Control Chart Analysis of E k /M/1 Queueing Model

Transcription

1 Intenational OPEN ACCESS Jounal Of Moden Engineeing Reseach (IJMER Contol Chat Analysis of E /M/1 Queueing Model T.Poongodi 1, D. (Ms. S. Muthulashmi 1, (Assistant Pofesso, Faculty of Engineeing, Pofesso, Faculty of Science, Avinashilingam Institute fo Home Science and Highe Education fo Women, Coimbatoe, India Abstact: Queueing poblems ae most common featues not only in ou daily-life situations such as ban countes, post offices, ticet booing centes, public tanspotation systems, but also in moe technical envionments such as in manufactuing, compute netwoing and telecommunications. Fo any queueing system aveage queue length, aveage system length, aveage waiting time in the queue and aveage waiting time in the system ae the main obsevable pefomance chaacteistics. Contol chat is a gaph used to study how a pocess changes ove time and it is also used to contol ongoing pocesses. In this pape contol limits ae established to study the behavio of E /M /1 queueing model using the pefomance chaacteistics. Numeical esults ae given to highlight its applications. Keywods: queue length, waiting time, Elang aival, exponential sevice, contol limits. I. INTRODUCTION Queueing model with Elang aival has been discussed by Goss and Hais (1998 and seveal othes. Contol chat is a quality contol technique evolved initially to monito poduction pocesses. Montgomey (5 poposed a numbe of applications of Shewhat contol chats in assuing quality in manufactuing industies. Shoe ( developed contol chat fo andom queue length of M /M /s queueing model by consideing the fist thee moments and also Shoe (6 developed Shewhat-lie geneal contol chats fo G/G/S queueing system using sewness. Khapade and Dhabe (1 constucted the contol chat fo andom queue length of M/M/1 queueing model using method of weighted vaiance. Poongodi and Muthulashmi (1 analyzed numbe of customes in system of M/E /1 queueing model using contol chat technique. The Elang aival queueing system is applicable to many eal-life situations: (i In militay ecuits a ecuit fist lines up to have blood test at one station, an eye examination at the next station, mental test by a psychiatist at the thid, and is examined by a docto fo medical poblems at the fouth and so on. Recuit needs to pass all phases befoe enteing fo selection. (ii In a typical polling system, votes have to pass though many phases lie showing identity cads, signing, getting in ma etc., befoe voting. (iii In an ailine counte, passenges ae expected to chec in which consists of many phases, befoe enteing into the plane. and so on. This motivated the autho to study the constuction of Shewhat contol chats fo numbe of customes in the queue, numbe of customes in the system, waiting time in the queue and waiting time in the system of E /M/1 queueing model. II. E K / M/1 MODEL DESCRIPTION Conside a queueing system in which the inte-aival times follow -Elang distibution with mean1/ in which an aival has to pass though phases, each with a mean time 1/ pio to enteing the sevice. Sevice times follow an exponential distibution with mean1/ µ. Let p n be the pobability that thee ae n customes in E /M/1 queueing system. Then p n n -1 jn (P j p (1 whee p j (P is the pobability of completion of j phases and is given by p j (P λ p μ (P j whee ϵ (,1 is the oot of the chaacteistic equation µ z +1 (+µ z + (j ( IJMER ISSN: Vol. 4 Iss. 5 May

2 (P 1 since p taing /µ, equation ( becomes, j Contol chat analysis of E /M/1 Queueing model p (P j (1 Theefoe equation (1, educes to p n (1- ( n-1 which is a geometic distibution. Contol chat analysis of E /M/1 queueing model is caied out in the following sections. III. NUMBER OF CUSTOMERS IN THE QUEUE Let L q denote the numbe of customes in the queue. The expected numbe of customes in the queue, E(L q is given by E(L q (1 (n -1 p (3 and E(L q (n -1 pn Then vaiance of L q is (1 (1 (1 n Va(L q E(L q (E(L q (4 (1 Uppe contol limit (UCL, cental line (CL and lowe contol limit (LCL of Shewhat contol chat, unde the assumption that the numbe of customes in the queue follows nomal distibution, ae given by UCL E (L q + 3 Va(L q CL E (L q LCL E (L q - 3 Va(L q The paametes of the contol chat ae obtained by using (3 and (4 in (5 as UCL CL LCL (1 3 3 IV. NUMBER OF CUSTOMERS IN THE SYSTEM Let L s denote the numbe of customes in the system (both in queue and in sevice. The expected numbe of customes in the system is given by E(L s np n n (1 - ( (5 IJMER ISSN: Vol. 4 Iss. 5 May

3 Contol chat analysis of E /M/1 Queueing model (1 (6 and E(L s n pn (1 Then vaiance of L s is (1 (1 n ( Va(L s E(L s (E(L s (1 (1 The paametes of Shewhat contol chat, unde the assumption that the numbe of customes in the system follows nomal distibution, ae given by UCL E (L s + 3 Va(L s CL E (L s LCL E (L s - 3 Va(L s Using (6 and (7 in (8, the paametes of the contol chat ae obtained as UCL CL LCL 3 (1 3 (1 (1 V. WAITING TIME IN THE QUEUE Distibution of waiting time of a custome in the queue fo the model unde study is given by W q (t 1 e - µ (1- t, t. Then the pdf of waiting time is w q (t µ (1- e - µ (1- t Let w q denote the waiting time of customes in the queue. The mean of w q is (7 (8 E(w q μ μ(1 t e μ t and E(w q μ (1 t e dt dt μ t (9 IJMER ISSN: Vol. 4 Iss. 5 May. 14 6

4 Contol chat analysis of E /M/1 Queueing model μ (1 Then vaiance of w q is Va(w q E(w q (E(w q ( μ (1 (1 The paametes of Shewhat contol chat, unde the assumption that the waiting time of customes in the queue follows nomal distibution, ae given by UCL E (w q + 3 Va(w q (11 CL E (w q LCL E (w q - 3 Va(w q The paametes of the contol chat ae obtained by using (9 and (1 in (11 as UCL 3 μ(1 ( CL μ(1 LCL 3 μ(1 ( VI. WAITING TIME IN THE SYSTEM Let w s denote the waiting time of customes in the system. The expected waiting time of customes in the system, E(w s is given by E (w s E(L s / λ λ (1 Also the vaiance of w s is given by Va(w s Va(L s / λ (1 λ (1 The paametes of Shewhat contol chat, unde the assumption that the waiting time of customes in the system follows nomal distibution, ae given by UCL E (w s + 3 Va(w s (14 CL E (w s LCL E (w s - 3 Va(w s The contol chat paametes ae obtained by using (1 and (13 in (14 as CL UCL λ (1 3 (1 λ - IJMER ISSN: Vol. 4 Iss. 5 May (13

5 Contol chat analysis of E /M/1 Queueing model LCL 3 (1 λ - VII. NUMERICAL ANALYSIS Numeical analysis is caied out to analyze the pefomance of queueing system with efeence to the paametes λ, µ and. As LCL values ae negative fo the selected values of the paametes, they ae consideed as zeo and theefoe not shown in the table as a sepaate column. Table gives the contol chat paametes fonumbe of customes in the queue, numbe of customes in the system, waiting time of customes in the queue and waiting time of customes in the system fo cetain selected values of λ, µ and. TABLE - CONTROL CHART PARAMETERS FOR L Q, L S, W Q AND W S Numeical values in the table eveal the following featues: Fo all the pefomance measues of E /M/1 queueing system (i incease in the aival ate λ, inceases the paametes and UCL fo fixed values of µ and. (ii incease in the sevice ate µ, deceases the paametescl and UCL fo fixed values of λ and. (iii incease in the numbe of phases, deceases the paametescl and UCL fo fixed values of λ and µ. VIII. CONCLUSION In this pape contol chat technique is applied to analyze the system chaacteistics of E /M/1 queueing system. These chaacteistics povide the aivals to decide whethe to join the system o not. In addition it gives the infomation on the waiting time. REFERENCES [1] D. Goss and M. Hais,Fundamentals of queueing theoy( 5 th edition, John Wiley & Sons, Inc.,1998. [] D.C. Montgomey,Intoduction to statistical quality contol( 5 th edition, John Wiley& Sons, Inc.,5. [3] H. Shoe, Geneal contol chats fo attibutes, IIE tansactions,, 3, [4] H. Shoe, Contol chats fo the queue length in a G/G/S System, IIE Tansactions, 6, 38, [5] M.V. Khapade ands.d. Dhabe,Contol chat fo andom queue length N fo (M/M/1: ( /FCFS Queueing model, Intenational Jounal of Agicultual and Statistical sciences,1, Vol.1, [6] T. Poongodi. ands. Muthulashmi, Random queue length contol chat fo (M/E /1: ( /FCFS queueing model, Intenational Jounal of Mathematical Achive- 3(9,1, IJMER ISSN: Vol. 4 Iss. 5 May. 14 6