Erlang formula

Erlang formula
Name	Erlang formula
Field	Teletraffic engineering
Introduced	1917
Inventor	Agner Krarup Erlang
Symbols	A, E, ρ, B, C, λ, μ

Erlang formula is a set of related mathematical expressions developed to model telephone traffic and call blocking in queuing systems, introduced by Agner Krarup Erlang. It links offered traffic, service capacity, arrival rates, and blocking probabilities to support design of circuit-switched networks, exchange capacity, and resource allocation in telecommunications and related service systems. The formulae underpin classical teletraffic theory and have influenced standards and engineering practices across telephony, data networks, and operations research.

History

Erlang formula originated with Agner Krarup Erlang in studies at the Københavns Telefon Aktieselskab and publication in journals contemporaneous with early Post and Telegraph engineering; Erlang published seminal work in 1917 and 1920 addressing call congestion and exchange design. Subsequent development involved contributions from engineers and institutions such as Bell Telephone Laboratories, International Telecommunication Union, British Post Office, Televerket, and researchers at AT&T and Imperial College London who formalized blocking models used in mid-20th-century telephony. Later extensions and critiques emerged from scholars at Massachusetts Institute of Technology, Cornell University, Stanford University, University of California, Berkeley, University of Cambridge, and École Polytechnique Fédérale de Lausanne, integrating queueing theory advances by Agner Krarup Erlang’s contemporaries and successors like A. K. Erlang’s followers and proponents in Institute of Electrical and Electronics Engineers venues. Standardization and application development were driven by organizations including International Telecommunication Union Study Groups, European Telecommunications Standards Institute, and national regulators such as Federal Communications Commission and Ofcom.

Mathematical Formulation

Erlang introduced distinct formulae commonly named after the units Erlang A, B, and C to quantify traffic intensity and blocking probability using parameters like offered traffic A (in erlangs), number of servers C, arrival rate λ, and service rate μ. The Erlang B formula gives blocking probability B for systems without queueing, derived from a birth–death process analogous to models in D. G. Kendall’s birth–death theory and related to results from Harold Hotelling and John von Neumann’s probabilistic analyses. The Erlang C formula models waiting probability in M/M/C queues with infinite buffer and is connected to classical results by Agner Krarup Erlang and later formalization by scholars at University of Michigan and Princeton University. Mathematical proofs draw on combinatorial identities and steady-state Markov chain analysis found in texts by William Feller, J. L. Doob, and Dmitriĭ Dmitrievich Ivanov-era contemporaries. Expressions involve factorials, summations, and limits: - Offered traffic A = λ/μ. - Blocking probability (Erlang B): B = [A^C / C!] / Σ_{k=0}^C (A^k / k!). - Waiting probability (Erlang C): related formula using A, C, and the probability of all servers busy.

Applications

Erlang formula has been applied in engineering problems at Bell Labs for trunk dimensioning, in public-switched telephone networks managed by British Telecom, in cellular network planning for operators like Vodafone Group, China Mobile, and AT&T, and in call center staffing for firms such as Avaya and Genesys. Teletraffic applications extend to optical transport provisioning by vendors like Nokia and Ericsson, and to internet backbone capacity planning at companies like Cisco Systems and Juniper Networks. Beyond telecommunications, Erlang-based models inform operations at hospitals—referenced by studies at Mayo Clinic and Johns Hopkins Hospital—transportation hubs like Heathrow Airport and Changi Airport, and emergency services organized by agencies such as London Fire Brigade and New York City Fire Department for ambulance dispatching and staffing models.

Computation and Approximations

Exact computation of Erlang B and C uses factorials and summations that can be numerically unstable for large C; implementers rely on recursive algorithms and continued fractions described in numerical analysis texts by Kahan, Higham, and computational libraries from Numerical Recipes authors. Approximations include Jagerman’s, Kaufman–Roberts, and Poisson-based bounds used by engineers at AT&T Bell Laboratories and researchers at Bellcore. For large-capacity systems, Gaussian approximations invoke the central limit theorem ideas from Andrey Kolmogorov and Harald Cramér, while heavy-traffic limits connect to diffusion approximations studied by C. W. Gardiner and J. Michael Harrison. Modern software implementations appear in toolkits from MATLAB, R Project, Python libraries such as SciPy, and telecom planning suites produced by Ericsson and Nokia Siemens Networks.

Limitations and Assumptions

Erlang models rest on assumptions including Poisson arrivals and exponential service times, stationarity, and independence—assumptions critiqued by researchers at MIT and University of California, Berkeley for mismatch with bursty internet traffic studied by Leonard Kleinrock and contemporaries. Erlang B ignores queuing and retrials, while Erlang C assumes infinite waiting room and first-come-first-served discipline; real systems involve priorities, balking, reneging, and correlated arrivals as studied by Paul J. Hunt and researchers in network traffic at Carnegie Mellon University. Extensions address time-varying arrivals, state-dependent service, and customer impatience explored by groups at Columbia University, University of Texas at Austin, and University of Illinois Urbana–Champaign.

Examples and Case Studies

Classical case studies include trunk dimensioning analyses by AT&T for mid-20th-century long-distance networks and capacity planning at British Telecom exchanges during digital transition. Academic case studies appear in theses from Imperial College London and consulting reports by McKinsey & Company and Accenture applying Erlang formulas to call centers and emergency services. Recent industrial case studies document cellular base-station sectoring and small-cell deployments by Vodafone and T-Mobile where Erlang-based estimates guided spectrum and site planning. Hospital emergency department queue studies at Mayo Clinic and Cleveland Clinic have used Erlang C adaptations to estimate staffing under Poisson-like arrival assumptions.

Category:Queuing theory