A.K. Erlang:

The Man and the Measure

 

A Presentation for

 

The Second International Symposium on

Telecommunications History

 

 

 

August 5-6, 1994

Brookline, Massachusetts

 

By

 

Roger Clery

 

 

 

° A corporate telecommunications manager calculates the required number of tie trunks to interconnect the PBX switch at the home office to the PBX at the new overseas offices.

 

° A Communications engineer is calculating the required size of microcells for the new Personal Communication Systems.   These systems will allow communications anywhere and anytime the user desires 

 

°  An operations manager at a Local Exchange Carrier does a study of the delay time requirements of Signal System 7.  Although this system uses packet data to control the voice network, traffic tables developed for voice traffic can be used to estimate delay factors. 

 

INTRODUCTION

 

What these people are doing is traffic engineering.   They will be using tables and formulas derived and proven long ago.   They will probably give little thought to the people from the past and their work that has  that has made this possible. This paper explors the life and work of the foremost telephone traffic engineer.

 

October, 1946 at a plenary meeting of the C.C.I.F  (the forerunner of the CCITT) the members decided that the erlang will be the international unit of telephone traffic.  The erlang, a measuring unit with no physical properties, would take its place in the world along with the volt, ohm, watt and newton.  Who is this man Erlang?  And, what is this unit of measure?

 

Agner Krarup Erlang

 

THE BEGINNING

 

Denmark, a country in northern Europe on the Jutland peninsula and adjacent islands, January 1,  1878, two years after the invention of the telephone, Agner Krarup Erlang was born.  His father was a schoolmaster and his mother was from a family of ministers and ministers' wives.  He is said to have had a happy childhood and a proper upbringing.   He attended the village school, and with some tutoring by the assistant schoolmaster passed his Præliminæreksamen, (sort of a GED) at the age of 14.   He had to receive special permission because of his age but he passed with distinction.   He learned French and Latin and in two years took the Studentereksamen, the university entrance examination.  He passed with distinction.   As a boy he is said to have been quiet and peaceable, preferring reading to playing.

 

COLLEGE

 

Five years later, after beginning the university  he graduated with a master’s degree.   His major was in mathematics and he had minors of astronomy, physics and chemistry.   He was not particularly sociable and so his friends nicknamed him “the private person.”  During his college years he became friends with H.C. Nybølle. Nybølle later became professor of statistics at the University of Copenhagen.

 

In 1903 he entered a contest of mathematics -- winning an award for Huygen’s solution of infinitesimals problems.   Erlang was a member of the Mathematicians’ Association and it was at the association’s meetings that he became aquatinted with Dr. Jensen, chief engineer of the Copenhagen Telephone Company.    Jensen introduced Erlang to Dr. Johannsen, managing director of the company.   It pays to know the right people, then and now.

 

THE TELEPHONE COMPANY

 

Dr. Johannsen introduced probability theory into telephony.  He wrote two major works Waiting Times and the number of calls and Busy.

Dr. Johannsen decided to set up a technical / scientific laboratory for the company.    In 1908 the lab was established and surprise, surprise,  Erlang was named head of the laboratory.  By 1909 Erlang had written The Theory of Probability and Telephone Conversations.   This gave exact solutions to the problems that Dr. Johannsen had stated earlier.

 

In 1913 Erlang and the new chief engineer P.V. Christensen published a paper on The Number of Selectors in Automatic Telephone Exchanges.   This paper used probability theory and had tables showing the probability of losses.

 

THE B TABLE

 

In 1917 Erlang published his most important work, Solutions of some Problems in the Theory of Probabilities of Significance in Automatic Telephone Exchanges.  It is here that Erlang has his famous B table.  This table equates  of grade of service, number of servers and traffic density.

 

Erlang published other papers, invented a tester for measuring AC current and did many practical things to help the company.   An example of his practical  work was his study of stray currents (improper grounds) that he did when the lab was beginning.   He had no assistants at that time so he went around Copenhagen, followed by a worker carrying a ladder.  He would use the ladder to go down manholes to check the current in the lead sheaths of telephone cables.   

 

PERSONAL FACTS

 

He never liked social gatherings or parties.  He never smoked of drank.  He never married and lived very frugally.  He donated much of his money to charity and helped others when possible.   He seldom gave direct answers to problems, preferring to enter into lengthy conversations, with the attempt to lead the questioner to discover the answer for himself.

 

He worked from early morning to late at night, never missed a day of work for almost 20 years.   In 1929 he became ill and died a few days later at the age of 51.

 

 

HIS WORK

 

His work is difficult to read; first because of its brief style, and second, because of  the omission of proofs and intermediate steps.

Even those with statistical and mathematical backgrounds will have difficulty.   Also, it was originally in Danish, although all of Erlang’s works have now been translated to English.    At the time for there writing, the 1910s and 20s Dr. Thorton C. Fry of Bell Labs and Dr. A.E. Vaulot in France were said to have learned Danish just to read Erlang’s works.  

 

In the Autumn of 1943 the editors of the journal Tekniska...Telegrafstyrelsen in Sweden had a contest for naming the natural unit of telephone traffic. The Danish Post & Telegraph Office, the Copenhagen Telephone Company and the Royal Danish College of Engineering Suggested erlang for this natural unit of telephone traffic.  Also, 8 out of 26 Swedish entrees desired erlang.   Scandinavian countries have been using erlang since then.

 

The erlang is one hours worth of traffic-3600 call seconds.  Ten people each talking 6 minutes equal one erlang.  120 people each talking for 3 minutes is 6 erlangs.   Note that one line can give a maximum of one erlang in one hour's time.

 

The Bell system used and still uses CCS for a unit of traffic.   CCS is Hundred (C) Call Seconds.   36 CCS equals one erlang so that one measure can be converted to the other.   But, as divestiture has changed all Bell things, erlang has become acceptable.  Papers written by AT&T people or RBOC people more often than not use erlang.

 

 

The Natural Unit of Measure

 

BEFORE ERLANG

 

The Science of Probability and Statistics can be traced back to the 1700s.   Jacob Bernoulli  with sampling and error; S. D. Poisson with the law of large number and the Poisson distribution laid the groundwork for Erlang’s work.  Some understanding of waiting line or queuing theory is most helpful to understand Erlang’s work.

 

QUEUING THEORY

 

Waiting in line at the checkout counter in the grocery store is a good example.  1) There is an input process with arrivals into the system it is assumed that arrivals are random and that the average arrival rate can be measure.   You may know that the arrival rate is one every five minutes, but you do not know when any particular customer will arrive.   2) There is a service mechanism,  i.e., the checkout counter.  You know the average service time, but you do not now how long the next customer arriving will take. Say that the average service time is four minutes, some take longer some shorter most are about four minutes.    3) There is a queuing mechanism -- a line to get service and a way to leave the system.   l will be the arrival rate = 12 per hour minutes. m will be the service rate = 15 per hour.  h is the utilization rate = l/m = 12/15 = .8      h must always be < 1 or else the line just gets longer and longer.    From this information we can calculate the following:

            Average number of customers in system = h/(1-h) = .8/(1-.8)=4

            Average number of customers in line = h2/(1-h)=.64/.2=3.2

            Average time in the system = 1/( m-l)=1/(15-12)=1/3 hour

            Average time in line = h/( m-l)= .8/(15-12)=.26 hour

        The probability that there will be exactly 4 arrivals during any half hour time period

 =( (l·t)x · e-lt )/(x!)  = ((12·.5)4 · 2.7182818-12·.5)/(4·3·2·1)=

((6^4)*2.7181818^(-6))/(4*3*2*1) = .13385

 

FIRST MAIN PROBLEM

 

From the above formula Erlang deduced a now famous formula for telephone traffic.   Desiring to know the probability of loss, that is the probability that a call is blocked is designated by the letter B. Given the number of service units (these could be trunks or dial tone units or whatever) x  and the traffic intensity y.  y is to be in natural time units -- this unit is now known as the erlang.   Given some factors such as 1) calls arrive at random  2) calls have constant holding time 3) the system is at statistical equilibrium, that is the average number of calls in not increasing of decreasing 4) the number of potential customers/ arrivals is large 5) time period for service is small 6) customers that are blocked disappear from the system, then erlang’s formula give exact results.

 

 

THE FORMULA

 

                                    yx / x!                                     

B = _______________________________

            1 + y/1! + y2/2! +...+ yx/x!

 

This formula gives the B  value which is the probability of blockage -- the grade of service.   Note that higher values are bad; lower values are good.    .10 = 10% = 1 in 10 chance of blockage.    .05 = 5% = 5 in 100 chance of blockage.

 

 

TRAFFIC ENGINEERING

 

Normally traffic intensity varies by time of day, day of week, month of year, etc.     Traffic engineers select the busiest hour of the busiest day of the busiest week and use the average traffic of that hour for use with the erlang formula or tables based on that formula.   Blockage at non-busy hours would of course be less.  Suppose that 2 erlangs of traffic are carried over  5 trunks what is the grade of service? Using the Erlang B Table it can be determined that the grade of service is about .04    If the traffic remains at 2 erlangs and the desired grade of service is .01 then 7 lines are required.

 

Offered load in erlangs using the Erlang B Formula

 

x       B=

.005

.01

.02

.03

.05

.10

1

.005

.011

.021

.031

.053

.113

2

.106

.153

.224

.282

.382

.602

3

.349

.456

.603

.716

.900

.1.275

4

.702

.870

1.093

1.259

1.525

2.052

5

1.132

1.361

1.658

1.876

2.219

2.896

6

1.662

1.909

2.276

2.543

2.961

3.751

7

2.158

2.501

2.936

3.250

3.738

4.678

8

2.703

3.128

3.637

3.987

4.543

5.613

9

3.333

3.783

4.345

4.748

5.371

6.560

10

3.961

4.462

5.084

5.530

6.216

7.502

11

4.611

5.160

5.842

6.328

7.077

8.504

12

5.279

5.876

6.615

7.141

7.950

9.472

13

5.964

6.608

7.402

7.967

8.835

10.474

14

6.664

7.352

8.201

8.804

9.730

11.473

 

Some factors should be noted doubling the number of lines x more than doubles the traffic carried.   Or, another way of stating that same idea is that 8 lines are better then two groups of 4 lines.   The efficiency of small number of lines is low.  The incremental difference at B =.005 from 1 lines to two is only .101 erlangs, but from 13 to 14 lines is .7 erlangs. 

 

ERLANG C

 

Other tables commonly used are the Erlang C table for Automatic Call Distributors (ACDs.)  With this formula is is assumed that calls do not disappear, but are held in a queue until serviced.  Also calls lengths must be exponentially distributed -- More short calls few long calls.     The use of the Erlang C table is  more complex that the B table.   Erlang didn’t discover all the traffic formulas used.  Poisson, Binomial, Engset, Crommelin-Pollaczek, Neal-Wilkinson and other have devised formulas for various purposes.  The two most used are Poisson and Erlang B.

 

WHY STUDY TRAFFIC FORMULAS?

 

Why would anyone spend time studying traffic engineering formulas and tables?  I became started because the solutions given in textbooks and technical books supplied the wrong solutions to the tie line/trunk problem given at the beginning of this paper.  Dr. Richard Born of Northern Illinois University and I wrote a paper on the selection of the correct number of tie lines.   Alternate route tables based on the Erlang B formula are necessary.  I developed a method of using electronic spreadsheets to solve the problem.   While many are intimidated by complex formulas or computer programs,  breaking formulas down into spreadsheet cells makes them less so.   Telecommunications students need some experience with these traffic engineering tables and formulas, if for no other reason than being able to select the correct table and then use it correctly.

 

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