Erlang C Staffing Calculator

1 The Problem

Every contact centre faces the same planning question: given a forecast of incoming calls, how many agents must be scheduled so that, say, 80% of callers are answered within 20 seconds? Understaffing degrades service quality; overstaffing increases labour costs. The classical answer comes from queueing theory, in the form of the Erlang C model, and this project implements it end to end as an interactive Python calculator: from raw inputs (contact volume, handle time, shift shrinkage) to a staffing recommendation, with plots and a Monte Carlo analysis of the sensitivity of that recommendation to uncertain demand.

2 The Queueing Model

2.1 Offered Load

The workload is summarised by the offered load (traffic intensity), the dimensionless product of the arrival rate $\lambda$ and the average handle time. With $\mu = 1/\text{AHT}$ the service rate,

A=\frac{\lambda}{\mu}.

Intuitively, $A$ is the number of agents that would be busy at all times if calls arrived perfectly smoothly.

Erlang C models the centre as an $M/M/c$ queue, which rests on five assumptions: Poisson arrivals, exponentially distributed (memoryless) call durations, $c$ identical agents, an infinite queue with no abandonment, and first-come-first-served ordering. These assumptions make the model tractable at a cost: real centres see time-varying arrival rates, call durations closer to log-normal, and callers who hang up. These departures motivate the Monte Carlo layer in Section 5.

2.2 The Erlang C Probability of Waiting

For $c$ agents serving Poisson arrivals with exponential service times (an $M/M/c$ queue), the probability that an arriving call finds every agent busy and must wait is the Erlang C formula:

P_{\text{wait}}= \frac{\dfrac{A^{c}}{c!}\,\dfrac{c}{c-A}} {\displaystyle\sum_{n=0}^{c-1}\frac{A^{n}}{n!} +\frac{A^{c}}{c!}\,\frac{c}{c-A}}, \qquad A<c.

2.3 Service Level and Abandonment

Given $P_{\text{wait}}$ , the probability that a call is answered within a target time $t$ follows from the exponential distribution of waiting times:

\text{SL}(c,t)=1-P_{\text{wait}}\, e^{-(c-A)\,\mu\,t}.

The same structure yields a first-order estimate of abandonment: replacing the target answer time with the callers' average patience $\tau$ gives the fraction of calls still waiting when patience runs out,

P_{\text{abandon}}\approx P_{\text{wait}}\, e^{-(c-A)\,\mu\,\tau}.

2.4 Occupancy and Waiting Time

Two further indicators complete the planner's dashboard: the occupancy $\rho=A/c$ , the fraction of time agents are busy (desirable when high, but unsustainable when too high), and the mean wait of a queued call,

W_q=\frac{P_{\text{wait}}}{c\mu-\lambda}, \qquad c\mu>\lambda,

which links the same quantities directly to the customer's experience.

3 From Model to Staffing Decision

The theoretical model only speaks about agents actually on the phones. The calculator adds the operational layer that planners live with:

Shrinkage. Breaks, meetings, and training make a fraction $s$ of paid time unavailable. The tool either computes $s$ from the shift breakdown or accepts it directly, and converts scheduled staff $c$ into effective agents $c_{\text{eff}}=\lfloor c\,(1-s)\rfloor$ .
Occupancy cap. Sustained utilisation above ~85–90% burns agents out, so solutions with $A/c_{\text{eff}}$ above a configurable maximum are rejected.
The search. The calculator then finds the smallest scheduled headcount whose effective agents meet the required service level while respecting the occupancy cap, and reports staff, occupancy, and the estimated abandon rate together, alongside a plot of service level versus agents staffed against the target line.

4 Numerical Stability

Evaluating the Erlang C formula naively overflows quickly, since $A^{c}$ and $c!$ both exceed the floating-point range for realistic call volumes. The implementation works entirely in log-space: factorials via the log-gamma function, $\log\Gamma(n+1)$ , and sums via the numerically stable log-sum-exp identity

\log\sum_i e^{x_i}=m+\log\sum_i e^{x_i-m}, \qquad m=\max_i x_i,

so the calculator remains accurate for large contact centres where a direct translation of the textbook formula would fail.

5 Monte Carlo Uncertainty Analysis

A point forecast of demand is always wrong in practice, so the second half of the tool asks a sharper question: how does the staffing plan behave when the inputs fluctuate? Call volume and handle time are drawn from normal distributions around their forecast means, and the full pipeline (offered load, Erlang C, service level) is re-evaluated across thousands of simulated scenarios. The output is a distribution of achieved service levels for the recommended headcount, visualised as a histogram, rather than a single number. A plan that looks fine deterministically can reveal a heavy tail of poor-service scenarios under modest demand variability, which is information that is needed before a schedule is fixed.

The two approaches also delimit each other's usefulness: for small centres (fewer than about 20 agents) the deterministic model and the simulation give nearly identical recommendations, while for large centres (beyond roughly 60 agents) the Monte Carlo approach becomes preferable for sizing the workforce, as variability compounds with scale.

6 Implementation

The tool is written in Python, with NumPy for the vectorised simulations and Matplotlib for the plots. The calculator runs as an interactive command-line session: it prompts for the shift structure (or a known shrinkage percentage), the demand forecast (e.g. 400 contacts per 30-minute period, 257 s average handle time), the service target (e.g. 80% in 20 s), the occupancy cap, and caller patience, then prints the deterministic staffing recommendation and runs the Monte Carlo study around it.