15.1 PSAP Staffing Models: Erlang C & NENA Staffing Calculator

15.1 PSAP Staffing Models: Erlang C & NENA Staffing Calculator

Quick Answer: Modern emergency communications center staffing must be driven by empirical data using approaches such as NENA's PSAP Staffing Guidelines Report (NENA-REF-001-2003) and staffing worksheet, rather than obsolete citizen-to-telecommunicator ratios. Determining authorized agency headcount requires a three-step mathematical process: first, calculating required seated console positions per hour using Erlang C queuing models based on call arrival rates, Average Handle Time (AHT = talk time + hold time + wrap-up time), CAD event volume, and peak radio channel occupancy; second, computing Net Available Work Hours (NAWH) by subtracting shrinkage (paid leave, training, FMLA, administrative time: typically 20%–35%) from gross contractual hours (2,080 hours); and third, deriving the Shift Relief Factor (SRF = 8,760 ÷ NAWH). Multiplying seated positions by the SRF (typically 4.8 to 5.6) establishes the true Full-Time Equivalent (FTE) requirement to maintain 24/7/365 operations without chronic understaffing or catastrophic overtime spirals.

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1. NENA Staffing Guidance & Multi-Dimensional Workload Analysis

Historically, public safety answering points (PSAPs) relied on arbitrary staffing metrics inherited from law enforcement patrol formulas, such as "one dispatcher per 10,000 residents." Modern 9-1-1 governance rejects this approach. NENA's PSAP Staffing Guidelines Report (NENA-REF-001-2003) and its staffing worksheet steer agencies toward comprehensive, data-driven workload analysis that accounts for the non-linear, multi-channel responsibilities of modern telecommunicators.

┌─────────────────────────────────────────────────────────────────────────────┐
│                     COMPREHENSIVE WORKLOAD ANALYSIS INPUTS                  │
├──────────────────────────┬──────────────────────────┬───────────────────────┤
│    TELEPHONY WORKLOAD    │       CAD WORKLOAD       │    RADIO / LMR LOAD   │
├──────────────────────────┼──────────────────────────┼───────────────────────┤
│ • Inbound 9-1-1 & Ten-Dgt│ • Dispatched events      │ • Push-to-talk (PTT)  │
│ • Outbound callbacks     │ • Self-initiated events  │ • Airtime duration    │
│ • Call Arrival Dist.     │ • NCIC / CJIS queries    │ • Channel occupancy   │
│ • Average Handle Time    │ • Unit status updates    │ • Secondary patching  │
│   (Talk + ACW + Hold)    │ • Mutual aid coordination│ • Tactical talkgroups │
└──────────────────────────┴──────────────────────────┴───────────────────────┘

Call Arrival Distributions & Diurnal Patterns

Emergency calls do not arrive in a uniform, predictable stream. They follow a Poisson arrival distribution, exhibiting pronounced diurnal (hourly), day-of-week, and seasonal peaks:

• Hourly Diurnal Curves: Across North American emergency communications centers, call volume drops to its nadir between 0300 and 0600 hours, begins climbing steadily at 0700, and reaches sustained peak plateaus between 1400 and 2100 hours.
• Day-of-Week Fluctuations: Weekend evenings (Friday and Saturday nights) typically exhibit sharp increases in high-acuity law enforcement and emergency medical calls compared to mid-week periods.
• Seasonal and Meteorological Spikes: Summer months correlate with increased outdoor activity and traffic volume, while severe weather events (hurricanes, blizzards, severe convective storms) generate immediate exponential surges in call volume.

Sizing to Peak vs. Average Traffic

Sizing a communications center based on average 24-hour call volume rather than 85th or 90th percentile arrival distributions guarantees systemic failure. Because 9-1-1 volume peaks sharply during afternoon hours and weekend periods, staffing to an "average" level produces surplus capacity at 0400 hours and catastrophic queue delays, abandoned calls, and dropped 10-second service levels during afternoon rush hours and severe weather incidents.

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2. Average Handle Time (AHT) Formulation

Staffing models must never rely solely on total call counts; they must account for the duration of time a telecommunicator is actively committed to processing each call transaction. Average Handle Time (AHT) represents the complete operational duration of a call event:

$\text{AHT} = \text{Average Talk Time (ATT)} + \text{Hold Time} + \text{After-Call Work (ACW)}$

                     AVERAGE HANDLE TIME (AHT) STRUCTURE
┌───────────────────────┬───────────────────────┬─────────────────────────────┐
│   Average Talk Time   │       Hold Time       │    After-Call Work (ACW)    │
│         (ATT)         │   (Transfer / Queue)  │          (Wrap-Up)          │
├───────────────────────┼───────────────────────┼─────────────────────────────┤
│ • Caller interrogation│ • Secondary agency    │ • CAD narrative entry       │
│ • Address verification│   bridging (EMS/Fire) │ • Unit status updates       │
│ • Calming techniques  │ • Language line       │ • Geocode rebidding         │
│ • Pre-arrival EMD/EFD │   conference delays   │ • External agency callbacks │
└───────────────────────┴───────────────────────┴─────────────────────────────┘

• Average Talk Time (ATT): The active voice, text, or video duration between the telecommunicator answering the call and call disconnect.
• Hold / Transfer Time: The time a caller is held while bridging to secondary agencies (e.g., secondary EMS or fire dispatch) or waiting for third-party interpretation services.
• After-Call Work (ACW / Wrap-Up): The non-voice time immediately following disconnect during which the call-taker enters supplementary notes into CAD, re-bids location data, notifies external utility companies, or initiates administrative notifications. During ACW, the call-taker is unavailable to answer incoming calls, making ACW a direct operational driver of queue depth.

Traffic Intensity in Erlangs

Workload in telecommunications traffic engineering is expressed in dimensionless units called Erlangs, representing continuous hours of call traffic within a given evaluation window (typically one hour):

$A = \lambda \times s$

Where:

• $A$ = Traffic intensity (in Erlangs)
• $\lambda$ = Call arrival rate (calls per hour)
• $s$ = Average Handle Time (AHT, expressed as a fraction of an hour; e.g., 150 seconds = $150 / 3,600 = 0.0417$ hours)

Example: If a PSAP receives 120 calls during its peak hour with an average handle time of 150 seconds (2.5 minutes, or 0.0417 hours):

$A = 120 \times 0.0417 = 5.0 \text{ Erlangs}$

This means there are exactly 5.0 solid hours of continuous call-handling work generated in that 60-minute window. To prevent infinite queue growth, the center must deploy strictly more than 5 seated positions.

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3. Telephony Queuing Theory: Erlang B vs. Erlang C

To translate raw workload volume into required operational console positions per hour, public safety relies on queue modeling formulated by Danish mathematician A.K. Erlang.

                             ERLANG C QUEUING MODEL
Incoming Calls (λ) ──────► ┌──────────────────────┐ ──────► Answered Immediately
(Poisson Arrival)          │  Operational Queue   │         (Within 10 Seconds)
                           │ (Calls Wait in Line) │
                           └──────────┬───────────┘
                                      │ (If all c positions busy)
                                      ▼
                           ┌──────────────────────┐
                           │   c Seated Consoles  │
                           │ (Service Rate: μ=1/s)│
                           └──────────────────────┘

Erlang B vs. Erlang C Comparison

The Erlang C Delay Formula

The Erlang C formula calculates the probability that an incoming call will encounter all $c$ agents busy and be forced to wait in the queue, designated as $P(W > 0)$ or $C(c, A)$:

$P(W > 0) = \frac{\frac{A^c}{c!} \frac{c}{c - A}}{\sum_{k=0}^{c-1} \frac{A^k}{k!} + \frac{A^c}{c!} \frac{c}{c - A}}$

Where $A$ is traffic intensity in Erlangs and $c$ is the number of seated agents ($c > A$). Once $P(W > 0)$ is derived, the probability that a caller waits longer than a designated threshold time $t$ (such as 15 or 20 seconds) is calculated via exponential decay:

$P(W > t) = P(W > 0) \cdot e^{-(c - A) \frac{t}{s}}$

Answering Targets Used in Staffing Models

NFPA 1225 and NENA-STA-020 set the answering targets that staffing models test against:

• 90% of 9-1-1 calls answered within 15 seconds ($P(W > 15\text{s}) \le 0.10$).
• 95% of 9-1-1 calls answered within 20 seconds ($P(W > 20\text{s}) \le 0.05$).

Worked example. A center receives 120 calls in its peak hour with an average handle time of 150 seconds. Traffic intensity is $A = \lambda \cdot s = (120 \div 3600) \times 150 = 5.0$ Erlangs, with $s = 150\text{ s}$.

Nine seated positions is the first staffing level that meets both the 90%/15-second and 95%/20-second targets. Because Erlang C ignores abandonment, multitasking, and radio workload, planners treat the result as a floor and then add radio positions and minimum staffing rules.

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4. Assumptions & Limitations of Erlang C in Public Safety

While Erlang C is the universal engine behind workforce management software, candidates must understand its core mathematical assumptions and operational limitations:

1. Assumption of Infinite Queue Patience (No Abandonment): Erlang C assumes callers will hold indefinitely until an agent answers. In reality, panicked citizens hang up and repeatedly redial after 15 to 30 seconds of ringing, generating multiple phantom call records that distort queuing metrics.
2. Assumption of Dedicated Single-Task Agents: Erlang C assumes agents do nothing but answer inbound telephone calls. While this holds true for call-taking specialist pools in large metropolitan agencies, it fails completely in small-to-medium centers.
3. Multi-Tasking in Consolidated Centers: In centers where telecommunicators simultaneously answer 9-1-1 calls, monitor 10-digit administrative lines, process CAD calls, and dispatch radio channels, telephony represents only a fraction of their cognitive load.
4. Radio Channel Occupancy Limits: A common planning rule of thumb (not a codified APCO or NENA standard) is to keep peak radio channel airtime occupancy below roughly 50% to 60%. As occupancy climbs past that range, field responders encounter severe transmission queue delays and the probability of simultaneous transmissions ("clobbering" or heterodyne squeal) rises exponentially, endangering officers and firefighters in peril.
5. Small Center Distortion: In a center with 2 or 3 console positions, mathematical Erlang C might indicate that 1.8 agents can handle the call volume. However, if a single high-acuity incident occurs (e.g., active vehicle pursuit or pediatric CPR), one telecommunicator is entirely monopolized by radio traffic while the second is committed to the CPR caller. A third incoming call will ring unanswered. Small centers must therefore establish minimum staffing safety floors based on operational roles rather than pure telephony Erlang equations.

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5. Net Available Work Hours (NAWH) & Shrinkage Computation

Once the required seated console positions per hour are established across the 24-hour diurnal cycle, managers must calculate how many actual human beings are required to fill those seats. An employee hired for a 40-hour workweek does not provide 40 hours of seated console time.

                   GROSS CONTRACTUAL HOURS (2,080 HRS)
                                    │
         ┌──────────────────────────┴──────────────────────────┐
         ▼                                                     ▼
NET AVAILABLE WORK HOURS (NAWH)                            SHRINKAGE
• Active Call-Taking Console                              • Vacation / Annual Leave
• Active Radio Dispatch Console                           • Sick Leave & Medical
• Tactical Incident Monitoring                            • Mandatory Training / CE
                                                          • Statutory Paid Holidays
                                                          • FMLA / Military Leave
                                                          • QA/QI Reviews & Briefings

Defining Shrinkage

Shrinkage represents the total proportion of paid time during which an employee is unavailable to sit at an operational console to answer calls or dispatch units. Shrinkage is divided into external shrinkage (paid time off) and internal shrinkage (on-duty non-operational time). In public safety communications, shrinkage typically ranges between 20% and 35% (averaging 23% to 28%).

Mathematical Formulation of NAWH

Gross contractual hours for a standard full-time employee (FTE) working 40 hours per week across 52 weeks equal 2,080 hours per year ($52 \times 40 = 2,080$).

$\text{NAWH} = \text{Gross Annual Hours (2,080)} - \sum \text{Shrinkage Hours}$

Baseline Calculation: $\text{NAWH} = 2,080 \text{ Gross Hours} - 480 \text{ Shrinkage Hours} = 1,600 \text{ Hours}$

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6. Shift Relief Factor (SRF) & Total Line Headcount Sizing

To continuously staff a single console position 24 hours a day, 7 days a week, 365 days a year, the center must provide 8,760 total operational coverage hours ($365 \times 24 = 8,760$; or 8,784 in a leap year: $366 \times 24$).

The Shift Relief Factor (SRF)—also known as the Coverage Factor—defines how many budgeted FTEs must be employed to guarantee that exactly one seated position is occupied around the clock without scheduled overtime.

The SRF Formula

$\text{SRF} = \frac{\text{Annual Operational Hours for One 24/7 Seat (8,760)}}{\text{Net Available Work Hours (NAWH)}}$

Using our baseline NAWH of 1,600 hours:

$\text{SRF} = \frac{8,760}{1,600} = 5.475 \approx 5.48 \text{ FTEs per Seat}$

• If an agency experiences high leave usage resulting in NAWH of 1,500 hours: $\text{SRF} = 8,760 / 1,500 = 5.84$ FTEs per seat.
• If an agency operates with minimal shrinkage resulting in NAWH of 1,720 hours: $\text{SRF} = 8,760 / 1,720 = 5.09$ FTEs per seat.

Sizing Total Authorized Agency Headcount

To determine total authorized line-telecommunicator staffing, management multiplies the minimum continuous seated console positions determined by Erlang C and radio workload modeling by the agency's SRF:

$\text{Total Required Line FTEs} = \text{Minimum Seated Positions} \times \text{SRF}$

Operational Scenario: A consolidated 9-1-1 center determines that to meet NENA call-answering benchmarks and handle law/fire dispatch channels, it requires a continuous minimum of 6 operational consoles active 24/7/365. The agency's NAWH is calculated at 1,620 hours (SRF = $8,760 / 1,620 = 5.41$).

$\text{Required Line FTEs} = 6 \times 5.41 = 32.46 \approx 33 \text{ Authorized FTEs}$

Note on Supervisory & Administrative Overhead: This calculation accounts strictly for line-level console coverage. Operational supervisors, quality assurance specialists, training coordinators, IT/GIS technicians, and executive leadership must be budgeted in addition to line FTEs.

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7. Operational Traps & The Overtime "Death Spiral"

                     THE OVERTIME DEATH SPIRAL
┌──────────────────────────────────────────────────────────────────┐
│                     Understaffed Core Baseline                   │
│                 (Budgeted without full shrinkage)                │
└─────────────────────────────────┬────────────────────────────────┘
                                  ▼
┌──────────────────────────────────────────────────────────────────┐
│             Escalating Mandatory Overtime (Holdovers)            │
└─────────────────────────────────┬────────────────────────────────┘
                                  ▼
┌──────────────────────────────────────────────────────────────────┐
│            Chronic Sleep Debt, Fatigue, Severe Burnout           │
└─────────────────────────────────┬────────────────────────────────┘
                                  ▼
┌──────────────────────────────────────────────────────────────────┐
│           Spike in Unplanned Sick Leave & Medical Leaves         │
└─────────────────────────────────┬────────────────────────────────┘
                                  ▼
┌──────────────────────────────────────────────────────────────────┐
│             Accelerated Resignations & Early Retirement          │
└─────────────────────────────────┬────────────────────────────────┘
                                  ▼
┌──────────────────────────────────────────────────────────────────┐
│          Deepening Deficit (Back to Mandatory Overtime)          │
└──────────────────────────────────────────────────────────────────┘

The "Budget FTE" Fallacy

A pervasive administrative trap occurs when local government budget officials compute coverage by dividing annual seat hours (8,760) by gross contractual hours (2,080), yielding an artificial ratio of 4.21 FTEs per seat ($8,760 / 2,080 = 4.21$). Budgeting 4.21 FTEs per seat completely ignores shrinkage, creating a built-in structural deficit of 20% to 25%. In a 6-seat center, budgeting at 4.21 yields 25 positions instead of the required 33—guaranteeing 16,640 hours of uncovered console time that must be forced onto staff via mandatory overtime.

The Training Pipeline Lag

Telecommunicator vacancies cannot be filled immediately. The hiring and training pipeline spans 9 to 18 months (recruitment, background investigation, polygraph/psychological screening: 3–6 months; classroom academy: 2–3 months; 1-on-1 CTO floor training: 4–8 months). Training attrition is also common, and many centers lose a significant share of each trainee class. If an agency only hires after an employee departs, it operates in permanent operational deficit.

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8. ENP Exam Watch

• Erlang B vs. Erlang C: Erlang B sizes physical network trunks and circuits (blocked calls cleared); Erlang C sizes call-taking personnel (blocked calls queued).
• AHT Calculation: AHT must always include After-Call Work (ACW/wrap-up) and hold time, never talk time alone.
• Budget Fallacy (4.21): Dividing 8,760 by 2,080 yields 4.21 FTEs per seat, which falsely assumes zero shrinkage. True operational public safety SRF values range from 4.8 to 5.6.
• Radio Channel Rule of Thumb: Planners commonly keep peak LMR channel occupancy below about 50% to 60% to avoid clobbering and transmission delays; it is a planning guideline, not a national standard.