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 Lq in units?
A)0.4
B)0.133
C)4.167
D)4.583
E)6

Question

Quizplus · Accepted Answer

The M/D/1 system can be analyzed using Little's Formula which states L = λW, where L is the average number of units in the system, λ is the arrival rate, and W is the average time a unit spends in the system.
Since we only want to find Lq, we can use the formula Lq = λWq, where Wq is the average waiting time in the queue.
To find λ, we can use the fact that units arrive every 10 minutes on average, which gives us an arrival rate of 6 units per hour (since there are 60 minutes in an hour).
To find W, we need to consider both the service time and the waiting time in the queue. Since the system is in steady-state, the average waiting time in the queue is equal to the average time spent in the system minus the service time. Thus, W = Wq + 4 minutes.
Let's use Little's Formula to find W: 
L = λW
L = 6 * W
We know that L = Lq + ρ, where ρ is the average number of units being served (in this case, it's 1 since there's only one server). Since the system is in steady-state, Lq is the same as L - ρ. Thus,
L - ρ = λW
Lq = λW - ρ
Lq = 6 * (Wq + 4) - 1
Lq = 6Wq + 23
Now we need to find Wq. Since the system is M/D/1, we can use the formula Wq = (ρ^2) / (1 - ρ) * (1 / μ), where μ is the service rate (in this case, it's 1/4 units per minute).
ρ = λ / μ = 6 / (1/4) = 24
Wq = (24^2) / (1 - 24) * (1/4) = 0.133333...
Therefore, Lq = 6Wq + 23 = 23.8 or around 0.133 units on average are waiting in the queue. 
The best choice is B.

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