Stage 04 · Routing: Your Lead Queue Obeys Erlang, Not Optimism

Ariadne's routing stage exists because the handoff from "an inquiry arrived" to "a specific competent human is now on it" is where most systems have no policy at all. The default policy — whoever notices — is a queue discipline, just a bad one.

The one identity you can always rely on

Little's Law states that the average number of items in a queueing system equals the average arrival rate times the average time an item spends in the system Little 1961:

The conditions are mild: the three means are finite, the processes are stationary, and the arrival process is metrically transitive with nonzero mean Little 2011. No assumption about the distribution of arrivals or service times. This is why it is the first thing to compute about any queue.

For routing, the useful corollary is offered load. If inquiries arrive at rate $\lambda$ and each consumes mean handling time $1/\mu$, then the number of people continuously occupied is

If you receive 20 inquiries a day and each takes 30 minutes of genuine attention, $R = 10$ person-hours per day. If your team has 6 hours of uncommitted attention per day, the queue is not a scheduling problem. It is unstable, and no amount of discipline will fix it.

How many people do you actually need

Once $R$ is known, Erlang C gives the probability that an arriving inquiry has to wait at all, with $s$ servers and $\rho = R/s$ van Leeuwaarden et al. 2019:

and mean delay follows as $E[W] = C(s,\lambda)\,/\,\bigl((1-\rho)s\bigr)$.

The formula is unwieldy; the design rule extracted from it is not. Square-root staffing says to staff at

where $\beta$ is a service-grade parameter Whitt 1992. This sits on the Halfin–Whitt heavy-traffic limit, in which the delay probability converges to a non-degenerate value that depends only on $\beta$, exactly when $\sqrt s\,(1-\rho) \to \beta > 0$ Halfin & Whitt 1981.

Erlang C is the wrong model, and everyone uses it anyway

Erlang C assumes an infinite-patience customer: everybody waits forever. Nobody waits forever.

Garnett, Mandelbaum and Reiman introduced the corrected model and named it Erlang A — A for abandonment — analysing M/M/N + M with exponential patience of mean $1/\theta$, and stated plainly that Erlang C "is the model most often used in call-centre practice" despite this defect Garnett et al. 2002. Their framework yields a clean identity relating abandonment to waiting:

Gans, Koole and Mandelbaum's survey of the whole field makes the same methodological point at length Gans et al. 2003, and Zeltyn and Mandelbaum generalised the result to arbitrary patience distributions, which matters because real patience is not exponential Zeltyn & Mandelbaum 2005.

How much does abandonment matter empirically? Brown and colleagues analysed a full year of call-by-call data from an Israeli bank — more than 1,200,000 calls, of which roughly 450,000 requested an agent. About 80% were served and about 20% abandoned before being served Brown et al. 2005. Mean service time was 201 seconds with a standard deviation of 248; mean patience was 446 seconds.

A model that predicts zero abandonment will systematically tell you that you are fine when a fifth of your demand is walking away.

Turning β into money

Borst, Mandelbaum and Reiman closed the loop by pricing the service-grade parameter. If $c$ is the hourly cost of an agent and $a$ the hourly cost of a waiting customer, the optimal staffing takes the form

and across their tested parameter range this is exactly optimal or within one agent Borst et al. 2004.

For a service firm, $a$ is not abstract. It is the contribution margin of a deal, multiplied by the rate at which waiting destroys the chance of winning it — which is the five-minute decay from the earlier post in this series.

What routing looks like when it is built

The mathematics converts into four concrete design decisions:

1. Compute $R$ honestly, using real handling time including follow-up, not the optimistic version.
2. Staff with explicit safety capacity, accepting that a small team must look underloaded to be responsive.
3. Assign deterministically — a named person per segment and hour, with a fallback — rather than by whoever notices.
4. Model abandonment, because your service level should be measured against inquiries that arrived, not inquiries that waited.

None of this requires software you do not have. It requires knowing which queue you are in.

References

1. Little, J. D. C. (1961). A proof for the queuing formula: L = λW. Operations Research, 9(3), 383–387. https://doi.org/10.1287/opre.9.3.383
2. Little, J. D. C. (2011). OR Forum — Little's Law as viewed on its 50th anniversary. Operations Research, 59(3), 536–549. https://doi.org/10.1287/opre.1110.0940
3. Gans, N., Koole, G., & Mandelbaum, A. (2003). Telephone call centers: Tutorial, review, and research prospects. Manufacturing & Service Operations Management, 5(2), 79–141. https://doi.org/10.1287/msom.5.2.79.16071
4. Halfin, S., & Whitt, W. (1981). Heavy-traffic limits for queues with many exponential servers. Operations Research, 29(3), 567–588. https://doi.org/10.1287/opre.29.3.567
5. Whitt, W. (1992). Understanding the efficiency of multi-server service systems. Management Science, 38(5), 708–723. https://doi.org/10.1287/mnsc.38.5.708
6. Garnett, O., Mandelbaum, A., & Reiman, M. I. (2002). Designing a call center with impatient customers. Manufacturing & Service Operations Management, 4(3), 208–227. https://doi.org/10.1287/msom.4.3.208.7753
7. Borst, S., Mandelbaum, A., & Reiman, M. I. (2004). Dimensioning large call centers. Operations Research, 52(1), 17–34. https://doi.org/10.1287/opre.1030.0081
8. Brown, L. D., Gans, N., Mandelbaum, A., Sakov, A., Shen, H., Zeltyn, S., & Zhao, L. (2005). Statistical analysis of a telephone call center: A queueing-science perspective. Journal of the American Statistical Association, 100(469), 36–50. https://doi.org/10.1198/016214504000001808
9. Zeltyn, S., & Mandelbaum, A. (2005). Call centers with impatient customers: Many-server asymptotics of the M/M/n + G queue. Queueing Systems, 51(3–4), 361–402. https://doi.org/10.1007/s11134-005-3699-8
10. van Leeuwaarden, J. S. H., Mathijsen, B. W. J., & Zwart, B. (2019). Economies-of-scale in many-server queueing systems: Tutorial and partial review of the QED Halfin–Whitt heavy-traffic regime. SIAM Review, 61(3), 403–440. https://doi.org/10.1137/17M1133944