An analysis on Kendall Lee queueing system with non-preemptive priority at BRI Ahmad Dahlan Yogyakarta

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1 Journal of Physcs: Conference Seres PAPER OPEN ACCESS An analyss on Kendall Lee ueueng system wth non-preemptve prorty at BRI Ahmad Dahlan Yogyakarta To cte ths artcle: S D Ayunngtyas and N Bnatar 217 J. Phys.: Conf. Ser Vew the artcle onlne for updates and enhancements. Ths content was downloaded from IP address on 19/8/218 at 9:34

2 An analyss on Kendall Lee ueueng system wth nonpreemptve prorty at BRI Ahmad Dahlan Yogyakarta S D Ayunngtyas and N Bnatar Department of Mathematcs Educaton, Faculty of Mathematcs and Natural Scence, Yogyakarta State Unversty, Jl Colombo No.1 Depok Karangmalang Sleman Yogyakarta Indonesa E-mal: nkenash@uny.ac.d Abstract. Ths research was amed to determne the Kendall Lee model, measure the Performances and analyze the optmal condton of ueue at tellers lne at BRI Unt K.H Ahmad Dahlan Yogyakarta. Frst, the model was constructed based on Kendall Lee n whch the parameter values were obtaned from the ftness test of the data dstrbuton. Next, the Performances measurements were counted usng Lttle's Formula whle the optmal condton was hold based on the total cost. Snce the addton of the number of servers wll nfluence the total cost, the optmal condton s reached for a number of servers wth lowest total cost and satsfyng steady state condton. The result shows that the ueue system n teller counter at BRI Ahmad Dahlan Yogyakarta uses M/M/2 model wth non-preemptve servce dstrbuton. The optmum number of the server due to ts total cost was found to be two servers. By these, t could be concluded that the system has been usng optmal condton. 1. Introducton In our everyday lfe, the phenomenon of the ueue s very common. A ueue s a part of the crcumstances that occur n a seres of random operatonal actvtes n a servce faclty [1]. The ueue occurs because the need for a servce exceeds the capacty avalable for the servce. Therefore, to ncrease the system s performance, the servce provders provde multple servers. Moreover, some system dffers the customers nto dfferent prortes such as n customer servce centers, arport securty checkponts, hosptal emergency rooms, cloud computng systems or processor management n certan computer operatng systems [2]. In a prorty ueue, each customers has a relatve prorty to be served [3]. Based on ts prorty level, there are two types of prorty servce dscplne. The frst one s the non-preemptve prorty. If durng the servce tme, one wth hgher prorty customer arrves, t must wat untl the tem of lower prorty has been served [4]. The second one s the preemptve prorty. If durng the servce tme, one wth hgher prorty level arrves, t mmedately dsplaces the lower prorty customer. In ths case, prorty servce dscplne brngs new problem on the ueueng system. The ueue theory s one of the branches of appled mathematcs that can be used to solve ueung problems. Accordng to Hezer and Render, ths has been very usefull for a servce or manufacturng company [5]. One model of ueue system was ntroduced by Kendall, whch later was developed by Lee. The lne model of Kendall Lee s a model lne that s used to specfy the characterstcs of a ueue ncludes the dstrbuton of arrval and departure of customers, the number of servers, servce Content from ths work may be used under the terms of the Creatve Commons Attrbuton 3. lcence. Any further dstrbuton of ths work must mantan attrbuton to the author(s) and the ttle of the work, journal ctaton and DOI. Publshed under lcence by Ltd 1

3 dscplne, the capacty of the system and the number of customers who want to enter the ueue system as the source. The used Kendall Lee lne model s denoted by (a / b / c) : (d / e / f) [6], where f the system capacty and callng source are both nfnte then t s smplfed by a / b / c / d. A research study on the system wth prorty servce dscplne has been done by numerous researchers. For the sngle server, t has been dscussed n [3]. Specfcally, a system wth two prorty classes has been solved n [4] whle the watng tme for markovan multserver wth non-preemptve prorty M / M / c was dscussed n [7]. Research on the system of servce prorty ueue wth nonpreemptve mult-server was done by Gal, Hantler, and Taylor [8]. From the research, they have derved the Performances measurements of the system and the watng tme for each prorty class. A research conducted by Kao and Wlson [9] also dscusses the customer watng tme for the system of non-preemptve prorty ueue multple servers wth two prorty classes. Another study that dscusses ueung system wth a prorty of servce was done by N'amah and Sugto [1]. A ueue wth non-preemptve prorty was mplemented by a servce provder namely BRI KH Ahmad Dahlan Yogyakarta at teller s lne. The abundance of customers from the customer servce s lne and surveyor s lne to teller s lne has a hgher prorty to be served than the drect customers. By these, they found ther drect customer servce s delays are lnng up to get servce. The servce delays occur whenever there s an abundance of customer servce. Therefore, an analyss of ths ueueng system should be done. In ths research, t wll be dscussed the Kendall Lee model wth non-preemptve prorty occurred n BRI Unt KH Ahmad Dahlan Yogyakarta. The purpose of ths study was to analyze the model lne for Kendall Lee, to measure the Performances of the servce system at teller lne, and to fnd the number of server whch optmze the total cost. 2. Method The followng wll explan the methods and research desgn, locaton and tme of the study, data collecton technues and desgn, and data analyss technues Method and research desgn The method used n ths research was descrptve research method. Descrptve research method ams to descrbe the character of a varable, group or socal phenomena that occur n socety [11]. The desgn used n ths research was the case and feld research desgn Data collecton technue The research was conducted at PT Bank Rakyat Indonesa Unt KH Ahmad Dahlan Yogyakarta whch s located at KH Ahmad Dahlan st No. 8 Yogyakarta. The process of retreval data was obtaned n the customer's watng room, fgure 1. The data was collected usng the prmary data collecton technue n two ways, by usng ntervew and feld observaton. The data are the nterarrval tme of abundance customers from customer servce, the average rate of customer arrval of the ueue drectly to the teller and the servce tme. Fgure 1. Queue system. 2

4 2.3. Data analyss technue Customer data analyss technues n teller lne BRI Unt KH Ahmad Dahlan n Yogyakarta s as follows: 1. The prmary data for the number of drect customer arrval, abundance customer arrval tme and the servce rate are grouped per 15 mnutes and obtaned for 4 hours of research per days. 2. Posson dstrbuton test for the data was conducted usng the Kolmogorov-Smrnov test. Accordng to Segel [12], the calculaton steps for Kolmogorov-Smrnov test are (a) defne the - theoretcal cumulatve functon, e the expected cumulatve dstrbuton under H, (b) observe S X, the cumulatve freuency dstrbuton whch the values by parng each nterval N s obtaned from a random sample of N observatons, wth ntervals F ( X ), the cumulatve dstrbuton functon under the H assumpton, and then subtract each levels. (c) count the test statstc, N F X wth S ( ) N X for D max F X S ( X ). (d) determne the Dtable value whch s determned by referrng to the crtcal values from the one sample Kolmogorov- Smrnov table. 3. The prosedure to conduct a one sample Kolmogorov-Smrnov test was: Null hypothess H : the data follows the Posson dstrbuton, Alternatve hypothess H 1 : the sata do not follows the Posson dstrbuton. The level of sgnfcance : Crtcal value : the null hypothess, H, s rejected f the test statstc D s greater or euals than the crtcal value obtaned from the table, D D tabel. 4. Determne the ueue model of the customers ueueng system n teller lne at Bank Rakyat Indonesa (Persero) Branch Offce Unt KH Ahmad Dahlan Yogyakarta. Accordng to Taha [6], Kendall Lee's Queue Model s denoted as follows: a / b / c: d / e / f. where a : Dstrbuton of the arrval rate b : Dstrbuton of servces or departures tme rate c : Number of server n parallel, c = 1, 2, 3,... d : Servce dscplnes, such as FCFS, LCFS, SIRO e : The maxmum capacty of the system f : The callng source 5. The traffc ntensty for the -th prorty customer s, where s the average rate of c customer servces, s the average of customer arrval rate on th prorty, and c s the number of tellers to serve. Examne the steady-state soluton, 1 2 It s exst f the value s less than one. In other words, the system has reached a stable condton. 6. Calculate the Performances of the ueung system. The measures of Performances are the expected value of customer watng tme on all prortes ( W ) and the expected number of -th () customers n the ueue system () W s. Accordng to Gross et al [13], W and W s can be calculated successvely usng euaton (1) and euaton (2). 2 W W (1) 1 3

5 and where W () W ( ) ( ) 1 s W, (2) c nc c c1 c!1 c c n n! and k, (3) 6. Optmze the ueue system by consderng ts total cost. The cost analyss has taken a sgnfcant place n the ueueng theory [14]. The total cost was defned as the sum total of the total cost of servces per unt tme, and the cost of the watng customers per unt tme [5]. In mathematcal symbol, the cost model can be counted usng the followng formula: ETC c ESC c EWC() c (4) where c : the number of servers ETC() c : the total costs for c servers per unt tme. ESC() c : the expected servce cost per unt tme for c servers. EWC() c : the expected watng cost per unt tme for c servers. Accordng to Gross et al [13], to calculate the value of the expectatons of many customer types are n the ueue system, t can be usng the euaton 5: L () s 1 k 1 nc c c1 c! 1 c c n n! Research and Dscusson The ueueng n teller lne at BRI Unt KH Ahmad Dahlan follows prorty dscplne. The abundance customer from the customer servce lne and the surveyor lne to the teller lne has hgher prorty to be served. Therefore, the drect customer n the teller lne should wat untl the abundance customer are acompletely servced. In otherwse, f the arrval abundance customer occurs when all the teller are beng servng the customers, then the abundance customer should wat untl the servce s completely done. Ths type of prorty dscplne s called a nonpreemptve prorty servce dscplne. The study of ths research deals only for teller servce. Once customer completely served from the teller lne, t s assumed that the customer left the ueueng system. Therefore, the ueue follows sngle phase mult channel model. The ueue and servce flow model can be seen from fgure 2. (5) Fgure 2. Queue model. 4

6 3.1. Dstrbuton Match Test Dstrbuton sutablty test was used to test the data drectly teller lne customer arrval, arrval overflow customers, and customer servce s the Kolmogorov-Smrnov test. It was obtaned that the drect customer arrval per 24 mnutes was Posson dstrbuted wth parameter 1 8 the abundance customers arrval was followng Posson dstrbuton wth 2 1 and the servce was exponentally dstbuted, Kendall Lee s Queue Model Teller lne systems n PT Bank Rakyat Indonesa (Persero) Branch Offce Unt KH Ahmad Dahlan Yogyakarta has a model lne of Kendall Lee M / M / 2 : NPD / /. Ths model means arrvals and customer servce Posson dstrbuted wth 2 servers. Servce dscplne on the model M / M / 2 : NPD / / contans non-preemptve prorty rule wth unlmted system capacty and callng source. Fgure 3. State transton dagram Steady State Check The steady state condton s reach f the traffc ntensty s less than one From ths result, the ueung system wth two server has reached steady state condton Performances measurements The system characterstcs could be recognsed by lookng at the expected rate of watng tme of the customer n ueue lne for both prorty, W, and the expected rate of tme spent by -th type of () customer on the system, W s. By substtutng the parameter s values nto euaton (1) and euaton (2), the performances measurements from the system are as presented n table 1. Table 1. Performance s measurements Measurements Value (mnute) (mnute) (mnute) (mnute) (mnute) Analyss Queue In order to provde a better servce, management could perform three ways, add the number of servers, decrease the servce tme or reduce the system capacty. Ths research only consdered wth the number of servers by usng the cost model. The cost model can be used to evaluate and determne the optmum number of servers n the ueue system [15]. The operatng costs to be consdered n ths 5

7 research were the cost of watng for the customer, the salary and the logstcs costs for addng a number of tellers. Further nformaton, teller n PT Bank Rakyat Indonesa (Persero) Branch Offce Unt KH Ahmad Dahlan Yogyakarta work for seven hours each day and fve days n a week. There was an extra cost for servcng f there s an addtonal teller. The expected servce cost per unt tme s euals to the number of server whch s multpled wth the servce cost for one server. In other word, the formula for the expected servce cost can be wrtten as euaton (6). ESC() c c C (6) 1 Now, we consdered to the value of the servce cost for each server. To serve the customer, the management should pay for the salary and the logstc cost. The salary for a teller per month s Rp 3,5, and the logstc cost to buy a computer, desk, chars, and prnter are Rp 4,,, Rp 1,1,, Rp 7, and Rp 1,7,, respectvely. Assumed that the lfe perod of the logstc was fve years, then the servce cost per mnute s gven n table 2. Type Table 2. of servces Unt per Month (Rp) per Day (Rp) per Hour (Rp) per Mnute (Rp) Salary Person , 25, 417 Computer Set 67, 3, Table Unt 19, Chars Unt 12, Prnter Unt 29, Total 3,627, 181, From the table, t can be concluded that the servce cost per mnute for one server s Rp 434. Next, the expected watng cost s consdered to the customer s dsadvantage for beng n the system. Suppose that n order to be served, the customer should leave from work. In other words, the cost of watng euals the cost of the customer f t s not gong to work.therefore, the watng costs for customers are related to the per capta ncome of Indonesa. Accordng to the Central Bureau of Statstcs per capta ncome of Indonesa n 216 was Rp mllon. Therefore, the cost of watng customer per mnute, C 2, was Rp 476. The expected watng cost per unt tme EWC() c Ls C2. (7) Table 3 presents the three scenaro of number of tellers, whch are one, two and three, and the traffc ntensty,. Table 3. The expected number of customer Number of Teller Steady State (customer) 1 1 No - 2 Yes 1,423 3 Yes 1,988 Based on the table 3, for one teller, the system wll not reach a stable condton. It means that the teller wll have no tme for dle or rest. Snce the optmal condton s reach f the traffc ntensty s less than 1, a one teller ueue system s not an optmal condton. Next, the system s stablty was found to be n mnmum two tellers. The greater number of servers, the less expected number of 6

8 customers n the system. Therefore, we determned the optmum number of servers regardng to the number satsfed steady state condton but also mnmze the total cost. Table 4 shows the total needed cost f there s an addtonal teller. Total Teller Servce Total Servce Table 4 Total cost per mnute Watng Number of cost for Customers n Customers the System Total Watng c C 1 c x C 1 C 2 L s L s x C 2 Total (c x C 1 ) + (L s x C 2 ) 2 Rp 434 Rp 868 Rp Rp 677 Rp Rp 434 Rp 1.32 Rp Rp 524 Rp Based on the calculaton of the total cost per mnute n table 4, the optmal number of tellers was two tellers because t reures the lowest total cost, accounted to Rp 1,545. Snce the number of tellers at the moment wass 2 systems, we can conclude that the ueue system was already under optmal condtons. 4. Concluson Based on the results, the ueue system for teller servce at PT Bank Rakyat Indonesa (Persero) Branch Offce Unt KH Ahmad Dahlan n Yogyakarta follows multple channels sngle-phase lne models Kendall Lee ( M / M / 2 / NPD ). It means that the arrvals and servce are Posson dstrbuted, two servers are avalable and the appled servce s dscplne s a non-preemptve prorty. In addton, the capacty of ueung systems and callng sources s unlmted. The sze of the Performances of the ueue system n the company ncludes the average expected value of the customer's watng tme durng mnutes wth an average value of the expected tmng of the overflow customers and clents drect lne tellers are n the ueue untl the system had been served consecutvely for mn and mnutes. Teller ueue system n the enterprse can acheve optmal condtons currently served by two tellers. Therefore, the ueue system s already under optmal condtons. References [1] Kakay T J 24 Dasar teor antran untuk kehdupan nyata (Yogyakarta: And) [2] Brandwajn A and Begn T 217 Mult-server preemptve prorty ueue wth general arrvals and servce tmes Performance Evaluaton (Amsterdam: Elsever) [3] Mller R G 196 Prorty Queues. Ann. Math. Statst [4] Chang W 1965 Queung wth nonpreemptve and preemptve-resume prortes. Operatons Research [5] Hezer J and Render B 211 Operatons Management: 1 (New Jersey: Pearson). [6] Taha H A 27 Operaton Research and Introducton (Upper Saddle Rver: Pearson Educaton, Inc) [7] Kella O and Yechal U 1985 Watng tmes n the non-preemptve prorty M/M/C ueue. Stochastc Models, [8] Gal H R, Hantler S L et al 1988 Analyss of a non-preemptve prorty multserver ueue Advances n Appled Probablty [9] Kao P C and Wlson S D 1999 Analyss of non-preemptve prorty ueues wth multple servers and two prorty classes. European Journal of Operatonal Research [1] N'amah D and Sugto 211 Queung system wth prorty servce Proceedngs of the Natonal Semnar on Statstcs (Semarang: UNDIP) [11] Martono N 211 Quanttatve research methods (Jakarta: Rajawal Pers). [12] Segel S 1956 Nonparametrc Statstcs for the Behavoral Scences (New York: Mc Graw-Hll) [13] Gross D 28 Fundamentals of Queung Theory: 4. (New Jersey & Sons, Inc: John Wley). 7

9 [14] Mshra S S, Yadav D K 21 Computatonal approach to cost and proft analyss of clocked ueueng networks Contemporary Engneerng Scences [15] A Muhajr, N Bnatar 217 Queueng System Analyss of Mult Server Model at XYZ Insurance Company n Taskmalaya Cty AIP Conference Proceedngs