STAT4438: Simulation and Modeling

RSTAT4438: Lab Exercise IV

Lab Exercise IV: Single Server Queuing System

Given queuing system for 10 customers with the following inter-arrival time (ITA) and service time (ST):

ST	4	1	4	3	2	4	5	4	5	3
ITA	-	8	6	1	8	3	8	7	2	3

IAT = c(0,8,6,1,8,3,8,7,2,3)
ST = c(4,1,4,3,2,4,5,4,5,3)
AT= cumsum(IAT)
TSB = c()
TCW= c()

TSE=c()
TCSS=c()
idl=c()
for (i in 1:length(IAT)) {
  TSB[i]= max(AT[i],TSE[i-1])
  TSE[i]= TSB[i]+ST[i]
  TCW[i]= abs(TSB[i]-AT[i])
  TCSS[i]= abs(TSE[i]-AT[i])
  idl[i] = ifelse(i==1,0,ifelse(AT[i]>TSE[i-1],AT[i]-TSE[i-1],0))
}

data.frame(IAT,AT,ST,TSB,TCW,TSE,TCSS)

##    IAT AT ST TSB TCW TSE TCSS
## 1    0  0  4   0   0   4    4
## 2    8  8  1   8   0   9    1
## 3    6 14  4  14   0  18    4
## 4    1 15  3  18   3  21    6
## 5    8 23  2  23   0  25    2
## 6    3 26  4  26   0  30    4
## 7    8 34  5  34   0  39    5
## 8    7 41  4  41   0  45    4
## 9    2 43  5  45   2  50    7
## 10   3 46  3  50   4  53    7

• Assume the first customer arrives at time t= 0 minute. Write an R code using ITA and ST to construct a matrix or a dataframe including the following columns:

  • inter-arrival time (ITA)

  • Arrival time (AT)

  • Service time (ST)

  • Time service begins (TSB)

  • Time a customer waits in the queue (TCW)

  • Time service ends (TSE)

  • Time customer spent in the system (TCSS)

  • Idle time of the server (Idle)

• Use the above variables to calculate the following:

1. Average waiting time
2. Probability that a customer has to wait in a queue.

mean(TCW>0)

## [1] 0.3

ST

##  [1] 4 1 4 3 2 4 5 4 5 3

3. Average service time
4. Average time between arrivals
5. Average waiting time of those who wait
6. Average time a customer spend in a system
7. Idle rate
8. Utilization rate

N.B Submit your answers as a html or pdf file.

set.seed(123)
id= sample(seq(1,100,1),5,replace = F)
id

## [1] 31 79 51 14 67

id= c(60,30,42,9,65)
IAT = function(id){
  IAT= ifelse(id<=50,5,ifelse(id<=80,10,15))
}
ITA=IAT(id)
ST=c(7,7,7,7,7)
AT= cumsum(ITA)
TSB = c()
TCW= c()

TSE=c()
TCSS=c()
idl=c()
for (i in 1:length(ITA)) {
  TSB[i]= max(AT[i],TSE[i-1])
  TSE[i]= TSB[i]+ST[i]
  TCW[i]= abs(TSB[i]-AT[i])
  TCSS[i]= abs(TSE[i]-AT[i])
  idl[i] = ifelse(i==1,0,ifelse(AT[i]>TSE[i-1],AT[i]-TSE[i-1],0))
}

data.frame(id,ITA,AT,ST,TSB,TCW,TSE,TCSS)

##   id ITA AT ST TSB TCW TSE TCSS
## 1 60  10 10  7  10   0  17    7
## 2 30   5 15  7  17   2  24    9
## 3 42   5 20  7  24   4  31   11
## 4  9   5 25  7  31   6  38   13
## 5 65  10 35  7  38   3  45   10

tempdir()

## [1] "C:\\Users\\dimaa\\AppData\\Local\\Temp\\RtmpW4hW5t"

# [1] "C:\Users\XYZ~1\AppData\Local\Temp\Rtmp86bEoJ\Rtxt32dcef24de2"
dir.create(tempdir())

## Warning in dir.create(tempdir()):
## 'C:\Users\dimaa\AppData\Local\Temp\RtmpW4hW5t' already exists

id= c(60,30,42,9,65)
IAT = function(id){
  IAT= ifelse(id<50,5,ifelse(id<80,10,15))
}
ITA=IAT(id)
ST=c(7,7,7,7,7)
AT= cumsum(ITA)
TSB = c()
TCW= c()

TSE=c()


for (i in 1:length(ITA)) {
  TSB[i]= max(AT[i],TSE[i-1])
  TSE[i]= TSB[i]+ST[i]
  TCW[i]= abs(TSB[i]-AT[i])
}

id2= c(18,74,13,30,52)
AT2 = TSE
ST2 = function(id2){
  ifelse(id2<30,4,ifelse(id2<80,8,12))
}
ST2= ST2(id2) 
TCB2 = c()
TCW2=c()
TSE2=c()
TSS=c()
for (i in 1:length(id2)) {
  TCB2[i]= max(TSE2[i-1],AT2[i])
  TCW2[i]= TCB2[i]-AT2[i]
  TSE2[i]= TCB2[i]+ST2[i]
  TSS[i]= TSE2[i]-AT[i]
}
data.frame(id,ITA,AT,ST,TSB,TCW,TSE,id2,TCB2,TCW2,ST2,TSE2,TSS)

##   id ITA AT ST TSB TCW TSE id2 TCB2 TCW2 ST2 TSE2 TSS
## 1 60  10 10  7  10   0  17  18   17    0   4   21  11
## 2 30   5 15  7  17   2  24  74   24    0   8   32  17
## 3 42   5 20  7  24   4  31  13   32    1   4   36  16
## 4  9   5 25  7  31   6  38  30   38    0   8   46  21
## 5 65  10 35  7  38   3  45  52   46    1   8   54  19