Dimensioning and Efficiency

By definition, if we were to dimension a route or estimate the required number of servicing channel, 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 time for call setup (i.e., making the connection) or for switch processing time. In practice, if we sized our trunks, trunk routes, or switches this way, there would be many unhappy customers.

On the other hand, we do, indeed, want to dimension 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

Table 4.2 Trunk-Loading Capacity, Based on Poisson Formula, Full Availability

Grade of

Grade of

Grade of

Grade of

Grade of

Service

Service

Service

Service

Service

1 in 1000

1 in 100

1

in 50

1

in 20

1

in 10

Trunks

UC

TU

UC

TU

UC

TU

UC

TU

UC

TU

1

0.1

0.003

0.4

0.01

0.7

0.02

1.9

0.05

3.8

0.10

2

1.6

0.05

5.4

0.15

7.9

0.20

12.9

0.35

19.1

0.55

3

6.9

0.20

16

0.45

20

0.55

29.4

0.80

39.6

1.10

4

15

0.40

30

0.85

37

1.05

49

1.35

63

1.75

5

27

0.75

46

1.30

56

1.55

71

1.95

88

2.45

6

40

1.10

64

1.80

76

2.10

94

2.60

113

3.15

7

55

1.55

84

2.35

97

2.70

118

3.25

140

3.90

8

71

1.95

105

2.90

119

3.30

143

3.95

168

4.65

9

88

2.45

126

3.50

142

3.95

169

4.70

195

5.40

10

107

2.95

149

4.15

166

4.60

195

5.40

224

6.20

11

126

3.50

172

4.80

191

5.30

222

6.15

253

7.05

12

145

4.05

195

5.40

216

6.00

249

6.90

282

7.85

13

166

4.60

220

6.10

241

6.70

277

7.70

311

8.65

14

187

5.20

244

6.80

267

7.40

305

8.45

341

9.45

15

208

5.80

269

7.45

293

8.15

333

9.25

370

10.30

16

231

6.40

294

8.15

320

8.90

362

10.05

401

11.15

17

253

7.05

320

8.90

347

9.65

390

10.85

431

11.95

18

276

7.65

346

9.60

374

10.40

419

11.65

462

12.85

19

299

8.30

373

10.35

401

11.15

448

12.45

492

13.65

20

323

8.95

399

11.10

429

11.90

477

13.25

523

14.55

21

346

9.60

426

11.85

458

12.70

507

14.10

554

15.40

22

370

10.30

453

12.60

486

13.50

536

14.90

585

16.25

23

395

10.95

480

13.35

514

14.30

566

15.70

616

17.10

24

419

11.65

507

14.10

542

15.05

596

16.55

647

17.95

25

444

12.35

535

14.85

572

15.90

626

17.40

678

18.85

26

469

13.05

562

15.60

599

16.65

656

18.20

710

19.70

27

495

13.75

590

16.40

627

17.40

686

19.05

741

20.60

28

520

14.45

618

17.15

656

18.20

717

19.90

773

21.45

29

545

15.15

647

17.95

685

19.05

747

20.75

805

22.35

30

571

15.85

675

18.75

715

19.85

778

21.60

836

23.20

31

597

16.60

703

19.55

744

20.65

809

22.45

868

24.10

32

624

17.35

732

20.35

773

21.45

840

23.35

900

25.00

33

650

18.05

760

21.10

803

22.30

871

24.20

932

25.90

34

676

18.80

789

21.90

832

23.10

902

25.05

964

26.80

35

703

19.55

818

22.70

862

23.95

933

25.90

996

27.65

36

729

20.25

847

23.55

892

24.80

964

26.80

1028

28.55

37

756

21.00

876

24.35

922

25.60

995

27.65

1060

29.45

38

783

21.75

905

25.15

951

26.40

1026

28.50

1092

30.35

39

810

22.50

935

25.95

982

27.30

1057

29.35

1125

31.25

40

837

23.25

964

26.80

1012

28.10

1088

30.20

1157

32.14

41

865

24.05

993

27.60

1042

28.95

1120

31.10

1190

33.05

42

892

24.80

1023

28.40

1072

29.80

1151

31.95

1222

33.95

43

919

25.55

1052

29.20

1103

30.65

1183

32.85

1255

34.85

44

947

26.30

1082

30.05

1133

31.45

1214

33.70

1287

35.75

45

975

27.10

1112

30.90

1164

32.35

1246

34.60

1320

36.65

46

1003

27.85

1142

31.70

1194

33.15

1277

35.45

1352

37.55

47

1030

28.60

1171

32.55

1225

34.05

1309

36.35

1385

38.45

48

1058

29.40

1201

33.35

1255

34.85

1340

37.20

1417

39.35

49

1086

30.15

1231

34.20

1286

35.70

1372

38.10

1450

40.30

50

1115

30.95

1261

35.05

1317

36.60

1403

38.95

1482

41.15

there were 47 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 $1.00 per erlang-hour. The very upper limit of service on the route is 47 erlangs, and the telephone company would earn $47 for the busy hour (much less for all other hours) 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 satisfied with service as possible?

Remember from Table 4.1, with an excellent grade of service of 0.001, that we relate grade of service to subscriber satisfaction (one element of quality of service) and that 47 trunks could carry 30.07 erlangs during the busy hour. Assuming the route did carry 30.07 erlangs, let's say at $1.00 per erlang, it would earn $30.07 for that hour. From a revenue viewpoint, that would be the best hour of the day. If the grade of service were reduced to 0.01, 47 trunks would bring in $35.21 (i.e., 35.21 erlangs) for the busy hour. Note the improvement in revenue at the cost of reducing grade of service.

Here we are relating efficiency on trunk utilization. Trunks not carrying traffic do not bring in revenue. If we are only using some trunks during the busy hour only minutes a day to cover BH traffic peaks, the remainder of the day they are not used. That is highly inefficient. As we reduce the grade of service, the trunk utilization factor improves. For instance, 47 trunks will only carry 30.07 erlangs with a grade of service of 1 in 1000 (0.001), whereas if we reduce the grade of service to 1 in 20 (0.05), we carry 41.54 erlangs (see Table 4.1). Efficiency has improved notably. Quality of service, as a result, has decreased markedly.

4.2.4.1 Alternative Routing. One method to improve efficiency is to use alternative routing (called alternate routing in North America). Suppose we have three serving areas, X, Y, and Z, served by three switches (exchanges), X, Y, and Z, as illustrated in Figure 4.4. Let the grade of service be 0.005 (1 in 200 in Table 4.1). We find that it would require 48 trunks to carry 34.25 erlangs of traffic during the BH to meet that grade of service between X and Y. Suppose we reduce the number of trunks between X and Y, still keeping the BH traffic intensity at 34.25 erlangs. We would thereby increase efficiency on the X-Y route at the cost of reducing grade of service. With a modification of the switch at X, we could route traffic bound for Y that met congestion on the X-Y route via switch Z. Then Z would route that traffic on the Z-Y link. Essentially this is alternative routing in its simplest form. Congestion would probably only occur during very short peaking periods in the BH, and chances are that these peaks would not occur simultaneously with peaks

Figure 4.4 Simplified diagram of the alternative (alternate) routing concept. (Solid line represents the direct route, dashed lines represent the alternative route carrying the overflow traffic from X to Y).

Figure 4.4 Simplified diagram of the alternative (alternate) routing concept. (Solid line represents the direct route, dashed lines represent the alternative route carrying the overflow traffic from X to Y).

Figure 4.5 Traffic peakedness, the peaks are carried on alternative routes.

of traffic intensity on the Z-Y route. Furthermore, the incremental load on the X-Z-Y route would be very small. The concept of traffic peakedness that would overflow onto the secondary (X-Z-Y) is shown in Figure 4.5.

4.2.4.2 Efficiency Versus Circuit Group Size. In the present context a circuit group refers to a group of circuits performing a specific function. For instance, all the trunks (circuits) routed from X to Y in Figure 4.4 make up a circuit group irrespective of size. This circuit group should not be confused with the "group" used in transmission engineering of carrier systems.6

If we assume full loading, we find that efficiency improves with circuit group size. From Table 4.1, given a grade of service of 1 in 100, 5 erlangs of traffic require a group with 11 trunks, more than 2:1 ratio of trunks to erlangs, and 20 erlangs requires 30 trunks, a 3:2 ratio. If we extend this to 100 erlangs, 120 trunks are required, a 6:5 ratio. Figure 4.6 shows how efficiency improves with group size.

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