Question

For a single server with Poisson arrival and exponential service time, the arrival rate is 12 per hour. Which one of the following service rates will provide a steady state finite queue length?

• A. 6 per hour
• B. 10 per hour
• C. 12 per hour
• D. 24 per hour

Answer 1

The correct option is D 24 per hour

$L_q=\text{queue length}=\frac{{\rho }^2}{1-\rho }$

= finite only when $\rho <1$

so using options (d),

$\rho =\frac{\lambda }{\mu }=\frac{\text{Arrival rate}}{\text{Service rate}}=\frac{12}{24}=\frac{1}{2}<1$

For other options $\rho \ge 1$ & $L_q$ can not be finite because when arrival rate is more than service rate queue will keep on increasing.