A single-phase waiting-line system meets the assumptions of constant service time or M/D/1. Units arrive at this system every 12 minutes on average. Service takes a constant 8 minutes. The average length of the queue Lq is approximately
A) 0.67
B) 2.5
C) 4.5
D) 5.0
E) 7.5

Question

Quizplus · Accepted Answer

The M/D/1 queue model applies here, where arrivals are Markovian (M), service times are deterministic (D), and there is one server (1). The formula for the average length of the queue, $L_q$, in an M/D/1 system is $L_q = \frac{\rho^2}{2(1-\rho)}$, where $\rho = \frac{\lambda}{\mu}$, $\lambda$ is the arrival rate, and $\mu$ is the service rate. Given $\lambda = \frac{1}{12}$ arrivals per minute and $\mu = \frac{1}{8}$ services per minute, $\rho = \frac{1/12}{1/8} = \frac{2}{3}$. Substituting $\rho$ into the formula gives $L_q = \frac{(2/3)^2}{2(1-2/3)} = \frac{4/9}{2/3} = \frac{4/9}{2/3} = \frac{4/9 \times 3/2} = \frac{2}{3} = 0.67$.

A Single-Phase Waiting-Line System Meets the Assumptions of Constant Service