Homework Two - Simio and Simulation: Chapter 2

Complete problem 1 and attempt with Simio

Problem 1

For an M/M/1 queqe with mean interarriaval time 1.25 minutes and mean service time 1 minute, find all five of \(W_{q}\), W, \(L_{q}\), L, and \(\rho\). For each, interprete in words. Be sure to state all of your units (always!), and the relevant time frame of operation.

This system is an expontital interval time, exponential service time, and single server system.

\(L = \lambda * W\) \(L_{q} = \lambda * W_{q}\) \(W = W_{q} + E(S)\)

In this case, we have \(\lambda = 1/1.25 = 0.8\) min and \(E(S) = 1\) min. Expressing the formula in a different way, we can get L, and use that to calculate W, \(W_{q}\), \(L_{q}\), and \(\rho\).

\(L = \frac{\lambda}{1/E(S) - \lambda} = \frac{0.8}{1/1 - 0.8} = 4\) \(W_{q} = \frac{L}{\lambda} - E(S) = 4/0.8 = 5 - 1 = 4\) \(W = W_{q} + E(S) = 4 + 1 = 5\) \(L_{q} = \lambda * W_{q} = 0.8 * 4 = 3.2\)

\(\rho = \lambda / (c * \mu) = 0.8/(1*1) = 0.8\)

For this problem, we have an average total number of 4 people in the system, with a 5 minute average time in the system. The average time in queue is 4 minutes, and average people in queue is 3.2. Utilization is 80%.

arrivaltime <- 1.25
servtime <- 1
lamb <- 1/arrivaltime
mu <- 1/servtime
rho <- lamb / (1 * mu)
L <- lamb / (mu - lamb)
W_q <- L / lamb - servtime
W <- W_q + servtime
L_q <- lamb * W_q
  
data.frame('lambda'=lamb, 'mu'=mu, 'rho'=rho, 'L'=L, 'W'=W, 'L(q)'=L_q, 'W(q)'=W_q)
##   lambda mu rho L W L.q. W.q.
## 1    0.8  1 0.8 4 5  3.2    4

Simio

Simio Simulation Set-up

Simio Simulation Set-up

Simio Simulation Results

Simio Simulation Results