EFFICIENT TRAFFIC LOADING OF TELEPHONE LINE GROUPS 
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Author:     Jock Mackirdy 
            Business Advisory Services 
            Luton, UK 
CompuServe: 100121,1355 
Phone/fax:  (44)-1582-597878 
Created 15 March 1994 
Revised 12 April 1994 
Revised 28 January 1995 (revisions marked "*") 
Revised for upload to TELECOMMUNICATIONS Forum 
 23 March 1995 (revisions marked "@") 
 
All comments on content and suggestions for improvement will be  
gratefully received. 
 
(*) Disclaimer: Telephone traffic theory is based on probability 
and chance, so no solution will be mathematically exact. 
 
(@) Equally, in measuring traffic, minute accuracy is not needed.  
 
1.  Purpose (* - new para.) 
    ------- 
Efficient arrangement of telephone lines in a multi-line 
group (exchange lines, inter-PBX links etc.) maximises the 
number of calls carried by the group and minimises the risk of 
call collisions. Improved customer service is allied to minimum 
operating cost. 
 
2.  Background information 
    ---------------------- 
I have applied the standard probability theory of pure chance 
telephone traffic to the specific case of calls to and from a 
multi-line subscriber's installation (usually a switching system 
of some sort). 
 
The basic data is from graphs in Atkinson's "Telephony", vol 2, 
published 1950 by Pitmans.  The mathematics supporting the  
graphs is, as far as I know, in the public domain.  It applies  
to pure chance traffic only and becomes less accurate for very  
long call holding times (e.g. dial-up modem links regularly  
connected for more than a few minutes) and installations which  
are regularly overloaded. 
 
(*) A subscriber's originating traffic is most easily derived by  
analysing telephone bills.  Itemised bills (itemising all calls) 
should be analysed to determine the busiest hour of the busiest 
day in the billing period, with all lines taken together. 
For each line, the calls coomencing during this hour should then 
be grouped into five-minute bands. 
 
There is no need for greater precision, since we are dealing with 
random events (the start and end of individual calls). 
 
2.  Basic tests for a sound installation 
    ------------------------------------ 
The most efficient arrangement is a single group of lines with  
a single telephone number and automatic sequential hunting for  
a free line, with incoming and outgoing calls starting at opposite  
ends of the group of lines.  
(@) It is also the most cost-effective for per-line call discounts, 
since the vast majority of calls will use the very early choice lines. 
 
The least efficient arrangement, with the greatest chance of a  
busy line or a call collision, is "one number per line" incoming  
combined with a free choice of outgoing line.   
 
Case 1  One telephone number per line 
------ 
A customer dialling any of the published numbers has a high  
probability of finding that number engaged, by either an  
outgoing or incoming call.  This can also happen to some extent  
on an installation with "bypass" numbers superimposed on a  
hunting group. 
 
Case 2  One number for all calls 
------ 
The public exchange (US. "office") directs calls to the first  
free line in a pre-determined order. All lines have to be found  
busy before busy tone is returned. 
 
Case 3  More than one number, in hunting groups 
------ 
This is a half-way stage between Cases 1 and 2.  It is markedly  
less efficient in usage of lines than Case 2, because a small  
group of lines exhibits the symptoms of Case 1 in not absorbing  
traffic peaks. 
 
Case 4  Pick any line for outgoing calls 
------   
The risk of call collisions is highest (i.e. accidentally answering 
an incoming call during the 2-sec. (U.K.) silent period of ringing).   
(*) It is also the least effective way of bulking calls for per-line 
discounts. 
 
Case 5  Sequential hunt for a free outgoing line 
------ 
Combined with Case 2, this is the most efficient way to use  
lines. It is important that the subscriber's switch starts  
looking at the LAST incoming line first (the line with the  
greatest chance of being free), and then works backwards in  
sequence. 
 
3.  Improving call-handling efficiency 
    ---------------------------------- 
Step 1  Arrange your installation to apply Case 5 and Case 2. 
------ 
 
Step 2  Examine the number and distribution of calls on the last 
------  year's telephone bills to assess how many lines are needed.  
        There are three possibilities; not enough, the right number  
        or too many. 
 
Not enough lines   
---------------- 
A subjective test is that incoming calls regularly meet engaged  
tone and/or the business has difficulty finding a free outgoing  
line, even outside peak traffic periods. 
 
The first objective test is that with reverse hunting (Case 5)  
applied, the first choice outgoing line is not clocking up the 
bulk of calls (at least 75-80% on a 5-line or smaller group, at  
least 50-60% on a larger group up to 10 lines). 
  
The second objective test is that with Cases 2 and 5 both  
applied, significant numbers of outgoing calls (more than 5% in  
total) are appearing on late choice outgoing lines. 
 
Right number of lines 
--------------------- 
If the volume of incoming and outgoing calls is roughly equal,  
the lines required to carry the outgoing calls are mirrored with 
the rest to give a symmetrical arrangement, in which the  
"middle" line carries no more than 5% of outgoing calls. 
 
Too many lines 
-------------- 
If you improve a badly-designed installation by implementing  
both cases 2 and 5, as many as 50% of the lines can be ceased. 
(*) The "middle" line should be carrying around 5% to 10% of  
outgoing calls during the very busiest hour.  If 50% or more 
of lines are carrying no outgoing calls (apart from the odd few 
during peak demand), then lines can be ceased to achieve the 
symmetrical pattern described above. 
  
4.  Calculator for total lines required 
    ----------------------------------- 
(* totally revised to use the "busy hour" analysis method *) 
 
One Erlang is the equivalent of a single call lasting one hour. 
Note: I have allowed approx. 1 min. extra per call for dialling 
and the time spent listening to ringing tone and for no-reply calls. 
 
Total mins.    %age o/g calls  Lines required  Outgoing traffic 
in busy hour   on o/g line 1   out      total  (Erlangs) 
------------   --------------  ---      -----   --------------- 
   0 -  5         95%           1         2       0-0.1 
   5 - 10         85%           2         3-4       0.2 
  10 - 20         80%           2         4-5       0.3 
  20 - 25         75%           3         5-6       0.4 
  25 - 28         70%           3         6         0.5 
  28 - 34         63%           4         7-8       0.6 
  34 - 40         59%           4         8         0.7 
  40 - 50         56%           4         8         0.8 
  50 - 55         53%           4         8-9       0.9 
  55 - 60         50%           4         8-9       1.0 
 
You will see how much more efficient a larger group of lines is  
(4 outgoing lines can handle ten times as many calls as 1 line,  
with much better protection against short-term peaks). 
 
For larger installations (10 lines and above, or bills for over  
25,000 units per quarter), I suggest you seek specialist help.   
I can provide an initial diagnosis from your telephone bills,  
with suggestions for potential improvements and areas to  
investigate further. 
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