A single-phase waiting-line system meets the assumptions of constant service time or M/D/1.Units arrive at this system every 10 minutes on average.Service takes a constant 4 minutes.What is the average length of the queue L_q in units? A)0.4 B)0.133 C)4.167 D)4.583 E)6

Question

Quizplus · Accepted Answer

We can use the formula for the expected number of customers in a single-server queue with constant service time: L = (λ/μ) * (1/1 - ρ), where λ is the arrival rate, μ is the service rate, and ρ = λ/μ is the utilization factor.

Plugging in the given values, we get:
λ = 1/10 = 0.1 per minute
μ = 1/4 = 0.25 per minute
ρ = λ/μ = 0.1/0.25 = 0.4
L = (0.1/0.25) * (1/1 - 0.4) = 0.4 * (1/0.6) = 0.667

However, this formula gives us the expected number of customers in the system, including those being served and those waiting in line. Since we are only interested in the length of the queue, we need to subtract the expected number of customers being served from L. Since this is a M/D/1 system, there is only one customer being served at a time. Therefore, the expected number of customers being served is simply the utilization factor ρ.

Lq = L - ρ = 0.667 - 0.4 = 0.267
Finally, since we are asked for the average length of the queue in units (i.e. customers), we need to divide Lq by the average number of units per customer, which is 1. Therefore, the answer is:
Lq/1 = 0.267/1 = 0.267, which is closest to answer choice B) 0.133.

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