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Decision tree; infinite source blocking formulas

Short answer

The decision tree uses source type (infinite/finite) and blocked call handling (cleared/held/delayed). For infinite sources: Erlang B for cleared calls B = A^N/N! / Σ_{k=0}^N A^k/k!; Poisson for held calls P = e^{-A} Σ_{k=N}^∞ A^k/k!; Erlang C for delayed calls C = (A^N/N! · N/(N-A)) / (Σ_{k=0}^{N-1} A^k/k! + (A^N/N! · N/(N-A))).

What are the formulas used in telecommunication traffic engineering decision tree? Describe the blocking formulas used in infinite sources.

Formulas in the traffic engineering decision tree

The correct formula is chosen by two questions: are the sources infinite or finite, and what happens to blocked calls — cleared (LCC), held (LCH) or delayed (LCD)?

                 Traffic sources
           +-------------+-------------+
       Infinite                     Finite
   +-------+-------+          +-------+-------+
  LCC     LCH     LCD        LCC     LCH     LCD
Erlang B Poisson Erlang C   Engset  Binomial Finite
                                            queue
SourcesBlocked callsFormula
InfiniteClearedErlang B
InfiniteHeldPoisson
InfiniteDelayedErlang C
FiniteClearedEngset
FiniteHeldBinomial
FiniteDelayedFinite queue

Blocking formulas for infinite sources

Assumptions: Poisson arrivals, exponential holding times, N servers, full availability, offered traffic AA in Erlangs.

1. Erlang B (lost calls cleared)

B=AN/N!∑k=0NAk/k!B = \frac{A^N/N!}{\sum_{k=0}^{N}A^k/k!}

Blocked calls vanish. Most widely used for trunk dimensioning (Erlang B tables).

2. Poisson (lost calls held)

P=e−A∑k=N∞Akk!P = e^{-A}\sum_{k=N}^{\infty}\frac{A^k}{k!}

Blocked calls are assumed to stay in the system (keep retrying) for a holding time; gives slightly higher blocking.

3. Erlang C (lost calls delayed)

C=ANN!NN−A∑k=0N−1Akk!+ANN!NN−AC = \frac{\dfrac{A^N}{N!}\dfrac{N}{N-A}}{\sum_{k=0}^{N-1}\dfrac{A^k}{k!} + \dfrac{A^N}{N!}\dfrac{N}{N-A}}

Probability that a call has to wait; mean wait =C h/(N−A)= C\,h/(N-A). Used for call centres and common-control equipment.

✓ Answer checked by Claude Opus 5.5 (high effort)

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