Call Center Queuing Models: Erlang & Applicability in Manufacturing

The Beginner-Friendly Guide to Queueing Models

From call-center Erlang formulas to aircraft-factory “quality callboards”

1. Why queues matter in the real world

Wherever customers (or jobs) arrive randomly and wait for help, queueing theory gives you a crystal ball:

If you can describe how fast calls come in and how long each one takes, the math will tell you how many servers you need.

2. The building blocks of any queue

1. Arrival rate (λ) – average calls per hour.

2. Service time distribution – how long a tech spends on a job (minutes).

3. Number of servers (s) – technicians on duty.

4. Queue discipline – usually “first in, first out.”

5. Customer behavior – will they wait forever, leave after a while, or get blocked completely?

Change #5 and you switch Erlang models.

3. Meet the Erlang family - B, C, and A

Think of them as successive realism upgrades:
Erlang B ↗ C (adds queue) ↗ A (adds impatience).

4. A factory example: the “quality callboard”

Scenario

• Mechanics finish a step and post a digital “QA verify” request.

• Goal: a quality technician arrives within 15 minutes 80 % of the time.

• Service time (hands-on inspection + travel): triangular distribution

  • min = 15 min, most likely = 35 min, max = 120 min.

4.1 Converting the triangular time to an average rate

Mean of a triangular (a, m, b) = (a + m + b)/3
⇒ (15 + 35 + 120) / 3 ≈ 56.7 min per job
Service rate μ ≈ 1 / 56.7 min ≈ 1.06 jobs /hour per tech.

(You can still feed the full distribution into simulation, but the mean is enough for a first Erlang pass.)

4.2 When to use each model

Most shops land between C and A: they queue, but impatience grows.

5. How many technicians do we need? (Rule-of-thumb walk-through)

You only need four inputs: arrival rate λ, service rate μ, servers s, and patience (for Erlang A).

1. Estimate λ – count calls last month and divide.

   • Suppose mechanics posted 10 calls per hour during peak.

2. Pick a model – Erlang A (calls can cancel if wait > 20 min).

3. Use a calculator – free tools like Erlang-for-Call-Centers, qtools in R, or any online Erlang A spreadsheet.

   • Plug in λ = 10, μ = 1.06, mean patience ≈ 20 min.

   • Vary s until:

     • P(wait > 15 min) ≤ 20 %

     • Abandonment rate acceptable (< 5 %)

4. Interpret – typically you’ll see something like:

   • s = 9 techs → 18 % wait > 15 min, 4 % abandon. ✅

   • s = 8 techs → 28 % wait > 15 min, 7 % abandon. ❌

(Numbers depend on λ; the method stays the same.)

6. Practical tips & common pitfalls (QA edition)

7. Where to go next

• Simulate your exact triangular times with a discrete-event tool (SimPy, AnyLogic, SIMIO).

• Blend roles: maybe 2 techs float between two production lines—adjust λ accordingly.

• Automate: nightly script pulls yesterday’s callboard log → recomputes λ, μ → emails staffing suggestion.

Take-aways

1. Erlang B, C, and A are progressively more realistic call-center models.

2. Even simple averages can guide first staffing decisions.

3. For an aircraft factory’s QA callboard, Erlang A matches reality best because mechanics will eventually cancel a pending request.

4. A 15-minute SLA isn’t magic—track arrivals & service times, pick the right model, and the math will tell you how many techs to roster. Remember, achieving the SLA comes at the cost of Utilization.