Week4

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Complete Chapter 2 Problem 9 from Kelton followed by Chapter 4 Problem 15.

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Common notations as per page number 26 and formulas on page 28

• Wq = The steady-state average time in queue
• W = The steady-state average time in system
• Lq = The steady-state average number of entities in queue
• L = The steady-state average number of entities in system
• p = The steady-state utilitization of a server ( or group of servers)

Problem 9 from chapter 2

Multiserver Queueing

MM1<-function(Arrival_Rate,Service_Rate,servers)
{
  
  lambda <- 1/Arrival_Rate 
  mu <- 1/Service_Rate
  
  p <- lambda / (mu*servers)
  L <- p/(1-p)
  W <- L/lambda
  
  es <- 1/mu
  
 
  Wq <- W-es
  Lq <- lambda*Wq
  df<-data.frame(p,L,W,Wq,Lq)
  return (df)
}

MD1<-function(mu,lambda)
{
    n1 <- mu - lambda
    list1=list(
      p <-lambda / mu,
      Lq = ( lambda * lambda ) / ( 2 * mu * n1 ),
      L = ( lambda * lambda ) / ( 2 * mu * n1 ) + lambda / mu,
      Wq=lambda / ( 2 * mu * n1 ),
      W=( lambda / ( 2 * mu * n1 )) + (1 /  mu )
    )
    return(list1)
}

MD1((1/0.9),1)

## [[1]]
## [1] 0.9
## 
## $Lq
## [1] 4.05
## 
## $L
## [1] 4.95
## 
## $Wq
## [1] 4.05
## 
## $W
## [1] 4.95

MM1(0.9,1,1)

##          p   L  W  Wq        Lq
## 1 1.111111 -10 -9 -10 -11.11111

As we can see queues from MM1 exceed that of MD1.

Problem 15 from chapter 4

Question 15

Here in simio chosen time is 10 hours and 1 hour warming up period.

Below is scrren of simio result.

Simio output