Given:
System type: M/M/1 queue
Mean interarrival time: 1.25 mins
Mean service time: 1 min
Find:
\(W_q\)
\(W\)
\(L_q\)
\(L\)
\(\rho\)
Mean arrival rate:
\[\lambda=\frac{arrival rate}{interarrival time}\]
\[\lambda=\frac{1 \quad customer}{1.25 \quad min}=0.8\quad customers/min\]
Mean service rate:
\[\mu=\frac{1}{meanservicetime}\]
\[\mu=\frac{1 }{1}=1 \quad customers/min\]
Server utilization:
\[\rho=\frac{\lambda}{\mu}\]
\[\rho=\frac{0.8}{1}=0.8\]
Customers in queue:
\[L_q = \frac{p^2}{1 - p}\]
\[L_q = \frac{0.8^2}{1 - 0.8} = 3.2 \quad customers in queue\]
Customers in system:
\[L= \frac{\lambda}{1 - \lambda}\]
\[L= \frac{\lambda}{1 - lambda}\]
\[L= \frac{0.8}{1 - 0.8} = 4 \quad customers in system\]
Total time in system:
\[W = \frac{L}{\lambda}\]
\[W = \frac{4}{0.8} = 5 \quad mins\]
Time in queue:
\[W_q = W-E(s) = W - (1/\mu)\]
\[W_q = 5 - 1 = 4 \quad mins\]