First Simio Models

p = The steady-state utilitization of a server

L = The steady-state average number of entities in system

W = The steady-state average time in system

Wq = The steady state average time in queue

Lq = The steady-state average number of entities in queue

2.9

MD1<-function(mu,lambda){
  temp = mu - lambda
  rList=list(
    p =lambda / mu,
    Lq = ( lambda * lambda ) / ( 2 * mu * temp ),
    L = ( lambda * lambda ) / ( 2 * mu * temp ) + lambda / mu,
    Wq=lambda / ( 2 * mu * temp ),
    W=( lambda / ( 2 * mu * temp )) + (1 /  mu )
  )
  return(rList)
}

MM1<-function(mu,lambda){
  p.v <- lambda / mu
  L.v <- p.v/(1-p.v)
  W.v <- L.v/lambda
  Wq.v <- W.v-(1/mu)
  Lq.v <- lambda*Wq.v
  rList=list(
    p = p.v,
    Lq = Lq.v,
    L = L.v,
    Wq = Wq.v,
    W = W.v
  )
  return(rList)
}

#arrival rate 1 per minute
#service rate 1/0.9 per minute

MD1((1/0.9),1)
## $p
## [1] 0.9
## 
## $Lq
## [1] 4.05
## 
## $L
## [1] 4.95
## 
## $Wq
## [1] 4.05
## 
## $W
## [1] 4.95

Looking at these two results, we can compare by setting a variable to each function return and checking the difference. We notice that queues and times from MM1 exceed that of MD1.

md1Var<-MD1((1/0.9),1)
mm1Var<-MM1((1/0.9),1)
mapply('-',md1Var,mm1Var,SIMPLIFY=FALSE)
## $p
## [1] 0
## 
## $Lq
## [1] -4.05
## 
## $L
## [1] -4.05
## 
## $Wq
## [1] -4.05
## 
## $W
## [1] -4.05

Moving into Simio. An experiment was set-up at 10 hours with 10 replications and a 1 hour warm-up period to reduce start-up bias. There is a similarity of values, however a more refined simulation could have used an increased number of replications and length of simulation (days not hours.)

Simio Results

Simio Results