6MM004Advanced Techniques in Operational Research Queuing Theory.docx

1、6MM004 Advanced Techniques in Operational Research Queuing Theory6MM004: Advanced Techniques in Operational ResearchGroup Assessment: Queuing TheoryTuesday 25th March 2014Ryan Smith Leyi Xiao Rebecca Louise Lloyd Li Liu ContentIntroduction & Background 3The Queuing System 3Assumptions & Consideratio

2、ns 4Observed Data 5Hand Simulation 8Calculations 16DENNIS Analysis 17Comparison to Observed Data 17Additional Servers 18GPSS Simulation 21Comparison to Observed Data 22Comparison to DENNIS 22Additional Servers 23 Conclusion 24Introduction & BackgroundThis is a group report based on the subject of qu

3、euing theory. As a group, we have observed and recorded a two-stage queuing system, analysed the data we acquired and found the average arrival rate, the average service rate, the average number queuing, and the average waiting time at each stage. We then compared our analysis results with DENNIS, a

4、 computer programme which analyses queuing data in a similar way. After that, we created a GPSS simulation of our queuing system and compared it with our other results. We have concluded our results at the end of this report.The Queuing SystemIn order to observe a two-stage queuing system, we chose

5、to gather data in the Wolverhampton Argos; Wulfrun Square, The Wulfrun Centre. The data gathering involved the observation of two stages. The first stage involved customers queuing and paying for their items, with two servers. The second stage involved the customers collecting their items from the c

6、ollections points. Customers would come into the store and either went to the catalogue tables to find what they want or pick something up from the shelves in the queuing area or have reserved items that need paying for and collected. Whichever of the methods the customer used, they would join the q

7、ueue to pay for the item(s). After they had paid the customer would walk to the collection area to wait for their item(s) to be delivered to the collection points and finally collect it/them and leave the store.The observation was scheduled to last approximately 60 minutes and took place at 15:06 on

8、 24th February 2014 and was concluded at 16:24, having observed 50 customers in total. Below shows the layout of the location used for gathering the required data.Assumptions & ConsiderationsBefore looking at the data gathered and begin to explain the analysis, it is important to identify some assum

9、ptions and considerations about the data gathering exercise.Permission before we started to collect our data, we had to ask for a managers permission to observe the queuing system in their store.Time of day the data gathering occurred on a mid week day after lunch time, we decided on this time as an

10、 earlier time many not have been as busy and any later may have effect the amount of customers we could observe. Later in the report, we identify what may have occurred if an additional server had been added. The effect of adding an additional server on and at this particular time of day may not be

11、the same as if we added an additional server on say a weekend when queues will invariably be larger.Servers the two servers in the first stage werent constant. They would float from the service to other roles dependant on the number of customers in the system. For DENNIS and GPSS it will be assumed

12、that the servers are always constant, this may result in results be lower than our observed. Rogue Customers during the observations, there were a few occasions whereby people joined the first stage and left without joining the second stage. The simplicity of our data gathering meant when such occur

13、rences occurred there was no difficulties in recording the other data. These customers are therefore involved in our first stage data but not in the second stage, leaving the second stage data smaller.Context within the first stage there is a waiting and service time whereas in the second stage ther

14、e is just a collection time (waiting and service time combined). This is because the service time would hardly vary customer to customer. After paying for your items, within the first stage, you join the collection queue, within the second stage. In the collection queue you wait until you are called

15、 to collect your items, meaning that customer waiting time would vary but the service time would hardly change as items are just handed over.Poisson distribution when performing our GPSS simulation we are assuming a Poisson distribution, as the arrival of customers did not occur uniformly. This mean

16、s we can recycle the Poisson distribution for both stages. Observed Data The table below shows details of the data we gathered on the day.The first column, Customer, shows the order of customer as they join the queue, with a total of 50 customers observed. The second column, Queue (In/Out), shows th

17、e time at which customers joined the queue and left Queue for Till.The third column, Till (In/Out), shows the time at which customers joined Till and left Till to collect their products.The fourth column, Collection (In/Out), shows the time at which customers joined and left Collection.CustomerQueue

18、TillCollectionInOutInOutInOut1A069691091092561B701091092852855751C1432342342712714561D3123133135755756841E3804354355115117561F4115115115955959161G6356436437347349881H70773473477277210731I72677277284384310291J73378178183583510541K73583583589489410721L8418438438768761M8788868869239231N9479569561080108

19、011821O1184119711971334133415521P1215124512451355135515331Q1246133413341367136720461R1273135513551390139015071S1278136713671403140315891T1350139013901453145316271U1367140314031456145616521V141814531453151815181W1553155415541603160317391X179918091809276627661Y1864203420342106210623051Z207221062106218

20、5218525232A2078218521852352235230312B2137235223522463246325122C2138246324632570257028181D2151257025702616261627061E2254261626162649264929411F2304264926492693269328501G2306269326932806280629921H2309276627662812281231842I2624280628062846284631672J2656281228122871287134172K2656284628462928292832222L272

21、9287128713241324134602M2890292829282972297232982N2902297229722985298534942O2916298529853057305733242P3111321632163352335235992Q323832413241327932792R3303331733173677367737372S3458353935393566356637112T3550356635663655365537432U3846386038603909390939742V3883390939093964396440502W391039643964402240224

22、1922X404040904090432843284481The second table shows more details of the data we gathered on the day. The first column, Customer, shows the order of customer as they join the queue.The second column, Arrivals, shows the time at which a new customer joined the Queue, after the previous customer.The th

23、ird column, Waiting Time, shows the waiting time for each customer in the Queue until joining the TillThe fourth column, Service Time, shows the total service time each customer spent at the Till.The fifth column, Collection Time, shows a combined time, waiting and service, each customer spent at Co

24、llection. CustomerArrivalsWaiting TimeService TimeCollection Time1A069401471B70391762901C7391371851D16912621091E6855762451F31100843211G2248912541H7227383011I1946711861J748542191K2100591781L1062331M378371N6991241021O237131372181P31301101781Q3188336791R2782351171S589361861T7240631741U1736531961V513565

25、1W1351491361X246109571Y65170721991Z20834793382A61071676792B59215111492C13251072481D1341946901E103362332921F50345441571G23871131861H3457463722I315182403212J32156595462K0190822942L731423702192M16138443262N1270135092O1469722672P1951051362472Q1273382R6514360602S15581271452T921689882U2961449652V372655862

26、W2754581702X13050238153Hand SimulationHaving observed the queuing system and recorded the data, we created a hand simulation of how the two-stage queuing system actually worked in practiceTime CustomerStage 1Stage 2NotesQueueQueueTillQueueCollectionSizeStartEndStart AEnd AStart BEnd BSizeStartEnd01A

27、10-0-1A ENTERS QUEUE691A0-69-0-1A LEAVES QUEUE691A0-69-0-1A ENTERS TILL A701B170-0-1B ENTERS QUEUE1091A0-109-0-1A LEAVES TILL A1091A0-1109-1A ENTERS COLLECTION1091B0-109-1-1B LEAVES QUEUE1091B0-109-1-1B ENTERS TILL A1431C1143-1-1C ENTERS QUEUE2341C0-234-1-1C LEAVES QUEUE2341C0-234-1-1C ENTERS TILL B2561A0-