352. 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.
352.1. Setup
- : arrival rate (Poisson)
- : service rate per server (exponential)
- : number of servers
- : abandonment rate (exponential patience time, mean )
352.2. 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.
352.3. Approximate formulas
Exact Erlang A is nonelementary — uses incomplete gamma function. The most useful results:
Abandonment probability:
(Specific formulas are technical; key insight is monotonicity: more → faster abandonment → less waiting.)
Expected wait given service:
(Approximate; sharper formulas in queueing texts.)
352.4. Garnett-Mandelbaum-Reiman scaling
In heavy traffic (), Erlang A admits a clean scaling limit (Halfin-Whitt regime extended). Lets you size:
- Quality-driven regime (): wait > 0 rare; small abandonment
- Efficiency-driven regime (): wait probable, abandonment substantial
- Quality-and-efficiency-driven (QED, ): balanced
Used in modern call-center capacity planning.
352.5. 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.
352.6. 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 |
| 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.
352.7. 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.
352.8. See also
- Erlang C — without abandonment
- Erlang B — no queue at all
- Square-Root Staffing