Prostok

library(readr)

## Warning: package 'readr' was built under R version 4.5.2

# 1. Memuat data
data <- read_csv("C:/Users/Msi user/Downloads/Call Center Data.csv")

## Rows: 1251 Columns: 9
## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## chr  (2): Answer Rate, Service Level (20 Seconds)
## dbl  (4): Index, Incoming Calls, Answered Calls, Abandoned Calls
## time (3): Answer Speed (AVG), Talk Duration (AVG), Waiting Time (AVG)
## 
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.

ic   <- data$`Incoming Calls`

# 2. Statistik deskriptif
cat('N        =', length(ic), '\n')

## N        = 1251

cat('Min      =', min(ic), '\n')

## Min      = 5

cat('Max      =', max(ic), '\n')

## Max      = 1575

cat('Mean     =', round(mean(ic), 4), '\n')

## Mean     = 198.5396

cat('Median   =', median(ic), '\n')

## Median   = 177

cat('Std Dev  =', round(sd(ic), 4), '\n')

## Std Dev  = 156.5342

cat('Variance =', round(var(ic), 4), '\n')

## Variance = 24502.95

cat('Q1       =', quantile(ic, 0.25), '\n')

## Q1       = 123

cat('Q3       =', quantile(ic, 0.75), '\n')

## Q3       = 233

# 3. Pembentukan state (berbasis kuartil: Q1=123, Q3=233)
assign_state <- function(x) {
  if (x < 123)        return('S1')   # Rendah
  else if (x < 233)   return('S2')   # Sedang
  else                return('S3')   # Tinggi
}
states <- sapply(ic, assign_state)
cat('\nDistribusi State:\n')

## 
## Distribusi State:

print(table(states))

## states
##  S1  S2  S3 
## 311 625 315

print(round(prop.table(table(states)) * 100, 2))

## states
##    S1    S2    S3 
## 24.86 49.96 25.18

# 4. Matriks frekuensi transisi
n     <- length(states)
N_mat <- table(states[-n], states[-1])
cat('\nMatriks Frekuensi Transisi:\n')

## 
## Matriks Frekuensi Transisi:

print(N_mat)

##     
##       S1  S2  S3
##   S1 106 143  61
##   S2 156 414  55
##   S3  49  67 199

# 5. Matriks probabilitas transisi
P <- N_mat / rowSums(N_mat)
cat('\nMatriks Probabilitas Transisi:\n')

## 
## Matriks Probabilitas Transisi:

print(round(P, 6))

##     
##            S1       S2       S3
##   S1 0.341935 0.461290 0.196774
##   S2 0.249600 0.662400 0.088000
##   S3 0.155556 0.212698 0.631746

cat('Verifikasi jumlah baris:', rowSums(P), '\n')

## Verifikasi jumlah baris: 1 1 1

# 6. Probabilitas n-langkah
mat_pow <- function(M, n) {
  result <- diag(nrow(M))
  for (i in seq_len(n)) result <- result %*% M
  result
}
for (n in c(2, 3, 5, 10)) {
  cat('\nP^', n, ':\n')
  print(round(mat_pow(P, n), 6))
}

## 
## P^ 2 :
##       
##              S1       S2       S3
##   [1,] 0.262667 0.505144 0.232189
##   [2,] 0.264371 0.572629 0.163000
##   [3,] 0.204551 0.347019 0.448430
## 
## P^ 3 :
##       
##              S1       S2       S3
##   [1,] 0.252017 0.505159 0.242823
##   [2,] 0.258682 0.535931 0.205387
##   [3,] 0.226315 0.419603 0.354082
## 
## P^ 5 :
##       
##              S1       S2       S3
##   [1,] 0.249416 0.500838 0.249746
##   [2,] 0.251644 0.508835 0.239521
##   [3,] 0.242804 0.477472 0.279724
## 
## P^ 10 :
##       
##              S1       S2       S3
##   [1,] 0.248881 0.499007 0.252112
##   [2,] 0.248966 0.499310 0.251724
##   [3,] 0.248630 0.498120 0.253249

# 7. Distribusi steady state
A  <- t(P) - diag(3)
A  <- rbind(A, rep(1, 3))
b  <- c(0, 0, 0, 1)
pi_ss <- qr.solve(A, b)
names(pi_ss) <- c('S1 (Rendah)', 'S2 (Sedang)', 'S3 (Tinggi)')
cat('\nDistribusi Steady State:\n')

## 
## Distribusi Steady State:

print(round(pi_ss, 6))

## S1 (Rendah) S2 (Sedang) S3 (Tinggi) 
##    0.248860    0.498935    0.252205

cat('Verifikasi (pi*P harus = pi):\n')

## Verifikasi (pi*P harus = pi):

print(round(pi_ss %*% P, 6))

##       
##             S1       S2       S3
##   [1,] 0.24886 0.498935 0.252205