Why do airlines overbook flights? Why do banks not keep cash equal to all deposits? Why do cellular networks sell more capacity than they have?

The answer in every case is the same fundamental insight: most resources are never used at full capacity simultaneously. By carefully analyzing usage patterns and accepting small probabilities of congestion, shared resources can serve far more users than dedicated alternatives at dramatically lower cost per user.

This principle—statistical multiplexing gain—is the quantitative foundation of modern communications. Understanding it explains why Internet access costs dollars per month instead of thousands, why mobile phones work despite limited spectrum, and why cloud computing is economically viable.

This page develops the mathematics and engineering behind efficiency gains, giving you tools to analyze and optimize shared communication systems.

Utilization is the fundamental efficiency metric: the fraction of channel capacity actually carrying useful data. Let's compare dedicated and shared approaches.

Dedicated Channel Utilization

In a dedicated channel system, each user receives exclusive capacity whether they're using it or not. Consider voice calls:

• A dedicated 64 kbps voice channel is allocated for an entire call duration
• Active speech occupies only ~35% of call time (due to silence, listening, pauses)
• Effective utilization: ~35%
• The remaining ~65% is paid for but carries no data

For bursty data applications, the situation is worse:

• A dedicated 100 Mbps Ethernet port to a workstation
• The user actively transfers data perhaps 5 minutes per hour
• Effective utilization: ~8%
• 92% of capacity sits permanently idle

Shared Channel Utilization

When multiple sources share a channel, their independent utilization patterns combine favorably:

• Individual sources may be 10% active, but different ones are active at different times
• Aggregate utilization can reach 70-80% while each individual can burst to full speed when needed
• The key insight: unused capacity from idle sources is immediately available to active sources

The Mathematics of Aggregation

Let's formalize why aggregation improves utilization. Consider n independent sources, each with:

• Peak rate: R (e.g., 10 Mbps)
• Average rate: a × R, where a is the activity factor (e.g., 0.1 × 10 Mbps = 1 Mbps)

Dedicated System:

• Total capacity required: n × R
• Total average utilization: n × a × R
• Efficiency: a (e.g., 10%)

Shared System:

• Aggregate average load: n × a × R
• Required capacity: depends on acceptable congestion probability
• Efficiency: approaches the target utilization limit

The shared system doesn't need capacity for all peaks simultaneously—only enough for the statistically likely maximum. As n grows large, this maximum becomes increasingly predictable.

Statistical multiplexing gain is the ratio of users a shared system can support compared to a dedicated system with the same total capacity. This gain derives from probability theory and grows with the number of sources.

Derivation Using Central Limit Theorem

Consider n independent, identically distributed sources, each with mean μ and variance σ². The aggregate load X is the sum:

X = X₁ + X₂ + ... + Xₙ

By the Central Limit Theorem, as n grows:

• E[X] = n × μ (mean scales linearly)
• Var[X] = n × σ² (variance scales linearly)
• StdDev[X] = √n × σ (standard deviation scales as square root)

The coefficient of variation (relative variability) is: CV = StdDev[X] / E[X] = (√n × σ) / (n × μ) = σ / (μ × √n)

Crucially, relative variability decreases with √n. With 100 sources, CV is 10× lower than with 1 source. The aggregate becomes increasingly smooth and predictable.

Calculating Required Capacity

To support n sources with shared capacity C, we need the probability of aggregate demand exceeding C to be below some acceptable threshold ε:

P(X > C) < ε

Using the normal approximation:

C = n × μ + z_ε × √n × σ

Where z_ε is the z-score corresponding to probability ε (e.g., z = 2.33 for ε = 1%).

Multiplexing Gain Calculation

The multiplexing gain G is the ratio of dedicated to shared capacity:

• Dedicated capacity (for n users, each needing peak rate R): n × R
• Shared capacity (from above): n × μ + z_ε × √n × σ

G = (n × R) / (n × μ + z_ε × √n × σ)

As n → ∞, the z_ε × √n × σ term becomes negligible compared to n × μ, and:

G → R / μ = 1 / a (the inverse of activity factor)

Example: 1000 Voice Sources

• Peak rate R: 64 kbps
• Mean rate μ: 22.4 kbps (35% activity)
• Standard deviation σ: 25 kbps (assume)
• Acceptable blocking ε: 1% (z = 2.33)

Dedicated: 1000 × 64 = 64,000 kbps = 64 Mbps Shared: 1000 × 22.4 + 2.33 × √1000 × 25 = 22,400 + 1,842 = 24,242 kbps ≈ 24.2 Mbps

Gain: 64 / 24.2 ≈ 2.6×

The shared system needs only 38% of the dedicated system's capacity while achieving 99% service probability.

For circuit-switched systems like traditional telephony, the Erlang model provides precise tools for dimensioning shared resources. Developed by Agnar Erlang in the early 1900s, these formulas remain fundamental to telecommunications engineering.

Traffic Load Measurement: The Erlang

The Erlang is the standard unit of telecommunications traffic intensity:

• 1 Erlang = One circuit occupied continuously
• 0.5 Erlangs = One circuit occupied 50% of the time
• 100 Erlangs = 100 circuits worth of average demand

Traffic A (in Erlangs) = λ × h, where:

• λ = arrival rate (calls per unit time)
• h = average holding time (duration per call)

Example: 600 calls per hour, average duration 3 minutes

• λ = 600 calls/hour = 10 calls/minute
• h = 3 minutes
• A = 10 × 3 = 30 Erlangs

Erlang B Formula: Blocking Probability

For systems where blocked calls are cleared (caller hangs up and may retry later), the Erlang B formula gives the probability of blocking:

P(blocking) = B(A, n) = (Aⁿ/n!) / Σₖ₌₀ⁿ (Aᵏ/k!)

Where:

• A = offered traffic in Erlangs
• n = number of available circuits

This formula assumes:

• Poisson arrivals (random, independent)
• Exponential holding times
• Blocked calls cleared (no waiting)
• Infinite population of potential callers

Erlang C Formula: Waiting Probability

For systems where blocked calls wait (call center queues), the Erlang C formula gives the probability of waiting:

P(waiting > 0) = C(A, n) = [n × B(A, n)] / [n - A × (1 - B(A, n))]

This is used for dimensioning systems with queues, like call centers or packet networks.

Using Erlang Tables for Capacity Planning

Erlang tables (or calculators) determine required circuits for given traffic and blocking targets:

Example: Support 50 Erlangs with ≤2% blocking:

• From Erlang B tables: Need 62 circuits
• With dedicated circuits (one per Erlang): Would need the equivalent of ~143 circuits at peak
• Multiplexing gain: 143/62 ≈ 2.3×

The gain increases with traffic volume. Higher Erlang loads allow proportionally fewer circuits per Erlang.

Packet switching takes statistical multiplexing further than circuit switching by sharing capacity at the packet level rather than the call level. This enables dramatically higher efficiency for bursty traffic.

Circuit vs. Packet Efficiency

In circuit switching:

• Resources allocated for entire call duration
• Silence and pauses waste allocated capacity
• Efficiency limited by activity factor (~35% for voice)

In packet switching:

• Resources allocated only when packets are ready
• No capacity consumed during silence
• Efficiency limited only by aggregate load and queuing tolerance

The Bursty Data Advantage

Data applications are far more bursty than voice:

• Web browsing: Burst when loading page, idle while reading (activity ~1-5%)
• File download: Full speed during transfer, nothing otherwise (activity varies wildly)
• Video streaming: Constant during viewing, nothing during pauses (activity ~30-50%)

Packet switching handles this burstiness naturally. A 100 Mbps link serving 1000 users at 1% average activity can let any user burst to 100 Mbps when idle users aren't transmitting.

Queuing Theory for Packet Networks

Packet network efficiency analysis uses queuing theory. The key relationship is the utilization factor:

ρ = λ / μ

Where:

• λ = average packet arrival rate
• μ = packet service rate (link rate / average packet size)

For stability, ρ must be < 1 (arrivals must be slower than service on average).

Wait Time and Utilization Tradeoff

For M/M/1 queues (Poisson arrivals, exponential service, single server), average waiting time W is:

W = 1 / (μ - λ) = (1/μ) / (1 - ρ)

As ρ approaches 1:

• ρ = 0.5: W = 2 × service time
• ρ = 0.9: W = 10 × service time
• ρ = 0.99: W = 100 × service time

This nonlinear relationship explains why packet networks can't run at 100% utilization—queuing delays would become unbounded. Practical targets are 70-80% utilization to balance efficiency against latency.

Statistical Multiplexing in Practice

Consider a 1 Gbps link serving 10,000 users:

• Each user: 10 Mbps peak, 1 Mbps average (10% activity)
• Aggregate average: 10,000 × 1 = 10 Gbps of offered demand
• Link capacity: 1 Gbps
• Oversubscription ratio: 10:1

If all users transmitted simultaneously (all 10% activity aligned), the link would be overwhelmed. But statistically, this never happens—the law of large numbers ensures aggregate demand stays near the 1 Gbps average with high probability.

Efficiency gains translate directly to cost savings. Understanding this relationship helps justify investment in multiplexing infrastructure and informs pricing models.

Capital Cost Savings

Multiplexing reduces infrastructure investment proportionally to multiplexing gain:

• Transmission equipment: Fewer links means fewer transceivers, amplifiers, regenerators
• Rights of way: Fewer cables means lower installation and lease costs
• Central office space: Consolidated equipment takes less physical space
• Power and cooling: Fewer devices means lower operational costs

Example: Metropolitan Fiber Network

The multiplexed approach costs 1/9th as much while serving the same customers equally well (for typical usage patterns).

Pricing Implications

Multiplexing efficiency enables consumption-based and shared pricing models:

Committed Information Rate (CIR): Customers purchase guaranteed capacity less than their peak access rate, paying premium only for firm guarantees.

Burst Pricing: Customers pay base rate for normal usage, premium for occasional bursts—matching cost to actual resource consumption.

Best-Effort/Oversubscription: Residential Internet is highly oversubscribed (often 20-50:1 on aggregation links) because most users are inactive most of the time.

Cloud Computing Economics

Cloud providers apply multiplexing principles to compute, storage, and networking:

• Aggregate virtual machines from thousands of customers on shared servers
• Most VMs are idle most of the time
• Oversubscription ratios of 10-20:1 are common
• Result: Cloud costs 10-20× less than dedicated infrastructure for typical workloads

Higher efficiency comes at the cost of statistical guarantees. Understanding this tradeoff is essential for system design that balances cost against user experience.

The Fundamental Tradeoff

Shared resources achieve efficiency by pooling unused capacity from idle users. But this means:

1. No absolute guarantees: If everyone transmits simultaneously, not all can succeed
2. Probabilistic service: We design for "99% of users get good service 99.9% of the time"
3. Peak degradation: During congestion, service quality drops

Quantifying the Tradeoff

For a given multiplexing gain G and number of users n, there's an associated congestion probability P_c:

• Higher G → Higher P_c (more efficiency means more congestion risk)
• Larger n → Lower P_c for same G (more users smooth demand)

Service Level Targets

Different applications tolerate different congestion levels:

• Emergency services (911): <0.01% blocking (essentially dedicated)
• Business voice: <1% blocking
• Residential voice: <2% blocking
• Business Internet: <5% congestion periods per month
• Residential Internet: Best-effort (significant oversubscription acceptable)

Quality of Service Mechanisms

To serve multiple service levels on shared infrastructure, QoS mechanisms provide differentiated treatment:

Priority Queuing: Critical traffic (voice, video) gets served before bulk traffic (downloads, backups). During congestion, low-priority traffic waits while high-priority traffic proceeds.

Weighted Fair Queuing: Each class gets proportional share of capacity. Business traffic might get 70% weight, residential 30%—so business users experience less congestion.

Rate Limiting: Each user's traffic is constrained to purchased capacity despite sharing the aggregate link. Prevents any user from monopolizing shared resources.

Admission Control: For circuit-like services (voice), new requests are rejected if accepting them would overload the network. Protects existing sessions at the cost of blocking new ones.

Let's examine how these efficiency principles apply in actual systems.

Example 1: Residential Internet Service

A cable ISP serving 100,000 homes:

• Access capacity: 1 Gbps per household (DOCSIS 3.1)
• Aggregate purchased capacity: 100,000 × 1 Gbps = 100 Tbps if dedicated
• Actual backbone capacity: 400 Gbps
• Oversubscription ratio: 250:1

This works because:

• Average household uses ~15 Mbps sustained (1.5% of peak)
• Peak concurrent usage is ~20% of households at prime time
• Statistical multiplexing smooths aggregate demand
• Capacity: 400 Gbps vs. actual demand: ~300 Gbps peak = adequate

Result: Infrastructure costs 1/250th of dedicated approach, sufficient for 99%+ of situations.

Example 2: Cellular Network

A 5G cell sector serving 1,000 active devices:

• Sector capacity: 1 Gbps shared (typical mmWave)
• Per-device peak capability: 100 Mbps
• If dedicated: 1,000 × 100 Mbps = 100 Gbps needed
• Actual: 1 Gbps provides ~1 Mbps average per device

Multiplexing gain: 100× This enables mobile broadband at consumer prices despite spectrum scarcity.

Example 3: Cloud Data Center

A cloud provider with 10,000 VMs on 1,000 servers:

• VM-to-server ratio: 10:1
• Each VM has virtual 1 Gbps NIC
• Physical server NIC: 100 Gbps
• NIC oversubscription: Only 10:1 (physical supports virtual)
• ToR-to-spine oversubscription: Additional 4:1

Most VMs are idle or low-bandwidth. The few high-bandwidth VMs statistically spread across servers. Total efficiency allows 10,000 × 1 Gbps (10 Tbps) of virtual capacity on ~250 Gbps of physical backbone.

We've developed the quantitative framework for understanding multiplexing efficiency. Let's consolidate the key insights:

What's Next:

With the efficiency gains of multiplexing established, the next page explores the types of multiplexing—the specific techniques (FDM, TDM, WDM, CDM, statistical) that implement channel sharing. Each technique offers different tradeoffs suitable for different applications.