Case Study for Shuruchi Restaurant Queuing Model

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1 IOSR Joural of Buie ad Maagemet (IOSR-JBM) e-issn: X, p-issn: Volume 19, Iue 2. Ver. III (Feb. 217), PP Cae Study for Shuruchi Retaurat Queuig Model Md. Al-Ami Molla Md. Al-Ami Molla, ecturer, Departmet of Natural Sciece, Daffodil Iteratioal Uiverity, Dhaka, Bagladeh Abtract: Waitig lie ad ervice ytem are idipeable part of our daily life. Every retaurat would like to avoid loig their deired cutomer due to a log wait i the queue. Providig more waitig chair oly, would ot olve a problem whe the cutomer withdraw thi retaurat ad go to aother. We thik ervice time eed to be improved, which how to maage the ytem ad improve the queuig ituatio, we ca apply queuig model. Thi paper aim to how the igle chael queuig model M/M/1. We have obtaied the oe moth daily cutomer data from a retaurat, amed Shuruchi at Savar i Dhaka city. Uig ittle theorem ad the igle chael waitig lie model M/M/1. We have determied the arrival rate λ, ervice rate µ, the utilizatio rate ρ ad the average waitig time i the queue before gettig ervice. We have alo determied the probability of impatiet cutomer to igore the ytem. At Shuruchi retaurat, the arrival rate i 1.43 cutomer per miute (cpm) ad the ervice rate i 1.45 cutomer per miute (cpm), the average umber of cutomer i 64, cutomer waitig time i 14 miute ad the utilizatio rate i.98. We have dicued about the beefit of applyig queuig model to a buy retaurat i our cocluio part. Keyword: Queue, ittle theorem, retaurat, M/M/1 queuig model, waitig time. I. Itroductio Queuig theory i the mathematical tudy of queue or waitig lie. Queue or waitig lie are familiar pheomea to all. Thi theory i ued to model ad predict waitig time ad the umber of cutomer arrival. Thi paper ue queuig theory to tudy the waitig lie i Shuruchi retaurat at Savar i Dhaka. It i the buiet retaurat i Savar. It provide 1 table of 6 people. There are 4 to 5 waiter workig at ay oe time. Durig weekday (Suday to Thurday), there are over 3 cutomer come to the retaurat ad durig weeked (Friday ad Saturday), over 5 cutomer at luch hour 1:pm-16:3pm. There are everal determiig factor for a retaurat to be coidered a good or a bad oe. Tate, clealie, the retaurat layout ad ettig are ome of the mot importat factor. Thee factor, whe maaged carefully, will be able to attract plety of cutomer [1]. Waitig for ervice i a retaurat i a part of our daily routie, epecially durig luch ad dier time. So, queuig theory i perfectly applicable i retaurat. Reearcher have previouly ued queuig theory to model the retaurat operatio [2], reduce cycle time i a buy fat food retaurat [3], a well a to icreae throughput ad efficiecy [4]. II. Backgroud The queuig theory or waitig lie theory wa iitially propoed by Daih telephoe egieer amed A. K. Erlag. I the early 193, he took up the problem o cogetio of telephoe traffic. The complexity wa that durig buy period, telephoe operator were uable to hadle the call the momet they were made, reultig i delayed call. A. K. Erlag directed hi firt effort at fidig the delay for oe operator ad later of the reult were exteded to fid the delay for everal operator. The field of telephoe traffic wa further developed by Moli i 1927 ad Thorto D-Fry i It wa oly after World War II that thi early work wa exteded to other geeral problem ivolvig queue. Hi work ipired egieer, mathematicia to deal with queuig problem uig probabilitic method. Queuig theory became a field of applied probability ad may of it reult have bee ued i operatio reearch, computer ciece, telecommuicatio, traffic egieerig ad reliability theory. It hould be emphaized that i a livig brach of ciece where the expert publih a lot of paper ad book. III. Baic Characteritic The baic characteritic of queuig pheomeo are Uit arrive at regular or irregular iterval of time, at a give poit called the ervice ceter. For example, pero arrivig a ciema hall, hip arrivig a port, patiet eterig the doctor chamber ad o o. All thee uit are called the arrival of cutomer. At a ervice ceter there are oe or more ervice chael or ervice tatio. If the ervice tatio are empty, the arrivig cutomer will be erved immediately, if ot will the arrivig cutomer wait i lie util the ervice i provided. Oce the ervice ha bee completed the cutomer leave the ytem. DOI: 1.979/487X Page

2 Cae Study for Shuruchi Retaurat Queuig Model IV. Characteritic of a Queuig Sytem Cutomer ad erver are the mai elemet of queuig model. Cutomer are ormally called uit. It may be a pero, machie, vehicle ad partie etc. Server i the ytem which perform the ervice to cutomer. Thi may be igle or multi-chael. Cutomer are geerated from a ource. O arrival at the facility, they ca tart ervice immediately or wait i a queue if the facility i buy. Whe a facility complete a ervice, it automatically pull a waitig cutomer, if ay, from the queue. If the queue i empty, the facility become idle util a ew cutomer arrive. A queuig model i pecified completely by ix characteritic: 1. Arrival Fahio: It repreet the arrivig patter of cutomer i the ytem. Cutomer do t come at a fixed regular iterval of time, their arrival are radom fahio, it ted to be clutered or cattered radomly. I a give time, the umber of arrival i etimated by uig a dicrete probability ditributio (DPD), uch a Poio ditributio. 2. Departure (ervice) Ditributio: It repreet the patter i which the umber of cutomer leave the ytem. It may alo be repreeted by the ervice time, which i the time period betwee ucceive ervice. It may be cotat or variable but kow, or radom (variable with oly kow probability). It i idepedet of the iter-arrival time. It i decribed by the expoetially probability ditributio. 3. Service Chael: The waitig lie ytem may have multi ervice chael ad igle ervice chael. Arrivig cutomer may form oe queue ad get erviced, a i a doctor cliic. The ytem may have a umber of ervice chael, which may be arraged i parallel or i erie or a complex combiatio of both. A queuig model i called igle chael model, whe the ytem ha oe erver oly ad multichael model, whe the ytem ha a umber of parallel chael each with oe erver. 4. Queue Diciplie (ervice diciplie): Thi repreet the order i which cutomer are elected from a queue, i a importat factor i aalyzig queuig model. The mot commo diciplie i firt come, firt erve (FCFS), e.g. railway tatio, bak ATM, doctor cliic etc. Other diciplie are lat come, firt erve (CFS) a i big godow ad ervice i radom order (SIRO) baed o priority. 5. Sytem Capacity: I the ytem the maximum umber of cutomer ca either be fiite or ifiite. I limited facilitie, oly a fiite umber of cutomer are allowed i the ytem ad ew arrivig cutomer are ot allowed to joi the ytem ule the umber become le tha the fixed umber. 6. Populatio: The ource from which cutomer are geerated may be fiite or ifiite. A fiite ource limit the cutomer arrivig for ervice. V. Kedall Notatio I 1953, Kedall propoed a otatio for ummarizig the characteritic of the queuig ituatio, which i writte a Where a / b/ c: d / e/ f a Arrival fahio, b Service time ditributio, c Number of parallel erver ( 1,2,3,...), d diciplie, e Maximum umber of uit allowed i the ytem, f Size of the callig ource. Queue VI. ittle Theorem ittle theorem [5] decribe the relatiohip betwee throughput rate (i.e., arrival ad ervice rate), cycle time ad work i proce (i.e., umber of cutomer/job i the ytem). Thi relatiohip ha bee how to be valid for a wide cla of queuig model. The theorem tate that the expected umber of cutomer (N) for a ytem i teady tate ca be determied uig the followig equatio. (1) Here, i the mea arrival rate for cutomer comig to the ytem, W ytem. Three fudametal relatiohip ca be derived from ittle theorem [6]: Icreae if or W icreae. Icreae if icreae or W decreae. W i the expected waitig time i the W Icreae if icreae or decreae. DOI: 1.979/487X Page

3 Cae Study for Shuruchi Retaurat Queuig Model VII. Shuruchi Retaurat Model (M/M/1: FCFS/ / ) We have obtaied oe moth daily cutomer data from thi retaurat through iterview with the retaurat maager a well a data collectio through obervatio at the retaurat. We have come to kow about the capacity of it ad the umber of waiter ad waitre. Baed o the above iformatio we have decided that the queuig model that bet illutrate the operatio of thi retaurat i M/M/1: FCFS/ /. Thi mea that the arrival are Poio ditributed ad ervice time i expoetially ditributed, retaurat ha oe erver oly. Aumptio: Cutomer come from a ifiite populatio, They follow the Poio ditributio, Cutomer behaviour are treated o a ervice diciplie Firt Come Firt Serve (FCFS) ad do ot balk or reege, Service time follow the expoetial ditributio, Utilizatio factor ρ<1 that i the average ervice rate i fater tha the average arrival rate. For aalyzig the Shuruchi retaurat M/M/1 queuig model the followig variable will be ivetigated [6] o The mea cutomer arrival rate. o The mea ervice rate. o o ; The utilizatio factor. Probability of zero cutomer i the retaurat p 1 (2) p o : The probability of havig cutomer i the retaurat p (1 ) p (3) o : Average umber of cutomer i the retaurat 1 o : The average umber of cutomer i the queue q q o W : q 2 1 The average waitig time i the queue W q q o W : The average waitig time pet i the retaurat W 1 Fig-1: chematic repreetatio of M/M/1 queuig ytem (4) (5) (6) (7) DOI: 1.979/487X Page

4 Cae Study for Shuruchi Retaurat Queuig Model VIII. Obervatio Ad Dicuio The oe moth daily cutomer data were hared be the retaurat maager, a how i Table-1 Table-1 Su Mo Tue Wed Thu Fri Sat 1 t week d week rd week th week From the above Fig-2, the umber of cutomer o weeked day i greater to the umber of cutomer durig weekday. Durig luch time, o weeked, i the buiet period for the retaurat. IX. Calculatio There are o average 3 cutomer come to the retaurat i 3.5 hour durig luch time. So, the arrival rate cutomer per miute (cpm). 21 From the obervatio ad dicuio we have foud out that o average each cutomer ped 45 miute ) i the retaurat, queue legth i aroud 2 cutomer ), o average ad waitig time i aroud 15 miute. (q Now, miute (uig equatio 6 ) W q 1.43 It i oticed that the actual waitig time doe ot vary by much compared to the theoretical waitig time. The average umber of cutomer i the Shuruchi retaurat uig ittle theorem (1) W cutomer. We ca derive the ervice rate uig (4) (1 ) 1.43( ) Hece, With very high utilizatio rate.98 durig luch period, the probability of havig zero cutomer i the ytem, uig (2) p The queuig theory provide the formula to etimate the probability of havig cutomer i the retaurat, a follow: p p (1 ) (1.98) It i aumed that, impatiet cutomer will tart to balk; whe they ee more tha 1 cutomer are already waitig for gettig ervice i the retaurat. It i alo aumed that a patiet cutomer ca tolerate maximum 25 cutomer i the queue. The capacity of the retaurat i 6 cutomer. Now, we ca calculate the probability of havig 1 cutomer i the queue. DOI: 1.979/487X Page (W

5 p Probability of cutomer goig away (more tha 1 cutomer i the queue) (more tha 7 cutomer i the retaurat) p p p % Cae Study for Shuruchi Retaurat Queuig Model X. Evaluatio The utilizatio factor i directly proportioal with the mea umber of cutomer which mea that the mea umber of cutomer will icreae a the utilizatio icreae. I the buy retaurat Shuruchi, the utilizatio factor i very high.98. I cae the cutomer waitig time i lower or i other word we waited for le tha 15 miute the umber of cutomer that are able to be erved per miute will icreae. Whe the ervice rate i higher the utilizatio will be lower, which make the probability of the cutomer goig away decreae [1]. XI. Beefit Sice Shuruchi retaurat i the buiet retaurat, o thi reearch ca help thi retaurat to eure their QoS (Quality of Service) by forecatig, if there are may cutomer i the queue. It may become a model to aalyze the curret ituatio ad to improve the ytem a retaurat ca ow, etimate how may cutomer will wait i the queue, by aticipatig the huge umber of cutomer comig to the retaurat with a view to gettig ervice. The formula provide mechaim to model the retaurat queue that i impler tha the creatio of imulatio model [7] XII. Cocluio Thi reearch paper ha dicued the applicatio of igle chael M/M/1 queuig model i Shuruchi retaurat. Applyig thi model we have obtaied that, the cutomer arrival rate i 1.43 cpm ad the ervice rate i 1.45 cpm. If there are 1 cutomer i the queue waitig for ervice, the probability of buffer flow i 6.23%. The probability of buffer overflow i the probability that cutomer will ru away, becaue may be they are impatiet to wait i the queue [1]. We expect our reearch ca help doig the bettermet of the retaurat. A our future work, we will develop a imulatio model for the retaurat. By developig a imulatio model we will be able to cofirm the reult of the aalytical model that we develop i thi paper. I additio, a imulatio model allow u to add more complexity o that the model ca mirror the actual operatio of the retaurat more cloely [8] Referece [1] Mathia Dharmawirya, Erwi Adi, Cae Study for Retaurat Queuig Model, 211 Iteratioal Coferece o Maagemet ad Aritificial Itelligece, IPEDR vol.6, Idoeia. [2] D.M. Bra ad B.C. Kulick, Simulatio of retaurat operatio uig the Retaurat Modelig Studio, Proceedig of the 22 Witer Simulatio Coferece, IEEE Pre, Dec. 22, pp [3] S. A. Curi, J. S. Voko, E. W. Cha ad O. Timhoi, Reducig Service Time at a Buy Fat Food Retaurat o Campu, Proceedig of the 25 Witer Simulatio Coferece, IEEE Pre, Dec. 25. [4] A. K. Kharwat, Computer Simulatio: a Importat Tool i the Fat Food Idutry, Proceedig of the 1996 Witer Simulatio Coferce, IEEE Pre, Dec. 1996, pp [5] J. D.C. ittle, A Proof for the Queuig Formula: =λw, Operatio Reearch, vol. 9(3), 1961, pp [6] M. agua ad J. Marklud, Buie Proce Modelig, Simulatio ad Deig. ISBN X. Pearo Preice Hall, 25. [7] T. C. Whyte ad D. W. Stark, ACE: A Deciio Tool Retaurat Maager, Proceedig of the 1996 Witer Simulatio Coferece, IEEE Pre, Dec. 1996, pp [8] T. Altiok ad B. Melamed, Simulatio Modelig ad Aalyi with ARENA. ISBN Academic Pre, 27. [9] V. K.. Kapoor (July, 1999), Operatio Reearch (4 th Editio). [1] Cooper R. B. (198), Itroductio to queuig theory (21 th Editio). North Hollad. [11] H. A. Taha (27), Operatio Reearch (8 th Editio). [12] Frederick S. Hillier & Mark S. Hiller (25), Itroductio to Maagemet Sciece. A Modelig ad Cae Studie Approach with Spreadheet. [13] Prem Kumar Gupta & D. S. Hira (23), Operatio Reearch. [14] Chae R. ET Aquilao N. (1973), Productio ad Operatio Maagemet. Iraw. [15] ee A. M. (1971), Applied Queuig Theory. New York. [16] More Phillip. (1971), Method of Operatio Reearch. odo. [17] Wager H. M. (1975), Priciple of Operatio Reearch. Iteratioal Editio. [18] Wito Wayae. (1991), Operatio Reearch, Applicatio ad Algorithm (2 th Editio). [19] Huliad Tao Yag, (2), Theory ad methodology queue with a variable umber of erver, Europea Joural of Operatio Reearch, vol. 124, pp [2] N. K. Tiwari & Shihir K. Shadilya. (29), Operatio Reearch (3 rd Editio). [21] Nita H. Shaha, Ravi M. Gor, Hardik Soi. (21), Operatio Reearch (3 rd Editio). [22] S. K. Dhar, Tazia Rahma, (213), Cae Study of Bak ATM Queuig Model, IOSR Joural of Mathematic, IOSR-JM, 7(1), 1-5. DOI: 1.979/487X Page

6 Cae Study for Shuruchi Retaurat Queuig Model [23] V. S. Selvi ad M. Nihathi, Mathematical Appliatio of Queuig Theory i Call Ceter, Iteratioal Joural of Scietific & Egieerig Reearch, IJSER, vol. 3, ISSUE 11, November-212. [24] Bhavi Patel Parvi Bhathawala, (212), Cae Study for Bak ATM Queuig Model, Iteratioal Joural of Egieerig Reearch ad Applicatio, IJERA, 2(5), , ISSN: [25] M. S. R. Murthy, M. Puhpa atha & K.G. R. Deepthi (Jauary-14), Miimizig The Waitig Time at Bak ATM for ervice with Queuig Model, Iteratioal Joural of Scietific Reearch, IJSR, ad vol. 3, ISSN: [26] Mohammad Shyfur Rahma Chowdhury, Queuig Theory Model Ued to Solve The Waitig ie of a Bak- A Study o Ilami Bak Bagladeh imited, Chawkbazar Brach, Chittagog, Aia Joural of Social Sciece & Humaitie, vol. 2 No. 3 Augut 213. DOI: 1.979/487X Page

Queueing Theory (Part 3)

Queueing Theory (Part 3) Queueig Theory art 3 M/M/ Queueig Sytem with Variatio M/M/, M/M///K, M/M//// Queueig Theory- M/M/ Queueig Sytem We defie λ mea arrival rate mea ervice rate umber of erver > ρ λ / utilizatio ratio We require