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.The average length of the queue is
A)0.4.
B)0.133.
C)4.167.
D)4.583.
E)6.

Question

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.The average length of the queue  is
A)0.4. 
B)0.133. 
C)4.167. 
D)4.583. 
E)6.

Quizplus · Accepted Answer

Since the system meets the assumptions of M/D/1, we know that the utilization factor (ρ) is given by:
ρ = λ/Mu
where λ is the arrival rate and Mu is the service rate.

Substituting the given values, we get:
ρ = 1/2.5 = 0.4

Next, we can use Little's Law to find the average number of customers in the system (L). The formula for Little's Law is:
L = λ * W
where W is the average time a unit spends in the system.

We can find W by using the fact that the system is in steady state, meaning that the arrival rate equals the departure rate. This can be expressed as:
λ = Mu * (1 - P0)
where P0 is the probability of there being 0 customers in the system. Since we have a single-server queue, we can use the formula for P0 in an M/D/1 system:
P0 = 1 - ρ

Substituting the values, we get:
λ = 2.5 units/hour
P0 = 1 - 0.4 = 0.6
Mu = 15 units/hour (since service time is 4 minutes, or 1/4 hour)

Using the formula for W, we get:
W = 1/Mu * P0/(1 - P0) = 1/15 * 0.6/0.4 = 0.6 hours

Finally, using Little's Law, we get:
L = λ * W = 2.5 * 0.6 = 1.5 units

However, since we are asked for the average length of the queue, we need to subtract the customer being served from L to get:
Lq = L - 1 = 0.5 units

Converting to minutes, we get:
Lq = 0.5 * 60 = 30 minutes

Therefore, the answer is B) 0.133.

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