Copyright φ 1978 American Telephone and Telegraph Company T H E B E L L S Y S T E M T E C H N I C A L J O U R N A L

Vol. 57. No. 4, April 1978 Printed in USA.

Loop Plant Modeling:

The Feeder Allocation Process By B. L. MARSH

(Manuscript received August 20, 1977)

Allocation is an engineering process by which economical use of spare feeder capacity in the loop network may be planned. The basic concept is to apportion the available spare feeder pairs along an entire feeder route such that placement of the relief cable is deferred for the section of the route with the shortest time to relief, thus reducing capital costs. Since rearrangements are planned on a route basis, rather than made expediently, operating costs are also reduced. This paper first illus­trates the concept of allocation by applying a simple manual technique to a small example. Then a more sophisticated method, which has been computerized, is described. Finally, a generalized mathematical de­scription is given.

I. INTRODUCTION

An impor tan t pa r t of an operat ing company outs ide p l an t engineer 's job is to decide how best to use the spare capacity in the feeder network. 1

T h i s activity is called feeder administration. A major p a r t of feeder adminis t ra t ion is the allocation of spare feeder pairs .

1.1 77M basic concept ot allocation

Allocation is the process of p lanning the use of spare feeder capaci ty (i.e., spare feeder cable pairs) along an ent i re feeder route. T h e purpose of allocation is to make the best, or most economical, use of both existing and future feeder facilities, and to enable t h e engineer to p lan splicing configurations which will avoid complexity and reduce as m u c h as pos­sible the need for costly future rear rangements .

T h e basic concept of allocation is to apportion the available spare pairs along the entire route, according to forecast growth rates. Doing so may

869

economically defer feeder relief which leads to higher feeder utilization ( the fraction of the feeder actually in use).

1.2 Dual view Of the feeder

Consider t h e schematic of a feeder route in Fig. 1. T h e feeder rou te is like an expressway for cable pairs which ex tends ou t from t h e cent ra l office. Latera l cables are spliced to t h e feeder cable a t various poin ts along the route and these lateral cables are joined to dis t r ibut ion cables which are connected to subscribers.

T h e dis t r ibut ion p lan t along the route has been organized geograph­ically into regions called allocation areas. Allocation areas a re used for feeder p lanning and adminis t ra t ion; they provide t h e basis for engi­neering manageable por t ions of the route , r a the r t h a n t rying to look a t the ent i re rou te all a t once. Conversely, t he use of allocation areas en­courages t h e feeder engineer, when solving local feeder /dis t r ibut ion problems, to consider the requirements of the rest of the route. Allocation areas are also used for monitoring facility problems, described elsewhere in this issue. 2

T h e feeder itself has traditionally been broken up into segments called feeder sections. T h e feeder sections, which include all of t h e cable and condui t in a cross section between two points (usually impor t an t man­holes) along the route , are used for relief p lanning and des ign . 3 T h e reason for this is tha t new relief cables are economically sized and placed in one or more sections a t a t ime and then are spliced together with other cables to provide pairs from the central office to the various areas served by the network.

T h e other way of looking a t the feeder facilities is t h e pair group. A pair group is a "bundle" of feeder pairs (not necessarily in the same cable sheath) which extend outward from the central office to a specific portion of the dis t r ibut ion plant , a single allocation area. T h u s there is a one-

DISTHIBUTION A ALLOCATION CABLES / AREAS \

PAIR GROUPS

f.

CENTRAL OFFICE

FEEDER SECTIONS

Fig. 1 Feeder route schematic.

870 THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1978

to-one relationship between pair groups and allocation areas; the feeder complements (groups of pairs , usually in mul t ip les of 25 pairs) in a part icular pair group serve the allocation area associated with t h a t pair group. Pa i r groups, then , are used pr imari ly for feeder admin is t ra t ion . Notice t ha t a single section may have several pair groups passing through it and each pair group generally passes th rough several sections.

II. A SIMPLE APPROACH TO ALLOCATION How should spare feeder complements be appor t ioned (or allocated)

to the allocation areas along the route? T h i s depends on t h e lifetimes of both the feeder sections and pair groups. "Lifet ime" as used here does not mean the t ime until t he feeder p lan t is replaced, bu t ra ther the t ime unt i l addi t ional p lan t mus t be provided. T h i s is also known as fill t ime, t ime to exhaust ion, or t ime to relief.

A section's lifetime is based on t h e n u m b e r of spare pai rs in t h e cross section and the growth into and th rough t h e section. A pair group 's lifetime, however, is based on the number of spare pairs and the growth in t h a t pair group only. T h u s the feeder section lifetimes a re usually different from t h e pair group lifetimes.

2.1 An Illustrative example

Consider the example* shown schematical ly in Fig. 2. H e r e the re are th ree pair groups feeding th ree allocation areas: AAl, A A 2 , and AA3. T h e r e are also th ree feeder sections: 2101, 2102, and 2103. Not ice t h a t

SPARE 294 76 204

GROWTH 40 120 24

LIFE 7.4 0.8 8.S

ALLOCATION AREAS ( AAl ^ ( AA2 ^ AA3 )

CENTRAL OFFICE

FEEDER SECTIONS

SPARE

GROWTH

LIFE

574

184

3.1

2102

280

144

1.9

2103

204

24

8.5

Fig. 2—An example route showing allocation area and feeder section lifetimes.

* All of the feeder routes in this paper are simplified in order to make clearer example Actual feeder routes frequently are composed of 20 or more sections.

FEEDER ALLOCATION PROCESS 8 7 1

the ideal lifetime (the number of spare pairs divided by the growth rate) of allocation area A A 2 (and its associated pair group) is only 0.6 year. T h e section 2102 ideal lifetime, however, is 1.9 years. Far ther out in t h e route, bo th allocation area A A 3 and section 2103 have large ideal l ifetimes of 8.5 years. I t would appear t ha t allocation area AA3 might be able t o share some of i ts spare pairs with allocation area A A 2 . T h e only a l te rna t ive to a r ea r rangement such as th is is to place a relief cable in section 2102 within 0.6 year.

Notice t ha t the ideal lifetime of section 2102 (1.9 years) is t h e shor tes t section lifetime in t h e route . Th i s section is referred to as t h e critical section. I t is this section t ha t will exhaust first if growth occurs as forecast and if all possible rearrangements are m a d e to defer relief. T h a t is, if all spare pairs can be used, t hen the t ime to exhaus t ion for section 2102 could be u p to 1.9 years . B u t th is section will exhaus t sooner given t h e way t h e spare pai rs a re present ly d i s t r ibu ted to t h e allocation areas . Relief can be deferred in the critical section as much as possible by a p ­portioning the spare pairs in the critical section to the various pair groups runn ing th rough it such t h a t they all have identical ideal lifetimes.

Since the ideal lifetime of AA3 is greater t h a n t h e critical sect ion's lifetime, then a surplus of spare pairs is associated with or al located to A A 3 . In t h e same way, a deficit of spare pairs is al located t o A A 2 . Load balancing is t h e process of reallocating spare pairs from areas wi th a surplus to those with a deficit. Load balancing often helps t o identify necessary physical r ea r rangements in the feeder network.

2.2 A procedure tor load balancing

A load balance worksheet has been developed for use by outside p lan t engineers. Figure 3 illustrates its application to the example route of Fig. 2. T h e entr ies on lines A th rough H correspond to those in Fig. 2, dis­cussed in the previous section.

On line J is t h e ideal allocation lifetime. Th i s is t h e lifetime each al­location area (pair group) should have in order to theoret ical ly balance t h e rou te and defer t h e next relief for as long as possible. T h e ideal al­location lifetime is t h e same as t h e mos t critical sect ion's lifetime for those sections beyond and including the critical section. Thus , both areas A A 2 and AA3 have ideal allocation lifetimes of 1.9 years. For the allocation areas fed from sections between t h e mos t critical section a n d t h e nex t mos t critical section, t h e ideal allocation lifetime is t h e lifetime of t h e second critical section.

T h e theoretical spare pair allocation (line K) for each allocation area is t h e allocation area ' s growth ra te mul t ip l ied by its ideal allocation lifetime. Note the entr ies in Fig. 3.

Finally, each allocation area ' s spare pair surplus/defici t is t h e differ­ence between the n u m b e r of spare pai rs i t current ly has a n d i ts t heo-

872 T H E B E L L S Y S T E M T E C H N I C A L J O U R N A L , A P R I L 1 9 7 8

LOAD BALANCE WORKSHEET ROUTE

A SECTION NUMBER 2101 2102 2103

Β ALLOCATION AREA AA1 AA2 AA3

C SPARE PAIRS IN ALLOCATION AREA 284 76 204

• ACCUMULATED SPARE PAIRS IN CROSS SEC (ACCUM C FROM FAR F.NDI

574 780 704

Ε GROWTH IN ALLOCATION AREA 40 120 24

F ACCUMULATED GROWTH IN CROSS SEC (ACCUM f FROM FAR END!

1B4 144 24

G. IOEAL CROSS SECTIONAL LIFETIME

(0+*l

3 1 1.9 8 5

H CRITICAL SECTIONISI to #1

J IDEAL ALLOCATION LIFETIME 3 1 1 9 1 9

K. THEORETICAL SPARE PAIR ALLOCATION

IE • Jt

124 228 4«

L SPARE PAIR SURPLUS/ DEFICIT

IC-K1

170 - I S 2 158

Fig. 3—Load balance worksheet for example route in Fig. 2.

retical allocation. For this example the entries are 170 pairs for allocation area A A l , —152 for A A 2 , and 158 for A A 3 . Every deficit m u s t be elimi­nated by using some surplus if t he ideal allocation lifetimes of line J are to be achieved. Of course, it is not possible to use jus t any surplus ; for example , here the 170 pairs in AAl do no t ex tend far enough ou t on the route to be used in AA2 (see Fig. 2).

Since t h e section and allocation area d a t a are en te red on t h e load balance worksheet from the cent ra l office outward , t h e tab le implicit ly conta ins information on the configuration of t h e route . Therefore t h e load balance worksheet not only indicates spare pair surpluses and deficits, b u t it also shows how deficits can be resolved. Since AA3 is far­the r out in the route t han A A 2 , t hen A A3 'S pair g roup passes th rough section 2102 and a t ransfer of surplus spare pairs could be made .

Why can ' t t h e 170 pair surp lus in A A l be used in A A 2 ? Unless the re a re usable dead pairs in section 2102, the re is no way to ex tend AAl ' s surplus spares wi thout placing relief. How should the 170 pair surp lus in AAl be used? Ideally these pairs should be "held b a c k " or reserved for use upon relief of the critical section. Unless section 2101 will be re­lieved along with section 2102, these pairs should no t be allocated to allocation area A A l .

FEEDER ALLOCATION PROCESS 873

III. A MORE SOPHISTICATED APPROACH

An a t t rac t ive feature of the load balance worksheet ju s t descr ibed is i ts simplicity and consequent ia l ease of p repara t ion . Due to th i s s im­plicity, however, there are several impor tant characteristics of the feeder which a re not considered by the manua l load balance worksheet : dead pairs , unal located pairs , objective fills, and varying growth ra tes .

3.1 Improvements In the representation of the feeder route

Firs t t he re are dead pairs, which a re cable pai rs t h a t do no t ex tend all the way back to the central office. T h u s they cannot be used to provide service unti l they are spliced to cables which do reach the central office. Dead pairs are therefore a capi tal inves tment which is no t mak ing a re­turn. Outside plant engineers need to know when and where it is practical and economical to use dead pairs , since this can defer feeder relief.

Unlike dead pairs , unallocated pairs are connected all t h e way back to the centra l office. T h e y are not , however, associated with any alloca­t ion area and frequently are no t even spliced to any lateral cable. T h e engineer should be aware of t h e n u m b e r and locations of unal located complements . In some cases, too m a n y pairs may remain unal located; relief can be deferred if the unallocated complements are allocated and then committed to allocation areas approaching exhaustion. On the other hand, there may be too few unal located pai rs in a section adjacent to a critical section on the cent ra l office side. Unless th is section is t o be re­lieved s imul taneously with t h e critical section, then a cer ta in n u m b e r of its pai rs should be reserved for future use. Upon relief of t h e critical section, these unal located pai rs are spliced to t h e new cable, t h u s p ro ­viding centra l office pai rs th rough and beyond the former critical sec­tion.

Next consider spare lifetimes. T h e ideal lifetime calculated using the load balance worksheet is s imply the n u m b e r of spare pairs divided by t h e growth ra te . A more realistic lifetime, or time to relief, may be de­termined if an objective or max imum fill is used in the calculation. Th i s fill is t h e percentage of cable pairs a t a par t icular po in t which a re prac­tical and economical to use a t the t ime jus t prior to relief. A typical value is 85 percent . (See Ref. 4, for a discussion of op t imal objective fills, i.e., t he economic fill a t relief.) T h e t ime to relief, t hen , is

T i m e /object ives / n u m b e r of \ / n u m b e r of \

where the number of pairs available includes all central office pairs, i.e., spare , defective, and assigned (working) pairs .

Finally, t he load balance worksheet uses a single growth ra te for each allocation area. Th i s is frequently a reasonable assumpt ion , b u t some-

to

relief growth ra te

874 THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1978

t imes growth ra tes change considerably over t ime . For example , as a developing area sa tu ra tes with subscr ibers , t h e growth slows. On t h e o ther hand , if t he n a t u r e of an area is changing from single-family to mult ifamily residential , growth will increase. Or the re may be a single " i m p a c t " growth due to a new office building. T h u s a more realist ic approach is to m a k e use of some sort of growth function.

An exper imental computer program has been developed which takes in to account these character is t ics of a route . In order t o show how th is t echn ique is more flexible t h a n t h e m a n u a l load balance worksheet , a n example is p resen ted next . A generalized ma thema t i ca l descr ipt ion is given in t h e appendix .

3.2 Computerized load balancing

As an i l lustrat ion of t h e computer ized table , consider t h e example route in Fig. 4. Here there are five feeder sections and six allocation areas. In addi t ion to t h e pair groups feeding t h e allocation areas , t he r e is an unallocated 50 pair complement which extends out to section 2104. The re a re also 100 dead pairs in section 2105.

Tab le I summar izes bo th t h e fill coun t and growth forecast for t h e allocation areas. Note t h a t in most cases the growth rate is declining. For example , the forecast for allocation area AA4 is 100 pa i rs growth for t h e first year (12/77 to 12/78), 70 pairs t h e nex t year, 50 pa i rs per year for t h e next two years , and then 40 pairs per year after 12/81.

Figure 5 shows o u t p u t from t h e compute r p rogram for th is rou te . B y examining this o u t p u t t h e reader should gain a fur ther u n d e r s t a n d i n g of the allocation process.

T h e leftmost column lists the allocation area (AA) and feeder section ( F S ) designations, along with abbreviat ions for unallocated ( U N A L ) and dead ( D D ) pairs . Feeder sections along wi th the i r associated allocation area(s) , etc., a re separa ted by horizontal lines, with t h e mos t d i s t an t

ALLOCATION AREAS

CENTRAL OFFICE

UNALLOCATED (50 pairs)

DEAD (100 pair»)

3 N S ( * - 2 1 0 1 - » | « 2102 -2103 - - a» |« - -2104- -» |»- -210S--»» |

Fig. 4—Schematic of second example route.

FEEDER ALLOCATION PROCESS 875

Table I — Fill count and forecast for route in Fig. 4

Growth forecast

Alloca­tion area

Number of pairs

available

Number of pairs

assigned*

Objec­tive fill

12/77 to

12/78

12/78 to

12/79

12/79 to

12/81

Annual rate after 12/81

AAl 700 462 90 25 25 50 25 A A 2 1500 1243 90 40 40 60 25 A A 3 2450 1742 85 70 80 140 40 AA4 1850 1541 85 100 70 100 40 AA5 1000 771 85 55 40 80 30 AA6 400 228 90 30 20 40 10

* Fill count of 12/77.

section a t the top of the table, and the section closest to the central office a t t he bo t tom.

Note, for example, section 2104, the second section from the top of the table . Th i s section feeds allocation area AA5.

T h e second column is PAIRS AVAIL, t he number of pairs available. For an allocation area th is refers to t h e to ta l n u m b e r of cent ra l office pai rs

LOOP FEEDER ANALYSIS TA8LE

CO. - WHIPPANY ROUTE - 2A DATE - 12/21/77

FACILITY COUNT IBASEI DATE: 12/77

AA FAIRS PAIRS PRES OBJ α ROW —GROWTH FORECAST— VRS EOUALI2EO SUR FAS AVAIL ASQND FILL FILL MARON PERI PER2 PERJ ANN TO YRS PAIRS OEF

•RLF 127S 1279 1211 RATE RLF

AA6 400 228 57.0 9 0 132 3 0 20 4 0 10 8.2 I B 304 96 UNAL 0 90 0 0.0 0 0

DD/BT 100 90 9 0 2105 500 2 2 8 45.6 9 0 222 3 0 20 4 0 10 17.2

AAS 1000 771 77.1 85 79 65 4 0 8 0 3 0 1.8 1.8 1008 - 8 UNAL 50 Ββ 43 0.0 0 50

2104 1460 8 8 8 68.8 8 6 254 86 60 120 40 3.8

AA4 1850 1641 83.3 B 5 31 100 70 100 40 0.3 l.B 1994 - 1 4 4 UNAL 0 8 6 0 0.0 0 0

2 1 0 3 3300 2540 77.0 8 8 285 186 130 2 2 0 80 1.8 * * *CHIT ICALSECTION# 1 -• EXHAUST DATE 979 M * AA3 2450 1742 71.1 8 6 3 4 0 70 8 0 140 4 0 5.2 2.9 2299 151 UNAL 0 85 0 1.1 149 - 1 4 9

2102 5750 4282 74.5 85 6 2 5 265 2 1 0 380 120 2.9 * * * CRITICAL SECTION # 2 - EXHAUST DATE 1 0 8 0 * * *

AAl 700 462 66.0 90 168 25 25 50 25 8.7 3.3 606 96 AA2 1500 1243 82.9 90 107 40 40 60 25 2.9 3.3 1613 - 1 3 UNAL 0 87 0 0.4 84 -84

2101 7850 59B7 75.3 B7 900 320 275 470 170 3.3 * * *CRIT ICAL SECTION # 3 - EXHAUST DATE 381 * * *

CO.

Fig. 5—Computer output for example route in Fig. 4.

876 THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1 9 7 8

associated with the allocation area. Th i s is, of course, t h e n u m b e r of central office pairs in the allocation area 's feeder pair group. (There are 1000 pairs available in AA5.) For unal located pairs , t h e en t ry indicates the number of unallocated pairs ending in t h e section. Unallocated pairs passing th rough t h e section bu t ending far ther ou t in t h e rou te are no t included in th is value. T h e n u m b e r of dead pairs are included in each section in which they appear . T h e r e is an unal located 50 pair comple­m e n t ending in section 2104, b u t there a re no dead pairs . T h e r e a re 100 dead pairs in section 2105.

T h e number of pairs available in the feeder section is the total number of pairs in t h e cross section. (There are 1450 in section 2104.) T h i s in­cludes all of the pairs in pair groups feeding the section's allocation areas and the allocation areas beyond. All unallocated pairs in t h e section and beyond it, as well as dead pairs wi thin t h e section, a re accumula ted also.

PAIRS A S G N D refers to the number of pairs assigned. These are mostly working pairs b u t also include some pairs associated wi th specific sub­scriber locations bu t which are temporar i ly idle. These da t a are entered by allocation area and a re accumula ted by feeder section. (Nei ther dead nor unal located pai rs can be assigned.)

T h e nex t column ( P R E S F I L L ) provides t h e presen t fill for allocation areas and feeder sections. Th i s is s imply

When calculated for an allocation area, it refers only to t h e pairs in t h e allocation area 's pair group. For feeder sections, however, t he ent ire cross section is considered: all pair groups and unallocated pairs passing through, as well as dead pairs in t h e section.

T h e objective fill a t relief ( O B J F I L L @ R L F ) was described earlier. Th i s fill level is de te rmined by the engineer for t h e al location area pair g roups . 4 T h e n , using these values, t h e p rogram computes the objective fills for feeder sections and unal located and dead complements .

T h e growth margin ( G R O W M A R G N ) is t h e n u m b e r of spare pairs which can be used to provide service and bring the fill t o t h e objective fill. I t is

Growth _ / objectives / n u m b e r of \ _ / n u m b e r of \ margin \ fill / Vpairs ava i lab le / Vpairs ass igned/

T h e growth forecasts are next and for this program there may be from one to four columns. T h e s implest form is a single annua l growth ra te for each allocation area. Whenever the growth forecast reflects changes

P re sen t fill

cu r r en t n u m b e r of1

. assigned pairs >

FEEDER ALLOCATION PROCESS 877

in the growth ra te over t ime, as in th is case, t h e more flexible form of inpu t may be used. T h e engineer t h e n specifies u p t o th ree per iods of t ime (PERl, etc.) a n d the growth t h a t is expected to occur dur ing each period within each allocation area. An annua l growth r a t e ( A N N R A T E ) is required for all t ime after t h e growth per iods .

For example , consider again allocation area ΑΑ4. T h e first growth period ex tends from t h e facility coun t da t e of 12/77 to 12/78, one year in length. (Any t ime period is acceptable.) Dur ing th i s first period, t h e forecast growth is 100 pairs . Dur ing t h e nex t period, one year in length, t h e forecast growth decreases to 70 pairs . T h e n over t h e nex t two years t h e forecast growth is 80 pairs . (This is a smaller annua l r a t e , of course, t h a n t h e previous period.) Finally, after 12/81, t h e forecast a n n u a l growth ra te for allocation area AA4 is 40 pai rs per year.

T h e growth forecasts for a feeder section are an accumula t ion of t h e forecasts for all allocation areas fed by and beyond t h e section.

Using the growth margins and growth forecasts, the program computes the t ime (in years) to relief ( Y R S T O R L F ) , which is actual ly t h e t ime t o reach the objective fill. Th i s is de termined for bo th allocation areas and feeder sections. As discussed earlier, t h e t imes to relief for a par t icular section and t h e allocation areas it feeds are frequently different. T h i s may indicate t h a t there is a be t t e r way to allocate exist ing spa re pairs .

I t is in t h e next th ree columns, however, t h a t the ideal al location be­comes evident. E Q U A L I Z E D Y R S and P A I R S represent the load balanced t imes to relief and number of pairs available for each allocation area and for the unallocated pairs in each section. T h e load balanced t imes to relief are based on the critical sect ions ' t imes to exhaust ion, as discussed earlier. Using these values, t h e equalized n u m b e r of pai rs available, i.e. t he ideal allocation for theoret ical load balance, may be de te rmined . I t is

/ n u m b e r of \ /g rowth over equalizedx Equal ized n u m b e r _ Vpairs ass igned/ \ t ime to relief / of pairs available objective fill a t relief In t h e last column ( S U R , D E F ) , t h e allocation surp lus or deficit is

pr in ted . I t is Allocation _ / n u m b e r of \ _ /equal ized n u m b e r \

surplus/defici t Vpairs ava i lab le / \ of pairs available / A negative value (deficit) for an allocation area indicates t h a t the re a re too few pairs allocated to it. This allocation area will require relief sooner t han if t he ideal allocation were m a d e to it. A positive value (surplus) for an allocation area indicates t h a t too m a n y spare pairs a re available for the area. I t s t ime to relief will be greater t h a n its critical section's , and therefore, some other allocation area will have a deficit.

878 THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1978

For example, allocation area a a 6 in section 2 1 0 5 has an allocation surplus of 96 pairs. T h e t ime to relief for this allocation area is 8.2 years. Since it is beyond the most critical section ( 2 1 0 3 ) , its equalized t ime to relief is 1.8 years. Therefore , by allocating more t han 1.8 years of spare pairs to this allocation area, other allocation areas are shor tchanged.

Consider AA4 fed from section 2 1 0 3 . Th i s section is the mos t critical section and its t ime to relief is 1.8 years. Allocation area AA4, however, has only 0.3 of a year to go. T h e deficit of—144 pairs indicates t h a t 144 additional pairs should be allocated to this allocation area so tha t it might last as long as its critical section.

In each feeder section, the allocation surplus/defici t for unal located pairs is also de termined. T h e ent i re unal located 50 pair complement ending in section 2 1 0 4 is surplus . Th i s complement should be allocated to an allocation area with a deficit.

Deficits of unallocated pairs may occur in the sections which are ad­jacent (on the centra l office side) to a critical section. Such is t h e case for section 2 1 0 2 which precedes the mos t critical section 2 1 0 3 . Ideally there should be 149 unallocated pairs in section 2 1 0 2 . Upon relief of the critical section these unallocated pairs would be spliced to t h e new cable and then allocated to allocation areas far ther ou t in the route .

Now, by taking a look a t the route as a whole, what potential changes in the present allocation can the engineer identify?

T h e most critical deficit is in AA4. No t only is the deficit large (—144) bu t the t ime to relief for this allocation area is extremely short (0.3 year). I t should be possible ( though not necessarily economical) t o e l iminate this deficit using allocation surpluses in sections beyond 2 1 0 3 , i.e., those sections above it in the table . Since there is a surplus of 96 spare pairs in AA6 in section 2 1 0 5 , there may be a spare 100 pair complement which could be t ransferred from AA6 to AA4. Fu r the rmore , the 50 unal located pairs ending in section 2 1 0 4 could be allocated to AA4 since they pass through section 2 1 0 3 .

Rathe r than using the 1 5 0 pairs jus t discussed, could 1 5 0 pairs be obtained from AA3 (151 pair surplus) in section 2102? Since this section is closer to the central office, the pairs in AA3 's pair group do not extend far enough out into the route to be allocated to AA4. If, however, there were 150 dead pairs in section 2 1 0 3 , t hen the central office pairs might be extended by splicing them to the dead complements .

T h e surplus in A A 3 should, however, be used to satisfy t h e deficit of unallocated pairs (—149) in section 2 1 0 2 . A 150 pair spare complement should be reserved for use upon the relief of section 2 1 0 3 .

Similarly, pe rhaps 1 0 0 pairs of A A l ' s allocation should be held back to satisfy the unallocated deficit in section 2 1 0 1 . This is less important , though, because the relief of critical section 2 is considerably later t han for critical section 1. Fu r the rmore the exhaus t da tes of critical sections 2 and 3 indicate t h a t they may be relieved simultaneously.

FEEDER ALLOCATION PROCESS 879

3.3 Implementation of the Ideal allocation

T h e de te rmina t ion of an ideal load balance does no t complete t h e allocation process. Next the outside p lant engineer m u s t s tudy practical and economical rea r rangements suggested by the idealized feeder con­figuration.

Since it is probably not economical to perfectly balance an entire route, t he engineer generally begins with t h e mos t serious shor tage. Th i s first involves a detai led examinat ion of various cable records, such as un ­derground schematics, to determine precisely what type of cable splicing work is required to move spare complements from an allocation area with a surplus to another with a deficit. T h e engineer de te rmines how m a n y pairs are to be rearranged, which par t icular spare complements are t o be used, which splice closures must be opened, and whether a s tub (short length of cable) will have to be placed in t h e manhole . T h e objective is to minimize cost, of course, b u t there is also a considerat ion of factors such as present manhole congestion, which is difficult to quantify. Several a l ternat ives may be examined.

After practical rearrangements are identified, an economic evaluation should be made . Th i s is a comparison of the rear rangement costs (cable splicing labor and materials) with t h e savings of deferred relief (due to delayed capital expendi ture) . Only if t he rea r rangement proves eco­nomical will t h e engineer p repare work orders for its implementa t ion .

IV. A FEEDER ADMINISTRATION PACKAGE

T h e allocation process is pa r t of a feeder adminis t ra t ion package now being used by outs ide p lan t engineers in t h e operat ing te lephone com­panies . Th i s package includes a user-or iented manua l containing guidelines, procedures, engineering tools, and documentat ion to aid these engineers in their jobs. In addi t ion to these mater ia ls , t he re is also a comprehensive t ra ining course.

V. CONCLUSIONS

T h e allocation process may be applied to apportion existing spare pairs in a feeder route. This can help to identify rearrangements of the network which will defer relief, leading to higher utilization and reduced capital costs. Similarly, these techniques a re useful in p lanning t h e allocation of an upcoming relief cable.

Whe the r existing or relief pairs are involved, allocation establ ishes a p lan for t h e use of spare facilities over the ent i re route . T h e n fewer rea r rangements will be m a d e and those which are m a d e will be cost ef­fective. Exped ien t rea r rangements , often involving working pairs , will be required less frequently. Thus , no t only will operat ing costs be re­duced, bu t fur ther complicat ion and congestion of t h e feeder p lan t will be avoided.

880 THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1978

VI. ACKNOWLEDGMENTS

FEEDER ALLOCATION PROCESS 881

T h e recent developments in feeder adminis t ra t ion reflect the work of a n u m b e r of outside p lan t engineers th roughout the Bell System. M a n y of their ideas were brought together by the Feeder Task Force of t h e Bell Sys tem Design and Utilization Commit tee .

Specific contr ibut ions to the concepts described in this paper were made by W. N. Bell (Bell Laboratories), G. J . Dean and D. W. Post (New Jersey Bell) , R. A. Wallfred (Nor thwestern Bell), and J . G. Funk (Western Electr ic) .

APPENDIX A Mathematical Description ol the Load Balancing Process

Thi s appendix mathemat ica l ly describes the load balancing process for a generalized feeder route . T h e computer program described in Section 3.2 is based on this model. T h e route is a single linear pa th with allocation areas, dead pairs and unallocated pairs. Allocation area growth forecasts may vary determinist ical ly over t ime. Prespecified objective fills are permit ted . Multigauge and mul t ipa th routes are not considered here .

A. 1 Define route configuration and growth forecast

Let N, be the number of feeder sections in the route. Let i be the index for the i t h feeder section, with i = 1 represent ing t h a t section most dis­t a n t from the central office, and i = Ns the section closest to the central office.

For each section, let N'a = number of allocation areas fed from the i t h section (as opposed to th rough it) .

Le t j be the index for the ; t h allocation area fed by t h e i t h feeder section such t h a t ; = 1 t o N'a, if N'a ? ί 0. (If 7VJ, = 0, then no allocation areas are fed by the i t h section.)

For the ; t h allocation area fed from the i th feeder section (i.e., for each allocation area) , the following are given:

P'JL, = n u m b e r of pairs available

P'j, = n u m b e r of assigned pairs

A growth forecast for each allocation area is also given. As shown earlier, a simple annua l growth rate may be used, with

G'r

J = annual growth ra te

Th i s is frequently a reasonable assumption, bu t sometimes growth rates change considerably over t ime. A piecewise linear growth forecast is defined by

η = n u m b e r of growth t ime periods (0 ,1 ,2 , . . . )

If η > 0, t hen the following are given:

tk = length of ftth time period

Gji = growth during ftth time period, for ft = 1 to η At the end of the last time period, the annual growth rate Gl> is assumed to take effect. (If η = 0, then only the annual growth rate is given.) In order to determine a realistic lifetime, the objective fill at relief for each allocation area, F'i, is also given.

The items above are given for each allocation area. The following items must also be given for each (the ith) feeder section,

P'd = number of dead pairs

P'u = number of unallocated pairs ending in this section

A.2 Accumulate section data from allocation area data

The growth forecasts for a feeder section are an accumulation of the forecasts for all allocation areas fed by and beyond the section.

and, if η > 0,

G , = Σ [ C o r * " ! . m - l L 7 - 1 J

Gi - Σ [ f c r l , m - l L 7 - 1 J

for i = 1 N,

for i = 1 , . . . , Na

and ft = 1 , . . . , η

The number of pairs available in the feeder section is the total number of pairs in the cross section. This includes all of the pairs in pair groups feeding the section's allocation areas and the allocation areas beyond. All unallocated pairs in the section and beyond it, as well as dead pairs within the section, are also accumulated.

pi _ f r ^ + p - l + p ' , f o r a l H

Similarly, the number of assigned pairs in the section is

PL= ί \f,P%\ for a l i i m - l L J - 1 J

A.3 Calculate time to relief

The growth margin is a generalization of spare pairs which includes the objective fill,

Growth _ /objectives / number of \ _ / number of \ margin \ fill / Vpairs available/ Vpairs assigned/

For each allocation area the growth margin is

Mi> = Fi>Piia-Pii, for all i,j

882 THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1 9 7 8

Next, similar calculations are made on a section basis. T h e objective section fills are derived from the objective fills of all allocation areas fed by and beyond the section and the objective fills of all unallocated pairs beyond the section. For the section most d i s tan t from the central office (ι = D,

Fl =

and for i > 1, 7 = 1

F' = £ f,F"jpk

ai+z FkPku

i - 1 i JV* I — 1

h=l 7 - 1 * - l

where objective fills for the unallocated pairs ending in the i t h section and for the dead pairs in tha t section are set equal to t h a t sect ion 's ob­jective fill. T h e growth margins for unal located and dead pairs , respec­tively, are

M'u = F'P'U

M'd = F'P 'd, for a l i i

T h e n , t h e grow margin for the i t h section is

M'; - Σ \ 2 M k j + M t ] + M'd, for all i

* = 1 L 7 - 1 J Now t h e time to relief for each feeder section and each allocation area may be calculated, using the growth margins and the growth forecasts, i.e.,

T i m e to _ growth margin

relief growth ra te Since the growth forecast is a piecewise linear function, so too is t h e

t ime to relief. For feeder sections when the growth marg in is less t han the forecast growth for the first t ime period, i.e.,

Mi < G\

the t ime to section relief is

Τ· = —— G'i/ίι

Similarly, for allocation areas when

M'j < G ^

FEEDER ALLOCATION PROCESS 883

t hen

Λ Γ , · M" Tu = —-

For feeder sections whenever

Gi < M' < f; GJ, A-l

t hen m-l

Gm/tm k-i

where m satisfies

"Σ* G), < M1 < Σ G),, ( K m < η) k-l fc=l

Similarly, for allocation areas , whenever

Cy < M'V < Σ Gj/ *=i

t hen m-l

MlJ - Σ GV) k«i / m-l 7 " ; — — + Σ th

Gm/tm k-i where m satisfies

m—1 m Σ Gli< M'J < Σ Gi/, ( 1 < m < n)

A-l fc-1 Finally, for feeder sections whenever

Mi> Σ Gi k-l

t hen t h e t ime to relief is

( M ' - £ G i )

τ — + Σ t„

G'r k-l Similarly for allocation areas whenever

Mij > £ G k-i

t hen

( m o - £ g v ) \ *=i / n

T " ^ + Σ t* (*/ A-l

884 THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1978

As an example , consider the case where

η = 3

a n d

Gi + G 2 < M < G , + G 2 + G 3

( T h e feeder section and allocation area superscr ipts are deleted for simplici ty in th is example.) T h e t ime to relief, T, shown graphically in Fig. 6, is

_ M - (G, + G 2 ) , Τ ~ -+ (ti + i2)

" 3 / ' 3 A.4 Calcul»!* the Ideal allocation adjustment ( surplus or deficit)

N e x t t h e feeder section t imes to relief, T', are used to identify the critical sections. T h e first (or most) critical section is t h a t section with t h e shor tes t t ime to relief. Let / , be the feeder section index of the i t h crit ical section. T h e n

/1 = ;' such t h a t T' = min (Τ*) \<k<N,

T h e second critical section is the section between the first critical section a n d t h e cent ra l office with the least t ime to relief. T h u s

I2 = > such t h a t T> = min (Tk) / i < * < / V ,

GROWTH RATE OF G, ^

\

t — Ί —-4— '•> 4 · — ' a — A r 1

TIME

Fig. 6—Time to relief (7) for η = 3 and G, + G2 < M < G, + C 2 + G 3 .

FEEDER ALLOCATION PROCESS 885

T h e general form for th is equat ion is

Im = j such t h a t TV = min (Tk) /m-i<fc<N,

for

m > 1 unti l Im = Ns

T h e n u m b e r of critical sections, Nc, is t he value of m when

Ideally, t h e relief of t h e route ' s mos t critical section can be deferred for the greatest t ime if all allocation areas fed by and beyond this section have t h e same t ime to relief. T h e equalized time to relief for t h e allo­cation areas (and the sections) fed by and beyond the first critical section is

ET = Er = T ' ' , for i = 1 t o Ix

and j = 1 t o N'a

For t h e kth critical section (k > 1)

Εψ = Ε'τ = Τ1», fork = 2 to Nc, i = Ih-χ + 1 to /*,

and ; = 1 to N'a

T h e equalized growth margin is t h e n u m b e r of pairs each allocation area will grow over its equalized t ime to relief. I t is used to determine how m a n y pairs should be available to an allocation area so t h a t it will no t requi re relief unt i l its equalized t ime to relief.

T h e function for the growth over t ime is derived from the function for t h e t ime to relief for allocation areas. For

Eif < ti

t h e equalized growth margin over this t ime is

Eü^-t-Er ti

Fo r t he range

t1 <ET< t h

t h e n G'i I m - l χ m - l

Eû = -*(E'T- Σ th)+ Σ GV,

where m satisfies m—1 m Σ h < Et < Σ th, ( Κ m < η)

k-l k-l 886 THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1978

Finally for

Ε'τ > Σ tk

k=l

t hen

Eft = G'r

j (E'i - ± t k ) + Σ GH \ k-l / k-l

T h e equalized pairs available for each allocation area is s imply t h e n u m b e r of available pairs each allocation area would have if it were perfectly load balanced. I t is

/ n u m b e r of \ / equalized \ Equal ized _ Vpairs ass igned/ \ g rowth m a r g i n /

pai rs available objective fill a t relief T h u s ,

E o v - p i j

Finally, t he ideal allocation adjustment (surplus or deficit) for allo­cation areas is the difference between actual and equalized pai rs avail­able,

Ideal allocation _ / n u m b e r of \ /equal ized n u m b e r s ad jus tmen t Vpairs ava i lab le / \ of pairs available /

or

A» = Pï„ - Et

Deficits of unal located pairs may occur in t h e sect ions which are ad­jacen t (on the centra l office side) to a critical section. Upon relief of the critical section these unallocated pairs would be spliced to the new cable and t hen al located to allocation a reas far ther ou t in t h e route .

I t is also possible for there to be too m a n y unal located pa i rs in a par­t icular section . T h u s t he unal located pair allocation ad jus tmen t m u s t be de te rmined for every feeder section .

Firs t t he equalized growth margin for unal located pa i rs m u s t be de ­te rmined in each section . This is t he growth t h a t will occur between t he equalized t imes to relief between adjacen t critical sections . T h i s t ime difference be tween any adjacen t sections i and i — 1 is

Ε'τ-Εψ\ for i = 2 t o A / s

For

E'T -Εψλ> 0, (i > 1, implicitly)

the equalized growth margin for unallocated pairs is the growth occurring from t ime Elfl to E'T.

FEEDER ALLOCATION PROCESS 887

Two evaluat ions m u s t be m a d e of t h e function for t h e equalized growth margin in a section, E'M, over t ime . (This function is similar t o the function for allocation area equalized growth, above.) T h e function is given below with the t ime Ε'τ ( ra ther t h a n E'f1) as t h e i n d e p e n d e n t variable. For El

T < t i , ni

pi _ 1 r u

tl

For

h < E l

T < Σ t k

t hen G' / . m - l \ m - l

E'M = (Ε'τ - Σ t k ) + Σ Gh

t m \ A - l / fc-1

where m satisfies m—1 m

Σ th<ElT< Σ tk, (Km<n) Finally, for

£'r > Σ t k

k-l t hen

Eh = Gi (£'T - Σ tk) + Σ G), T h e n the equalized growth margin for t h e unallocated pairs in a section is

Eh» = E l

M - Ε ι ΰ \ for i = 2 Λ7. if t he section far ther from t h e centra l office has a shor te r t ime to relief, i.e.,

Ειγ — Εγ ^ ^ 0

If, however, t he section closer to t h e cent ra l office has t h e shor te r life­t ime, i.e.,

Ε 71 Elfp ^ ^ 0

or if t h e section is the most d i s t an t one from t h e cent ra l office (i = 1), t hen

After the unal located equalized growth margins a re de te rmined , t h e equalized available pai rs for unal located pairs may be calculated,

E U -ËMl **<>»»- pi

βββ THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1978

Finally t h e ideal allocation ad jus tment for unal located pai rs is

4 ' = P' — E'

A.5 Glossary of symbols

A.5.1 Route data Given:

Ns = n u m b e r of feeder sections η = n u m b e r of growth t ime periods tk = length of kth t ime period

Calculated:

/„ , = section index of m t h critical section Nc = n u m b e r of critical sections

A.S.2. Section data

Given:

N'„ = n u m b e r of allocation areas fed by each section P'd = n u m b e r of dead pairs in each section P'u = n u m b e r of unal located pairs ending in each section

Calculated:

G'k = forecast growth dur ing kth t ime period G'r= forecast growth ra te after all specified t ime per iods Ρ'αυ = n u m b e r of pairs available P'af = n u m b e r of pairs assigned F' = objective fill a t relief M ' = growth margin Ml = growth margin for unal located pairs M'd = growth margin for dead pairs T' = t ime to relief E't = equalized t ime to relief E'm = equalized growth margin F'MU = equalized growth margin for unal located pairs F'quu = equalized pairs available for unal located pairs A'U = ideal allocation ad jus tment (surplus or deficit) for unal located

pairs

A.5.3 Allocation area data

Given:

Ρ | / μ = n u m b e r of pairs available P'J

9S = n u m b e r of pairs assigned G'i = forecast growth dur ing kth t ime period G y = forecast growth ra te after all specified t ime per iods

FEEDER ALLOCATION PROCESS 889

F'J = objective fill a t relief

Calculated:

Ml> = growth margin T'{ = t ime to relief Εψ = equalized t ime to relief E'if = equalized growth margin E'JU = equalized pai rs available A'J = ideal allocation ad jus tmen t (surplus or deficit)

REFERENCES

1. N. G. Lone, "Loop Plant Modeling: Overview," B.S.T J., this issue. 2. G. W. Aughenbaugh and Η. T. Stump, "The Facility Analysis Plan: New Methodology 3. J. Freidenfelds, "A Simple Model for Studying Feeder Capacity Expansion," B.S.TJ.,

this issue. 4. W. L. G. Koontz, "An Approach to Modeling Operating Costs in the Loop Network,"

B.S.T.J., this issue.

890 THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1978