The Short Answer
An erlang is a unit of traffic. One erlang is one channel kept fully busy for the whole measurement period, which is almost always the busy hour. It is a ratio, so it has no units of its own.
The traffic a group of users offers is easy to work out. Multiply the number of calls in the busy hour by the average length of a call in hours:
Traffic (erlangs) = calls per hour × average call length in hours
Thirty calls an hour at two minutes each is 30 × (2 ÷ 60) = 1 erlang, which is the same load as one channel that never goes idle.
Knowing the offered traffic is only half the job. The other half is deciding how often a user is allowed to find every channel busy, which is the grade of service. Erlang B takes the offered traffic and the grade of service and returns the number of channels you need. Get those two inputs right and the channel count, and from it the repeater count, follows directly rather than from a rule of thumb.
What Is an Erlang?
The erlang is named after Agner Krarup Erlang, the Danish engineer who worked out how to size telephone exchanges in the early twentieth century. The same maths sizes a modern trunked radio system, a dispatch console or a call centre, because the underlying question never changes. It is how many shared channels a pool of users needs so that calls rarely block.
One erlang means one circuit occupied for one hour, or two circuits each busy for half an hour, or any other combination that adds up to one channel hour of use. Because it is a ratio of busy time to available time, a single channel can never carry more than one erlang. A group of ten channels can carry up to ten erlangs in theory, though in practice it carries far less if you want blocking to stay low.
Offered Traffic, Carried Traffic and the Busy Hour
Three terms do most of the work in traffic engineering, and it is worth keeping them separate.
- Offered traffic is the demand the users generate, whether or not the system can serve it.
- Carried traffic is the demand the system actually handles. It is always a little less than the offered traffic, because some calls arrive when every channel is busy and are turned away.
- The busy hour is the single busiest hour of a normal working day. You size for this, not for the daily average, because the average hides the peak that actually breaks the system.
The gap between offered and carried traffic is the whole point of the exercise. A well sized system keeps that gap small, so almost every call offered is a call carried.
Grade of Service: How Much Blocking You Accept
Grade of service, usually shortened to GoS, is the probability that a call arrives to find every channel busy. A GoS of 0.02, or two per cent, means two calls in a hundred block during the busy hour. The rest connect straight away.
There is no single correct figure. A commercial dispatch fleet might design to two per cent, while a public safety or mine emergency network might insist on one per cent or better, because a blocked call there is a safety event, not an inconvenience. The tighter the grade of service, the more channels the same traffic demands, so GoS is a genuine cost decision rather than a formality.
Erlang B, Erlang C and Engset: Which Model to Use
The right formula depends on what happens to a call that cannot be served immediately.
- Erlang B assumes a blocked call is cleared. The caller gets a busy signal and the attempt is gone. This is the correct model for most conventional and trunked radio systems, where a user who finds no free channel simply keys up again a moment later. It is the default for channel and repeater sizing.
- Erlang C assumes a blocked call is queued and waits for the next free channel. This suits dispatch consoles, contact centres and any system with a call queue. It answers a different question, which is how long callers wait, and it always asks for more channels than Erlang B for the same traffic.
- Engset is a refinement of Erlang B for small user populations. When the number of users is less than about ten times the number of channels, Erlang B quietly overstates the traffic and Engset gives the more accurate, and usually smaller, channel count.
Choosing the wrong model is a common and expensive mistake. Sizing a cleared radio system with Erlang C buys channels you never needed, while sizing a queued console with Erlang B leaves callers waiting far longer than the design intended.
For a system with N channels carrying A erlangs of offered traffic, the blocking probability B is:
B(N, A) = (Aᴺ ÷ N!) ÷ Σ (Aᵏ ÷ k!) for k = 0 to N
Erlang B rests on a few assumptions worth stating. It treats call arrivals as independent and random, and holding times as exponentially distributed, assumptions that have been shown to model radio dispatch traffic well across the busy hour. It also assumes a blocked call is cleared rather than queued, which is exactly how a conventional or trunked radio user behaves when every channel is busy.
You rarely evaluate this by hand. In practice you invert it, fixing the grade of service and solving for the smallest N that meets it. The table below shows the maximum offered traffic each channel count can carry at a two per cent grade of service, which is the everyday working version of the formula.
| Channels (N) | Max traffic at 2% GoS (erlangs) | Traffic per channel |
|---|
| 1 | 0.02 | 0.02 |
| 2 | 0.22 | 0.11 |
| 4 | 1.09 | 0.27 |
| 8 | 3.63 | 0.45 |
| 12 | 6.61 | 0.55 |
| 16 | 9.83 | 0.61 |
| 20 | 13.18 | 0.66 |
Worked Example: Sizing a Mine Radio Fleet
A surface mine runs 120 field radios on a shared P25 system. In the busy hour, which lines up with shift change, each user makes about six short press to talk calls averaging 18 seconds.
First find the traffic each user offers:
6 × (18 ÷ 3600) = 0.03 erlangs per user
Then the offered traffic for the whole fleet:
120 × 0.03 = 3.6 erlangs
Reading down the table to a two per cent grade of service, seven channels carry 2.94 erlangs and eight channels carry 3.63 erlangs. The fleet needs eight channels. Seven would push blocking well past the target at the busy hour, and the shift change is exactly when a blocked call matters most.
The last step is to turn channels into hardware, which depends on the technology. P25 Phase II carries two voice paths per repeater, so eight channels means four repeaters. A quick sanity check confirms the model choice: 120 users across eight channels is a ratio of 15 to 1, comfortably above the threshold of 10, so Erlang B is the right tool and Engset is not needed here.
From Channels to Repeaters: Voice Paths per Repeater
A channel count is not a repeater count. How many repeaters you buy depends on how many voice paths each repeater carries, which is set by the air interface.
| Technology | Voice paths per repeater |
|---|
| Analog FM | 1 |
| P25 Phase I | 1 |
| DMR Tier II | 2 |
| DMR Tier III | 2 |
| P25 Phase II | 2 |
| TETRA | 4 |
The eight channel fleet above therefore needs four P25 Phase II repeaters, or eight analog repeaters, for the same traffic. One caveat catches people out. In a simulcast system, several repeaters on the same frequencies transmit together and behave as a single RF site from the user’s perspective, so the hardware count and the channel count are no longer a simple one to one or two to one relationship.
Trunking Efficiency: Why Bigger Systems Carry More per Channel
Look again at the traffic per channel column. One channel carries 0.02 erlangs at two per cent blocking, but each of twenty channels carries 0.66 erlangs at the same grade of service. The larger pool is more than thirty times as efficient per channel. The picture below shows why.
Four separate 2-channel systems One shared 8-channel pool
[A] ██ [B] ██ ████████
[C] ██ [D] ██ any user, any free channel
A call on system A cannot use A call takes any free
a free channel on system B. channel in the pool.
More channels needed for the Same traffic carried
same grade of service. on fewer channels.
This is trunking efficiency, and it is one of the most useful ideas in capacity planning. When many users share a large pool of channels, the chance that all of them are busy at once falls sharply, so each channel spends more of its time carrying traffic. It is why combining several small radio groups onto one trunked system usually needs fewer channels in total than the separate groups did, and why doubling the users on a system rarely doubles the channels required.
Design for the Worst Busy Hour, Not the Average Day
The single biggest source of undersized networks is averaging away the peak. A site might sit near idle for most of the day and then surge at shift change, during an incident or across a scheduled blast window. If you size for the daily mean, the network blocks exactly when it is needed.
The safe method is to define the realistic worst case busy hour, with its own user count, call rate and hold time, and size to that. It is worth modelling two or three named scenarios, a normal peak and an emergency peak, and letting the worst one set the procurement figure. The radios on the order should be the count that survives the hardest hour, not the count that suits an ordinary afternoon.
The N minus 1 Check
For any network that cannot be allowed to fail, size the traffic and then ask a second question. What happens if one repeater is out of service?
The N minus 1 check recalculates the grade of service with one channel removed. If the eight channel mine fleet above loses a repeater, it drops to six channels carrying the same 3.6 erlangs, and the blocking climbs steeply. If the degraded figure is unacceptable, the honest answer is to add a repeater so the system still meets its target with one unit down. For public safety and critical infrastructure this is not optional, because the failure and the emergency that stresses the network often arrive together.
Common Erlang Mistakes
- Sizing on the daily average. The busy hour is the only hour that matters for capacity, and it can be several times the average load.
- Using Erlang C for a cleared radio system. Queued and blocked systems are different problems. Mixing them either wastes channels or leaves callers waiting.
- Ignoring finite populations. With a small user group, Erlang B overstates the traffic. Engset gives the leaner and more accurate answer.
- Forgetting the technology multiplier. A channel count is not a repeater count. Translate through the paths per repeater for the actual air interface before ordering hardware.
- Skipping the redundancy check. A network that only meets its grade of service with every repeater working has no margin for the day one fails.
Frequently Asked Questions
What is an erlang in simple terms? An erlang is a measure of how busy a channel is. One erlang means one channel is occupied for the entire hour. Traffic in erlangs is the number of calls in the busy hour multiplied by the average call length in hours.
How do you calculate offered traffic? Multiply the number of calls made in the busy hour by the average duration of a call in hours. For example, 40 calls an hour at 30 seconds each is 40 × (30 ÷ 3600) = 0.33 erlangs.
What is a good grade of service? Two per cent blocking is a common commercial target, meaning two calls in a hundred find no free channel. Public safety and mission critical networks often design to one per cent or tighter, because a blocked call there is a safety risk.
What is the difference between Erlang B and Erlang C? Erlang B assumes a blocked call is cleared and the caller tries again, which fits most radio systems. Erlang C assumes a blocked call is queued and waits, which fits dispatch consoles and call centres. Erlang C always asks for more channels than Erlang B for the same traffic.
Why does a bigger system need fewer channels per user? Because of trunking efficiency. When many users share a large channel pool, the chance that every channel is busy at the same moment falls, so each channel carries more traffic at the same grade of service.
How do I turn channels into repeaters? Divide by the voice paths each repeater provides for your technology. P25 Phase I and analog give one path per repeater, while P25 Phase II, DMR Tier II and III give two, and TETRA gives four.
Why Excel Erlang Tables Are No Longer Enough
A printed Erlang B table or a single spreadsheet cell answers one narrow question: how many channels for one traffic figure at one grade of service. Real systems ask more than that at once.
- Multiple busy hour scenarios. A normal peak and an emergency peak can demand very different channel counts, and the worst one should set the order.
- N minus 1 resilience. The design has to still meet its grade of service with one repeater out of service, which a static table never checks.
- Finite user populations. Small fleets need Engset, not Erlang B, or the channel count comes out too high.
- Mixed technologies. Turning channels into repeaters differs for analog, P25 Phase II, DMR and TETRA, and many sites run more than one.
- Future growth. The plan should hold a margin for the users the network will carry in a few years, not just today.
- Automatic frequency assignment. Once the channels are counted they still have to be placed so their intermodulation products miss every receiver on the site.
Each of these is a separate calculation in a spreadsheet, and keeping them consistent by hand is where errors creep in. That is why noIM₃ runs these calculations together and keeps them in step as the inputs change.
Build it in noIM₃
The Erlang B calculator sizes channels and repeaters for blocking radio systems, cross checks the result against Engset and Extended Erlang B, and runs the N minus 1 and busy hour scenario comparisons for you. When callers queue rather than clear, the Erlang C calculator answers the waiting time question instead. Once the channel count is set, the frequency plan optimiser and frequency coordination tool help place those channels cleanly on air.
Key Takeaway
An erlang is one channel kept busy for the hour, and traffic in erlangs is just calls per hour times the average call length. Feed the busy hour traffic and a grade of service into Erlang B and it returns the channels you need, which you then translate into repeaters through the paths per repeater for your technology. Size for the worst busy hour rather than the average day, use Engset for small fleets and Erlang C for queued systems, and always run the N minus 1 check on a network that cannot be allowed to fail. Do that and the capacity is a calculation, not a guess that turns into complaints a few months after go live.
References and Further Reading
The methods and figures in this article draw on the standard teletraffic and land mobile radio literature. These are pointers to the primary sources rather than a claim that every number here is quoted verbatim, and the worked examples are our own. Always check the current, in force version of any standard before relying on it in a design.
- A. K. Erlang, original work on the probability of loss in telephone traffic, first published in 1917 and 1918, which introduced the loss formula that carries his name.
- ITU-T E.500 series Recommendations on teletraffic engineering, including E.501 on the estimation of offered traffic, together with the related guidance on grade of service and circuit dimensioning.
- V. B. Iversen, Teletraffic Engineering and Network Planning, Technical University of Denmark, a widely used and freely available modern textbook covering Erlang B, Erlang C and Engset.
- TIA TSB-88, Wireless Communications Systems Performance in Noise and Interference Limited Situations, for land mobile and trunked radio channel performance and loading.
- TIA-102 (Project 25) standards suite, which defines the P25 Phase I and Phase II air interfaces behind the voice paths per repeater figures.
- ETSI TS 102 361 (DMR) and ETSI EN 300 392 (TETRA), which define the timeslot structures behind the DMR and TETRA capacity figures.
- Motorola Solutions system capacity guidance and APCO International channel loading guidance, for practical vendor and public safety grade of service planning.
