A call center has an arrival rate of 12 calls/hour. The average waiting time in the queue is 8 minutes, and the average service time is 4 minutes. What is the average number of customers in the system (queue + in service)?

Question

BlackTom AI · Accepted Answer

To solve this, I’ll start by extracting the given queueing parameters and converting times to consistent units.
- Arrival rate λ = 12 calls per hour.
- Average waiting time in the queue Wq = 8 minutes = 8/60 hours = 1/7.5 hours ≈ 0.1333 h.
- Average service time Ts = 4 minutes = 4/60 hours = 1/15 hours ≈ 0.0667 h. The corresponding service rate μ is the reciprocal, so μ = 1 / Ts = 15 calls per hour.

With λ = 12 and μ = 15, the system utilization is ρ = λ/μ = 12/15 = 0.8. This helps cross-check results, but the directly useful relationships are:
- Average time in system W = Wq + Ws, where Ws (time in service) = 1/μ = 1/15 h ≈ 0.0667 h.
- Therefore W = 0.1333 h + 0.0667 h = 0.2 h (which is 12 minutes).
- The average number in the system L is given by L = λ × W = 12 × 0.2 = 2.4 customers.

Now I’ll analyze each answer option:

Option 0.8: This number might appear if one mistakenly uses the utilization ρ as the average number in the system. However, ρ = 0.8 is the fraction of time the server is busy, not the average number of customers present. The average number in the system is not simply ρ; it also involves the queue length term. This option undercounts because even when the server is busy, there can be others waiting, so L > ρ in most cases including this scenario.

Option 2.4: This matches the calculation L = λ × W with W = Wq + Ws, or equivalently L = Lq + Ls where Ls = λ/μ = 0.8 and Lq = λ × Wq = 12 × (1/7.5) ≈ 1.6, giving L = 1.6 + 0.8 = 2.4. This is the correct aggregation of customers in the queue and in service.

Option 12.0: This would imply that there are 12 customers on average present, which would be far larger than the arrival rate and exceeds the server capacity in this setup (since μ = 15/hour, you wouldn’t typically have 12 in system if the times imply a much smaller average). It conflicts with both the derived W and L values and with the basic Little’s law relations used here.

Option 1.6: This equals the average number of customers in the queue Lq = λ × Wq = 12 × 0.1333 ≈ 1.6. While this is a legitimate component of the total in-system population, it does not account for the customers currently being served. Therefore it’s not the total in the system, only the queue portion.

In summary, calculating the total number in the system requires combining both the queue and the service components, either via L = λW or by summing Lq and Ls. The other options correspond to either the queue-only count, a utilization-like figure, or an overlarge total that conflicts with the system rates. The structured approach above leads to the correct total being 2.4 customers in the system.

A call center has an arrival rate of 12 calls/hour. The average waiting time in the queue is 8 minutes, and the average service time is 4 minutes. What is the average number of customers in the system (queue + in service)?

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