Modeling a Small Scale Business Outfit With M/M/1 Markovian Queuing Process and Revenue Analysis
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1 ISSN (Paper ISSN 5-05 (Onlne Vol.4, No.8, 014 Modelng a Small Scale Busness Outft Wth M/M/1 Markovan Queung Process and Revenue Analyss Nwankwo, Steve.C. School of Scence and Industral Technology Department of Statstcs, Aba State Polytechnc, Aba, PMB 7166 Ngera ABSTRACT Emal: stevenwankwo50@yahoo.com, Phone: Ths artcle presents a queung model on the operaton of a small busness whch are often vsted wth a lot of set- backs and uncertantes. Probablty models were derved for two system characterstcs namely the customers' nter-arrval rate (I t, the servce tme per customer (Y t, and the Servce charge (C t per customer was also recorded. Ths data collected were subjected to sngle factor analyss of varance wth fxed effect to confrm whether the data came from the same populaton dstrbuton. However t was observed that the nterarrval tme for the customers was exponental, servce rate also exponental wth a traffc ntensty of = 0.30 and the server was dle for about 70% n every one hour. Wth an average servce charge of 3.5 per customer the busness makes N1, per year. Key words: stochastc model, queue, nter-arrval rate, servce rate, Traffc ntensty. 1.0 INTRODUCTION Most of us spend part of our walkng hours watng for somethng - to pay at the checkout counter at a market, mal a letter n the post offce, get food n a cafetera, borrow a book from lbrary, land at an arport etc. Frustrated, we may ask can't enough checkout clerks be provded, enough toll booths, landng strps etc? Why s there always nsuffcent capacty? To make optmum decsons, t becomes necessary to predct the operatng characterstcs of watng lne systems such as the average number of people watng, people beng served, and attendants dlng etc. Ths paper ams at reportng such analyss carred out from a photocopyng centre based n Aba. When a watng lne system s desgned, management s usually nterested n overall performance and economc consequences as tme goes by. Thus decsons are based on performance characterstcs such as: ( ( ( (v The average (expected number of customers arrvng per hour. The average tme to servce a customer. The number of customers n the queue. The number of customers beng served, etc are all random varables. Thus we are dealng here wth probablty processes and so we need probablty models..0 LITERATURE REVIEW The lterature on queung or watng lne model and ts applcaton are extensve. There are many stuatons where queues are experenced, such as Arcrafts watng to land, fles n the "n-tray", cars awatng for routng servce etc. Therauf etal(1995 observed that the theory of watng lne emanated from the work of A.K. Erband. 86
2 ISSN (Paper ISSN 5-05 (Onlne Vol.4, No.8, 014 He explaned that Erland started hs work n 1909 when he expermented on a problem dealng wth the congeston of telephone traffc. Smth(1977 opned through hs schematc approval to the system that queung system comprses of several components such as, source or callng populaton, callng unt, servce dscplne and servce faclty. Frank(1963 Opned that when a customer arrves at a servce faclty, the customer often fnds the server busy and he thus has to wat, whle he s watng other customers may arrve, a farly long watng lne or queue may develop. Attahru (003 n hs work ttled "Comparatve queue analyss" argued that most work on queue analyss assumed ndvdual nter-arrval tmes. Ryan Berry (00 n hs paper queung theory, emphaszed that the frst paper on queung theory was publshed by A.K. Erlang where the problem of determnng how many telephone crcuts were necessary to phone servce. Honda etal (008 n ther paper Solvng of watng tmes models n the arport usng queung theory model and lnear programmng observed that watng lnes and servce systems are mportant parts of the busness world. Alexander (006 consdered a model where multple queues are served by servers whose capacty vares randomly. 3.0 METHODOLOGY The perod for data collecton spanned four weeks. The busness opens by 8.30am and closes 5.30pm each day, Monday to Saturday. For accuracy, a stop-watch and a calculator were used. The varables nter-arrval (I t, the servce tme (Y t, the watng tme (W t and the servce charge (C t per customer were computed and recorded. The summary statstcs of the computed operatonal varables I t, Y t, W t and C t are presented n appendx table 1. The data were further put to test, to test for equalty of mean on the data for nter-arrval tme (I t, servce tme (Y t, and watng tme (W t. We pcked a random sample of sze 10 (usng systematc samplng from each of the three varables data (I t, Y t, W t for each of the four weeks. Factor consdered were the four weeks. We used the method of sngle factor analyss of varance wth fxed effect. The model s Where Y j the jth observato n of theth week the unversal constant e j theeffect due to theth week theerror due to the jth observaton of the th week 87
3 ISSN (Paper ISSN 5-05 (Onlne Vol.4, No.8, 014 SST j1 Y j T ; ( kn k 4, n 10 SS 4 1 Y n T ( kn 0 0 Hypotheses 1 : j 0 : j Level of sgnfcance, Crtcal Regon F ; k 1, N k.e. F0.05,(3,36. 9 Test Statstc: F-rato = 0.78, and 1.81 for the three varables. The computatons and analyss are presented n appendx II table Decson Rule: We do not accept H 0. f Fcal F Concluson: From appendx table t shows (*ns not sgnfcance at 5% for all the varables. We concluded that the sample means of the varables I t, W t and Y t were equal. We note that equalty n mean only suggests, but do not guarantee equalty n the populaton dstrbuton of the varables. 4.0 ANALYSIS Based on the above conclusons we now had the bass to pool the data on the varables for fttng probablty models on them. ( Fttng probablty model on nter-arrval tme data (It From appendx table 1, the average customers nter-arrval tme (pooled = X hr arrvalrate ( A X customers per hour test for exponental ft (goodness-of-ft on the nter-arrval data. See appendx table 3 Hypotheses 88
4 ISSN (Paper ISSN 5-05 (Onlne Vol.4, No.8, 014 Level of sgnfcance, Crtcal Regon : (, k 1 (0.05, Test statstc: c ( o e e Decson: Snce We do not reject H 0 : but concluded that the I t data were Exponentally dstrbuted. (See appendx table 3 ( Fttng Probablty model on servce tme data (Yt From appendx table I, the Average servce tme per customer (pooled X 5.9 1hr 60mn Servce Rate ( customers per hour X test for exponental ft on the servce tme data. Hypotheses: ( ( Snce We do not reject H 0 : that the Y t Data are exponentally dstrbuted wt. Hence, we concluded n both cases that the nter-arrval rate for customers was exponentally dstrbuted wth 3 customers arrvng per hour, also the servce tme was exponentally dstrbuted wth customers recevng servce n every 1 hour. ( Determnng the Traffc Intensty From the appendx table 1, the average nter-arrval tme (I t s mnutes, therefore arrval rate customers per hour. 89
5 ISSN (Paper ISSN 5-05 (Onlne Vol.4, No.8, 014 Also from the appendx table 1, the average servce tme (Y t s 5.9, therefore average servce rate customers per hour. Hence, for the system we adopted a queung model of type m/m/1. Meanng: Exponental nter-arrval, Exponental servce rate wth one server (.e. queue of type m/m/1. Therefore the traffc ntensty Where = arrval rate = servce rate That s the system s busy approxmately 30% of every 1 hour and dle 70% or 4 mnutes n every 60 mnutes. (v Revenue Analyss Notatons: Let S = rate of servce per hour h = total hours open for servces n a month E(N = expect number of customers served n a months C t = Servce charge E(R = the Expected Revenue. The center renders servce for 4 days n a month, the center renders servce for 10 hours n a day. Therefore E(N = Servce rate(s X (4 days x 10 hours E(N = x 4 x 10 = 7 customers per month Gven from appendx table I that average servce charge C t =N3.5 per customer Then revenue model E(R = E(N * C t Expected revenue n a month E(R = 7 x 3.5 = 88, Whle, Expected revenue n a year E(R = E(N * C t.(1 months E(R = N88,519.44x 1 = N1,06,33.8 per year. 90
6 ISSN (Paper ISSN 5-05 (Onlne Vol.4, No.8, CONCLUSION/RECOMMENDATIONS The traffc ntensty (utlzaton factor of 0.30 reveals that the busness centre were only busy 30% of every one hour or 18 mnutes n every 60 mnutes n a day and the servers are dle for 70% of every one hour. Ths could be due to the locaton of the busness centre, and no other busness actvtes to keep the servers busy. We dscovered that each customer who calls to the centre pays an average of N3.5 and The estmated gross ncome revenue for the year endng was N1,06,33.8. We recommend that Any nterested nvestor on ths type of busness should do feasblty study on locaton before settngup the busness. The propretors of such busness should co-opt or ntroduce computer servces, spral bndng, ID-Card lamnaton servces to mop-up the dleness of the servers and ncrease the revenue generaton. Fnally, we strongly recommend popular spots lke schools premses, government secretarats etc for settng-up such busness. References Allexender L. Stolyar (006 Large Devatons of Queues under Qs Schedulng Algorthms. Bell labs Incent Technologes 600 Mountan Ave, c-3 Murray Hll. Attahru Sule (003 Dscrete Tme Analyss of MAP/PH/I Mult class General Preemptve Prorty Queue. A Journel of Naval Research Logstcs, vol. 50 (003 Frank.J. Wllams (005 Analyss of a Queung System Wth Overload control by Arrval Rates Journal of Appled Maths. vol. 18 pp Houda Mehr, Taoufk.D, Hchem.K (008, Solvng of watng lnes models n the Arport usng Queung theory model and lnear programmng Ryan Berry (00 Queueng Theory and Applcatons, nd Edton, PWS, Kent Publshng, Boston, Smth, D.E (1977. Quanttatve Busness Analyss, New York, John Wley and son, nc. Therauf, R.J. et al (1995. Decson Makng Through Operaton research. New York, John Wley and son, nc., pg Vazsony & Sprer (1987, Quanttatve Analyss for Busness. Prentce Hall of Inda Press, New Delh Vladmr & Dmtr, P.C (1968. Introducton to Mathematcal Modellng, Moscow, Government Press, pp
7 ISSN (Paper ISSN 5-05 (Onlne Vol.4, No.8, 014 Appendx 1. Table 1: SUMMARY STATISTICS OF DATA ON I t, Y t, W t AND C t FOR FOUR WEEKS VARIABLES WEEK1 WEEK WEEK3 WEEK4 AVERAGE Inter-arrval tme (I t S N Servce tme (Y t S N Watng tme (W t S N Servce charge (C t N N36.8 N31.63 N35.9 N6.87 N
8 ISSN (Paper ISSN 5-05 (Onlne Vol.4, No.8, 014 APPENDIX TABLE : COMBINED ANOVA TABLE FOR I t, Y t and W t Sources of varaton Sum of squares Df Ms f-rato F (0.05 Decson I t (ns Y t (ns W t (ns I t Y t W t I t Y t W t *(ns = not sgnfcant at for the three varables. Appendx Table 3: - Test for Exponental ft on It - Data Y F(y P F e ( f e e TOTAL
9 ISSN (Paper ISSN 5-05 (Onlne Vol.4, No.8, 014 Y class ntervals for I cumulatve the frequency of I - Data f ( y cumulatve dstrbuton value 1 e p f boundary dfference t t - Data of e Expected frequency ( np, ( n 337 y f ( y Table 4: Test for Exponental ft on Yt Data Y F F(y P ( Total
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