Consider a single-server system with exponential arrivals and service. The arrival rate is 0.2 customers per minute, and the average service time is 4 minutes per customer. Compute utilization, average waiting time, and the average queue length.

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BlackTom AI · Accepted Answer

Begin by identifying the system characteristics: we have an M/M/1 queue (exponential arrivals and service, single server). The arrival rate is lambda = 0.2 customers/min, and the average service time is 4 minutes, so the service rate is mu = 1 / 4 = 0.25 customers/min.

First, compute utilization: rho = lambda / mu = 0.2 / 0.25 = 0.8. This gives the server is busy 80% of the time.

Next, determine the key performance measures for an M/M/1 queue. The average number in queue is Lq = rho^2 / (1 - rho) = (0.8)^2 / (1 - 0.8) = 0.64 / 0.2 = 3.2 customers. The average waiting time in the queue is Wq = Lq / lambda = 3.2 / 0.2 = 16 minutes.

As a check, the average time in the system W = 1 / (mu - lambda) = 1 / (0.25 - 0.2) = 1 / 0.05 = 20 minutes, and the average number in the system L = lambda * W = 0.2 * 20 = 4, which aligns with the relationships L = Lq + lambda*Wq and rho = 0.8.

Now evaluate each answer option:
Option 1: 0.7; 12 min; 1.6 customers. Incorrect on all counts: rho is not 0.7 (it is 0.8), Wq is not 12 minutes, and Lq is not 1.6; this mixes wrong utilization with wrong queue metrics.
Option 2: 0.7; 12 min; 3.2 customers. The Lq value 3.2 matches the correct Lq, but rho is wrong at 0.7 (it should be 0.8), and Wq is listed as 12 minutes, whereas the correct Wq is 16 minutes. This shows confusion between utilization and queue metrics.
Option 3: 0.8; 16 min; 1.6 customers. Here rho = 0.8 is correct, and Wq = 16 minutes is correct, but Lq is 1.6, which is incorrect since Lq should be 3.2; so this option mixes correct time metrics with an incorrect queue length.
Option 4: 0.8; 16 min; 3.2 customers. All three values align with the M/M/1 results for rho, Wq, and Lq, respectively (rho = 0.8, Wq = 16 min, Lq = 3.2).

In summary, the correct combination is rho = 0.8, Wq = 16 minutes, and Lq = 3.2 customers, which corresponds to option 4.

Consider a single-server system with exponential arrivals and service. The arrival rate is 0.2 customers per minute, and the average service time is 4 minutes per customer. Compute utilization, average waiting time, and the average queue length.

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