Erlang's loss model

Erlang's loss model
Name	Erlang's loss model
Other names	Erlang B formula, blocking model
Field	Queueing theory
Introduced	1917
Inventor	Agner Krarup Erlang

Erlang's loss model. Erlang's loss model is a classical stochastic model in queueing theory describing call blocking in systems with limited servers and no waiting room. It originated in early telephone engineering and has become foundational across telecommunications, operations research, traffic engineering, and performance analysis. The model links arrival processes, service processes, and capacity planning via closed-form probability expressions used by engineers in AT&T, Bell Labs, British Post Office, and contemporary firms.

Introduction

Erlang's loss model was introduced by Agner Krarup Erlang to analyze congestion in the Copenhagen Telephone Exchange and later influenced designs at International Bell Telephone Company and Post Office Telephone Department. It assumes Poisson arrivals, exponential service times, and a fixed number of servers with immediate loss upon arrival if all servers are busy; this contrasts with systems studied by John von Neumann, Andrey Kolmogorov, and Norbert Wiener in related stochastic work. The model underpins engineering decisions at organizations like AT&T, Deutsche Telekom, and NTT and informs standards from ITU and ETSI.

Mathematical Formulation

Erlang's loss model, often called the Erlang B context in teletraffic science, models an M/M/c/c queue where c is the number of servers defined by infrastructure such as trunks in PSTN switches. The input is a Poisson process characterized by the parameter λ (offered load often expressed in erlangs), and service times are exponential with rate μ, familiar to researchers such as Agner Krarup Erlang and later formalized by A.K. Erlang's contemporaries. The blocking probability B(c, a) equals the probability that all c servers are busy given offered traffic a = λ/μ, derived from birth–death chains studied by Feller, Kolmogorov, and G. H. Hardy. The state probabilities follow a truncated Poisson distribution normalized over states 0...c, a structure analyzed by Srinivasa Ramanujan and used in calculations by Harold Hotelling in applied statistics.

Performance Metrics

Primary metrics include blocking probability, carried traffic, and lost traffic, which guide capacity decisions at organizations such as Cisco Systems, Ericsson, and Huawei. Blocking probability B(c,a) gives the fraction of arrivals denied service, while carried traffic equals a(1−B(c,a)), a quantity central to planning in the British Telecom and France Télécom eras. Other metrics—server utilization and occupancy variance—relate to studies by William Feller and Maurice Kendall on stochastic processes and are used in performance dashboards at Oracle and IBM.

Solving Methods and Algorithms

Closed-form evaluation of B(c,a) uses recursive relations and normalization techniques developed in the early 20th century and refined by researchers at Bell Labs and in textbooks by John G. Kemeny and Herbert Robbins. Numerical methods include recursion algorithms to compute blocking without overflow, continued fraction methods used in Harvard computational labs, and asymptotic approximations like the Erlang fixed-point and saddlepoint approximations influenced by Harold Jeffreys and R. A. Fisher. Modern computational libraries implemented by teams at MIT, Stanford University, and ETH Zurich provide stable routines, and iterative fixed-point methods are integrated into software from MathWorks and Wolfram Research.

Applications and Practical Use

Erlang's loss model applies to trunking in public switched telephone networks used by AT&T and Deutsche Telekom, call center staffing at Concentrix and Teleperformance, trunk dimensioning in cellular systems by Nokia and Qualcomm, and capacity planning for cloud telephony services from Twilio and Vonage. It guides spectrum allocation decisions in agencies like FCC and Ofcom and is used in transportation ticketing systems managed by Amtrak and Deutsche Bahn. In healthcare operations studied by NHS analysts and Johns Hopkins University researchers, variants inform bed occupancy and admission refusal rates, while logistics teams at UPS and DHL use analogous blocking models for resource contention.

Extensions and Generalizations

Extensions include models with retrials, priority classes, and non-Poisson arrivals studied by Lajos Takács, David Cox, and J. F. C. Kingman. The Erlang C model for waiting systems, the Engset model for finite source populations, and loss networks for circuit-switched networks by Frank Kelly generalize the basic assumptions. State-dependent service rates and vacations were explored by I. A. Glover and N. G. de Bruijn, while spatial and time-varying extensions connect to work by Claude Shannon on information capacity and by Richard Bellman on dynamic programming in resource allocation.

Historical Development and Impact

Historically, Erlang's loss model catalyzed the field of teletraffic engineering; its adoption by early telephone administrations such as the Copenhagen Telephone Exchange and London County Council shaped network planning practices used throughout the 20th century. Influential texts by Agner Krarup Erlang, A. J. C. B. Atkinson, and later monographs from Bell Labs codified techniques that influenced researchers at Princeton University and University of Cambridge. The model's impact spans standards in ITU-T, economic regulation at European Commission bodies, and academic curricula at institutions like Massachusetts Institute of Technology and University of California, Berkeley. Its principles remain relevant in modern digital networks, cloud services, and performance engineering at technology companies such as Google, Amazon, and Facebook.

Category:Queueing theory