# Dimensioning And Efficiency

By definition, if we were to dimension a route or estimate the required number of servicing channels, where the number of trunks (or servicing channels) just equaled the erlang load, we would attain 100% efficiency. All trunks would be busy with calls all the time or at least for the entire BH. This would not even allow several moments for a trunk to be idle while the switch decided the next call to service. In practice, if we engineered our trunks, trunk routes, or switches this way, there would be many unhappy subscribers.

On the other hand, we do, indeed, want to size our routes (and switches) to have a high efficiency and still keep our customers relatively happy. The goal of our previous exercises in traffic engineering was just that. The grade of service is one measure of subscriber satisfaction. As an example, let us assume that between cities X and Y there are 100 trunks on the interconnecting telephone route. The tariffs, from which the telephone company derives revenue, are a function of the erlangs of carried traffic. Suppose we allow a dollar per erlang-hour. The very upper limit of service on the route is 100 erlangs. If the route carried 100 erlangs of traffic per day, the maximum return on investment would be \$2400 a day for that trunk route and the portion of the switches and local plant involved with these calls. As we well know, many of the telephone company's subscribers would be unhappy because they would have to wait excessively to get calls through from X to Y. How, then, do we optimize a trunk route (or serving circuits) and keep the customers as happy as possible?

In our previous discussions, an excellent grade of service was 0.001. We relate grade of service to subscriber satisfaction. Turning to Ref. 20, Table 1-2, such a grade of service with 100 circuits would support 75.24 erlangs during the BH. With 75.24 erlangs loading, the route would earn \$75.24 during that one-hour period and something far less that \$2400 per day. If the grade of service was reduced to 0.01, 100 trunks would bring in \$84.06 for the busy hour. Note the improvement in revenue at the cost of reducing grade of service. Another approach to save money is to hold the erlang load constant and decrease the number of trunks and switch facilities accordingly as the grade of service is reduced. For instance, 70 erlangs of traffic at p = 0.001 requires 96 trunks and at p = 0.01, only 86 trunks.

### 8.1 Alternative Routing

One method of improving efficiency is to use alternative routing (called alternate routing in North America). Suppose that we have three serving areas, X, Y, and Z, served by three switches, X, Y, and Z as illustrated in Figure 1.8.

Let the grade of service be 0.005 (1 in 200 in Table 1-2, Ref. 20). We found that it would require 67 trunks to carry 50 erlangs of traffic during the BH to meet that grade of service between X and Y. Suppose that we reduced the number of trunks between X and Y, still keeping the BH traffic intensity at 50 erlangs. We would thereby increase the efficiency on the X-Y route at the cost of reducing the grade of service. With a modification of the switch at X, we could route the traffic bound for Y that met congestion on the X - Y route via Z. Then Z would route this traffic on the Z - Y link. Essentially, this is alternative routing in its simplest form. Congestion probably would only occur during very short peaky periods in the BH, and chances are that these peaks would not occur simultaneously with peaks in traffic intensity on the Z -Y route. Furthermore, the added load on the X - Z - Y route would be very small. Some idea of traffic peakiness that would overflow onto the secondary route (X + Z + Y) is shown in Figure 1.9.

One of the most accepted methods of dimensioning switches and trunks using alternative routing is the equivalent random group (ERG) method developed by Wilkinson [11]. The Wilkinson method uses the mean M and the variance V.

Figure 1.8. Simplified diagram of the alternative routing concept (solid line represents direct route, dashed line represents alternative route carrying the overflow from X to Y).

Figure 1.8. Simplified diagram of the alternative routing concept (solid line represents direct route, dashed line represents alternative route carrying the overflow from X to Y).

Overflow

Directly routed circuits

Time

Overflow

Directly routed circuits

Time