# An Introduction to Erlang B and Erlang C - Download as PDF

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```					 6                                                                                                                                   Telemanagement #187

An Introduction to                                                                                             that actually works — uses mathematical
formulas that form the basis of the disci-

Erlang B and Erlang C
pline called “trafﬁc engineering.” Some
of them are easy for anyone to use; others
require expert support, or at least special-
If you make decisions about networks, PBXs,                                                                    ized software. If you make conﬁguration
or call centres, you must understand these concepts                                                            decisions about networks, PBXs, or call
centres, you must understand the basic
IAN ANGUS
concepts involved, whether or not you plan
to do the calculations yourself.

“Hey, it’s simple arithmetic! We get 3,200 calls     And then they will bunch up during that                    Basic Concepts
a day. That’s 400 calls an hour. Each call lasts    hour, leading to times when the phones                     Until recently, anyone using trafﬁc engi-
three minutes, so each person can handle 20         are going crazy and times when many                        neering techniques had to either master
calls an hour. So we’ll need 20 incoming lines      lines are free.                                            the mathematics, or learn to use thick
and 20 people to answer the phones.”                   Then you have to take into consid-                      books of tables based on them. Personal
Does that sound familiar?                       eration the fact that no one can handle                    computers changed all that, ﬁrst by
Many telephone and call centre deci-            20 three-minute calls every hour without                   making trafﬁc programs widely afford-
sions are made using just that logic. It’s          burning out, and that there is usually post-               able, and now by making them available
simple, and it’s clear — and it’s absolutely        call work to do in addition to talking to                  on the Internet. Several sources are de-
wrong. The problem can be summed up                 callers.                                                   scribed in the box on page 7.
in three words: Calls Bunch Up.                        In short, the “simple arithmetic” ap-                       But remember the phrase, “garbage in,
If you get 3,200 calls in an eight-hour         proach will result in too few trunks, too                  garbage out.” Before you jump to your
day, and your calls follow a typical distri-        few people — and too many very unhappy                     browser, be sure you understand some
bution pattern, 550 or 600 of them will             callers.                                                   key concepts.
arrive during the busiest hour of the day.             The alternative — the only approach                         Erlang: The basic unit of telecom trafﬁc
intensity. Strictly speaking, an erlang is
what mathematicians call a “dimension-
It Started in Denmark                                                                                      less unit,” representing continuous use of
All modern methods for optimizing networks have their                                                      one circuit. However, since a single circuit
roots in work done by Agner Krarup Erlang, a scientist                                                     used continuously carries 60 minutes of
who joined the Copenhagen Telephone Company in                                                             calling in one hour, one erlang is usually
1908. He set out to solve the key problem in telephone                                                     deﬁned as 60 minutes of trafﬁc. If you
network design: how many trunks are needed to carry a                                                      receive 300 two-minute calls in an hour,
given amount of calling?                                                                                   then you received 600 minutes, or 10 er-
Imagine a village in which every home has a telephone                                                  langs, of trafﬁc in that hour.
connected to a local switch. How many trunks should                                                            (For many years, AT&T and Bell
the phone company install between that switch and                                                          Canada insisted on measuring trafﬁc in
the one in the next village? Erlang saw that there was                                                     CCS — 100 call seconds — instead of the
no absolutely correct answer. Rather, there is always                                                      more convenient but not-invented-here
a trade-off between service and cost. In the case of                                                       erlang. If you have data in CCS, divide the
A.K. Erlang, 1878-1929
the village, there are two extreme options, neither of                                                     numbers by 36 to get erlangs.)
which is acceptable:                                                                                           Busy Hour: Because “calls bunch up,”
! Provide just one trunk, and let callers wait until it’s available. The cost is low,                  all trafﬁc planning has to focus on peak
but the service is unacceptably poor.                                                                      periods. It isn’t acceptable to provide excel-
! Provide one trunk for every local phone line, so no call is ever blocked. The                        lent service most of the time and terrible
service is excellent, but the cost is much too high.                                                       service just when customers want to make
The problem was to convert that insight into hard numbers that would allow                             calls. Most commonly, we take the busiest
network planners and accountants to evaluate each possibility on the curves                                hour of each day for ﬁve or 10 days during
between low and high cost, poor and excellent service.                                                     the busiest time of year, then calculate
To do this, Erlang conducted the world’s first detailed studies of telephone traffic,                  the average of those hours’ trafﬁc load.
and then developed mathematical formulas to evaluate the trade-offs. His work was                          That “Average Busy Hour” ﬁgure is used
enormously influential worldwide. One Bell Labs researcher taught himself Danish                           to determine the maximum number of
just to be able to read Erlang’s papers in their original form. In 1946, the International                 trunks or people needed.
Consultative Committee on Telephones and Telegraphs (CCITT) honoured him by                                    Server: Something that handles calls.
adopting the name “erlang” for the basic unit of telephone traffic.                                        For example, in a call centre situation
there are two kinds of servers: the trunks

Reproduction in any form prohibited. For additional copies phone 905-686-5050.
July-August 2001                                                                                                                                           7

Finding Erlang On-line
Software has replaced the thick books of                                                                ! Erlang-Software: Another source for
tables that used to be basic equipment for                                                           PC traffic software, this one in Australia.
anyone involved in optimizing telephone                                                              Their Erlang-G program is said to include six
networks. The following are some useful                                                              common traffic formulas. \$40, with multiple-
resources. Telemanagement has not tested                                                             copy discounts. members.iinet.net.au/~clark/
all of these products, and does not endorse                                                          index.html.
any of them. All prices are in U.S. dollars.                                      ! Certis Technologies: Their ErlangCalc 1.2 program is
! Erlang Calculator 1.0: An Erlang B program for Palm                     available in three versions: Standard (Erlang B only, \$39); Pro
computers. This simple freeware program displays three                        (Erlang B, Extended Erlang B, Erlang C, Daily Traffic, \$69); Deluxe
blanks: Fail Rate (Blockage), Traffic, and Lines. You fill in two             (Erlang B, Extended Erlang B, Erlang C, Daily Traffic, Spreadsheet
and it calculates the third. Available from various sources,                  for Erlang B Batch-Processing, Graphical Presentation of Results,
including www.palmspot.com, www.visorvillage.com, and                         \$89). Free trial versions are available. www.certis.com/.
www.palmblvd.com.                                                                 ! ClienTel: Don’t be dismayed when you find that the home
! Westbay Engineers: A UK company that develops and sells                 page is in Dutch. Just click on Online Call Center Assistant for
traffic calculation software, ranging from Erlang for Excel (\$80) to          Erlang B and Erlang C calculators and a 24-hour call centre
complex network and call centre modeling tools. Their website                 scheduling program. www.clientel.nl/.
features the best on-line traffic calculators we’ve seen, including               ! Maths.org: Interested in the math behind Erlang’s formulas?
Erlang B, Erlang C, and call centre staffing. They also have an easy          There’s a tutorial, including some simple on-line calculators, at
! HTL Telemanagement: Turbo Tables (\$595) was the first                       ! Basic on-line calculators. When you need quick answers,
program to add traffic formulas to Excel, and remains one of the              these non-commercial sites can be useful. Most use few graphics
best. HTL focuses on call centre planning tools, using Hills A, a             and download quickly, a major advantage.
proprietary extension of Erlang C. www.htlt.com.                                  www.dcss.mcmaster.ca/~qiao/publications/erlang/erlang.html
! Teleopti: A Swedish firm that sells software for call centre                mmc.et.tudelft.nl/~frits/Erlang.htm
management and optimization. Their website includes a rather                      www.owenduffy.com.au/electronics/telecommunications.htm
complex Java-based Call Center Wizard for calculating trunk and                   www.cs.vu.nl/~koole/erlang.html
staffing requirements. www.teleopti.com.                                          persoweb.francenet.fr/~hilleret/

that carry the calls, and the agents who           ers, Trafﬁc, and Grade of Service. If you                 — how many trunks are needed to carry
answer them. With a voice mail or IVR              know any two of those, the formula will                   your toll-free calls, how many tie trunks
system, the servers are ports.                     calculate the third one.                                  between two ofﬁces, how many ports into
Grade of Service: The probability that             Erlang C: Use this formula when a                     your voice mail system, or some similar
all servers will be busy when a call attempt       blocked call is delayed — for example,                    question. Erlang B handles that relatively
is made. For example, on a trunk group:            when someone calls your call centre and                   easily, in four steps:
P.02 means that there is a 2% probability          must wait for an agent to take the call.                      1. Collect trafﬁc data. You need to
of getting a busy signal (being “blocked”)         It uses the same three variables, plus the                know how much trafﬁc will try to use the
when you have a given amount of trafﬁc             average length of each call, to calculate the             trunk group, each hour, for ﬁve or 10 busi-
and a given number of trunks. In a call            probability of being delayed and how long                 ness days. You may be able to use phone
centre, the same number would mean that            the delay is likely to be.                                bills, call detail reports, carrier trafﬁc stud-
there is a 2% probability of having to wait            These formulas only work if you have                  ies, or even manual counts — or you may
to speak to a human.                               a large number of independent sources                     just have to make educated guesses. The
of trafﬁc. For example, 10 people making                  objective is to produce an hour-by-hour
Probability Formulas                               outgoing calls, with no incoming calls,                   spreadsheet, showing the number of min-
There are many trafﬁc formulas, appropri-          will never need more than 10 trunks, no                   utes of trafﬁc in each hour. Divide those
ate to many different situations, but two          matter what the formula says! A math-                     numbers by 60 to get erlangs per hour.
of them, both developed by A.K. Erlang,            ematician will tell you that these formulas                   Bear in mind that the trafﬁc on your
cover the most common business telecom             require “inﬁnite sources,” but in practice                trunks may be greater than actual conver-
requirements.                                      they work very well if there are at least 10              sation time — you must allow for dialing
Erlang B: This is the formula to use           times as many possible sources (callers)                  time on outgoing calls, and for ringing
when a blocked call is really blocked —            as servers (trunks or agents).                            time on incoming calls, for example.
for example, when somebody calls your                                                                            2. Determine the Average Busy Hour.
phone number and gets a busy signal or             Erlang B: The Easy One                                    Select the busiest hour of each business
tries to access a tie trunk and ﬁnds it in         The most common trafﬁc engineering                        day, total the trafﬁc, then divide by the
use. It is built around three variables: Serv-     problem involves sizing a trunk group                     number of days.

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8                                                                                                                                   Telemanagement #187

3. Choose a target Grade of Service. In                                                                     time. That’s quite different from the
Erlang B:
most cases, a target of P.05 is acceptable,                                                                     number you must schedule to work each
P.10 is terrible, and P.001 is so good that                                                                     day, since almost no one can be on the
most callers will never get a busy signal.                                                                      phones 60 minutes an hour.
4. Use Erlang B. Calculate the number                                                                          4. How does waiting time affect
of trunks you need to carry that amount of                                                                      trunk load? The time a caller spends
trafﬁc with your target Grade of Service.               Erlang C:                                               on hold listening to music adds to your
At this point, you need to decide wheth-                                                                    trunk traffic — you can save money by
er the answer is acceptable. Usually, that                                                                      having fewer agents to answer calls, but
means deciding whether you can afford                                                                           that may require adding trunks and/or
the number or trunks or ports required.                                                                         increasing your toll-free service bill. You
If not, try reducing the number of trunks                                                                       may have to do several iterations to
and see what the effect is on the Grade of           complexity in at least four areas.                         determine the optimal mix of trunks,
Service. You may discover that the differ-               1. What’s included in call times? In a                 people, and delay.
ence is negligible — or you may decide               queuing system, trafﬁc includes not just                      These issues mean that anyone doing
that you have to get a budget increase or            “conversation minutes,” but also the time                  anything more than the simplest delay
ﬁnd some way to reduce the amount of                 agents spend doing “post-call work” re-                    calculations — above all anyone doing
trafﬁc. (Are those calls really necessary?)          lated to that conversation. Gathering ac-                  regular staff and conﬁguration planning
The best thing about the various PC              curate data on this can be much more dif-                  for a call centre — should consider buying
and Internet trafﬁc calculators is that they         ﬁcult than just looking at a trafﬁc study or               specialized call centre planning software.
make it very easy to do multiple calcula-            a toll-free service bill. In theory, your ACD              All such programs — despite the fact that
tions until you ﬁnd a balance between cost           reports provide the information — but                      every one claims to be absolutely unique
and service you can live with.                       that only works if everyone has been press-                — are ultimately based on A.K. Erlang’s
ing the right buttons at the right times.                  formulas from 90 years ago.
Erlang C: More Difﬁcult                                  2. What’s meant by delay? The “Aver-                      If you are comfortable with the math
Because Erlang B is so simple to use (insert         age Delay” might be the average of all calls,              and at creating spreadsheets, consider
two numbers, it calculates the third), many          including all the calls that didn’t wait at all,           buying one of the commercial programs
managers assume that Erlang C will be                or it might be the average of calls that actu-             that adds trafﬁc formulas to Excel, making
similarly easy. That’s a mistake — even              ally experienced a delay. The latter ﬁgure is              it much easier to do multiple “what if”
basic Erlang C calculations are difﬁcult,            usually more useful, but you must be very                  calculations.
and more complex ones can be daunting                clear which one you are concerned with.
indeed.
Erlang C is most commonly used to
calculate how long callers will have to wait
Also, using averages can conceal situa-
tions in which most delayed calls wait only
a few seconds, but some experience very
E    very telecom professional should have
at least a general familiarity with
Erlang B and Erlang C. However, whether
before being connected to a human in a               long delays. That can be a serious cus-                    you use them yourself or evaluate reports
call centre or similar situation. This adds          tomer service problem, even if the aver-                   produced by others, always keep three
ages look good.                                            things in mind.
Related Reading                                      Most call centres summarize their                         ! Using math is no substitute for using
! Call Center Management on Fast For-            delay objectives in a phrase like “answer                  your head. If the data you plug in to the
ward, by Brad Cleveland and Julia              80% of calls within 20 seconds,” but get-                  formulas isn’t valid, the answers won’t be
Mayben, includes a useful plain-Eng-           ting from the Erlang C formula to that                     any good either.
lish discussion of using Erlang C for          result can be difﬁcult.                                       ! All formulas make assumptions that
call centre planning. Order from                   3. What’s the hour-by-hour load? With                  simplify reality. Erlang B assumes that
www.incoming.com                               trunks, you don’t have the option of adding                callers who receive busy signals won’t im-
! Tables for Traffic Management and              or removing circuits every hour, so you                    mediately try again. Erlang C assumes
Design, by Ted Frankel, seems to be            must install and pay for the number                        that delayed callers will wait on hold in-
the only book of traffic tables still in       needed under peak load conditions. With                    deﬁnitely. For low blockage rates and short
print. Lee Goeller’s introduction and          people, the peak period determines maxi-                   queues, those assumptions don’t cause a
Frankel’s first three chapters consti-         mum stafﬁng, but you also need to calcu-                   problem — but when your service level is
tute an excellent brief course in traf-        late staff requirements for other times,                   poor, they can give misleading results.
fic engineering concepts. Order from           and plan staff scheduling accordingly.                        ! All trafﬁc formulas calculate prob-
www.abcteletraining.com.                       This usually means doing separate calcu-                   abilities, not absolutes. Erlang B and C
! Cisco’s website has useful tutorials.          lations for each half hour period in every                 predict what will happen, on average,
Go to www.cisco.com and type “Traf-            week you operate.                                          over many hours with similar trafﬁc. Your
fic Engineering” into the search                   And bear in mind that the Erlang C                     actual experience in any speciﬁc hour can
box.                                           forecast only tells you how many people                    be quite different.
must be answering phones at any given                                                               !

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