A Multi-Stage Queue Approach to Solving Customer Congestion Problem in a Restaurant

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1 Open Journal of Statstcs, 08, 8, ISSN Onlne: ISSN Prnt: 6-78X A Mult-Stage Queue Approach to Solvng Customer Congeston Problem n a Restaurant Adegoe S. Aboye, Kayode A. Samnu Department of Statstcs, The Federal Unversty of Technology, Aure, Ngera How to cte ths paper: Aboye, A.S. and Samnu, K.A. (08) A Mult-Stage Queue Approach to Solvng Customer Congeston Problem n a Restaurant. Open Journal of Statstcs, 8, Receved: October 9, 07 Accepted: Aprl 0, 08 Publshed: Aprl 3, 08 Copyrght 08 by authors and Scentfc Research Publshng Inc. Ths wor s lcensed under the Creatve Commons Attrbuton Internatonal Lcense (CC BY 4.0). Open Access Abstract The multstage queue model was developed for a stuaton where parallel and unrelated queues est at the frst stage only. These queues merged nto sngle queues at the remanng stages. The parallel queues offer servces that are dfferent from one another and customers arrve to on the queue that offer servces that they need. The mathematcal model was developed assumng an M/M/ queue system and the measures of effectveness were derved. The model was appled to solve the problem of customer congeston n a restaurant n the cty of Ibadan, Ngera that serves three dfferent local delcaces. The three local delcaces consttute three dfferent queues at the frst stage. The second stage conssts of only one queue whch s for purchase of drns and the thrd stage whch s the last stage s for payment. Every customer n the restaurant passes through the three stages. Utlzaton factors for the fve queues were determned and found to range from 70% to 97%. The average tme spent by customers n the system was found to be mnutes. A smulaton study usng what-f scenaro analyss was performed to determne the optmum servce confguraton for the system. The optmum confguraton reduced average tme for customers n the system from mnutes to 3.47 mnutes wthout hrng new servers. Keywords Multstage Queues, Customer Congeston, What-If Scenaro Analyss, Queue Networ. Introducton Queue s a common phenomenon. It occurs n all aspects of lfe. Queue nvolves watng n lne. We wat n lne n our cars n traffc ams or at toll booths; we wat on hold for an operator to pc up our telephone calls; we wat n lne at DOI: 0.436/os Apr. 3, Open Journal of Statstcs

2 A. S. Aboye, K. A. Samnu supermarets to chec out; we wat n lne at fast-food restaurants; and we wat n lne at bans and post offces. Customers do not, n generally, le these stuatons, and the managers of the establshments at whch they wat also do not le t snce t can lead to loss of busness through customer dssatsfacton. Why then do we have watng-n-lne? Gross et al. [] provdes an answer to the queston. Accordng to hm, demands for servces are usually more than facltes avalable for servce delvery. Ths s due to many factors. Some of the factors are shortage of avalable servers, nablty of a busness to provde the level of servce necessary to prevent watng-n-lne, lmtaton of space for servce facltes and poor arrangement of servers and number of stages a customer would have to pass through before he/she complete servce n the system may be another factors... Multstage Queue System/Networ Multstage queue system s a queue n whch customers are served at more than one staton, arranged n a networ structure. It s a collecton of nodes connected by a set of paths. In a queue networ, a group of servers operatng from the same faclty s dentfed as a node or a staton. Eamples of mult stage queues are found n Computer programmng, communcaton, bans and manufacturng systems. In a mult stage queue networ, customers demand servce from more than one server. Queue networs better represents the real world stuaton than a sngle system. A mult stage networ s represented dagrammatcally as shown n Fgure... Characterstcs of a Queue System Queue processes are descrbe by s basc characterstcs, whch provde adequate representaton of the queue system. The characterstcs are ) arrval pattern of customers, ) servce pattern of servers, 3) queue dscplne, 4) system capacty, 5) number of servce channels, and 6) number of servce stages []. Sundarapandan [] gave the followngs as the basc characterstcs of a queue system. ) the nput or arrval pattern ) the servce mechansm 3) the queue dscplne and 4) the system capacty. Nevertheless, these two researchers are talng about the same thng. Gross et al. [] provdes an ecellent eplanaton of Fgure. Multstage queue networ wth n statons. DOI: 0.436/os Open Journal of Statstcs

3 A. S. Aboye, K. A. Samnu these basc characterstcs..3. Performance Measures of a Queue System In ths wor we study a multstage queue system wth a certan number of ndependent parallel servers and multple queues at all or some of the stages..e. multple watng lnes wth ndependent parallel servers n seres. Ths research wor revews the applcaton of queue theory n analyzng performance measures of ths type of system and develop dscrete-event smulaton of customers flows n the system purposely to carry out what-f-scenaro analyss to mprove effcency of the system. There are varous performance measures of a multstage queue system. The ones we are nterested n are the total epected number of customers n the system, total epected watng tme of an arbtrary customer n the queues formed at all the stages and the total epected watng tme or lead tme (cycle tme) that an arbtrary customer would spend n the system. The man am of ths study s to analyze the customer-servce stuaton n a multstage queue system.. The problem arose n a restaurant n the cty of Ibadan, south-western Ngera that serves three classes of local delcaces durng lunch/ dnner servces. Because of lmted space and large customer pool, congestons frequently occur n the restaurant. A multstage queue model was desgned for the restaurant. The three-stage queue model has 3 servce ponts n stage and sngle servce pont n each of the remanng two stages. Fgure shows the dagrammatc llustraton of the system.. Methodology The multstage queue system conssts of varous statons through whch a customer requrng servce from such system would have to pass. Ths contans networs of queues whch can be descrbed as a group of nodes (say, of them) that s we have number of nodes, where each node represents a servce faclty Fgure. Dagrammatc llustraton of the three-stage queue model for the restaurant. DOI: 0.436/os Open Journal of Statstcs

4 A. S. Aboye, K. A. Samnu of some nd wth s servers at node,,,,. In the most general cases, customers may arrve from outsde the system at any node and may depart from the system from any node. Thus customers may enter the system at some node, traverse from node to node n the system, and depart from some node. Not all customers necessarly enter and leave at the same nodes, or tae the same path once they entered the system. Customers may return to nodes prevously vsted, sp some nodes entrely, and even choose to reman n the system forever []. A multstage queue system or queues n seres as descrbed n [3] are open networs where γ,,,, γ () 0, elsewhere and δ, ( + ) ( ),,,, 0 0, otherwse snce the nodes can be vewed as formng a seres system wth flow always n a sngle drecton from node to node. Ths s supported by a remarable theorem propounded by Jacson [3] that f ) nter-arrval tmes for a seres queue system are eponental wth rate γ ) servce tmes for each stage server are eponental and 3) each stage has an nfnte-capacty watng room, then nter-arrval tmes to each stage of the queue system are eponental wth rate γ, the arrval rate of each customer from node to node... Operatng Parameters and Notatons The followng system operatng parameters and notatons were used n ths research wor. N total number of customers n the system. β mean arrval rate at a sngle-channel, sngle server queue. ττ mean servce rate of a sngle channel, sngle server queue. ρρ system ntensty or load, utlzaton factor. P 0 steady-state probablty of an dle server. P n steady-state probablty of eactly n customers n the system. β mean arrval rate of customers at the th queue of the frst stage. γ effectve mean or epected arrval rate of customers at the th stage. π probablty that an arbtrary customer ons the th queue at the frst stage. τ mean servce rate of the th queue server at the frst stage. α mean servce rate of the th stage server. ρ system utlzaton factor of the th stage server. v total number of customers n the frst stage of the queue system. ω system utlzaton factor of the th queue server at the frst stage. δ transton probablty of a customer after completng servce at the (, ) th stage, proceedng to the th stage. () DOI: 0.436/os Open Journal of Statstcs

5 A. S. Aboye, K. A. Samnu T watng tme of a customer n the multstage queue system. T q total watng tme of a customer n queues formed at dfferent stages. T q watng tme of a customer n the th queue of the frst stage. T q watng tme of a customer n queue at the th stage. W q epected total watng tme of customers n queues formed at dfferent stages. W q average or epected watng tme of a customer n the th queue at the frst stage. W q average or epected watng tme of a customer at the th stage. W s epected watng tme of a customer n the multstage queueng system. P c probablty that there are eactly C customers at - stages. P v probablty that there are eactly V customers at the frst stage... The Mult-Stage Queue Model We assume there are stages n the queue model and stage has h parallel sngle-server queues whch are assumed to be ndependent. That s, they offer dfferent servces. It s also assumed that there s no balng, renegng or oceyng th n the system; the,,,, h queue follows a M/M/ queue system; the system s a feed-forward queue networ and bypassng s not allowed. Let the ar- th rval rate at the,,,, h queue be β and the servce rate be τ. Suppose the probablty that an arrvng customer ons queue s π. We want to determne the effectve rate of arrval and servce n the frst stage. Let γ epected customer arrval rate nto stage. Then h γ πβ β h We estmate π usng π, h π, β h Smlarly α πτ. The general traffc ntensty for stage one s gven as ω γ α. The arrval rate to stage s gven by γ δγ. Ths holds snce the system s a feed-forward networ and bypassng s not allowed. Recursvely, γ δ, γ,,,,. The ont equlbrum dstrbuton of the number of customers n the system s v n ( ) ( ). (3) P P ω ω ρ ρ vc, n The total epected number of customers n the system s L s ω + (4) ρ L s ω ρ The total epected watng tme of an arbtrary customer n the queues formed at all stages (W q ) s for,,,. β γ Wq + + (5) τ τ β γ τ β α α γ γ ( ) ( ) The total epected watng tme of an arbtrary customer n the system (W s ) s DOI: 0.436/os Open Journal of Statstcs

6 A. S. Aboye, K. A. Samnu derved as ρ Ws , (6) τ τ β γ τ β τ ρ α β ( ) for,,,. For detals of the dervatons of these measures go to Append..3. Smulaton Smulaton s a powerful tool for the analyss of new system desgns, retrofts to estng systems and proposed changes to operatng rules. Conductng a vald smulaton s both an art and a scence [4]. Stewart [5] defnes smulaton as mtaton of real-world scenaro systems. The system mght be manufacturng system, communcaton system, health-care system, restaurant system etc. There are many types and nds of smulaton namely: Monte Carlo-type smulaton, dscrete-event smulaton to menton but few. In ths research wor, we lmt ourselves to dscrete, stochastc process orented smulaton.e. a dscrete-event smulaton. Smulaton models are abstractons of a real-world, or proposed real-world, system. In other words, the models do not contan every aspect and detal of the system that they are attemptng to mtate [6]. The maorty of smulaton models could be descrbed as beng vsual nteractve smulatons (VIS). Ths was a concept frst ntroduced n [7]. The dea s that the model provdes a vsual dsplay showng an anmaton of the model as t runs. We use SIMUL8, a software pacage fortfed wth dynamc, powerful Graphcal User Interface (GUI) to carry out a vsual nteractve dscrete-event smulaton of customers flow n the multstage queue system. The what-f scenaro analyss to compare a number of system alternatves was also performed. It allows for the dentfcaton of problems, bottlenecs and desgn shortfalls before buldng or modfyng a system. 3. Results and Dscusson Data used for the analyss of the model were collected from Iyaduun Restaurant n the cty of Ibadan Ngera. The restaurant has about 5 tables and chars for customers and more than 0 servers wor on shft bass. At least 5 servers are on duty per shft. There are 4 to 5 waters and watresses whose obs are to secure comfortable places for customers to seated and clean the tables when customers leave. Data were collected on customer arrval tme on each of the three startng queues n the frst stage and the other queues n the second and thrd stages. The tme servce begns and tme servce ends n all the queues were also recorded. The queues are assumed to be nfnte capacty and customers vst all the stages. Data were collected for 5 days. The average arrval rates of customers and average servce rates, together wth the utlzaton factors of servers are dsplayed n DOI: 0.436/os Open Journal of Statstcs

7 A. S. Aboye, K. A. Samnu Table and Table respectvely. Total epected number of customers n the system s computed usng (.4) as customers The total epected watng tme of an arbtrary customer for a complete servce n the system who oned st, nd or 3 rd queue at the frst stage s obtaned usng (.6) as W s (0.04 mns, mns, mns) respectvely. 3.. Smulaton Runs A smulaton model was ftted to the system usng the nter arrval tmes and the servce tmes. Both are eponentally dstrbuted. The basc model as dsplayed n Fgure 3 was smulated for a tme equvalent of wees of operaton. The average or epected watng tme of customers at each node and average or epected lead-tme were obtaned and dsplayed n Table 3. Based on a two-wee smulaton run of the restaurant system, the average tme spent on the st queue at stage for local delcacy (Amala) by an arbtrary customer was 46 mns, lewse 3 mns on the nd queue for local delcacy and Table. Mean arrval rates of customers nto the system and mean servce rates at the queues. ST QUEUE AT STAGE ND QUEUE AT STAGE 3 RD QUEUE AT STAGE ND STAGE QUEUE 3 RD STAGE QUEUE MEAN ARRIVAL RATE (IN 0 MINUTES INTERVALS) MEAN SERVICE RATES (IN 0 MINUTES INTERVAL) Table. Utlzaton factors of servers. ST QUEUE AT STAGE ND QUEUE AT STAGE 3 RD QUEUE AT STAGE ND STAGE QUEUE 3 RD STAGE QUEUE UTILIZATION FACTOR Fgure 3. A Vsual Interactve dsplay of the smulaton for the basc model. DOI: 0.436/os Open Journal of Statstcs

8 A. S. Aboye, K. A. Samnu mns on the 3 rd queue for local delcacy 3. There were lumps or bottlenecs on the st queue of the st stage and last stage (Paypont) of the system. 3.. Performng What-If Scenaro Analyss We notced that there were bottlenecs at category I (local delcacy group ) servce pont of stage I and Paypont servce pont. We, therefore, perform a what-f scenaro analyss to compare alternatves and reduce both tme spent n queues and tme spent n the system as a whole. Ths was done by shftng the server attendng to customers nterested n local delcacy 3, beng the least utlzed actvty to on the one at Paypont where a bottlenec was observed. The server at local delcacy servce centre was assgned to attend to those who would be nterested n local delcacy. The model s shown n Fgure 4. The summary of the analyss s gven Table 4. The average tme spent n the system has been reduced from mns to 4.8 mns and the average queue tme at Paypont has also been reduced (Table 4). Ths s a great mprovement but we observed that large queue stll occurred at local delcacy servce pont. Tme spent here s stll 46 mn. To mprove on ths we used the confguraton n Fgure 5. The average tme spent by a customer n the system has reduced further to Table 3. Result of the smulaton for the basc model. Queue Stage 3 Stage Average Queue Tme 5 mns (Pay-pont) mn (Drns) Stage, Queue 46 mns (local Delcacy ) Queue 3 mns (Local Delcacy ) Queue 3 Average Tme n System mns (Local Delcacy3) mns Fgure 4. A vsual nteractve smulaton run for the frst what-f scenaro analyss. DOI: 0.436/os Open Journal of Statstcs

9 A. S. Aboye, K. A. Samnu Table 4. Average queueng tme the frst what f scenaro analyss. Name Paypont Drn Local delcacy 3 Local delcacy Local delcacy Average Tme n System Average Queueng Tme mns 3 mn mns 3 mns 46 mns 4.8 mn Fgure 5. A vsual nteractve smulaton run for the second what f scenaro analyss. Table 5. Average queueng tme after the second what-f scenaro analyss. Name Paypont Drn Local Delcacy & 3 Local Delcacy Average Tme n System Average Queueng Tme mn mn mns 0 mn 3.47 mn mns (Table 5). Also local delcacy now enoys zero watng tme Dscusson The analytcal results show that at any pont n tme as many as 76 customers are present n the system. The utlzaton factors for the fve queues n the system range from 70% to 97% (Table ). The tme spent by customers on queue, startng from stage and endng n stage 3 from any of the three queues of stage are respectvely 0.4 mnutes for local delcacy, 5.87 mnutes for local delcacy and mnutes for local delcacy 3. Customers requrng local delcacy DOI: 0.436/os Open Journal of Statstcs

10 A. S. Aboye, K. A. Samnu Fgure 6. Decrease n average tme spent n the system. spend longer tme on queue, reflectng the hgh demand for ths meal. The utlzaton factor for ths queue stands at 97% whch s the hghest. The smulaton study provdes dfferent operaton scenaros to the management of the restaurant. Through the study the average tme spent by customers n the system reduced from mnutes to as lttle as 3.47 mnutes. Ths was acheved by consderng dfferent confguratons of servers n the system wthout addng or subtractng any server. Ths means the reducton was acheved wth no addtonal cost to management. Yet as customers spend less tme n the restaurant more space are freed for addton customers to enoy the servces of the restaurant. Ths wll translate to hgher proft, mproved customer satsfacton and customer goodwll. Ths mprovement s llustrated n Fgure Concluson Ths study has shown an applcaton of queue theory n analyzng a multstage queue system consstng of M/M/ (Posson arrvals, eponental servce tmes, a sngle server and nfnte room capacty) queue models. Evaluaton of performance measures for the queues were done after dervng epressons for the effectveness measures. We assessed the level of servce delvery through utlzaton factors of each M/M/ queue. We found out that maorty of the servers are well utlzed as revealed by the values of the utlzaton factors obtaned. Also, dscrete-event smulaton that gave close mtaton of the system was developed and run to vsualze and ascertan the feasble operatons of the system. What-f scenaro analyss was carred out n order to mae vtal and necessary suggestons for decson-mang that wll mprove the effcency of the system. The method led to sgnfcant reducton n the watng tme of an arbtrary customer n the system by resolvng areas where bottlenecs were observed n the system. It aso provdes an effectve mean of managng the queues for mamum customer satsfacton at no addtonal cost. DOI: 0.436/os Open Journal of Statstcs

11 A. S. Aboye, K. A. Samnu References [] Gross, D., Shortle, J.F., Thompson, J.M. and Harrs, C.M. (008) Fundamentals of Queueng Theory. 4th Edton, John Wley and Sons Inc., New Jersey,. [] Sundarapandan, V. (009) Probablty, Statstcs and Queueng Theory. PHI Learnng Prvate Ltd., New Delh, [3] Jacson, J.R. (957) Networs of Watng Lnes. Operatons Research, 5, [4] John, S.C. (003) Introducton to Modellng and Smulaton. Proceedngs of the 003 Wnter Smulaton Conference, Maretta. [5] Stewart, R. (004) Smulaton: The Practce of Model Development and Use. John Wley and Sons Ltd., West Susse, -33. [6] Roger, B., Stewart, R. and Kathy, K. (0) Conceptual Modellng for Dscrete-Event Smulaton. CRC Press, Taylor & Francs Group, London, New Yor, -6. [7] Hurron, R.D. (976) The Desgn, Use and Requred Facltes of an Interactve Vsual Computer Smulaton Language to Eplore Producton Plannng Problems. PhD Thess, Unversty of London, London. DOI: 0.436/os Open Journal of Statstcs

12 A. S. Aboye, K. A. Samnu Append Dervaton of the Jont Equlbrum Dstrbuton of the Number of Customers n the System The ont probablty of N customers n the system s gven by P P N n P N n P N n P ( ) ( ) ( ) ( ) n 0 Let C N, for,3,, Recall that N V v, Therefore, Pn PPP v c ( 0) P PV vv, v,, V v PC c PC c P0 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 0) vc, P v P v P v P c P c P c P 3 where P ( 0) s a normalzng constant. From a sngle channel M/M/ queue model, where Pn P n Pρ 0 Followng ths, Equaton (3) becomes P 0 ρ, ( ) v v v c c3 c vc, ω ω ω ρ ρ3 ρ P 0 (7) Usng normalzng condton, v v v c c3 c ω ω ω ρ ρ3 ρ P( 0) v 0 v 0 v 0 c 0 c3 0 c 0 Mang P ( 0) v v v c c3 c ω 0 ω ω v 3 v 0 v 0 ρ ρ ρ c 0 c3 0 c 0 Snce each term of the denomnator s a geometrc seres, Therefore, P 0 ω ω ω ρ ρ ρ ( ) ( )( ) ( ) ( )( 3) ( ) ( ω) ( ρ) Substtutng RHS of Equaton (8) for P ( 0) n Equaton (7) We get, ( )( ) ( ) ( )( ) ( ) P ω ω ω ω ω ω v v v vc, c c3 c ρ ρ3 ρ ρ ρ3 ρ (8) v n ( ) ( ) (9) P P ω ω ρ ρ vc, n The Total Epected Number of Customers n the System Ths s done by decomposng the system nto ndvdual stages. Let Ls (,, 3,, ) be the epected number of customers at the th stage. Therefore, Ls L s DOI: 0.436/os Open Journal of Statstcs

13 A. S. Aboye, K. A. Samnu Let L s be the epected number of customers at the frst stage. Remember that N V V Therefore, where v the frst stage. L EV E V EV EV + EV + + EV ( ) ( ) s ( ) ( ) ( ) vp + vp + + vp v v v v 0 v 0 v 0 P s the probablty of V (,,, ) customers n the th queue of v v v s ( ω) ω + ( ω) ω + + ( ω) ω v 0 v 0 v 0 L v v v v v ( ) ( ) ( ) ( ) ( ) v v v v v v v ω ω ω + ω ω ω + + ω ω ω ( ) ( ) ( ω) ( ) ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω Snce each of the remanng stages has a sngle server wth sngle queue, then s ( ) n n 0 ρ L E N np 3 s3 ( 3) 3 n 3 n3 0 ρ3 L E N np ρ L E( N ) np ρ s n n 0 Therefore, the total epected number of customers n the system s L L L + L + + L s s s s ω ρ ρ ρ s + ω ρ (0) Total Epected Watng Tme of an Arbtrary Customer n the Queues Formed at All Stages (W q ) Startng wth the frst stage havng ndependent M/M/queues, we evaluate average or epected watng tme of an arbtrary customer n the th queue of the frst stage usng the followng relaton: where y+ y y y ( q ) ( q ) + ( ) ( ) E T E T E S E T y y W + q Wq + τ β DOI: 0.436/os Open Journal of Statstcs

14 A. S. Aboye, K. A. Samnu the watng tme of an arbtrary customer n the th queue. the epected watng tme of an arbtrary customer n the th queue. T the watng tme of the y th customer n the th queue. W epected watng tme of y th customer n the th queue. E S epected servce tme of the y th customer n the th queue. y T + q y W + q y q y q y ( ) y ( ) E T epected nterarrval tme between the y th customer and (y + ) st customer n the th queue We assume that there are V customers n the th (,,, ) queue and that servce has started before the arbtrary customer ons the th queue. Let L q be the queue length or epected number of customers n the th queue of stage Let ( ) v ( ) ( ) ( ) ( ) L EV v P v ω ω q v v v d v and substtute n the summaton. Thus, t follows that v d + ( ω ) dω ( ) ( d ) ω ω ω ω ( ω ) ( ω ) d d Hence we have, By Lttle s formulae, Therefore, and L W q m q ω β ω τ τ β L q ( ) γ W q Lq β γ τ τ β γ ( ) m ( ) E S m ( ) E T τ β Substtutng these three terms nto the relaton, we get the epected watng tme of an arbtrary customer n the th queue of the frst stage as: y β W + q + τ τ β γ τ β ( ) At the th stage, the arbtrary customer s the nth customer n the th queue DOI: 0.436/os Open Journal of Statstcs

15 A. S. Aboye, K. A. Samnu (,3,, ) staton. Therefore, L q, the queue length of nth customer n the th queue s gven as: ( ) ( ) L E N n P q n n where P n s the probablty of n customers at the th stage. It follows from equaton * that L q ρ γ ρ α α γ ( ) Hence, W q γ α α γ γ ( ) for,3,, The total average watng tme of an arbtrary customer n the queues formed at all the stages s obtaned as β γ Wq + + () τ τ β γ τ β α α γ γ ( ) ( ) for,,, The total epected watng tme of an arbtrary customer n the system (W s ) We obtaned equaton for the epected lead-tme E(T) (or cycle tme) for an arbtrary customer, ncludng watng tmes and servce tmes spent n the system from the frst arrval untl the fnal departure. By Lttle s formulae, the epected watng tme of a customer n a sngle channel system s gven as: Ws Wq + α Therefore, y+ ( ) ( q ) + ( ) + ( q ) + ( ) E T E T E S E T E S () DOI: 0.436/os Open Journal of Statstcs