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Traffic Engineering Basics

Traffic engineering is the unsung hero of telephone system design.  It’s the invisible hand that optimizes calling performance, prevents congestion, all while keeping operational costs in check. 

Imagine being tasked with interconnecting 35 exchanges with talking paths (trunk lines) such that the expected call load is met in a cost-effective manner. There are 10^1011 ways to do this, a 1,011 digit number (Appendix C). 
This section covers the what and how of traffic engineering. 

Exchanges are composed of interconnected switches to provide simultaneous talking paths for subscribers. The switches are connected in a fabric with “trunk lines” to carry the conversations between switches or offices. For example, in Fig 1 [Miller] we see each subscriber has a dedicated “line switch” (on the left) and these 100 switches access only 10 “line connector” switches (right side) via the 10 trunk lines (middle). A purpose of a line switch is to choose a free connector (or selector) for its associated subscriber when they go “off-hook.”  Line switches are discussed elsewhere on this site. 

Fig 1. 100-line step-by-step exchange with switches and trunks circa 1925 (redrawn) 

This means there can only be 10 simultaneous any-to-any conversations in this exchange. The 11th person going “off hook” must wait for an existing call to end. It’s economically impractical to design a system where 50 callers can speak to the other 50 subscribers—this will never happen in practice. This type of constrained access architecture was a practical and common design approach for telephone exchanges of that era.

The art of traffic engineering is to select the minimum number of trunks (talking paths) and connector switches to meet the practical needs of a community and make the system cost-effective to build and maintain. Is the design in Fig 1, with 10 connectors, practical using these criteria? The discussion to follow will help answer this type of question.

Quality by Design

In the field of telephony, Quality of Service comprises all aspects of a telephone call: time to dial tone, noise, crosstalk, echo, frequency response, loudness levels, and more. A subset of telephony QoS is Grade of Service (GoS). This comprises aspects of a connection relating to capacity and call loss only.  Its units are probability of loss (of a call being dropped). 

A typical GoS metric is, “Number of originating calls that failed out of 500 calls during the peak busy hour.”  It’s the proportion of lost (dropped) calls out of a group of calls/conversations in progress. GoS will be expanded on below. Most of the figures in this section are from [Atkinson] and this reference provides a good summary of telephone traffic engineering. 

Knowing the amount of expec
ted calling traffic is fundamental to designing an exchange with a target GoS value. Traffic engineers use statistics, queuing theory, the nature of traffic, and calling models to make predictions of traffic flow. A well-engineered exchange, based on traffic theory, can reduce installation and operational costs considerably compared to guessing. Guessing is not allowed!

Fig 2 shows typical traffic (calls per ½ hour) over a day. Telephone traffic is variable and statistical in nature.

Typical hour-to-hour variation for typical metro telephone exchange traffic

Fig 2. Typical hour-to-hour variation for typical metro exchange traffic

The “peak busy hour” (PBH) for new calls is between 9:30 and 10:30 AM for this example (~3,800 calls over 30 min or ~7,600 calls per hour). So, the equipment should be able to meet this traffic requirement, on average. Naturally, there will be occasional statistical peaks where a caller is denied service, and this is directly tied to the target GoS for the exchange or a portion of the switching fabric. Some calling loss is unavoidable, and desired economically, in any practical exchange design. The PBH is a critical metric when designing an exchange or trunking plans. 

To meet peak calling needs, the switching equipment must be designed and configured based on the target GoS value. This means the switching fabric and associated apparatus may sit nearly idle for many hours off-peak during the day. Seems a waste but that’s what it takes to support peak calling times. The statistical aspects of calling are quite non-linear with clusters of local peaks.  

Naturally, every subscriber wants zero call loss. However, the telephone operator needs a value (> 0) that yields a positive economic return coupled with subscriber satisfaction. No operator can assure zero call loss and stay financially solvent. Exchange design tradeoffs are needed to achieve practical, economically sound, exchange GoS values. 

Traffic metrics

Some of the most important metrics used by a traffic engineer are:
    •    Calling Rate (CR): The average number of calls originated by a single subscriber during the PBH.  This may range from .3 for a rural exchange to 1.5 in a busy metro area (1950’s data). 

    •    Holding Time (HT): Duration of a typical call including dialing and ringing phases. A commonly used value is 2.5 minutes (150 seconds). 

    •    Traffic Unit (TU): The average number of calls originating during a time equal to the holding time. There are several metrics for this including CCS (centrum call seconds).  1 CCS is 100 call-seconds. So, 1 CCS is one call lasting 100 seconds or, for example, two calls lasting for 50 seconds each. Another metric is the Erlang. Sixty minutes (3,600 seconds) for one call constitutes one Erlang. One Erlang is equal to 36 CCS. 

    •    Peak Busy Hour: Total traffic in calls during 1 hour. Most metrics are valid only during the PBH unless stated otherwise. 

    •    Grade of Service (GoS): A measure of connection integrity during a call’s life cycle. A lower value is always better. A common definition is, “Number of originating calls that failed out of K total calls during the peak busy hour.” K is 500 or 1000 or some other value. 

In the 1940-50’s, the British General Post Office (GPO, domestic telephone service) defined their standard GoS value as .002, or 1 in 500 calls lost per switching stage [Atkinson]. A call was lost, in most cases, because all the switches in a pool were fully occupied by other callers. So, for a 4-digit exchange (3 switches in talking path, end-to-end) the overall expected GoS should be about .006 (3 in 500). 

An exchange switching fabric is composed of switching stages (Ex: many 1st, 2nd, 3rd , … selectors).  A TU value can be applied to the entire exchange or any of the stages or parts of stages. So, for a small city office of 4K subs an exchange TU of 50-100 is common. On the other hand, a TU of 1-5 is more likely when considering only a portion of a switching stage. 

Traffic statistics examples

At a certain 10K subscriber exchange, the Calling Rate is .5 (50%). Hence, 5,000 calls are made during the peak busy hour. The average holding time is 2.5 minutes. So, the TU  = 5,000 *(2.5/60) = 208.3 Erlangs (defined above). This means that, on average, there are 208 simultaneous conversations at any given time during the peak busy hour. Knowing this, a traffic engineer can design the switch train (in stages) to support this load. But, how?

First, an equation is needed to compute the number of required pooled switches (and/or trunks) given values of TU and GoS. 

This takes the form:  S = F(TU, GoS)      Equ 1

    •    F is “a function of”
    •    S is the number of required pooled switches (or trunks) for a given stage(s). 
    •    TU is the expected traffic load for this stage (or stages) of switching. 
    •    GoS is the value of acceptable call loss (all switches/trunks in the pool are busy). 

Examining the full expression of (1) and its derivation is beyond our scope. Nonetheless, the full equation, and its variations, is beautiful and has enchanted mathematicians and engineers including Poisson (1837, for gambling), Erlang (1909) and Molina (1920’s). The Erlang formula is most often used for equation (1) by most traffic engineers. This 3-variable equation can be rearranged to solve for any one variable if the other two are provided [Atkinson, page 31]. 

Agner Krarup Erlang is considered the father of telephone traffic theory. He was a Danish mathematician, and engineer who invented the fields of traffic engineering and queueing theory. Erlang produced his famous equation, the Erlang distribution, in 1909. Interestingly, the Poisson distribution formula can be derived from the Erlang distribution formula.

The early traffic work of M.C. Rorty at Bell [Fagen] in 1903 stimulated the thinking of Edward C. Molina. He became a star engineer at Bell Labs for 41 years and was the first at Bell to fully apply traffic theory to determine how many trunks/switches are needed for a desired GoS.  See Appendix A. 

Fig 3 shows an example of equation (1) in action. 

Chart showing relationship between traffic loading in Erlangs, call loss and quality of service - QoS

Fig 3, GoS, TU, Trunks (Switches) relationships using Equ (1) by Erlang

Using the full expression of equation (1) above with different values of GoS and TU provides the number of trunks needed to support these conditions. Why trunks? Trunks in this context are the number of available paths one switch has to another switch in the fabric (see Fig 1). So, Equ (1) computes either trunks or switches depending on the context of the problem. 

Looking at the figure, for a TU of 4 Erlangs and a desired GoS of .005, 10 pooled trunks are needed. Using 9 switches results in an increased GoS (bad) and using 11 switches decreases the GoS but at a cost (bad).   

Figure 4 shows a sample implementation for a GoS of .001 for two groups of 100 subs (200 subs total). 

100 telephone subscribers accessing 10 switches each in two independent groups

Fig 4. 100 subscribers accessing 10 switches each in two independent groups

This example (Groups 1, 2 are each similar to Fig 1) uses “uniselector line switches”. Each subscriber has a dedicated uniselector (two shown in the figure) and a each uniselector has access to any of 10 selector switches (provides dial tone). Each uniselector switch hunts for the first idle selector when servicing a new call. The desired GoS is .001 for 100 subscribers. These subscribers share a pool of 10, 1st selectors. 

The TU is 3 Erlangs per group. So, 3 subscribers (in a group of 100) are simultaneously conversing at any time during the peak hour, on average. Using Fig 3, the GoS for this configuration is .001, as desired, for 10 switches. 

This example sheds light on how a traffic engineer would use a divide-and-conquer approach, relying on tables such as Fig 3, to design a new exchange.  Designing a large exchange with a target Grade of Service is an art and science and becoming proficient takes abundant training and experience. See Appendix C for more on this topic. 

The Distribution Function of Simultaneous Calls


Over the years, traffic engineers have developed reliable mathematical models to predict real-world measured values. One of the key models used is the Poisson statistical distribution. This computes the probability of having X number of simultaneous calls when knowing the average number of simultaneous calls, call this Xavg.  


The Poisson distribution is a good fit for modeling telephone calls because it accurately models the probability of receiving a certain number of calls during a fixed time interval. Fig 5 shows a probability curve given the average number of expected simultaneous calls is 8 (Xavg) and the number of switches (or trunk paths) available for use is limited to 10. For this example, the smooth theoretical curve is not precisely a Poisson distribution but nearly so as [Wilkinson] describes.   


The vertical lines are the “real-world” data values from measuring 1,500 telephone calls.  The dotted vertical lines show those proportions of observations when one or more calls are waiting. This is expected since there are only 10 available paths for the calls. About 11.5% of the time there will be 8 simultaneous calls and about 1% of the time there will be 2 or 20 simultaneous calls. The theory and observations are in good agreement and would be more so with, say, data from 3,000 calls.

traffic engineering telephone exchange- probability of number of simultaneous calls

    Fig 5. Probability of X simultaneous calls with Xavg= 8 and 10 paths (trunks)
              Theory versus the Throwdown simulator’s values. [Wilkinson

Curves such as these assist traffic engineers to determine the number of trunks (or switches) required based on the number of expected calls and tolerance for subscriber waiting time. 

Obtaining real-world calling results from a working exchange is a challenge since setting the desired measurement conditions is near impossible. For one, the most important statistics are gathered only during the busy hour. See Appendix B for more on the Throwdown simulator designed by Bell Labs to accurately model telephone traffic.  

Hands on with Poisson’s Equation

One form of the Poisson distribution is,

    P(X) = A^X /(e^A * X!)        Equ (2) 
    Where P(X) is the probability of X simultaneous calls occurring
                 A is the average expected (mean) number of simultaneous calls
                 e is the math constant 2.718…. 
                 X! is X factorial (5! = 5*4*3*2*1)
                 ^ is the exponent operator
This equation is plotted in Fig 6 for the values A= 10 (Traffic Units =10 Erlangs) on an exchange with 300 calls being made during the busy hour and Holding Time of 120 seconds per call. This plot has a log vertical scale and shows a more dynamic range than Fig 5.  

Equation (2) was derived assuming there are ample trunks and switches to prevent blocking which is a good approximation for some traffic scenarios.          

Mathematical probability of X simultaneous phone Calls  using Poisson equation

Fig 6. Mathematical probability of X simultaneous Calls with TU = 10

It’s fascinating that equation (2), so simple and elegant, can predict the traffic flow statistics in a working exchange. 

If you were inside an exchange, with the metrics of Fig 6, and studied the traffic flow (at Holding Time observation intervals) over many busy hour periods, you would see 10 simultaneously callers ~12.5% of the time. Or, for example, you would see 2 calls or 19 calls roughly .3% of the time. Equations like (2) are the bread and butter for traffic engineers. 

After Poisson created his distribution in 1837, gamblers used it to predict the probability of winning or losing at games of chance, such as roulette and dice. So, it has many uses.


The values of traffic engineering (TE) for exchange design are:

    •    Optimized Resource Allocation: TE ensures telephone exchanges have the appropriate number of trunks and switches to handle the expected call volume. 

    •    Cost-Effective Design: By accurately predicting traffic patterns, TE helps in designing a cost-effective telephone exchange while ensuring that adequate capacity is available to meet peak demand periods. 

    •    Enhanced Quality of Service: TE designs for a high level of service. It minimizes call blocking, delays, and congestion to an acceptable level of loss. 

Appendix A

Edward C. Molina


The text in this section is based on material from [Fagen].


Edward Molina was an extraordinary individual who left a lasting mark on the evolution of telephony. He joined the Western Electric Company in 1898 at the age of 21 with no formal education beyond high school and proved to be a versatile and prolific inventor, making important contributions to the development of machine switching.


Molina joined the Research Department of AT&T in 1901 and, during his long career in this company and with Bell Telephone Laboratories, contributed extensively to the development and application of traffic mathematics. This was made possible by a disciplined self-study of mathematics that led him to become a recognized expert in the work of the mathematicians Laplace, Poisson and Erlang.

                                                    Edward C. Molina 

He devised equations that provided relationships between traffic and loss as a function of the number of trunks (or switches).  Starting from 1908 his tables and methods were used by traffic engineers inside Bell and eventually worldwide. The so-called Molina Formula became the standard in the Bell System. It was several years later when Molina discovered that it had been derived earlier by the French mathematician Poisson, whose name has since then been associated with it [Molina1]. 

In recognition of this work and the encouragement and example it offered to young people for self-study, he received, in 1952, the Honorary Degree of Doctor of Science from Newark College of Engineering, an institution at which he taught mathematics until 1964, some 20 years after his "retirement" from the Bell System. 

                               Appendix B
                        The Throwdown Simulator

Materials in this section are derived from [Wilkinson]. 

Bell labs Throwdown exchange simulator

General view of #5 crossbar system simulator machine

To study the traffic-carrying characteristics of the No. 5 crossbar telephone exchange, a machine was built to simulate its operation. This machine, known as a throwdown machine, is controlled by a team of four operators. Its input is a statistically accurate representation of actual telephone traffic, and its output is a detailed record of the course of each call through the system. Setting up the conditions for busy hour statistics is straightforward. 

The word throwdown was chosen because in the very early days of paper and pencil traffic simulation, dice were thrown down to produce random values appropriate to call statistics. In Figure 5 of the main text, the vertical line values were produced by this simulator. 

Throwndown telephone traffic simulator in operation wtih Catherine Durnan operating

     Appendix C
    Efficient Trunking

As discussed in the main section, determining the optimum number of trunks between end points is essential for cost effective operations. Fig 1C [Atkinson] shows 35 interconnected offices in a large city (Glasgow, Scotland). Some exchanges are directly connected to other smaller nearby exchanges, but all connect to Glasgow Central.   Eleven exchanges are within a radius of just 2 miles. Each interconnection line represents one or more trunk lines. 

One way to partition subscriber numbering is to use 6 digits. The first two determine the exchange (office code) and the final four are for subscribers assigned to the office code. This permits 100 exchanges, each with a theoretical capacity of 10K subscribers. This would certainly work for the configuration in Fig 1C with 35 exchanges. 

There are (35^2 – 35)/2 = 595 number of possible ways to link 35 offices together with a single trunk line between any two. Notice there is no trunk path between Busby and Barrhead but there is between Busby and Giffhock. There are many fewer than 595 trunk paths in Fig 1C but it’s still quite a tangle. Keep in mind that some of the offices may have been serviced by manual boards or mixed with automatic. This also complicates operations. Also, trunks may be unidirectional or bidirectional (2 and 4-wire).  


Trunk network for Glasgow City area, Scotland, circa 1948

Fig 1C, Trunk network for Glasgow City area offices, Scotland, circa 1948 

Now, if the number of trunk lines per path ranged from 1 to 50 based on traffic engineering, then the total number of possible trunk line configurations in Fig 1C is 50^595 or approximately 10^1,011 (a 1,011 digit number). There are about 10^80 electrons in the universe so the number of possible trunk configurations is beyond staggering. 

What is the purpose of this example? To help appreciate the need for rigorous traffic engineering when designing a geographical area’s telephone network.  Using the tools discussed in the main text, a staff of competent traffic engineers could design the trunking plan for Fig 1C and settle on one configuration from the sea of 10^1,011 possible. Sure, as the subscriber count changed the number of optimal paths/trunks would change. So, lifetime employment was guaranteed for traffic engineers. 

Improving interconnectivity 

The trunking plan in Fig 1C was practical in 1948. Is there a better way to accomplish the task without the mesh of trunks?  Toll Offices came to the rescue. The No. 4 Crossbar Toll Exchange was the world's first four-wire (2 way) crossbar toll office. It was installed in Philadelphia, Pennsylvania in 1943. A toll office had no end point subscribers, only other offices. It was a routing machine. For more on this see – Trunking Plans. In the 1950’s toll offices became common worldwide, especially in areas like Greater London with 237 exchanges in 1948. 


Atkinson, J., Telephony Vol 2, A Detailed Exposition of the Telephone Systems of the British Post Office, 1950, Pitman publishers.

Fagen, M.D., A History of Engineering and Science in the Bell System, 1875-1925. 

Miller, Kempster, Telephone Theory and Practice, V3, 1933 Fig 34. 

(1) Molina, Edward C., The Theory of Probabilities Applied to Telephone Trunking Problems. Bell System Technical Journal, Vol 1, Number 1, July 1922
. (first issue of the BSTJ) 

(2) Molina, Edward C., MATHEMATICS IN THE TELEPHONE INDUSTRY, A talk to the Chicago Teachers' Mathematics Club, February 16, 1940, School Science and Mathematics Journal, May 1, 1940

Throwdown: G. R. FROST, WILLIAM KEISTER AND ALISTAIR E. RITCHIE, A Throwdown Machine for Telephone Traffic Studies, Bell System Technical Journal, March 1953.

Wilkinson, Roger, Working Curves for Delayed Exponential Calls Served in Random Order, Bell System Technical Journal, March 1953. 

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