6MM004Advanced Techniques in Operational Research Queuing Theory.docx
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6MM004AdvancedTechniquesinOperationalResearchQueuingTheory
6MM004:
AdvancedTechniquesinOperationalResearch
GroupAssessment:
QueuingTheory
Tuesday25thMarch2014
RyanSmithLeyiXiao
RebeccaLouiseLloydLiLiu
Content
Introduction&Background3
TheQueuingSystem3
Assumptions&Considerations4
ObservedData5
HandSimulation8
Calculations16
DENNISAnalysis17
ComparisontoObservedData17
AdditionalServers18
GPSSSimulation21
ComparisontoObservedData22
ComparisontoDENNIS22
AdditionalServers23
Conclusion24
Introduction&Background
Thisisagroupreportbasedonthesubjectofqueuingtheory.Asagroup,wehaveobservedandrecordedatwo-stagequeuingsystem,analysedthedataweacquiredandfoundtheaveragearrivalrate,theaverageservicerate,theaveragenumberqueuing,andtheaveragewaitingtimeateachstage.WethencomparedouranalysisresultswithDENNIS,acomputerprogrammewhichanalysesqueuingdatainasimilarway.Afterthat,wecreatedaGPSSsimulationofourqueuingsystemandcompareditwithourotherresults.Wehaveconcludedourresultsattheendofthisreport.
TheQueuingSystem
Inordertoobserveatwo-stagequeuingsystem,wechosetogatherdataintheWolverhamptonArgos;WulfrunSquare,TheWulfrunCentre.Thedatagatheringinvolvedtheobservationoftwostages.Thefirststageinvolvedcustomersqueuingandpayingfortheiritems,withtwoservers.Thesecondstageinvolvedthecustomerscollectingtheiritemsfromthecollectionspoints.
Customerswouldcomeintothestoreandeitherwenttothecataloguetablestofindwhattheywantorpicksomethingupfromtheshelvesinthequeuingareaorhavereserveditemsthatneedpayingforandcollected.Whicheverofthemethodsthecustomerused,theywouldjointhequeuetopayfortheitem（s）.Aftertheyhadpaidthecustomerwouldwalktothecollectionareatowaitfortheiritem（s）tobedeliveredtothecollectionpointsandfinallycollectit/themandleavethestore.
Theobservationwasscheduledtolastapproximately60minutesandtookplaceat15:
06on24thFebruary2014andwasconcludedat16:
24,havingobserved50customersintotal.
Belowshowsthelayoutofthelocationusedforgatheringtherequireddata.
Assumptions&Considerations
Beforelookingatthedatagatheredandbegintoexplaintheanalysis,itisimportanttoidentifysomeassumptionsandconsiderationsaboutthedatagatheringexercise.
Permission–beforewestartedtocollectourdata,wehadtoaskforamanager’spermissiontoobservethequeuingsystemintheirstore.
Timeofday–thedatagatheringoccurredonamidweekdayafterlunchtime,wedecidedonthistimeasanearliertimemanynothavebeenasbusyandanylatermayhaveeffecttheamountofcustomerswecouldobserve.Laterinthereport,weidentifywhatmayhaveoccurredifanadditionalserverhadbeenadded.Theeffectofaddinganadditionalserveronandatthisparticulartimeofdaymaynotbethesameasifweaddedanadditionalserveronsayaweekendwhenqueueswillinvariablybelarger.
Servers–thetwoserversinthefirststageweren’tconstant.Theywouldfloatfromtheservicetootherrolesdependantonthenumberofcustomersinthesystem.ForDENNISandGPSSitwillbeassumedthattheserversarealwaysconstant,thismayresultinresultsbelowerthanourobserved.
RogueCustomers–duringtheobservations,therewereafewoccasionswherebypeoplejoinedthefirststageandleftwithoutjoiningthesecondstage.Thesimplicityofourdatagatheringmeantwhensuchoccurrencesoccurredtherewasnodifficultiesinrecordingtheotherdata.Thesecustomersarethereforeinvolvedinourfirststagedatabutnotinthesecondstage,leavingthesecondstagedatasmaller.
Context–withinthefirststagethereisawaitingandservicetimewhereasinthesecondstagethereisjustacollectiontime（waitingandservicetimecombined）.Thisisbecausetheservicetimewouldhardlyvarycustomertocustomer.Afterpayingforyouritems,withinthefirststage,youjointhecollectionqueue,withinthesecondstage.Inthecollectionqueueyouwaituntilyouarecalledtocollectyouritems,meaningthatcustomerwaitingtimewouldvarybuttheservicetimewouldhardlychangeasitemsarejusthandedover.
Poissondistribution–whenperformingourGPSSsimulationweareassumingaPoissondistribution,asthearrivalofcustomersdidnotoccuruniformly.ThismeanswecanrecyclethePoissondistributionforbothstages.
ObservedData
Thetablebelowshowsdetailsofthedatawegatheredontheday.
Thefirstcolumn,Customer,showstheorderofcustomerastheyjointhequeue,withatotalof50customersobserved.
Thesecondcolumn,Queue（In/Out）,showsthetimeatwhichcustomersjoinedthequeueandleftQueueforTill.
Thethirdcolumn,Till（In/Out）,showsthetimeatwhichcustomersjoinedTillandleftTilltocollecttheirproducts.
Thefourthcolumn,Collection（In/Out）,showsthetimeatwhichcustomersjoinedandleftCollection.
Customer
Queue
Till
Collection
In
Out
In
Out
In
Out
1A
0
69
69
109
109
256
1B
70
109
109
285
285
575
1C
143
234
234
271
271
456
1D
312
313
313
575
575
684
1E
380
435
435
511
511
756
1F
411
511
511
595
595
916
1G
635
643
643
734
734
988
1H
707
734
734
772
772
1073
1I
726
772
772
843
843
1029
1J
733
781
781
835
835
1054
1K
735
835
835
894
894
1072
1L
841
843
843
876
876
1M
878
886
886
923
923
1N
947
956
956
1080
1080
1182
1O
1184
1197
1197
1334
1334
1552
1P
1215
1245
1245
1355
1355
1533
1Q
1246
1334
1334
1367
1367
2046
1R
1273
1355
1355
1390
1390
1507
1S
1278
1367
1367
1403
1403
1589
1T
1350
1390
1390
1453
1453
1627
1U
1367
1403
1403
1456
1456
1652
1V
1418
1453
1453
1518
1518
1W
1553
1554
1554
1603
1603
1739
1X
1799
1809
1809
2766
2766
1Y
1864
2034
2034
2106
2106
2305
1Z
2072
2106
2106
2185
2185
2523
2A
2078
2185
2185
2352
2352
3031
2B
2137
2352
2352
2463
2463
2512
2C
2138
2463
2463
2570
2570
2818
1D
2151
2570
2570
2616
2616
2706
1E
2254
2616
2616
2649
2649
2941
1F
2304
2649
2649
2693
2693
2850
1G
2306
2693
2693
2806
2806
2992
1H
2309
2766
2766
2812
2812
3184
2I
2624
2806
2806
2846
2846
3167
2J
2656
2812
2812
2871
2871
3417
2K
2656
2846
2846
2928
2928
3222
2L
2729
2871
2871
3241
3241
3460
2M
2890
2928
2928
2972
2972
3298
2N
2902
2972
2972
2985
2985
3494
2O
2916
2985
2985
3057
3057
3324
2P
3111
3216
3216
3352
3352
3599
2Q
3238
3241
3241
3279
3279
2R
3303
3317
3317
3677
3677
3737
2S
3458
3539
3539
3566
3566
3711
2T
3550
3566
3566
3655
3655
3743
2U
3846
3860
3860
3909
3909
3974
2V
3883
3909
3909
3964
3964
4050
2W
3910
3964
3964
4022
4022
4192
2X
4040
4090
4090
4328
4328
4481
Thesecondtableshowsmoredetailsofthedatawegatheredontheday.
Thefirstcolumn,Customer,showstheorderofcustomerastheyjointhequeue.
Thesecondcolumn,Arrivals,showsthetimeatwhichanewcustomerjoinedtheQueue,afterthepreviouscustomer.
Thethirdcolumn,WaitingTime,showsthewaitingtimeforeachcustomerintheQueueuntiljoiningtheTill
Thefourthcolumn,ServiceTime,showsthetotalservicetimeeachcustomerspentattheTill.
Thefifthcolumn,CollectionTime,showsacombinedtime,waitingandservice,eachcustomerspentatCollection.
Customer
Arrivals
WaitingTime
ServiceTime
CollectionTime
1A
0
69
40
147
1B
70
39
176
290
1C
73
91
37
185
1D
169
1
262
109
1E
68
55
76
245
1F
31
100
84
321
1G
224
8
91
254
1H
72
27
38
301
1I
19
46
71
186
1J
7
48
54
219
1K
2
100
59
178
1L
106
2
33
1M
37
8
37
1N
69
9
124
102
1O
237
13
137
218
1P
31
30
110
178
1Q
31
88
33
679
1R
27
82
35
117
1S
5
89
36
186
1T
72
40
63
174
1U
17
36
53
196
1V
51
35
65
1W
135
1
49
136
1X
246
10
957
1Y
65
170
72
199
1Z
208
34
79
338
2A
6
107
167
679
2B
59
215
111
49
2C
1
325
107
248
1D
13
419
46
90
1E
103
362
33
292
1F
50
345
44
157
1G
2
387
113
186
1H
3
457
46
372
2I
315
182
40
321
2J
32
156
59
546
2K
0
190
82
294
2L
73
142
370
219
2M
161
38
44
326
2N
12
70
13
509
2O
14
69
72
267
2P
195
105
136
247
2Q
127
3
38
2R
65
14
360
60
2S
155
81
27
145
2T
92
16
89
88
2U
296
14
49
65
2V
37
26
55
86
2W
27
54
58
170
2X
130
50
238
153
HandSimulation
Havingobservedthequeuingsystemandrecordedthedata,wecreatedahandsimulationofhowthetwo-stagequeuingsystemactuallyworkedinpractice
Time
Customer
Stage1
Stage2
Notes
Queue
Queue
Till
Queue
Collection
Size
Start
End
StartA
EndA
StartB
EndB
Size
Start
End
0
1A
1
0
-
-
-
-
-
0
-
-
1AENTERSQUEUE
69
1A
0
-
69
-
-
-
-
0
-
-
1ALEAVESQUEUE
69
1A
0
-
-
69
-
-
-
0
-
-
1AENTERSTILLA
70
1B
1
70
-
-
-
-
-
0
-
-
1BENTERSQUEUE
109
1A
0
-
-
-
109
-
-
0
-
-
1ALEAVESTILLA
109
1A
0
-
-
-
-
-
-
1
109
-
1AENTERSCOLLECTION
109
1B
0
-
109
-
-
-
-
1
-
-
1BLEAVESQUEUE
109
1B
0
-
-
109
-
-
-
1
-
-
1BENTERSTILLA
143
1C
1
143
-
-
-
-
-
1
-
-
1CENTERSQUEUE
234
1C
0
-
234
-
-
-
-
1
-
-
1CLEAVESQUEUE
234
1C
0
-
-
-
-
234
-
1
-
-
1CENTERSTILLB
256
1A
0
-
-
-
-
-
-
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