1. Introduction

In the extensive literature about the computation of automatic telephone exchanges, there is no attention, except for a few exceptional cases, for the application of Probability Theory to traffic problems, while the quantitative traffic problems themselves are almost completely ignored, even though their solution is of fundamental importance.

For the calculation of an automatic exchange, in the first place knowledge of the traffic volumes is required, before one can start to determine the required sizes of the different traffic paths based on these traffic volumes. In practice however, one usually has too little knowledge of the expected traffic, partially because there often is too little statistical data available about the existing traffic, and partially because, due to lack of experience or perhaps knowledge, one does not manage to draw from this data correct conclusions about the traffic densitities to be expected for the new exchanges.

Thus, the regrettable phenomenon occurs that one uses Probability Theory to dimension, rather precisely, the new traffic paths, but that the traffic volume data on which these calculations are based, are very inaccurate, sometimes even utterly wrong. Due to this, as is immediately obvious, the accuracy of the whole computation becomes problematic, because of the well known rule that the strength of a chain is determined by the weakest link. But furthermore one thus largely robs the usefulness from the meritorious work that has been done on the application of Probability Theory to telephony.

Furthermore, among telephone technicians one finds opinions about the use and the allowed manipulations on the given or computed traffic intensities which demonstrate a lack of correct insight in the meaning and nature of the traffic intensities.

The intention of this article is to once again draw attention to the traffic calculation itself, completely separate from the dimensioning of the traffic paths. For this purpose, a general overview of the nature of telephone traffic is given, the units of traffic intensity used and their meaning, with a more detailed discussion of the way in which statistical data should be used for determining expected traffic intensities and the computation of the traffic between the exchanges.

The ways in which, based on computed traffic intensities, the sizes of the traffic paths, waiting times, losses, etc. are determined, will thus not be considered.

2. Telephone traffic

In a modern telephone exchange any subscriber can, at any time of day or night, take the phone off the hook to set up a connection. The consequence is that telephone traffic has the typical property of being very irregular and showing large fluctuations.

If one has extensive and regular traffic statistics over an entire year, it turns out that in the end these fluctuations do have some regularity, despite the complete randomness of the starting time of each call.

If one makes a graph of the traffic amounts per hour, one will see fluctuations during the day (morning and afternoon), fluctuations with a duration of a week and long-stretched fluctuations over an entire year, with every now and then a pronounced peak.

At night the traffic drops to a very low value; also there is a dip at lunch and dinner time. The traffic is highest during the hours in which offices are open and people are at work.

Some days of the week can have more traffic than the other days. E.g., one gets the impression that on Saturday morning people already try to make up for the coming free afternoon.

Towns which have a peak season will during that season have a normal traffic which can be much higher than the traffic outside the season.

Also almost every exchange has some days during the year which show heavy traffic, such as the days before St. Nicolaas, Christmas and New Year.

All of this may be considered as generally known, but is mentioned here only because of some of the coming sections which discuss on which traffic values the dimensioning of an exchange must be based.

3. Units of telephone traffic

Two kinds of units of traffic are distinguished: one which simply indicates the number of calls, and a second one which takes into account the duration of the calls.

For this second unit one defines a unit call, and naturally multiple such units are in use. So far, we haven't introduced an international unit of traffic.

In the Scandinavian countries, people usually work with "traffic minutes". This unit is quite attractive and its use is increasing. It doesn't have the arbitrariness that affects other units and its size is such that calculating with decimals is not needed.

In America the unit call is based on an average holding time of 100 seconds. The (limited) advantage of this unit is that it leads to simple calculations, e.g. when multiplying numbers of calls by the average call duration, which is usually expressed in seconds.

Also the use of a traffic unit of 2 minutes is widespread. The choice fell on 2 minutes because town calls very often have approximately this as their average duration. This reason therefore only affects calculations concerning local networks, not inter-local traffic, which has a longer average call duration. For such traffic, sometimes a unit of 3 minutes is used.

Finally we want to mention the call unit of one hour. One could call this a theoretical unit, in contrast to the above practical units.

In the following table the ratios between the respective traffic units are given.

4. Traffic density

The number of calls on a traffic path is constantly changing. The number found at a certain instant is called the instantaneous traffic density. In analogy with electrical phenomena, the amount of traffic expressed in unit calls can be compared to the amounts of electricity, and the instantaneous traffic density to the electrical current.

The average value, around which the instantaneous traffic density fluctuates, is called the average traffic density or simply the traffic density. The period over which this average value is determined, is always one hour.

5. Units of traffic density

For the instantaneous traffic density there is obviously only one unit, namely the number of calls.

For the average traffic intensity there are two kinds of units, corresponding to the two units for telephone traffic, mentioned in Section 3. One unit is one call per hour and serves to indicate the traffic per hour, expressed in the number of calls, without considering the duration of the calls.

The second unit is a unit call per hour and thus one can speak of a number of call minutes per hour, a number of 2-minute calls per hour and a number of call hours per hour.

If the values for the average traffic density relate to the "busy hour", one adds this restricting clause and speaks of a number of "busy-hour calls" (B.H.C., busy hour calls), "busy-hour call minutes" (Sm, speaking minutes), "busy-hour unit calls" (E.B.H.C., equated busy hour calls) and "busy-hour call hours" (T.U., traffic unit; T.C., time calls; Belegungsstunden).

We do not see much justification for the abbreviations Sm., T.U. etc., firstly because they do not indicate that they refer to the "busy hour" and furthermore because then the simpler indications "minutes" and "hours" would suffice.

The unit "busy-hour call hours" could theoretically be called the unit of average traffic density, since it can be used without further derivation in the formulas of probability theory.

About the correct interpretation of the term "busy hour", more will be said in Section 9.

The goal of traffic calculation is determining the traffic densities of a new exchange to be expected for each of its traffic paths; we next want to devote some more words to the latter.

6. Traffic paths

Generally spoken, the traffic problem owes its existence to the fact that the means of communications use common, shared, traffic paths. In case of telephony the subscribers' calls are led through traffic paths that are common to larger or smaller groups of subscribers.

There are paths which are held only during part of the duration of a connection, such as is the case with control circuits and the personnel at the desks which, in the sense of traffic techniques, must be considered as a traffic path.

However, most traffic paths are continuously occupied during the entire duration of the conversation, such as the different groups of switches, connection wires, etc. By their nature they are the most important and therefore need most attention. Like all other traffic paths, they have nodes, where traffic comes together and goes apart again; they can be built for single-directional or bi-directional traffic, whichever is the economics of the operation requires.

7. Relation between traffic density and traffic path

The traffic paths are dimensioned based on the expected traffic densities using probability theory. Once the traffic densities are known, dimensioning the groups of traffic lines reduces in most cases to reading values off of curves which have been calculated for that. These curves have been computed based on formulas from Probability Theory and indicate in a simple way the relationship between traffic densities and the required numbers of traffic lines for the desired degree of service.

Several such curves exist, which may differ to some extent, depending on which assumptions and simplifications they have been based on, but which are quite similar in all cases, so that they give results which generally differ by only a few percent. The use of curves based on practical experience, dating back to the time when the theoretical treatment of such problems had not yet been completed, is not useful for the modern engineer.

8. Conditions for applying Probability Theory

If we allow the use of formulas from probability theory for telephony, the traffic data and the traffic itself must satisfy certain requirements.

In general, these conditions are not satisfied in telephony, from which it follows that there is no certainty that the outcomes of the calculations are correct. But based on experience it can be safely assumed that the approximation is quite accurate.

The first condition imposed on the traffic by probability theory is that there must not be a dependence among the calls. In this aspect, telephone traffic falls short, since there surely can be a relationship between successive calls of a subscriber. A clearly identifiable relationship is where a call is preceeded by one or more non-successful attempts which found the line of the desired subscriber busy. In general however it is assumed that, since the number of subscribers is relatively large, the effect of this error is small and can be neglected.

The second condition imposed by probability theory is that each call must have complete freedom to fall at any arbitrary time in the period under consideration. But as we have seen in section 2, the traffic density depends on the hour of the day. In the morning the instantaneous traffic density slowly increases to a maximum, and after some fluctuations decreases to a minimum in the evening. The starting times of the calls therefore are not totally free, but depend to some extent on time.

This circumstance demands more care than the previous one; it relates to the way the traffic data is determined which will serve as the basis for the traffic calculation of the exchanges. In cases where it is known that the traffic does not satisfy this second requirement of Probability Theory, we should set ourselves the goal to approximate it as close as possible by judicious use of the available statistical data and by suitable choice.

9. The "busy hour" terminology

In section 5, the word "busy hour" was already used, added as a restrictive clause to the units of traffic density. These "Busy hour traffic densities" serve as the basis for the further dimensioning of the exchanges and it is therefore important to define what is meant by this word. As will become clear in the following, a correct choice allows one to practically accommodate the second condition from probability theory as well as possible, even if it cannot be satisfied completely.

The term "busy hour" itself is not sufficiently determined and thus gives rise to random explanations and not allowed manipulations of the traffic quantities. After all, the expression "during the busy hour" does not specify whether one means the busy hour of a specific period, be it a day, week, month or year, or that one e.g. means the average traffic of the daily busy hours.

If we would base ourselves on the single busy hour of the entire year, then we would get as the result of the calculations such large groups of traffic lines, that even during that single busy hour there would be enough lines for the exceptionally high traffic of that moment, with as its consequence that a large part of the automatic material would have a very low efficiency during the rest of the year. This material would have to be bought and maintained, so the economics of the operation would be seriously harmed.

Such an interpretation of the term busy hour is therefore not acceptable. Thus the thought arises to use an average over several hours for the "busy-hour traffic density", just like the traffic density is already the average over one hour. In the choice of these hours we let ourselves be guided by the principle of satisfying as much as possible the second condition from probability theory mentioned in section 8. This condition is that the expectation and, by consequence, the traffic density, must not be a function of time.

To achieve this, out of the 8700 hours of a year, we only select a bit more than 300 hours for the further traffic calculations. Thus, from every day we only take one hour into consideration, and each such hour starts at the same moment of the day. We exclude Sundays and holidays.

Practically one works by adding the 300 values of the quarter from 9 till 9¼ and computing their average, doing the same for the quarter from 9¼ till 9½ and so on for the busy times of the day. We call the four consecutive quarters which give a maximum the busy hour. So it would be better to speak of an "average busy hour" taken over the year.

The idea behind the above is that the expectation that a subscriber makes a call at a specific moment of one day, is the same as that for the same moment of a random other working day over the entire year and that the variations over the year are random and independent of the time.

In this way, the condition of probability theory is approached, to the extent that that is within reach, although we know that even then non purely random deviations remain, such as the busy days before holidays, trade fair days, etc. For these days therefore no guarantee is given regarding the quality of service. This is nothing extraordinary and happens with every other kind of traffic paths, with only the difference that in the telephone operation there is the disadvantage that the public does not see the increased traffic demand with their eyes.

Furthermore there is still the non-random inaccuracy that the traffic tends to amass a bit in the middle of the hour. The influence of this is usually ignored, particularly since usually [toeslagen] are given for other reasons. Besides, this inclination is not of much importance, since the indicated average traffic curve is rather flat at the point of its maximum, so even rather important forward or backward moves of the busy hour cause only small changes in the traffic density.

Now the question arises whether it is permissible to apply the formulas from probability theory to such average "busy-hour traffic densities". This can in general be very firmly answered affirmatively. The formulas used most, such as those from Grinsted, Poisson, Erlang and Molina, all assume that the given traffic density is an average taken over an infinitely large number of hours.

10. Manipulations of traffic densities

In telephone exchanges there are often nodes where traffic paths come together and the amounts of traffic they carry merge, and other points where a traffic path and the traffic it carries are split.

Let's consider such a node, to study what relationship there is between the average traffic densities before and after the split.

If we neglect small differences, due to the selecting and searching of the switches, we can observe that at any random moment the number of calls before and after the split is the same, since these are the same calls. The instantaneous traffic density $v$ before the split is therefore equal to the sum of the instantaneous traffic densities after the split. Calling the latter $v_1, v_2, v_3, v_4,$ etc., then $$v = v_1+v_2+v_3+ \mbox{etc.}$$

These instantaneous traffic densities are entirely independent functions of the time. For their average values (traffic densities) we can write: \[ \begin{array}{rcl} V & = & \displaystyle \int_0^1 v \, dt, \\ V_1 & = & \displaystyle \int_0^1 v_1 \, dt, \\ V_2 & = & \displaystyle \int_0^1 v_2 \, dt, \;\; \mbox{etc.} \end{array} \]

The sum of the average traffic densities after the split equals \[ \begin{array}{rcl} V_1 + V_2 + V_3 \ldots \mbox{etc.} & = & \displaystyle \int_0^1 v_1 \, dt + \int_0^1 v_1 \, dt + \int_0^1 v_1 \, dt + \mbox{etc.} \\ & = & \displaystyle \int_0^1 (v_1 + v_2 + v_3 + \mbox{etc.} ) \, dt \\ & = & \displaystyle \int_0^1 v \, dt \\ & = & V \end{array} \]

Thus, when traffic paths split or join, the sum of the average traffic densities remains unchanged before and after the node.

This only holds if we take the different averages over exactly the same period, i.e., what we find as the average busy hour before the split is also the average busy hour for each of the traffic paths after the split.

If one has extensive statistical material at one's disposal, the collection of which is very expensive and therefore rather rare, one already notices upon casual observation that the daily busy hour of a traffic path is erratic, even more so as it carries less traffic. However, in determining the average busy hour of this traffic path, one will usually notice that it coincides quite accurately with the average busy hours of the other traffic paths of the exchange. Taking the average value over a large number of hours, as described in the previous section, thus has a strong leveling effect. Drawing from the statistical material at our disposal, it was checked for a specific net what fluctuations the busy hour of the different traffic paths, averaging over one month at a time, was subject to, but neither between the different months of the year, nor between different traffic paths, a significant deviation was noticed. This held both for the main traffic arteries and for the connection lines to smaller exchanges.

As goes without saying, this is not a generally valid conclusion and in practice differences may occur; however, these will always indicate an uneven structure of the net. In case this should happen in practice, one acts most easily by assuming for the calculation of the expected traffic densities that there is no such displacement, and then applying a "toeslag" at the end of the calculation, the magnitude of which one determines practically.

Among telephone technicians there is a rather widespread opinion, which is defended by theoreticians like dr. Lubberger and dr. Rückle, that when traffic densities merge, the combined traffic is less than the algebraic sum of the components, and that similarly at a split of a traffic path the sum of the traffic densities after the split is larger than the unsplit traffic before the split. On the basis of the above proof that traffic densities can be added algebraically, if their average busy hours coincide, and furthermore on the basis of what is observed in practice, namely that this is usually the case, we already can conclude that this opinion is not correct.
[Dr. Lubbberger, Die Wirtschaftlichkeit der Fernsprechanlagen für Ortsverkehr. Verlag von H. Oldenbourg. Chapter II G. Dr. Rückle and Dr. Lubberger. Der Fernsprechverkehr als Massenerscheinung mit starken Schwankungen. Verlag von Julius Sprinter. Chapter IX A.]

We want to briefly mention a few points based on which we feel that the authors of the cited books draw incorrect conclusions. In chapter IX.4 of the second book the following reasoning is used: $S$ subscribers together set up $C$ connections during the busy hour, each with a duration of $t$ seconds. The system is divided into groups of $s$ subscribers, which make a traffic of on average $C \frac{s}{S}$ connections per group. Some of these groups will carry fewer connections during the busy hour than the average, while other groups will carry more connections. The overloaded groups will then have a loss which is larger than what is assumed in the calculation, and the underloaded groups will have a smaller loss. The increase of the loss in the former groups will, as is known, exceed the decrease of the latter, so that together the groups will have a larger loss than what was expected based on their average traffic.

The error in this reasoning is that they assume that the traffic of the groups is variable, but that the total traffic of all groups is constant and equal to precisely $C$ calls. This error shows most clearly if one assumes that the number of groups is just two. According to the normal opinion the average traffic densities of of the groups must fluctuate independently of each other around their average values, while the authors tie the traffic of the second group to a value of $C$ minus the traffic of the first group. This is a totally arbitrary and inadmissible restriction, which is imposed on the traffic fluctuations and limits the free fluctuation.

A reasoning based on this is therefore not acceptable.

Furthermore it turns out that the cited authors do not base their theory on average values taken over a number of hours, but on higher values. This shows particularly clearly when they compare the outcomes of their theory to values observed in practice. For example, to support his theory dr. Lubberger compares, on page 26 in the first book mentioned above, maximum traffic values to each other, which he obtains from table II in O'Dell's writeup "The influence of traffic on automatic exchange design", which was published as no. 85 of the L.P.O.E.E.

It goes without saying that if one determines the size of the traffic paths on the basis of a traffic which is higher than the average, one can apply reductions, since the traffic peaks will not or only rarely coincide. Since however the traffic densities which serve to calculate the exchanges represent average values, we have to reject this theory based on its assumptions.

One question we would like to raise is why a reduction or "toeslag" is only applied when joining or splitting traffic densities, but not when it's about finding the product of variable quantities. Such a case occurs in the calculation of the traffic density from the number of calls and the average call duration. In appendix 1 we have shown, albeit superfluously, that also for this manipulation of variable quantities their average values must simply be multiplied. Should there be truth in a reduction theory, then it should also be extended to the multiplication of variable quantities.

11. Originating traffic

The three main pieces of data for designing a telephone exchange are: 1° the number of subscriber lines, 2° the number of calls per subscriber during the average busy hour, and 3° the average duration of the calls.

From these three numbers one calculates the total traffic which is sent by the subscribers and which we want to denote by the name of "originating traffic".

In a similar vein, we speak of "terminating calls" and "terminating-traffic densities", and in contrast to this we call the traffic that is received by the subscribers "end traffic".

The number of busy-hour originating calls per subscriber varies within rather wide bounds and depends on several well-known factors. One extreme is formed by the subscribers of exchanges which are in the business district of the city; the other by those of the rural exchanges.

The number of busy hour originating calls per subscriber for exchanges with call counters fluctuates in normal cases between the bounds 0.8 to 2.0. The latter value is only rarely exceeded and cases where such a high value occurs, must be checked carefully for correctness. For example, we know of a case where the value of 2.0 was significantly exceeded, but once the exchange was put into service, the imposed value of the originating traffic turned out to be a severe mistake, with the consequence that part of the invested capital was almost non-productive.

The cause of this must be found in interpreting traffic statistics with little experience and a lack of expertise, or also in applying safety coefficients improperly.

In case of uncertainty, there is a means to prevent the incorrect use of safety coefficients, which, however, is seldom or never applied, possibly because of unfamiliarity. This means consists of not applying the coefficients to the traffic per line, but to a number of lines. Let us suppose that a new exchange will have to have 1000 lines and that each of these lines has an average busy-hour originating traffic density of 2.2 minutes, in which a safety margin of 10 % is included. If one is uncertain whether this percentage is correct, it is to be recommended to design an exchange with 1100 lines, but with a traffic of 2 minutes per line. The purchase costs for the latter exchange are a little higher, but as far as traffic capacity is concerned, both are equivalent. The advantage achieved in the latter case however, is that if the 10 % safety margin turns out to be unneeded, one can now connect 10 % more lines to this exchange.

The originating traffic per line is determined using periodic observations in the way described in section 9. We assume the means for this are known. However, we do want to spend a few words on the factors which influence a "toeslag" on the originating traffic.

On the one hand, in a large net, where the number of connections per 100 inhabitants is still increasing, it becomes increasingly possible to reach a person by telephone, and as a consequence the occasions for a phone call increase. On the other hand however, in this increase also those people get involved in the telephone traffic who are not driven by necessity, but for whom being connected is more or less a luxury. These are the small-scale consumers.

Third, the automation of the telephone operations has a stimulating effect on the traffic and of course particularly there, where the quality of service previously left something to be desired. This factor is particularly influential in the automation of regional networks and of interlocal traffic. An incidental consequence of automation is, that the number of calls increases, but that the average duration of the calls decreases. The cause of this is that the handling of the traffic is made so much easier by the automation.

How these influences balance each other, is hard to say and must be determined by experience. In general however one can say, that as a rule, the "toeslag" must be taken larger as the traffic per line is less.

Furthermore, when determining the basis for the calculations, one must realise that the number of subscribers connected to an exchange in normal cases is at most 90 % of the full capacity. This because of the fact that a telephone number of a subscription that is cancelled, cannot be immediately assigned to a new subscriber. There's a safety factor hidden in this, if the calculation didn't take this into account yet.

Since call counters have already been deployed widely, we omit a discussion of the influence of their introduction on the traffic.

12. Terminating traffic

If in a town a single exchange provides the telephone connections, the terminating traffic is necessarily equal to the originating traffic, at least if one excludes prematurely disconnected connections, glowers, and interlocal traffic. If the originating traffic is given, then in such a case in principle the terminating traffic is known.

The situation is different in a net which comprises multiple exchanges. Knowledge of the terminating traffic is not determined by that of the originating traffic, but the traffic sent out by the subscribers of one exchange can be markedly different from the traffic that they receive.

It is a common phenomenon that an exchange draws more or less traffic to itself than it sends out to the other exchanges of the same net. In a certain sense one can call this a characteristic property of such an exchange. It also happens that this property is variable and reverses according to the hour of the day. Such a phenomenon can be the reason to advantageously install bidirectional lines.

So far, not much practical importance has been attached to knowledge of the terminating traffic, but in our opinion it has special value for the calculation of the traffic among the exchanges of a town or regional net. This will be demonstrated in the next section.

13. Incoming and outgoing traffic

In most cases, the outgoing traffic density from one exchange to another exchange turns out to be approximately equal to the incoming traffic density from that same exchange. Often however, quite significant differences occur between the traffic densities on the same route but in opposite direction and one definitely needs to take this fact into account when computing the future traffic densities. As said before, these differences are related to the characteristic properties of the exchanges, so they cannot be ignored.

A second point that we want to draw attention to, is that in distributing the traffic of one exchange among the other exchanges of the same net, the influence of the number of lines of each other exchange is only of secondary importance. Of primary importance is what could be called the traffic value of a line, by which we want to indicate that from a traffic-technical point of view not every line is equivalent to any random other line of the same net. A line which sends out more traffic will, concerning the incoming traffic, have to be counted heavier than a line which sends out less traffic. This principle is already used in the calculation of the groups of final selectors, which connect to the lines of home-exchanges. It is therefore not correct to base traffic calculations on the number of lines of the different exchanges, like dr. Lely does in his dissertation on "Waarschijnlijkheids-rekening bij automatische telefonie", but one must do this on the basis of the actual traffic values.

The theoretical traffic distribution of a net is of importance for judging the true distribution. It is also assumed as the traffic distribution, if little is known about the net. For this theoretical distribution one distributes the originating traffic densities of the exchanges over all exchanges of the net in proportion to their originating traffic densities. If a net consists of two exchanges, which have originating traffic densities of 1000 and 1500 min, then one exchange receives from itself a theoretical traffic of 1000 × 1000 : (1000 + 1500) = 400 min and from the other exchange 1500 × 1000 : (1000 + 1500) = 600 min. So, in this theoretical traffic distribution the originating traffic is always the same as the terminating traffic, while the same holds for the traffic density in both directions on each single route.

In the calculation of the mutual traffic densities between two exchanges, until now often use was made of so-called community factors. From Appendix 3b it turns out that the use of such factors gives rise to wrong outcomes and is therefore unusable in practice. After all, it turns out that when enlarging a net, the community factors are subject to change, which is at odds with the assumption on which they are based, which presupposes that they are constant. Because of this contradiction between the assumption and the result it is certain that the method of interest factors is useless for the calculation of the mutual traffic among a group of exchanges.

For those who are perhaps not familiar with the term "community factor" or perhaps use a different name for it, we want to note that they indicate the ratio between the true and the theoretical traffic between two exchanges.

Instead of the method of community factors, until recently we often used a method which we called the "method of ratios". This method gives quite reliable outcomes for the practical calculation of the mutual traffic densities to be expected. It is described extensively in appendix 3c of this article, but because it also has several errors and inconsistencies, we have abandoned it in favor of a newer and probably the best method, namely that of "double factors".

This original method is explained extensively in appendix 3d.

The general problem that shows up in the calculation of the traffic densities for the connection lines between several exchanges is that we, knowing the complete "traffic picture" for a certain period, search a corresponding picture for a new situation, which arises after several, usually not proportional, extensions or possibly even new exchanges are added to the net.

A purely theoretical treatment of this problem currently does not exist, but it is conceivable that it could be worked out using a function which indicates the affinity between subscribers. In this context we would like to casually note that proposition X of the PhD thesis of dr. Lely cannot be correct. This proposition says that the affinity between two random subscribers is inversely propotional to the distance separating them. Appendix 2 of this article shows, using a simple integration, that the affinity must be a function of at least the third power of the inverse of the distance.

The method of double factors, described in Appendix 3d, satisfies the following general requirements that we have formulated on the basis of practical experience.

a) Reversible, which means that starting from an initial situation using the method one calculates the final situation, and that one can, working backwards with the same method, from the final situation return to the original initial situation.

b) Partitionable, which means that the final situation is independent of the path followed, so that for example a random intermediate situation, derived from the initial situation, can be inserted.

c) Interchangeable, which means that if one reverses the direction of all traffic densities in the initial situation, one obtains, for the given values of the final situation, a final situation which is identical to the one obtained along the normal way, if one also there reverses the direction of all traffic densities.

d) Splittable, which means that we can combine or split exchanges from the initial situation without affecting the traffic densities not involved in this of the other exchanges. Theoretically the method of double factors does not satisfy this requirement, but practically the inaccuracy made is small.
It is unlikely that a method could exist which satisfies this requirement entirely.

14. Introduction of new exchanges

In the practice of traffic calculations it often happens that in a town network a new exchange is built, while one naturally does not have statistical data available for it. The data that one has are usually limited to knowledge of the kind of subscribers in traffic-technical sense, which will be connected to the new exchange, because this exchange usually serves to reduce the load on an existing one.

The method of community factors would give a solution to this, by for example assuming new factors for the new exchange after comparison to the factors of the existing exchanges. The outcome obtained this way is very inaccurate, unreliable and therefore unusable. In this case the method of double factors is again helpful, as described in Appendix 4.

This appendix discusses a new type of community factors, which however are based on different principles than the ones we rejected. We call them "base numbers".

15. Conclusion

In the above text and in the accompanying appendices we have tried to give an overview of the traffic calculation for telephony, a topic about which so far not much constructive has appeared.

A distinction has been made between the actual traffic calculation and probability theory, which are typically not distinguished in practice and in previous literature.

Some theories which consider certain manipulations with traffic densities defendable, have been demonstrated to be fundamentally wrong, while also the correct way of calculating has been shown.

Furthermore a new method is developed which serves the calculation of the mutual traffic densities to be expected, as needed when extending existing or introducing new exchanges.

First appendix

Multiplication of independent variable quantities

Let us assume some independent variable quantities $g_1, g_2, g_3, \mbox{etc.},$ which vary with the parameters $p_1, p_2, p_3, \mbox{etc.},$ and which during periods 0 to I, 0 to II, 0 to III, etc. fluctuate around their mean values $n_1, n_2, n_3,$ etc., then the average value of the product of the variables becomes: \[ G = \int_0^I \int_0^{II} \int_0^{III} \ldots g_1 . g_2 . g_3 \ldots dp_1 .\, dp_2;\, dp_3\, \ldots \] Since $g_1, g_2, g_3,$ etc., and $p_1, p_2, p_3,$ etc. are independent of each other, we may write \[ G = \int_0^I g_1 .dp_1 \times \int_0^{II} g_2 .dp_2 \times \int_0^{III} g_3 .dp_3 \;\;\;\mbox{ etc.} \] or: \[ G = n_1. n_2. n_3. \mbox{ etc.} \]

Therefore the average of the product of several independent variable quantities equals the scalar product of their average values.

Second appendix

Affinity

If we suppose that the affinity between subscribers of a theoretical telephone net of uniform density is inversely proportional to the distance, as dr. Lely assumes in proposition X of his PhD thesis, then one can write for the total traffic of one subscriber: \[ V = C \int_g^\infty \frac{1}{r} 2 \, \pi \, r \, . \, dr, \] where $C$ is a constant and $g$ is an inner limit of e.g. 750 m. By integration we find: \[ V = 2 \pi \, C \, I_g^\infty \, r. \] Since the value of the integral for the integration limit $r=\infty$ becomes infinity, the function which represents the value of the affinity must be of a higher power.

Further study shows that the affinity must be a function of at least the third power of the inverse of the distance between subscribers in such an idealised net.

Third appendix

Traffic among exchanges

In practice, the problem repeatedly occurs of calculating, starting from a known traffic distribution among a group of exchanges, the traffic distribution for a new situation to be expected at a future moment.

In the sequel we want to treat several generally known methods, such as the method of percentages and the method of community factors, and then proceed to an improved method, namely that of ratios, and finally describe a calculation method which does not have the deficiencies of the previous methods.

3a) Percentual method

One still encounters this deficient method all too frequently in practice. That we still give it some space here is mostly for putting forward some reasonable requirements that a good method must satisfy.

We start out from the assumption that the traffic picture of the initial situation is known. An example of such a "picture" is found in table 6. Now, using the method of percentages one calculates a second table, in which the various mutual traffic densities of the initial situation are expressed in percentages of the originating traffic densities of the sending exchanges. Next, one multiplies these percentages by the new originating traffic densities and assumes to thus have achieved a good picture of the new situation.

Only if the net expands evenly does this calculation method give reliable outcomes. But since this case rarely or never happens, it constantly leads to evidently incorrect results. A simple example will elucidate this sufficiently:

Be the initial situation for two exchanges A and B:

Based on these numbers we find that the traffic for both exchanges and for both directions is equal and 50% of the originating traffic densities.

Let us now assume for the new situation that only one exchange is extended and that one has good reasons to assume that the originating traffic from this office will double, while that of the second office will remain practically the same. I.e., a case where the old telephone building has no space for further expansion and all new lines need to be connected to the other office.

If we apply the method of percentages to this case, then one finds for the future traffic from A to B: $\frac{50}{100} \times 2000 = 1000$ min, and for that from B to A: $\frac{50}{100} \times 1000 = 500$ min. From these two traffic values it thus appears that, while originally the traffic in both directions was the same, this has now changed substantially. The equilibrium that was present originally, would now be highly broken. Although causing differences between incoming and outgoing traffic is not a criterion for the correctness of a method, it needs to be noted that something totally random occurred here. A shift has taken place for which no reasonable ground can be found.

The absurdity of what happens becomes most clear when one compares the traffic densities of both exchanges before and after the expansion. For one exchange the terminating traffic density per line turns out to decrease from 1.0 to 0.75, and for the other to increase from 1.0 to 1.5 per line. In practice this would mean that one could transport "eindkiezers" from one exchange to the other.

This concludes the discussion of the method of percentages. Calling it totally unusable is not an overstatement.

3b) Method of community factors

The method of community factors starts from the idea that the traffic density between two exchanges does not depend on the originating traffic density of only one exchange, but of both. It is based on the principle that the ratios between the observed and the so-called theoretical traffic densities remain unchanged upon expansion. These ratios are called community factors.

As theoretical traffic densities one denotes the traffic distribution which is obtained by distributing the originating traffic of one exchange over the different exchanges, in proportion to their originating traffic densities.

Let us set the observed originating traffic densities of exchanges 1, 2, 3, etc. to $_a V_1,\, _a V_2, \mbox{etc.}$ (total $V$); the observed traffic densities between exchanges to $v_{11}$ (from exchange 1 to 1), $v_{12}$ (from exchange 1 to 2), etc.; the theoretical traffic densities for the initial situation to $t_{11}, t_{12}, t_{13}, \mbox{etc.}$, then: \begin{equation} \label{eq:1} \renewcommand{\arraystretch}{2.2} \begin{array}{rcl} t_{11} & = & \displaystyle \frac{_a{V^2}_1}{V} \\ t_{12} & = & \displaystyle \frac{_aV_{1} \, . \, _a V_2}{V} \;\;\;\;\; \mbox {etc.} \end{array} \end{equation}

The community factors $c_{11}$ etc. give the ratio between the observed and the theoretical traffic density: \begin{equation} \label{eq:2} c_{11} = \frac{v_{11}}{t_{11}}, \;\; c_{12} = \frac{v_{12}}{t_{12}}, \;\; \mbox{etc.} \end{equation}

If we furthermore set the new originating traffic densities to $_a X_1,\, _a X_2, \mbox{etc.}$ (total $X$), then this method is based on the equality of the following ratios: \begin{equation} \label{eq:3} \renewcommand{\arraystretch}{2.2} \begin{array}{rclclclcl} c_{11} & = & v_{11} & : & \displaystyle \frac{_a{V^2}_1}{V} & = & x_{11} & : & \displaystyle \frac{_a{X^2}_1}{X} \\ c_{12} & = & v_{12} & : & \displaystyle \frac{_aV_{1} \, . \, _a V_2}{V} & = & x_{12} & : & \displaystyle \frac{_aX_{1} \, . \, _a X_2}{X} \;\;\;\;\; \mbox{etc.} \end{array} \end{equation}

From this formula the sought-for $x_{11}, x_{12}, \mbox{etc.}$ can be calculated.

In practice one does not work with formulas, but preferably makes use of a series of consecutive operations, whose outcomes one puts in tables each time.

The first table then contains the given traffic densities among exchanges, with the total corresponding originating traffic densities:

Among these values there is the relationship: \begin{equation} \label{eq:4} \renewcommand{\arraystretch}{1} \begin{array}{rcl} _a V_1 & = & v_{11} + v_{12} + v_{13} + \ldots \\ _a V_2 & = & v_{21} + v_{22} + v_{23} + \ldots \\ \mbox{etc.} \end{array} \end{equation}

From the given originating traffic densities one next calculates the theoretical traffic densities and again arranges them in the form of a table:

The $t$ values of Table 2 now satisfy the equations (1).

Now the community factors are calculated from Tables 1 and 2, by dividing the values from Table 1 by those of Table 2.

The $c$ values from Table 3 satisfy the equations (2).

Fourthly, one calculates from the given initial traffic densities of the new situation the corresponding theoretical mutual traffic densities, as follows:

In which: \begin{equation} \label{eq:5} u_{11} = \frac{_a{X_{1}}^2}{X}, \;\;\;\; u_{12} = \frac{_aX_{1} \, . \, _a X_2}{X}, \;\;\;\; \mbox{etc.} \end{equation}

Finally, one multiplies the values from table 3 by those of table 4 and obtains:

If all were well, these values from Table 5 should be the new mutual traffic densities, calculated using the community factors method. Upon applying it to numerical values it turns out that the sums of the values of the horizontal rows do not equal the given originating traffic densities of the new situation. After all, even though it holds that: \begin{equation} \label{eq:6} \renewcommand{\arraystretch}{1} \begin{array}{rcl} _a X_1 & = & u_{11} + u_{12} + u_{13} + \ldots \\ _a X_2 & = & u_{21} + u_{22} + u_{23} + \ldots \\ \mbox{etc.} \end{array} \end{equation} Then it does not necessarily hold that: \begin{equation} \label{eq:7} \renewcommand{\arraystretch}{1} \begin{array}{rcl} _a X_1 & = & c_{11} . u_{11} + c_{12} . u_{12} + c_{13} . u_{13} + \ldots \\ _a X_2 & = & c_{21} . u_{21} + c_{22} . u_{22} + c_{23} . u_{23} + \ldots \\ \mbox{etc.} \end{array} \end{equation}

Where this method is applied, one usually works by ascribing the error to the local traffic. By this rather arbitrary action, one satisfies the requirement that the sums of the values of a horizontal row must equal the total originating traffic, but it has as a consequence that all community factors change. And the method was based precisely on the principle that these factors are constant.

In summary, we can therefore say that working with community factors is fundamentally incorrect, since it contradicts itself. It therefore satisfies none of the requirements formulated in sections 13a through d.

3c) Method of ratios

The application of a simple correction factor to the method of community factors leads to the method of ratios. Thus, with this method one can work as described above for the method of community factors up to Table 5, and then apply a correction, which e.g. for the values of the first horizontal row equals: \[ \frac{_a X_1}{c_{11} . u_{11} + c_{12} . u_{12} + c_{13} . u_{13} + \ldots } \]

For the values of the second row: \[ \frac{_a X_2}{c_{21} . u_{21} + c_{22} . u_{22} + c_{23} . u_{23} + \ldots } \]

In this way one achieves that the sum of the horizontal rows becomes equal to the originating traffic densities. The error is now distributed proportionally over all values.

In summary the method of ratios gives, after introducing some simplifications: \begin{equation} \label{eq:8} \renewcommand{\arraystretch}{2} \begin{array}{rcl} x_{11} & = & v_{11} \displaystyle \frac{_a X_1}{_a V_1} \frac{_a X_1}{\displaystyle v_{11} \frac{_a X_1}{_a V_1} + v_{12} \frac{_a X_2}{_a V_2} + v_{13} \frac{_a X_3}{_a V_3} + \mbox{etc.} } \\ x_{12} & = & v_{12} \displaystyle \frac{_a X_2}{_a V_2} \frac{_a X_1}{\displaystyle v_{11} \frac{_a X_1}{_a V_1} + v_{12} \frac{_a X_2}{_a V_2} + v_{13} \frac{_a X_3}{_a V_3} + \mbox{etc.} } \\ \mbox{etc.} \end{array} \end{equation}

Rewritten in a different form: \begin{equation} \label{eq:9} \renewcommand{\arraystretch}{2} \begin{array}{rcl} x_{11} & = & c_{11} . {}_a X_1 \displaystyle \frac{_a X_1}{c_{11} . {}_a X_1 + c_{12} . {}_a X_2 + c_{13} . {}_a X_3 + \mbox{etc.} } \\ x_{12} & = & c_{12} . {}_a X_2 \displaystyle \frac{_a X_1}{c_{11} . {}_a X_1 + c_{12} . {}_a X_2 + c_{13} . {}_a X_3 + \mbox{etc.} } \\ \mbox{etc.} \end{array} \end{equation}

The values $c_{11} . {}_a X_1$, $c_{12} . {}_a X_2$ are called the ratios, from which the method derives its name.

In order to work quicker in practical applications, one only needs to follow the method of community factors until Table 3, and then proceed immediately to calculating the ratios. A fifth table then contains the traffic values of the new situation.

A general disadvantage of this method is that it is not transparent, but especially, that it does not satisfy the requirements given in sections 13 a to d. Stated succinctly, this method boils down to calculating a number of unknowns from a small number of givens and using some formulas of which we do not even know the meaning. We can escape this using the next method.

3d) Method of the double factors

This method is based on a symmetric treatment of the problem. While all previous methods were based on only the new originating-traffic densities, with this method we also introduce the new terminating-traffic densities as knowns.

Thus we get more control of the development of the problem, we achieve a symmetrical way of solution (e.g., we can reverse the direction of the traffic densities) and determine in advance the kind of subscribers, seen from a traffic-engineering standpoint.

Especially the latter is of fundamental importance.

Whether an exchange produces more traffic than it draws to it, is a typical property, which to some extent depends on the type of subscriber. Thus, for new subscribers that are to be involved in the telephone traffic, it needs to be determined in advance and in comparison to the characteristics of the already connected subscribers, to what extent they will increase or decrease the existing differences between originating and terminating traffic. All previous methods simply left this to chance, which in our opinion is inadmissible.

With the method of double factors we introduce a horizontal and a vertical relationship between the old and new traffic values, as follows: \begin{equation} \label{eq:10} \renewcommand{\arraystretch}{1} \begin{array}{rclrclrcl} x_{11} & = & p_1 . q_1 . v_{11} ; & x_{12} & = & p_1 . q_2 . v_{12} ; & x_{13} & = & p_1 . q_3 . v_{13} ; \\ x_{21} & = & p_2 . q_1 . v_{21} ; & x_{22} & = & p_2 . q_2 . v_{22} ; & x_{23} & = & p_2 . q_3 . v_{23} ; \\ \mbox{etc.} \end{array} \end{equation}

If the existing traffic picture is known, and the originating and terminating traffic densities of the new situation are given, it is possible to express the new traffic values among the exchanges in formulas. However, these formulas are rather complicated and their use is cumbersome, which is why we prefer to recommend a more practical method, which is demonstrated using the following example. It consists of progressively approximating the sought-for mutual traffic densities, starting from the given situation. Let the given traffic distribution be the following:

In the next table, one first writes the given originating and terminating traffic densities. Next one writes the individual values from Table 6 in the first horizontal row but multiplied by the quotient of the old and the new originating-traffic densities, i.e., $\frac{6\,000}{4\,000}$; for the second row this factor will become $\frac{4\,000}{3\,000}$ etc. However, the sums of the vertical rows of the new table will now differ from the terminating-traffic densities expected for the new situation. These sums have been added separately at the bottom of Table 7.

The second step of the approximation happens by multiplying the values of the vertical columns by the quotients of the demanded terminating-traffic densities and the summations found. Kolom 1 therefore will be multiplied by $\frac{6\,225}{5\,690}$, and column 2 by $\frac{4\,000}{4,030}$, etc.

This way the values of Table 8 are found. In this table the sums of the horizontal rows will now not match the values of the given originating-traffic densities. These sums have therefore been put beside the table.

This is repeated several times, until the differences between the sums and the originating and terminating traffic densities are so small that, with a minor rounding here and there, the following final situation is achieved: (See Table 9).

One sees from this example that the approximation goes quite quickly. If we compare this to the outcomes obtained using the other three methods, then their inadequacy shows clearly. In the next table, the top values of every group of three numbers represent the traffic values found using the method of percentages. The second series of numbers is calculated using the method of community factors, while the third number has been obtained using the method of ratios.

It turns out that for this particular case, the method of community factors gives the best outcome compared to the method of double factors (Table 9). However, this is only coincidence.

Its reason is that the values which we assumed for the various terminating-traffic densities happen to be close to the values found by the method of community factors.

Furthermore this example shows that the other methods give terminating-traffic values which are rather random. E.g., with the method of ratios, exchange 1 would not draw more traffic than it gives, as is the case in the given traffic picture. Exchange 2 would have taken over this role.

In section 13 of this article some requirements are given which a sound method should satisfy. The proof that the method of double factors satisfies them, is simple and is based on the symmetrical treatment of the problem. We therefore omit it.

Fourth appendix

Introducing new exchanges

By using the properties of the method of double factors, we can easily and reliably add one or more new exchanges to the net.

To do so, we use the property that any intermediate state can be inserted between the given and the sought-for situations. The intermediate state that we insert for this calculation, has values for the originating and terminating traffic densities which are equal in number of units to the number of exchanges.

Applied to the example from appendix 3 we find for the individual values, using the continued calculation method, the following intermediate state:

The values in this table are somewhat analogous to the community factors discussed in Appendix 3b. However, while we saw that these did not stay constant when the net is extended, those numbers are independent of that. The principle that one aimed for by introducing the community factors, namely searching for a few fundamental values, independent of the situation (state) and that would thus represent the intrinsic ratio between the exchanges, turns out to be realised with this.

In order to introduce a new exchange, one adds e.g. in the above case a fifth horizontal row and a fifth vertical column, and fills both of them with estimated values, chosen in comparison to the values of the existing exchanges and on the basis of local knowledge of the net. It goes without saying that then neither the horizontal nor the vertical rows will have a sum equal to the number of exchanges of the net in the new situation. The base numbers thus undergo a change upon introducing new exchanges. But that we have in principle not changed anything to the old base numbers and that these are still present hidden in the new situation, appears from the fact that one can go back from the new to the old situation by simply omitting the values of the fifth row and column, followed by the approximation method to return to the given situation.

This new situation is of course calculated from the extended table of base numbers using the method of double factors.

Thus, it turns out that the method is also reversible in this regard.

We would like to give to this new kind of "community factors" the name of "Base numbers".

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