ad P deote the steady state distributio of radom variable N give by P = P,! P,! () where P =,!! is the mea arrival rate ad is the mea servie rate. Let radom variable W S deote the waitig time spet i the system by the ustomer. This iludes both the waitig time ad servie time. The p.d.f. of radom variable W S is give by

3 Iteratioal Joural of Sietifi ad Researh Publiatios, Volume, Issue, Deember ISSN 5-5 ( ) WS f(w S ) = ( ) e, W S > () Thus W S follows a epoetial distributio with mea E(W S ) = = W S f(ws ) d WS ( )W S ( ) e d WS W S E(W S ) = ( ) () E(W S ) = ( ) (4) Usig (.) ad (.4), the variae is give by V(W S ) = ( ) (5) The distributio futio of W S is F() = P(W S ) = f(w ) d S W S = ( )W S ( ) e d W S = e ( ), >. F() = e ( ), >. The probability distributio of N is a geometri distributio with parameter ( ). The steady state distributio of the radom variable N depeds o two parameters ad oly through their ratio. Usig (), the average umber of ustomers ad variae of queue legth i the system is give by E(N) = = P + P = P +!! P

4 Iteratioal Joural of Sietifi ad Researh Publiatios, Volume, Issue, Deember 4 ISSN 5-5 = P )! ( )! ( Here P =!! E(N) = =!!!! (6) E(N ) = = ) P )P (( = )P ( + )P ( + P + P = ) (! P + ) (! P +! P +! P E(N ) = =!!!! (7) Usig (6) ad (7), the variae of queue legth i the system is give by V(N) =!!!!

5 Iteratioal Joural of Sietifi ad Researh Publiatios, Volume, Issue, Deember 5 ISSN 5-5!! )! ( )! ( (8) Usig (8), the stadard deviatio is alulated as = /!! )! ( )! (!!!! (9) E(N ) = = )P ) ( ) )( (( = P! ) )( ( +! ) )( ( +! ) ( +! ) ( +!! =!!!! ()

6 Iteratioal Joural of Sietifi ad Researh Publiatios, Volume, Issue, Deember 6 ISSN 5-5 Usig (), the third etral momet is give by =!!!! -!!!!! ) ( )! ( +!! =!!!! )! ( )! (!!!!!! () Let S k (N) deote the traditioal measure of skewess for radom queue legth amely, S k (N) =. where is the third etral momet.

7 Iteratioal Joural of Sietifi ad Researh Publiatios, Volume, Issue, Deember 7 ISSN 5-5 Usig (9) ad (), the skewess for radom queue legth is give by S k (N) = / )! ( )! (!!!! )! ( )! (!! )! ( )! (!!!!!! (). SHEWHART S CONTROL CHART C Whe it is possible to speify stadard values for the proess mea ad stadard deviatio, we may use these stadards to establish the otrol hart. The the parameters of the hart are UCL = + A CL = LCL = A where the quatity = A, say, is a ostat that depeds o. Usig epressios (6) ad (9), the otrol limits for (M / M / ) : ( / FCFS) model are alulated as follows.

8 Iteratioal Joural of Sietifi ad Researh Publiatios, Volume, Issue, Deember 8 ISSN 5-5 UCL =!! )! ( )! (!!!! A )! ( )! ( / CL =!! )! ( )! ( LCL =!! )! ( )! (!!!! A )! ( )! ( / TABLE I MEAN, STANDARD DEVIATION, UPPER CONTROL LIMIT FOR THE RANDOM VARIABLE N FOR DIFFERENT VALUES OF USING SHEWHART S CONTROL CHART C WITH A = AND C =

9 Iteratioal Joural of Sietifi ad Researh Publiatios, Volume, Issue, Deember 9 ISSN 5-5 S.No. CL UCL u ARL CONSTRUCTION OF CONTROL CHART C FOR RANDOM VARIABLE N Before ostrutig the hart, it is essetial to otie that the probability futio of a geometri distributio is a mootoously dereasig futio of ad sie otrol limits for queueig system are to be obtaied, there is o meaig i defiig lower otrol limit (LCL) for this distributio. Also upper otrol limit (UCL) obviously makes sese sie it will eable moitorig improbable high value of N.. CONTROL CHART C I the queueig model (M / M / ) : ( / FCFS), the radom variable N deote the umber of ustomers i the system both waitig ad i servie. The steady state distributio of radom variable N is give by epressio (). I order to obtai otrol limits usig Shore s method (), the first three momets of the radom variable N give by epressios (6), (8) ad () respetively, are eeded. The epressio () shows that skewess of the distributio of N depeds o,, ad. Takig ito aout the mea, stadard deviatio ad skewess of the uderlyig geometri distributio of N the epressio for the ew otrol limits for the umber of ustomers i (M / M / ) : ( / FCFS) system usig Shore s method () are as follows : Cotrol Limits for Cotrol Chart C whe S k <.5

10 Iteratioal Joural of Sietifi ad Researh Publiatios, Volume, Issue, Deember ISSN 5-5 UCL = E(N) + V(N) +.4 V(N) S k (N). CL = E(N) LCL = E(N) V(N) +.4 V(N) S k (N) +. Cotrol Limits for Cotrol Chart C whe S k >.5 UCL = E(N) +.64 V(N) V(N) S k (N). CL = E(N) LCL = E(N).64 V(N) + (.4) (.946) V(N) S k (N) +. Cotrol Limits for Cotrol Chart C for (M / M / ) : ( / FCFS) Model whe S k <.5 UCL = )! ( )! (!! +!!!! )! ( )! (!!

11 Iteratioal Joural of Sietifi ad Researh Publiatios, Volume, Issue, Deember ISSN !!!! )! ( )! (!!!!!!!! )! ( )! (!! + )! ( )! ( CL = )! ( )! (

12 Iteratioal Joural of Sietifi ad Researh Publiatios, Volume, Issue, Deember ISSN 5-5!! LCL = )! ( )! (!!!!!! )! ( )! (!! +.4!!!! )! ( )! (!!

13 Iteratioal Joural of Sietifi ad Researh Publiatios, Volume, Issue, Deember ISSN 5-5!!!!!! )! ( )! (!! + )! ( )! ( Cotrol Limits for Cotrol Chart C for (M / M / ) : ( / FCFS) Model whe S k >.5 UCL = )! ( )! (!! +.64!!!!

14 Iteratioal Joural of Sietifi ad Researh Publiatios, Volume, Issue, Deember 4 ISSN 5-5 )! ( )! (!! + (.946)!!!! )! ( )! (!!!!!!!! )! ( )! (!!

15 Iteratioal Joural of Sietifi ad Researh Publiatios, Volume, Issue, Deember 5 ISSN )! ( )! ( CL = )! ( )! (!! LCL = )! ( )! (!!.64!!!! )! ( )! (!! + (.4) (.946)!!!!

16 Iteratioal Joural of Sietifi ad Researh Publiatios, Volume, Issue, Deember 6 ISSN 5-5 )! ( )! (!!!!!!!! )! ( )! (!! + )! ( )! ( Table. shows that the miimum value of skewess is early. for high values of. TABLE II SKEWNESS, CONTROL LIMITS FOR THE RANDOM VARIABLE N FOR DIFFERENT VALUES OF USING CONTROL CHART C WITH L = ad C = S.No. Skewess CL UCL u ARL

17 Upper Cotrol Limit (UCL) Iteratioal Joural of Sietifi ad Researh Publiatios, Volume, Issue, Deember 7 ISSN Cotrol hart C Cotrol hart C Traffi itesity (ρ) FIGURE II :COMPARISON OF UPPER CONTROL LIMIT (UCL) OF CONTROL CHARTS C AND C WITH L = AND C = FOR DIFFERENT VALUES OF It is otied that the upper otrol limit of otrol hart C outperforms the otrol hart C for all values of. Upper otrol limit values i otrol hart C are very very large tha that of hart C. Therefore for skewed populatio the use of otrol hart C is advisable. This a be observed from Figure II

18 Iteratioal Joural of Sietifi ad Researh Publiatios, Volume, Issue, Deember 8 ISSN 5-5 FIGURE III: COMPARISON OF AVERAGE RUN LENGTH OF CONTROL CHARTS C AND C WITH L = AND C = FOR DIFFERENT VALUES OF CONCLUSON The urves of ARL values are displayed i Figure III for harts C ad C. The horizotal ais of spas.5 to.45, Figure III demostrates that the ARL produed by hart C is superior to the hart C. REFERENCES. Arold, A.O. (978). Probability, Statistis ad Queueig Theory with Computer Siee Appliatios, Seod Editio, Aademi Press Limited, New York.. Barrer, D.Y. (957). Queueig with Impatiet Customers ad Idifferet Clerks, Operatios Researh, 5, Khaparde, M.V. ad Dhabe, S.D. (). Cotrol Charts for Radom Queue Legth N for (M / M / ) : ( / FCFS) Model, Iteratioal Joural of Agriultural ad Statistial Siees,, Khaparde, M.V. ad Dhabe, S.D. (). Cotrol Charts for Radom Queue Legth N for (M / M / ) : ( / FCFS) Queueig Model Usig Skewess ad Power Trasformatio, Bulleti of Pure ad Applied Siees, E, Motgomery, D.C. (5). Itrodutio to Statistial Quality Cotrol, Fifth Editio, Joh Wiley ad Sos I. 6. Shore, H. (). Geeral Cotrol Charts for Attributes, IIE Trasatios,, Sudamai Ramaswamy, A.R. (). Statistial Aalysis of M / M / Model, Researh Highlights,, 7-.

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.