DATA 604 Week 2 Discussion

For an \(M/M/1\) queue with mean interarrival time 1.25 minutes and mean service time 1 minute, find all the five of \(W_q,\: W,\: L_q,\: L,\: and \: \rho\). For each, interpret in words. Be sure to state all of your units (always!), and the relevant time frame of operation.

Solution

m_int <- 1.25                 # Mean interval Time
m_serv <- 1                   # Mean Service Time
n_serv <- 1                   # Number of server($M/M/1$ queue)

lambda <- 1 / m_int           # Mean rate of arrival (lambda)
mu <- 1 / m_serv              # Mean service rate (mu)

rho <- lambda / mu            # Utilization of the server

W_q <- rho / (mu * (1 - rho)) # Mean wait in the queue

W <- 1 / (mu * (1 - rho))     # Mean wait in the system

L_q <- rho^2 / (1 - rho)      # Mean number of customers in the queue

L <- rho / (1 - rho)          # Mean number of customers in the system

\(W_q\) = 4

\(W\) = 5

\(L_q\) = 3.2

\(L\) = 4

\(\rho\) = 0.8

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