MMc Abandonment

355. MMc Abandonment

Erlang A (also written M/M/c+M): M/M/c queue with abandonment — customers leave after waiting too long. More realistic for call centers, where impatient callers hang up.

355.1. Setup

$𝜆$ : arrival rate (Poisson)
$𝜇$ : service rate per server (exponential)
$𝑐$ : number of servers
$𝜃$ : abandonment rate (exponential patience time, mean $\frac{1}{𝜃}$ )

355.2. Why it matters

Erlang C assumes infinite patience — every customer eventually gets served. Erlang A models the real situation where customers hang up at some rate.

Without abandonment: a slightly-overloaded system has unbounded waits. With abandonment: a slightly-overloaded system has bounded waits and some abandonment rate.

355.3. Approximate formulas

Exact Erlang A is nonelementary — uses incomplete gamma function. The most useful results:

Abandonment probability:

𝑃 (abandon) \approx 𝐶 (𝑐, 𝑎) \cdot \frac{𝜌 - 1}{\dots}

(Specific formulas are technical; key insight is monotonicity: more $𝜃$ → faster abandonment → less waiting.)

Expected wait given service:

𝐸 [𝑊_{𝑞}] \approx \frac{𝐶 (𝑐, 𝑎)}{𝑐 𝜇 - 𝜆 + 𝜃 (𝑐 - 𝜌 𝑐)}

(Approximate; sharper formulas in queueing texts.)

355.4. Garnett-Mandelbaum-Reiman scaling

In heavy traffic ( $𝜌 \approx 1$ ), Erlang A admits a clean scaling limit (Halfin-Whitt regime extended). Lets you size:

Quality-driven regime ( $𝛽 > 0$ ): wait > 0 rare; small abandonment
Efficiency-driven regime ( $𝛽 < 0$ ): wait probable, abandonment substantial
Quality-and-efficiency-driven (QED, $𝛽 \approx 0$ ): balanced

Used in modern call-center capacity planning.

355.5. Call-center application

Erlang A predicts service level (= P(answered within X seconds, before abandoning)) directly. For typical inbound centers:

Target: SL = 80% of calls answered within 20 seconds
Erlang A converts agent count + abandonment rate + call duration into SL

This is what tools like Genesys, Five9, and Erlang calculators compute.

355.6. Compared to Erlang C

	Erlang C (M/M/c)	Erlang A (M/M/c+M)
Abandonment	no	yes
Customers behavior	infinite patience	exponential patience with mean $\frac{1}{𝜃}$
Overload regime	unbounded queue	bounded by abandonment
Realism	optimistic on staffing	closer to reality
Computation	closed form	more complex, often approximated

Erlang A recommends fewer agents than Erlang C at the same target SL, because abandonment shortens queues — closer to reality.

355.7. Patience modeling

Real patience is rarely exponential — often:

Lognormal — most calls abandon quickly, but a tail waits long
Mixture — different customer segments with different patience
Time-varying — patience changes with wait

These need simulation. Erlang A with exponential patience is the analytical starting point.

355.8. See also

Erlang C — without abandonment
Erlang B — no queue at all
Square-Root Staffing