Simulasi Variabel Random

Studi Kasus: Simulasi Kedatangan Pelanggan dan Waktu Pelayanan di Minimarket

1. Buat simulasi untuk distribusi diskrit dan distribusi kontinu.

a. Simulasi Distribusi Diskrit (Poisson)

set.seed(123)

n <- 1000       # jumlah simulasi
lambda <- 20    # rata-rata kejadian

poisson_data <- rpois(n, lambda)

# Histogram
hist(poisson_data,
     breaks = 30,
     main = "Histogram Distribusi Poisson",
     xlab = "Jumlah Kejadian",
     col = "lightyellow")

# Rata-rata simulasi
mean(poisson_data)

## [1] 19.916

Histogram menunjukkan distribusi jumlah kejadian yang cenderung miring ke kanan, yang merupakan karakteristik distribusi Poisson.

Nilai rata-rata simulasi akan mendekati parameter λ (lambda).

b. Simulasi Distribusi Kontinu (Eksponensial)

Distribusi eksponensial digunakan untuk memodelkan waktu antara dua kejadian, misalnya waktu pelayanan pelanggan di kasir.

set.seed(123)

n <- 1000
rate <- 0.5

exp_data <- rexp(n, rate)

# Histogram
hist(exp_data,
     breaks = 30,
     main = "Histogram Distribusi Eksponensial",
     xlab = "Waktu",
     col = "lightcoral")

# Rata-rata simulasi
mean(exp_data)

## [1] 2.059959

Histogram menunjukkan distribusi yang miring ke kanan, yang berarti waktu yang kecil lebih sering muncul dibanding waktu yang besar.

2. Buat studi kasus sendiri yang melibatkan simulasi variabel random dari distribusi yang telah dipelajari.

      Sebuah minimarket melayani pelanggan setiap hari selama jam operasional. Dalam kegiatan sehari-hari, jumlah pelanggan yang datang serta waktu yang dibutuhkan untuk melayani pelanggan tidak dapat dipastikan secara pasti karena dipengaruhi oleh berbagai faktor, seperti waktu kedatangan pelanggan, jumlah barang yang dibeli, dan kondisi antrean di kasir. Oleh karena itu, fenomena ini dapat dimodelkan menggunakan variabel random dan dianalisis melalui simulasi statistik.
      Dalam studi kasus ini, jumlah pelanggan yang datang ke minimarket dalam satu jam diasumsikan bersifat acak dan dimodelkan menggunakan distribusi Poisson, karena variabel ini menyatakan jumlah kejadian dalam suatu interval waktu dan bernilai diskrit. Sementara itu, waktu pelayanan pelanggan di kasir dimodelkan menggunakan distribusi Eksponensial, karena variabel ini berupa durasi waktu yang bersifat kontinu.
      Simulasi dilakukan menggunakan R Studio untuk menggambarkan pola kedatangan pelanggan dan waktu pelayanan di minimarket. Hasil simulasi kemudian digunakan untuk menganalisis rata-rata jumlah pelanggan, rata-rata waktu pelayanan, serta beban pelayanan kasir dalam suatu periode waktu tertentu.

a. Simulasi Jumlah Pelanggan (Distribusi Poisson)

# Simulasi jumlah pelanggan per jam
set.seed(123)

n_hours <- 100
lambda_customers <- 25

customers <- rpois(n_hours, lambda_customers)

customers

##   [1] 22 30 16 25 33 27 18 16 31 26 27 25 22 31 29 24 21 19 23 19 21 24 19 19 23
##  [26] 18 29 29 29 28 27 24 23 23 21 29 24 31 20 22 20 24 26 24 24 31 23 32 17 19
##  [51] 26 26 22 35 18 26 27 25 29 35 22 20 21 28 24 21 25 24 25 26 23 28 23 26 30
##  [76] 27 23 30 29 27 26 21 31 22 32 23 19 33 25 30 22 24 21 27 23 31 33 27 23 25

Rata-rata Jumlah Pelanggan

mean_customers <- mean(customers)
cat("Rata-rata pelanggan per jam:", mean_customers)

## Rata-rata pelanggan per jam: 24.96

Probabilitas Jumlah Pelanggan Lebih dari 30

prob_busy <- sum(customers > 30) / n_hours
cat("Probabilitas pelanggan > 30:", prob_busy)

## Probabilitas pelanggan > 30: 0.13

Histogram

hist(customers,
     breaks = 20,
     main = "Distribusi Jumlah Pelanggan per Jam",
     xlab = "Jumlah pelanggan",
     col = "lightgreen")

      Jumlah pelanggan per jam dimodelkan dengan distribusi Poisson karena menghitung jumlah kejadian dalam interval waktu tertentu (1 jam), bernilai diskrit, dan kedatangannya dianggap acak serta independen. Dengan parameter λ = 25 selama 100 jam simulasi, diperoleh rata-rata 24,96, yang mendekati nilai λ sehingga hasil simulasi konsisten dengan teori 𝐸(𝑋)=𝜆.
      Dari simulasi juga diperoleh probabilitas pelanggan lebih dari 30 sebesar 0,13 (13%), yang berarti sekitar 13 dari 100 jam termasuk jam sibuk. Hal ini menunjukkan bahwa pada kondisi tertentu minimarket dapat mengalami peningkatan jumlah pelanggan sehingga mungkin diperlukan pengaturan antrean atau penambahan kasir.
      Histogram hasil simulasi menunjukkan bahwa jumlah pelanggan paling sering berada di sekitar 23–27 pelanggan per jam, dengan beberapa nilai yang lebih rendah maupun lebih tinggi hingga sekitar 35 pelanggan. Pola ini memperlihatkan adanya variasi kedatangan pelanggan dari jam yang relatif sepi hingga jam yang cukup ramai, namun tetap berpusat di sekitar nilai rata-rata yang diharapkan.

b. Simulasi Waktu Pelayanan (Distribusi Eksponensial)

# Simulasi waktu pelayanan pelanggan
set.seed(123)

n_customers <- 1000
rate_service <- 0.4

service_time <- rexp(n_customers, rate_service)

service_time

##    [1]  2.108643153  1.441525677  3.322637170  0.078943398  0.140527440
##    [6]  0.791253041  0.785568231  0.363167010  6.815591161  0.072883618
##   [11]  2.512075144  1.200536819  0.702534069  0.942794578  0.470710102
##   [16]  2.124465324  3.908008849  1.196901041  1.477337088 10.102529278
##   [21]  2.107874328  2.414678028  3.713189485  3.370111214  2.921322461
##   [26]  4.014630858  3.741857172  3.926631367  0.079419360  1.494624228
##   [31]  5.419599364  1.266539322  0.648894543  6.492230291  3.072564329
##   [36]  1.976704397  1.573200194  3.136602507  1.471711606  2.823225085
##   [41]  1.050912007 18.027518940  2.114304913  0.563855016  2.750847044
##   [46]  5.620764231  3.409335748  1.440979170  6.813189624  3.280407607
##   [51]  0.226478375  0.765509625  2.668032674  0.783790640  2.436600376
##   [56]  4.719558288  1.411471507  6.442403323  2.619239370  2.561103355
##   [61]  2.569674039  0.711671152  3.907629722  0.105220732  0.246577225
##   [66]  0.246423280  0.700962903  0.739467397  2.431074122  2.310068060
##   [71]  4.106011317  4.049781919  6.340363850  3.803874058  0.950035508
##   [76]  0.596282417  1.166218559  0.105678629  0.799192248  1.609027301
##   [81]  1.431407759  0.540138756 11.246683494  4.642726383  1.713646594
##   [86]  3.598631565  4.327884959  3.111958250  3.658251400  3.843484946
##   [91]  0.011497816  2.771913670  0.749925794  2.980007518  2.787321759
##   [96]  0.168439727  1.201671802  3.926135848  0.649865267  4.642305719
##  [101]  1.158049052  0.590089366  2.955248554  0.149178429  1.008096104
##  [106]  2.357349601  1.041451611  1.883046012  0.471715970  2.192134678
##  [111]  0.475094552  2.447695515  0.808439568  3.301193884  0.796151621
##  [116]  4.012668240  0.364335291  4.507823296  0.075148636  3.258601373
##  [121]  0.499480612  4.378640605  4.409170830  2.107608946  0.871881940
##  [126]  8.244562000  1.004265437  2.751754336  3.322307931  1.604756620
##  [131]  0.489584289  1.141972141  0.931875410  8.658987227  3.185069298
##  [136]  2.703712877  0.742397902  0.210401836  7.383502293  4.920630636
##  [141]  1.659637814  4.023480359  1.517092710  0.234130783  0.811588210
##  [146]  4.399705780  0.665276799  2.351565419  1.122771732  3.222266198
##  [151]  0.445722656  3.929315219  0.155351994  1.426458356  4.326952114
##  [156]  3.276859762  3.237182415  1.168276773  0.082224471  0.382595106
##  [161]  2.596180010  2.706114265  9.368227845  0.014065228  0.010918427
##  [166]  4.249960403  1.351670371  1.175921529  0.128898539  1.292965848
##  [171]  3.707857523  3.198713898  3.566170435  5.295735853  3.560400935
##  [176]  2.210285778  6.727467146  5.574767743  1.307011150  0.171626412
##  [181]  1.422876333 10.914777340  1.395329414  1.146953260  0.650659264
##  [186]  1.747104362  2.407970971  1.828698887  2.780698927  1.514061365
##  [191]  1.678544185  4.079147075  0.254815329  1.019960307  6.850871383
##  [196]  7.129693646  0.060433430  1.188987015  1.670271546  2.375575756
##  [201]  2.844437507  0.206912716  4.107269262  0.397251037  3.696539844
##  [206]  0.282579347  0.560314176  1.676276600  3.842076775  4.822054840
##  [211]  0.533930501  1.419732394  4.833309501  9.006869657  3.298946614
##  [216]  2.326910337  0.788642156  0.361955130  2.320736279  0.645856165
##  [221]  1.345969556  0.188385915  0.321094266  6.497100382  5.175477485
##  [226]  0.513452544  6.915313699  3.640856928  6.907305744  3.707085331
##  [231]  0.685260654  1.875834315  2.625993753  0.903937756  2.402360432
##  [236]  1.657842990  3.944093533  4.806202277  0.557507874  9.613014724
##  [241]  0.481317723  2.452241708  0.524894220  3.114914589  4.175237752
##  [246]  0.976721808  2.528342085  0.187459107  6.676436788  4.189051157
##  [251]  2.062796659  0.419361680  1.158538294  4.276154125  1.227865414
##  [256]  3.907418074  2.813437586  0.866662407  0.119113008  2.730885879
##  [261]  3.566574602  1.295347078  4.063970418  5.485043602  0.424682787
##  [266]  1.940648146  3.955881761  1.551990331  6.455268138  1.610150356
##  [271]  2.880694567  1.733674340  3.838591014  1.569668606  2.310506694
##  [276]  0.938713428  0.200630871  0.891119287  3.178774665  5.518162963
##  [281]  1.883523782  0.085843567  1.434636763  2.768928711  2.897282401
##  [286]  2.103869258  5.897520347  2.887410647  4.526012195  0.180268603
##  [291]  0.019743560  0.591464714  5.999851119  0.685627643  4.400604022
##  [296]  1.841432126  1.575407475  2.244293196  2.879578231  2.353995871
##  [301]  7.421068996  0.094024617  0.895067079  0.044234398  2.261744197
##  [306]  0.774656081  1.582390694  7.591407634  1.195158428  2.720062889
##  [311]  1.646254117  0.177706958  1.978321588  1.504741060  6.364391646
##  [316]  1.660538871  2.001418676  1.265550498  0.141170791  1.511191085
##  [321]  3.084603997  2.508841941  5.064509962  0.346913313  0.917924572
##  [326]  0.828675713  0.149462293  0.049214497  8.750016703  8.248270601
##  [331]  0.348446155  0.751307502  0.768833412  3.185405566  1.561524143
##  [336]  0.235117088  0.958820076  0.437306835  2.961537011  3.365922032
##  [341]  9.102593152  0.333670414  3.141434323  2.911827429  0.262249067
##  [346]  0.833525789  0.105327569  0.153121932  0.041853149  2.607817924
##  [351]  0.055402610  0.932451756  3.767226883  1.507147918  3.891377062
##  [356]  1.613586958  2.542846238  2.974561830  0.648727119  6.563953575
##  [361]  9.683334208  1.011888911  1.274327268  0.076501039  1.377037957
##  [366]  3.001032120  7.800115700  2.876978768  1.970380649  0.527619805
##  [371]  1.295676697  1.724173122  1.913222484  4.768917656  1.594776916
##  [376]  3.410638969  0.469192466  1.494117789  0.156516245  0.628718185
##  [381]  1.228181086  6.113742147  2.238044063  0.572810787  1.264660290
##  [386]  0.824412804  1.183101730  0.097702951  0.014541521  2.738315682
##  [391]  1.727780615  1.532175819  5.606293718  5.696873902  0.272351087
##  [396]  4.436090826  9.376181228  0.106569232  6.149719585  2.062891412
##  [401]  3.437216962  0.436390621  1.533472148  5.005640080  2.509440791
##  [406]  1.073774053  1.656109894  0.954146071  0.111678831  4.379701009
##  [411]  1.801802542  0.674002549  0.197804570  3.767806901  2.069985720
##  [416]  0.415152976  4.266253747  6.550648282  2.349355767  1.834192688
##  [421]  9.867458506  0.579796755  0.232131294  2.997312883  4.683196205
##  [426]  2.635418970  0.798106686  1.146804657  1.739252231  0.186953717
##  [431]  1.053354908  1.061866215  9.821089110  3.402013448  1.121766897
##  [436]  4.410237823  2.390995843  1.521281396  0.002064935  3.932783738
##  [441]  8.380615580  0.486767413  2.446152386  0.139645002  1.477110102
##  [446]  5.455907727  0.975103487  0.432188678  4.154445505  5.563806095
##  [451]  0.287187916  1.796471760  5.040612174  0.068905696  4.301425182
##  [456]  0.346763444  3.848782205  3.879911154  3.355699081  0.915051412
##  [461]  0.767139102  1.171228122  1.537495547  3.527024295  1.740653291
##  [466]  2.548337867  3.305560851  0.676837885  1.543060225  1.820879965
##  [471]  1.597335699 11.196699135  5.034077936  9.167254506  2.064194924
##  [476]  0.940135646  4.313849659  3.309174529  6.847066530  2.038519699
##  [481]  2.847530157  1.825316362  1.943016408  8.190966248  1.860830993
##  [486]  0.828002482  0.838201157  8.384719546  0.975526222  2.865483619
##  [491]  0.241221529  2.182960664  3.487391611  0.449508119  0.308378044
##  [496]  0.943605601  2.345576042  0.527934207  0.131450390  7.519876674
##  [501] 12.258517560  1.359547412  2.887834348  0.360368603  1.019064750
##  [506]  0.786105302  2.126389402  6.588182081  1.061768862  5.653925787
##  [511]  0.015907634  3.566627810  2.495199419  1.131994441  0.755613036
##  [516]  0.130216668  0.073405463  0.737210718  2.202155335  4.111641636
##  [521]  6.078351217  3.778659692  4.361116905  1.309926616  0.136974703
##  [526]  0.022598382  0.965994507  0.185927469  0.257493282  1.226709211
##  [531]  0.515914781  1.827134653  4.445940186  0.848017730  0.232613971
##  [536]  1.557318289  1.295834010  9.375294349  3.041904953  8.501880895
##  [541]  1.489514686  0.940251327  0.417814316  0.841618218  0.056572983
##  [546]  1.313752946  0.123781963 13.953396920  0.920862207  3.072234667
##  [551]  2.428400246  1.541931276  7.785031683  2.870294899  1.186124672
##  [556]  2.713018479  0.273409147  3.710608254  1.424657782  0.353671215
##  [561]  0.481814804  0.274986038  0.041049291  0.674380780  6.415636674
##  [566]  4.201920107  3.287032903  2.651012393  3.385442484  2.273392642
##  [571]  0.301402687  4.083218914  0.076188356  0.106160021  3.332327236
##  [576]  2.511965228  0.616306918  3.765294524  0.950097835  6.027554937
##  [581]  1.409844573  8.113595694  1.599223339  1.926952426  2.061372097
##  [586]  0.716107797  0.012127397  3.423325200  2.966137213  0.635553503
##  [591]  0.238890344  4.030833463  4.791979641  1.884007484  0.360743398
##  [596]  3.234559735  0.327326903  1.648119394  0.710569097  3.147855468
##  [601]  1.047899274  5.672755921  4.169432171  0.828114742  1.368460612
##  [606]  2.186868756  5.323728327  2.323662195  4.668542277  0.034988634
##  [611]  9.234904660  1.368430197  0.298325396  0.707313196  2.216725944
##  [616]  1.463691709  0.427827882  3.180457810  1.556186748  2.418498890
##  [621]  0.407236691  2.198668835  4.241315856  3.571498641  0.608629914
##  [626]  0.239971416  3.950620177  0.297729346  2.508725090  1.097936414
##  [631]  2.153014051  5.733037241  3.145818864 12.091836826  1.228352784
##  [636]  1.633577452  0.934727571  1.997266789  0.795989629  3.965263171
##  [641]  4.426596677  2.295980540  2.931940718  5.807466027  1.065750944
##  [646]  0.254613645  3.683043830  0.973807948  1.109292796  0.644638804
##  [651]  1.946957998  9.280261107  3.156180491  4.404472550  2.429289145
##  [656]  3.368418594  1.296678483  1.951770002  5.249440334  1.869774365
##  [661]  5.422865195  1.285492667  0.346420707  1.504589244  1.138813552
##  [666]  4.755402957  0.147028782  0.493192734  1.144615256  5.590067091
##  [671]  1.631657893  2.251208223  3.924053685  0.342843770  5.009427257
##  [676]  4.821848441  1.093058726  2.368046907  3.733538450  7.916326532
##  [681]  6.134796479  1.598412592  1.014897519  0.338741868  1.106546331
##  [686]  1.262221067  3.047093919  2.112970764  0.158416636  1.829328665
##  [691]  1.658951589  3.286506799  5.603880877  0.376350817  0.628341577
##  [696]  0.642024019  2.515254593  3.743925742  1.038188859  0.644590048
##  [701]  5.330936699  0.363898906  2.991302318  5.113045899  1.938542607
##  [706]  5.597369843  0.954949249  2.102444144  3.953985814  0.696290350
##  [711]  0.375446877  1.100252289  3.205328004  6.458659558  0.674482025
##  [716] 14.535820744  5.176200699  0.895114047  0.233930932  3.107850528
##  [721]  1.442359645  0.964568333  0.471364857  5.977374445 10.339975221
##  [726]  4.869164540  0.190271662  3.084853880  0.847530983  1.090459045
##  [731]  1.845073109  1.426743092  0.210941449  3.398341404  0.136846623
##  [736]  3.054659553  1.510913540  4.924208638  0.599004752  6.733881280
##  [741]  7.661932855  2.124606500 12.598492172  2.008468753  2.484899382
##  [746]  0.443531383  3.540452457  2.344558132  5.838321061  2.450308865
##  [751]  0.624527684  2.216829907  0.353565967  7.801712770  4.780085012
##  [756]  1.135432353  1.633450252  0.053605377  0.338630829 16.054710107
##  [761]  3.841297966  0.143067416  0.431009946  6.972057465  3.894206788
##  [766]  1.015364870  5.321032195  0.609035311  0.152309761 10.326053439
##  [771]  1.126139760  2.713987296  0.272815898  1.188261036  0.615645810
##  [776]  2.343152009  3.154832797  5.007485011  1.051556520  0.312584044
##  [781]  1.991858079  1.177792018  0.246398969  2.694138391  0.074828174
##  [786]  1.578839738  0.131362847  5.026536900  0.525622467  1.352985235
##  [791]  1.167242775  1.694529991  0.345244212 10.369273885  2.673262521
##  [796]  0.275569034  5.635179751  3.695781887  0.923043481  6.683481303
##  [801]  5.041454006  2.892669334  1.730978788  3.652207797  1.395344343
##  [806]  1.783924785  0.240216630  7.143593467  1.530547381  2.610396023
##  [811]  0.618818267  3.076653178  1.754103597  0.062874858  2.436389134
##  [816]  0.407165788  0.586722041  2.028094898  1.238932743  1.719573541
##  [821]  0.173795235  2.255220567  2.164433264  0.773133724  0.504408328
##  [826]  0.181799654  0.487259333 10.148062868 12.728788968  3.593300991
##  [831] 10.135076357  6.383781931  2.073648150  3.740770587  1.684889555
##  [836]  2.137484864  4.154162640  0.427835521  3.941295445  4.546927097
##  [841]  2.625083001  4.643774680  3.288283173  0.567798092  0.928356672
##  [846]  1.617628001  2.972689089  0.032633645  2.347567542  5.730718192
##  [851]  0.326163587  0.105424318  0.653828306  2.881702176  1.825159522
##  [856]  3.572699385  2.155904494  0.149325030  1.111994340  0.278933691
##  [861]  4.624549746  1.150917810  0.447628252  2.357691578  1.363971995
##  [866]  2.595849973  0.720392459  0.172154071  3.336243647  2.253342070
##  [871]  1.489863156  1.502553560  3.102124913  3.048764903  0.727578071
##  [876]  0.894724905  3.941367571  0.563829514  7.788817323  1.842201073
##  [881]  0.003358496  1.935298955  0.494010745  1.179188131  1.067889059
##  [886]  6.195211942  4.721815332  6.494381138  1.554447394  4.764090371
##  [891] 10.319344154  0.238491129  2.535119098  5.321599901  1.165199967
##  [896]  3.192183855  2.547320476  0.194265747  1.569252621  0.083722692
##  [901]  0.410719486  1.653670311 10.417173403  1.328448335  2.379494058
##  [906]  1.560058722  0.347889865  0.790741543  4.430306253  0.423550447
##  [911]  2.471373042  3.196154242  4.193510872  9.490875578  3.037347647
##  [916]  3.456352227  4.794257171  1.026646409 11.147061430  1.156200465
##  [921] 10.475729688  0.719200011  4.669675914  0.712216878  1.733017845
##  [926]  2.970538342  4.291178463  8.244672045  1.367655280  2.544041246
##  [931]  0.264382353  0.881652645  2.006211660  8.659926075  2.790122715
##  [936]  0.153332054  3.136187242  2.417862860  0.912875271  2.079017027
##  [941]  3.133815094  0.541356875  3.755076844  5.360999364  0.047279037
##  [946]  1.436924817  2.819470949 10.180814280  1.230467990  9.109040171
##  [951]  1.877834287  1.561964919  0.637512208  0.652862430  3.141766454
##  [956]  2.337004641  1.027353750  2.511088890  5.726178572  0.456936210
##  [961]  4.532471173  1.206750621  2.171630033  0.695929053  5.278029217
##  [966]  0.221822849  2.698579813  3.439168238  3.165714762  0.544235688
##  [971]  2.687705252  1.743390356  9.984641871  7.744263131  0.323852680
##  [976]  0.545304893  0.873526316  3.463150689  0.771906439  0.309127246
##  [981]  1.807892199  5.388419591  1.645358474  1.355231485  0.809175849
##  [986]  1.425914712  0.172765481  2.002482156  1.111110256  2.795919746
##  [991]  3.059986785  6.988963875  0.107226435  5.327956033  0.064424860
##  [996]  0.710349745  1.425391798  3.233495070  0.676714343  2.097084962

Rata-rata Waktu Pelayanan

mean_service <- mean(service_time)
cat("Rata-rata waktu pelayanan:", mean_service)

## Rata-rata waktu pelayanan: 2.574948

Probabilitas Waktu Pelayanan Lebih dari 5 menit

prob_service <- sum(service_time > 5) / n_customers
cat("Probabilitas waktu pelayanan > 5 menit:", prob_service)

## Probabilitas waktu pelayanan > 5 menit: 0.137