This paper is prepared Tor theBank's internal use and is notfor publication. The views arethose of the authors and notnecessarily those of the Bank.

INTERNATIONAL BANK FOR RECONSTRUCTTON AND DEVELOPMENT

INTERNATIONAL DEVELOPMENT ASSOCIATION

Economics Department Working Paper No. 29

TIIE OPTIMUM NUMBER OF BERTHS OF A TvA71OJmAl_

November 29, 1968 SVW P29

This paper presents a method of specifying therelationship between the total waiting time of shipsat a port and the number of ship arrivals, the numberof berths and the average service time per ship. Thisrelationship enables us to derive curves showing the

~ variation in waiting time costs with different numbersof ship arrivals.

The paper then describes how demand curves for portWfBG 1 services can be drawn for a series of years and how these

curves can be combined with the waiting-time cost curvesto determine the desirability and timing of investment inadditional berths. The paper's annexes spell out thestatistical assumptions on which the waiting-time modelis based and test these assumptions for the port of Callao.

6S, 51C)

Sector and Projects Studies Division

Prepared by:

Jan de WeilleAnandarup Ray

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TABLE OF CONTENTS

Page No.

PREFACE

I. Introduction and Outline........ ...... 12

II. The Simple Case ...... .. . ..... ..... 3

III. The Mlore Complicated Case ....... 6

IV. Cost-Benefit Analysis. The Optimum Invest-

ment Pattern .................. ...... ....... 13

V. Three Practical Topics .................... 0 18

A. Special Cases .................... 18

Berth Specialization 8s

One-Way Channels. ..... .... . . . ....... 19

B. Additional Investment Alternatives .19

Reduction in Service Time. 19

Containerization. . 20

Lighterage ....... .. 20

C. Distribution of Benefits ................. 22

VI. Possible Further Model-Testing and Research.. 24

ANNEX I Ship Operating Cost Data

ANNEX II The Two Basic Assumptions: Poisson and

Erlang Distributions

ANNEX III The Two Basic Assumptions Tested: The Port

of Callao

ANNEX IV Mathematics of Waiting Time Computation

ANNEX V Quantification of Waiting Time

Table of Contents - contd.

TABLES

II-1 Calculation of Waiting Time: An Example

CHARTS

III-1 Total Waiting Time of Ships, Number of Berths and

Average Service Time Per Ship

III-2 Total Waiting Time as a Function of Product of

Ship Arrivals and Average Servi.ce Time

IV-1 Supply Alternatives and Demand

V-1 Berth Versus Lighterage

ANNEY. TABLES

I-1 Ship Size and Operating Costs

II-1 Poisson Distribution for Selected Numbers of Ships

Arriving

II-2 Erlang (k = 1) Distribution for Selected Average

Service Time

V-1 Total Annual Waiting Time (in days)

V-2 Total Annual Waiting Time and Product of Annual

Ship Arrivals and Average Service Time

ANNEX CHARTS

I-1 Daily Operating Costs of Ships in Port

II-1 Assumed Arrival and Service Time Distribution

PREFACE

This paper deals with a problem in port economics. It tries

to answer the question: How many berths should a port have? Intackling this particular problem, the paper touches upon a number of

related problems in port economics. Deliberately we have made no

attempt in this paper to solve all these problems simultaneously. We

expect to deal with these problems one by one in other papers.2/

Ultimately, we hope to combine the various papers in one volume under

a general title such as "Port Economics".

We are grateful for the helpful comments and assistance we

received in preparing this paper from our colleagues in the TransportationProjects Department and the Statistical Services and Sector and Projects

Studies Divisions of the Economics Department, and from professionalsoutside the Bank. Miss Suzy Snell edited the paper. However, the usual

disclaimer applies: the authors alone are responsible for the views

expressed in this paper.

1/ Subjects for other papers are: The Optimum Depth of a Port (pre-

liminary draft prepared); The Optimum Productivity (service tuixe)of a Port; Demand for Port Services; Pricing of Port Services;

Quantification of Ship Operating Costs; Uncertainty Aspects in Port

Appraisal; Organizational Aspects of a Port; Containerization and

its Future; and finally a paper presenting an integrated view of

the various aspects dealt with in the individual papers.

THE OPTIMUM NU-DER OF BERTHS OF A PORT

I. INTRODUCTION

1. This paper deals with the question: how many berths should aport have? Given that the number of ship arrivals in the port changesover time, some port capacity (number of berths) exists which maximizesthe total benefits to ship owners and the Port Authority. 1/

2. On the one hand, if the Port Authority insists on 100 percentutilization of its capacity, it will minimize its costs per ship. Shipowners, however, will incur costs waiting outside the port for a berth. 2/On the other hand, if enough berths are constructed and maintained toavoid any waiting, the ship owners waiting time and costs will bereduced to zero, but the Port Authority will be faced with high costsfor construction and maintenance of berths. Its marginal berth mightconceivably handle only one ship per year.

3. Thus, the basic problem is one of trade-offs: more berths, morecosts for the Port Authority and less for the ship owners; less berths,less costs for the Port Authority and more costs for the ship owners. Theoptimum number of berths will be somewhere between the number of berthswhich would be fully utilized all year-round (lower limit) and the numberof berths required to avoid any waiting of ships (upper limit).

4. The general problem of reconciling capacity with changing demandis not restricted to ports. The power, airline, tourism, restaurant, and

1/ The service time should also be treated as a variable when theoptimum number of berths is in question. For simplicity's sake,this report considers service time given.

2/ Although estimates of the ship waiting time costs differ and nounanimity of opinion exists, an amount of some $1,200 per day fcra 10,000 ton ship might be considered indicative of the order ofmagnitude involved. Annex I to this paper, "Ship Operating Cost Data",presents some information on this subject.

railway industries are also familiar with this problem. All these

industries share the property that the services they produce are not

storable. 1/

Outline

5. Sections II and III of this paper focus on the question: howmuch waiting time will be incurred for given numbers of berths and given

numbers of ships arriving annually? Section II, "The Simple Case", assumes

that we know when the ships will arrive during the year and how much time

each ship will spend in berth. Section III, "The More Complicated Case",

does not assume such knowledge. Instead, a number of assumptions are made

and the total waiting time is computed on the basis of queuing theory.

Annex II briefly presents the two basic assumptions in quantitative terms

and Annex III shows for the Port of Callao (Peru) the results of a test

of the two basic assumptions. Annex IV presents the mathematics of the

queuing model used in this paper and Annex V presents the quantitative

results for a large range of situations. Chart III-2 (page 11 ) presents

these results in a nutshell.

6. Section IV, "Cost-Benefit Analysis", introduces the demand side

and in combination with the results of the preceding section discusses

the cost/benefit analysis. Chart IV-1 (page 15) provides a visualimpression of the supply alternatives in combination with demand.

Section V, "Three Practical Topics", focuses on a number of points which

we think pertinent to the subject of the paper. In the last sectionsome suggestions are made for possible further research on the subject.

1/ The manufacturing industry, on the other hand, is in a position to

produce for a future moment in time and by storing its output reconcilescapacity and demand for its services. The storability of the services

thus introduces an element of flexibility on the capacity side.

- 3 -

II. THE SIMPLE CASE

7. The simplest way to approach the problem is to assume that thenumber of ship arrivals for each day over a certain period, for example,

ten days, are knowin. This might be the case for a port receiving only

liner ships following a fixed schedule. For exanmple, on January 1, threeships are expected; January 2, no ships; January 3, five ships; etc. In

addition, it is assumed that the service time for each ship is known,

in our example, one day. Given then the number of berths available in theport, the total waiting time during the period can be computed as follows:

Table II-1: Calculation of Waiting Time: An Example

(1) (2) (3) (4) (5)

Date Ship Arrivals Four-Berth Port Five-Berth Port(number) berths ships berths ships

available waiting available waiting

1-1-1968 3 4 0 5 0

1-2-1968 U 4 0 5 0

1-3-1968 5 4 1 5 0

1-4-1968 7 4 4 5 2

1-5-1968 2 4 2 5 0

1-6-1968 2 4 0 5 0

1-7-1968 6 U 2 5 1

1-8-1968 3 4 1 5 0

1-9-1968 5 4 2 5 0

1-10-1968 1 4 0 5 0

Total 34 40 12 50 3

-4 -

8. As column 2 shows, overall capacity in a four-berth port duringthe 10-day period considered is 40 berth days. The demand for berth daysis slightly less: 34 (34 ships requiring 1 day each). Nevertheless, asthe third column indicates, on six out of the ten days ships are waitingto be admitted because of the irregular arrival pattern. The total numberof ship days spent waiting is 12.

9. If one more berth were available this number would be reducedto only 3. In other words, adopting the previous figure for the value ofone ship day ($1,200), the gross benefits during this 10-day period ofhaving one more berth would be $10,800 ($1,200 x 9). 1/

10. The example of Table II-1 covers too short a period on wjhich tobase a cost-benefit analysis. However, the principle is clear: if full

information on the timing of the ship arrivals is given and the servicetime for each ship is known, the total waiting time and costs can becalculated for different numbers of berths.

i1. Before moving to the next section, the term berth occupancy rate,frequently used in port project appraisal, should be introduced. Thisterm indicates the degree of utilization of available berths. Forexample, in the case of Table II-1 when 4 berths are available, theaverage berth occupancy rate amounts to 34, i.e., 85%o In case 5 berths

40are available, the occupancy rate would be -0, i.e., 68%. For 6 berths,

50it would be 34, or 57%. Once the optimum number of berths (say 5 berths in

60our example) has been determined, the optimum berth occupancy rate (68%)for the port in question follows automatically. Since optimum number ofberths and optimum berth occupancy rate reflect two sides of the samecoin, the discussion in this paper runs only in terms of the optimum

1/ This presupposes that no additional ships would be attracted by the portif an additional ber-th were available and waiting time would bereduced, i.e.,that demand is fully inelastic.

-5-

number of berths.

12. In general, for a one-year period,

number ofship arrivals

percentage = 100

berth occupancy (average number of ) (number)

(ships each berth ) X ( of ) X 365(can service per day) (berths)

The denominator in the expression is the total capacity of the port inthat year, expressed in terms of the total number of ships the port iscapable of servicing. This capacity will increase as the average numberof ships each berth can service per day and/or the nunber of berths in-crease. As can be seen from this definition, the optimum nrmaber of berthsimplies a corresponding optimum berth occupancy rate, given the averagenumber of ships that can be serviced per day per berth in the port.

- 6 -

III. THE HiORE COMPLICATED CASE

13. The previous section made the convenient assumptions that both

the number of ships arriving each day and the service time of each ship

are knowfn. On that basis it was possible to construct a table such as

Table II-1 and to determine without difficulty the ship waiting time

involved.

14. In practice, however, neither the timing of ship arrivals nor

the service times for the individual ships are known. Consequently, some

workable and realistic assumptions have to be made in order to enable

computation of the waiting time. This paper makes two assumptions with

regard to the ship arrivals:

(a) thiat the frequency distribution of the number of

ships arriving daily follows a Poisson distribution;l/

(b) that the timing of the daily ship arrivals is

random during the period under consideration.

Assumption (a) was adopted because the literature on the subject indicate3

that the Poisson distribution often fairly accurately reflects observed

freqaelncy distribution.2/3/ Assumption (b) only indicates that no rela-

tionship exists between the number of ships arriving on a certain day, the

day before, or the day after.

1/ See Annex II to this paper.

2/ Israel Ports Authority, Reports and Statistics Center: Yearbook of

Israel Ports Statistics, 1963/64, Haifa, January 19659 ppXI and XII.

The report states: "For the first time, the frequency of daily ship

arrivals has been analyzed with regard to Haifa port. This analysis

confirmed the hypothesis that most vessels arrive at the port at randon,

intervals, at frequencies that may be approximately described by the

Poisson distribution. Similar results have been reported by ports

abroad. (See Petter C. Omtvedt, Report on the Profitability of Port

Investments, Oslo, 1963 and F. Marques Da Si va, Boletin do Porto de

Lisboa, No. 150, 1963.) The Poisson distribution may therefore serve

as a convenient mathematical model for predicting frequency distri-

bution for ship arrivals."

3/ See also Annex III to this paper.

- 7 -

15. With regard to the service time of ships this paper assumes:

(a) that the frequency distribution of the servicetimes is reflected by an Erlang (k = 1) frequencydistribution; 1/

(b) that the specific service times are independentlydistributed.

Assumption (a) again was adopted because the literature indicates that theErlang distribution often provides a fair approximation of reality. 2/ 3/Assumption (b) only indicates that the service time of a certain ship isindependent of that of the ship just serviced and does not influence theservice time of the next ship. The reason why the service time is regardedas a random variable is because there are numerous factors affecting theaverage service time per ship. This assumption is realistic if thereare no strong functional relationships between the average service timeand a small number of variables.

16. Given these four assumptions and the number of ship arrivals peryear, the number of berths available, and the average service timeper ship, the total waiting time can be computed. For the mathematicsinvolved, see Annex IV to this paper. /

17. Annex V shows total annual ship waiting time for a range of annualship arrivals (100 - 2,000), average service times (one to five days) andnumbers of berths (one to fifteen). The abbreviation n.f. (not feasible)indicates that the combination listed would result in an endless queuebuilding up. Chart III-1, based on a sample of the data from Annex V,presents a visual impression of the relationships involved.

1/ See Annex II to this paper.

2/ Mettam, J.D., Cons, M., Forecasting Delays to Ships in Ports, (undated).

3/ See also Annex III to this paper.

4/ See also Sasieni, M., Yaspan, A. and Friedman, L., Operations ResearchMethods and Problems, New York, 1959, pp. 143-145,where the generalproblem is discussed.

Chart III-1: Tctal-Waitiug T1ae g Shb. Number of Berths and Average Service Time Per Ship

100 Ships 300 Ships 500 Ships 700 Ships 900 Ships 1500 Ships

A. Average Service Time: One Day

n. f

8)000-

6,oo

Io0o

2,000.1,000.

800I6400.

. 2 00 3

442t Ll

13 135 135 1 3357'-4

a, UN U OF BERM IN CT B. Average Service Time: Two Days

A.t

to 8,cooo6,000c

ak. 4~,OOD

o 2,000.

600.S- 400.

200.

1 1 5 3 5 _ 3 5 7 13 5 7 1 3 5 7 9

C. Average Service Time: Three Days

NOTE n..idiaentfasbe

0 8,000.4 6,000tn h1,000.

2,000.1,000.

.4 800-600.

200-0 .

1 35 35 3 57 1 3 579 1 3 579 1 3579 fl 13

NUMBER OF BERTHS IN PORT

NOTE: n.f. indicates not feasible.

18. As to be expected, Chart III-1 shows that by increasing numbersof ship arrivals (from left to right), or increasing average service times(f,rom top to bottom), the total annual waiting time increases sharply..For example, if average service time is one day and 100 ships arriveannually, total annual waiting time is close to zero for a one-berthport. If 200 arrive, the waiting time increases to about 1,000 ship days/annum. Whereas,if 500 ships arrive, the one-berth capacity would notsuffice any more because an endless queue would build up. Increasingthe average service time per ship for a given number of ship arrivals hasa similar impact on the total waiting time.

19. Looking diagonally across Chart III-1, an interesting interrela-tionship between the variables becomes clear. For example, total waitingtime is exactly the same for 1,500 ships arriving annually and an averageservice time of only one day, as for 500 ships arriving and an averageservice time of three days. Similarly, it is the same for 900 ships andan average service time of one day, as for 300 ships and an average servicetime of three days. Total annual ship waiting time is a function of theproduct of annual ship arrivals and average service time, i.e., total annualservice time required.

20. The chart also suggests that, if we reject the possibility of anendless queue building up, the optimum number of berths lies within anextremely limited range. In other words, our number of options isextremely limited: only two in most of the chart cases. In someinstances, the optimum number of berths follows automatically from thechart. For example, in the case of 500 ships per year and an averageservice time of two days, the choice is between three and four berths.In the case of 900 ships and three days service time the optimum is eightberths, in the case of 1,500 ships and service time of two (three) daysthe optimum number of berths amounts to nine (thirteen). We repeat thatthese conclusions are based on the assumption that we do not accept thepossibility of an endless queue building up.

21. Chart III-2 throws more light on this issue and shows that themaximum finite queue (or maximum annual waiting time, given that wereject the possibility of an endless queue) depends on the number of berths:

- 10 -

Number of Berths Maximum Finite Queuei/Approximate number of,ship days per annum2,

l . . . . . . . . . . . . 12,000

2 . . . . . . . . . . . . ll,'000

3 . . . . . . . . . . . 8,000

4 . . . . . . . . . . . . 800

5 . . . . . . . . . . . . 220

6 . . . . . . . . . . . . 60

7 . . . . . . . . . . . insignificant

8 . . . . . . . . . . . . zero

l/ The maximum finite queue, of course, is reached when the product ofaverage service time and annual ship arrival approaches the totalcapacity of the port in terms of berth days (in other words, whenthe berth occupancy rate - see paragraph 5 - approaches 100 percent).

2/ Arrived at by visual extrapolation of the calculated values ofChart III-2. As the number of berths increases, the maximum feasiblewaiting time sharply decreases.

L * ~~~Chart 111-2' Total Waiting Time as a Ftmotion of Product of Ship Arival asand Average Service Time

1- r 00 -

-%O00- Ahs~~~~~~~~~~~~~-

-7~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~K

21* -

-~~~~~~~~~~~~~~~~~~~~~b

I'm -.- '-.

*~~~~~~~~~1 - ~ ~ ~ ~ ~ ~ ~ ~ ~ -

100 200 300. bq0 500 6G0 70086 00010 io-lw1303m 1W T4-DIUW A 9n1% lo

__DCTO_TTLANUL HPARRIVAL ...... EVC T ~ ________ --- I----- - - - . _ _ _ __A D AV RAG _S --RVICE T,- _ _ _

- 12 -

22. The purpose of this and the previous sections was to determinehow much waiting time would be incurred given the number of berths andgiven the number of ships arriving annually.-Some combinations resultin extrenely high waiting times. In the case of infinite queues, forexample, some ships would never be serviced. In case of a maximumfinite queue and one berth (see table in previous paragraph) the waitingtime per ship would be 33 days i 12,000 ) if the average service

time is one day, and 66 days ( 65 ) if the average service time is two365

days, etc. The next section, "Cost-Benefit Analysis", considers how manyships will arrive given the waiting time per ship, by introducing thedemand function for port services.

- 13 -

IV. COST-BEMIEFIT ANALYSIS: THE OPTIMUM INVESTtIT PATTERN

23. The preceding section has provided us with one of the building

blocks for the cost-beniefit analysis, that is, we know the total waiting

time involved given the number of berths and the number of ship arrivals.

In principle, this is a purely technical relationship. A second inputrequired for the cost-benefit analysis is the value of waiting time of

ships. Annex I presents quantitative information on the subject andindicates possible sources for additional information.

24. Two more inputs required are estimates for the cost of constructingand maintaining additional berths and the number of ship arrivals in

future years. As to the costs of additional berths, we have not attempted

here to present orders of magnitude and we assume that the relevant infor-

mation will be available in each specific case.

25. As to the future number of ship arrivals, in transport projectevaluationl/ future demand is often estimated on the basis of some

growth rate (or some more complicated relationship with income, popu-

lation, etc.) and a unique estimate for each future year is presented.The question then becomes to determine the minimum cost solutionsatisfying this given demand.2/ This approach implies that the demand

function for each future year-is completely inelastic. In other words,a certain number of ships will arrive regardless of costs in wiaiting time.

1/ This not only holds true in case of transport projects but also for,e.g., power projects.

2/ See also van der Tak, H. G. and Ray, A., The Economic Benefits of

Road Transport Projects , Economics Department, EC-160, March 1, 1968,

page 3.

26. What is required, in principle, is knowledge of the demand curve

over the relevant interval in each future year. A pragmatic approach

to estimating these future demand functions would be to estimate first

the demand in each future year (on the basis of some assumed growth rate,

e.g.) given a specified cost (waiting time) level. This would give us

one point of each annual demand curve. Then assumptions could be made

for other points of the demand curves, e.g., there is a certain

elasticity between costs (waiting time) and volume.

27. Chart IV-1 presents schematically the supply alternatives and

demand. The vertical axis indicates the waiting time costs per ship and

the horizontal axis the annual volume of ships.l/ The annual demandcurves are indicated by the year they refer to T1968 to 1970) and show

how many ships will come to the port in each year given the level of con-

gestion (i.e., waiting time costs per ship). The demand curves in the

graph are presented as straight and parallel lines but this may not

necessarily be the case in practice.

28. The upward sloping curves indicate the relationship between

annual ship arrival and waiting time costs per ship, i.e., the supply

functions, for one, two, and three berths.2/ The curves slope upwardand become vertical at tlhe point where an endless queue forms,indicating

full capacity utilization and infinite waiting time and costs.

29. No problem exists at present (1967): all ships which arrive are

promptly serviced by the one available berth and no waiting time is incurred.

1/ This implies that the average service time is given. Assuming a

different average service time would merely result in a different

scale of the horizontal axis (see paragraph 19).

/ These curves can be derived from the curves presented in Chart III-2

in combination with the unit value of waiting time. It should be noted

that Chart III-2 refers to total waiting time of all ships, whereas

Chart IV-1 refers to the cost of waiting time per ship.

Chart IV - 1: Supply Alternatives and Demand

Waiting SUPPLY: SUPPLY:Time one twoCosts berth berthsperShip

DEKiND___

1970

DEMAN

O DEKAND 196919

G

I, ~~~~~~~~~~~~~~~~~~~~~~~~SUPPLY:

berths

L

0 A- D E H AnnualShip

B 1968 benefits from an additional berth Arrivals

@ 1969 benefits from an additional berth

- 16 -

The question of an additional berth does not arise yet.l/ In 1968,the demand curve has shifted sufficiently to the right to introducesome waiting time. If a second berth were available, waiting time wouldagain be reduced to zero. In 1969 we face three alternatives: one, twoor three berths with corresponding waiting times and annual shiparrivals, etc.

30. For 1968 a decision has to be made. Should a berth be added?The benefits of an additional berth would be to reduce the waiting timecost per ship to zero and increase the number of ship arrivals. Thegross benefits for 1968 would amount to the familiar rectangle OABC plusthe triangle ADB. The rectangle reflects the total reduction in waitingtime cost for the number of ships (OA) that would have used the portanyway ("normal'' traffic)2/ and the triangle reflects the consumersurplus of the additional traffic (AD) induced by the reduction inwaiting time ("tinduced"f traffic)2/. Similarly, in 1969 the benefitsof having available a second berth wiould amount to LXFG (WYFG reductionin total waiting time for the "normal" number of ship arrivals plusMKF the consumer surplus for "induced" ship arrivals)3/. In principle,the gross benefits over the entire lifetime of the second berth cannow be computed (1968:ODBC; 1969:LKFG; 1970:etc.) and the presentvalue of -the aggregate determined. A comparison with the present valueof the total costs of the second berth (construction and maintenancecosts) trill then indicate the desirability of the investment.

31. It should be added here that the benefits during the first yearof the project should at least equal the savings which could be realizedby postponing the investment by one year.4/ In the above example: thebenefits in 1968 of ODBC should at least equal the savings which wouldbe realized by building the berth in 1969 instead of 1968. Assuming,e.g., a berth construction cost of $1 million and a discount rate of 10%l

1/ To keep the argument as simple as possible, the aspect of theconstruction time involved is neglected here.

2/ See van der Tak/Ray, op. cit., p. 4.

3/ It should be emphasized here that the cost curves representaverage costs and not marginal costs. Consequently, no producersurplus exists. See also van der Tak/Ray, op. cit., page 30,paragraph 58 on this subject.

4/ See de Weille, J., The Optimum Timing of Investments , (draft),Economics Department, December 15, 1966.

- 17 -

the savings realized from a postponement of construction by one year

would amount to $1 million minus -$1 mllion. $901910 .i1.10

32. The optimum number of berths for a certain year thus is reachedwhen (a) the marginal berth has a positive net present value and (b) the

benefits of the margina' berth during that year equal at least the savingswhich could have been realized by postponing the construction of the

berth to the following year.

33. Much the same kind of analysis applies when we consider thealternative of varying the average service time. In the end, we want

to optimize simultaneously the number of berths and the average service time

per ship, yielding an optimum berth occupancy rate (see paragraphs 43-46).

34. It will be clear from the foregoing that any a priori and

general rule-of-thumb such as: "a berth should be added if the berth

occupancy rate is above 80%?", is strictly inappropriate.2/ Of course,

once we have analyzed a specific port and have determined the optimum

number of berths, the corresponding optimum berth occupancy rate can

be determined (see paragraph 11).

1/ Plus, of course, any maintenance costs which we would avoidbecause of the construction postponement.

2/ An example of such arbitrary criterion can be found in "Prediction

of 1aximum Practical Berth Occupancy"l, Journal of the Waterways and

Harbors Division, Proceedings of the American Society of CivilEngineers, June 1960, pp. 69-7b, by T. J. Tratar, A. A. Goodman and

A. E. Brant. The article deals with the same problem tackled in

this paper. They state:"?...A reasonable limit for berth occupancy is

believed to be that at which the number of vesselsdesiring to occupy berths at the terminal exceedsthe number of berths available on about 5% of thetime, or an average of about 18 days per year..."l.

It is clear that the criterion suggested above is not based on

economic analysis. A similar arbitrary criterion exists with respect

to roads: "the 30th hour peak traffic"l.

- 18 -

V. THREE PRACTICAL TOPICS

A. Special Cases

Berth Specialization

35. In the foregoing we implicitly assumed that the berths wereinterchangeable, i.e., that all berths have the same average servicetime for all ships. In practice, however, a port might have somespecialized berths, e.g., for the exclusive use of oil tankers. Inthat case, the tanker berths, tanker traffic, and tanker waiting timecosts should be dealt with separately from the other berths, traffic,and corresponding waiting time costs. In essence, we face then twoseparate and independent questions: What is the optimum number oftanker berths? And what is the optimum number of non-tanker berths?

36. The problem becomes much more complex if all berths areinterchangeable for one category of ships but not for another. Thismight be the case for a port which has both deep and shallow berths.The deep berths can acconmodate both large and small ships while theshallowy berths can only accommodate small ships, therefore the numbersof deep and shallow berths cannot be optimized independently. If wedid optimize the nurmbers of deep and shallow berths independently, wemiglht wind up with too few deep berths and a too high number of shallowberths because we have allowed for no spill-over of small boats intolarge berths.

37. Furthermore, smaller ships may have an average service timedifferent from that of the larger ships; and larger ships will probablyenjoy a preemptive priority over small ships when it comes to occupyingdeep berths. The problem becomes still more complex if we distinguishbetween three or more categories of berths and ships.

38. In all these cases of simultaneous optimization of inter-dependent classes of berths, the best approach to solve the problemwould seem to be the MIonte-Carlo routine. The Monte-Carlo routinewould offer us probability distributions of the waiting time involvedfor specific numbers of differing berths.l/ These probability distri-butions again would allow us to calculate the expected waiting timeinvolved in each case. Finally, comparison between the cost of onemore berth and the corresponding reduction in expected waiting timecost will provide us with the optimum number and combination of berths.

1/ To reduce the number of computer runs, upper and lower bounds shouldbe computed for the number of different berths to be considered.See also paragraph 36.

One-Way Channels

39. Inland ports are connected with the sea by a channel. Providedthis channel connection is sufficiently wide to allow two-way shiptraffic, it does not create special analytical problems in the contextof this paper. The optimum numnber of berths can be determined as indi-

cated before.

40. However, problems arise if the connecting channel is narrowenough to permit only one-way ship traffic over part or all of itslength. In that case traffic has to be policed; upstream traffic willhave to wait till downstream traffic has cleared the passage, and viceversa. Ships may face three periods of waiting: (a) on approach tothe port, for traffic from the port to clear the channel passage;(b) once arrived at the port, for an ermpty berth, and (c) on their wayout of the port, for traffic to the port to clear the passage.

41. Two, and possibly three, interdependent optimization issuesare involved here. What is the optimum number of berths? What is theoptimum channel width? And if the optimum channel still impliesregulation of traffic, what is the optimum system of traffic regulation?A geno.-al mathematical formulation of this case would quickly becomevery comiplex. The best approach here again would seem to be computersimulation of a number of altiernatives.

B. Additional Investment Alternatives

Reduction in Service Time

42. The focus of this paper is on the number of berths a portshould have. For that reason the service time of ships was consideredgiven. It is, of course, clear that a reduction in service time hasa similar effect on the port's capacity and corresponding total waitingtime (costs) as an increase in the number of berths. In each specificsituation, therefore, a comparison should be made between the costof increasing the number of berths and the cost of reducing the servicetime.

43. The tables in Annex V to this paper may be of use in thisrespect. They indicate the reduction in waiting time both for anincrease in the nunber of berths,given the service time, and for adecrease in service time, given the number of berths. Once therespective costs are known, an optimum combination of berths and servicetime can be determined.

- 20 -

44. If service time per ship could be functionally related toone or two variables, it would be possible to predict the variationsin service times with accuracy and to simplify the analytical problems.In reality, it is unlikely that such a functional relationship can beidentified, because average service time depends on the time taken totie up, to discharge cargo, to clear berth, to clear cargo from sheds;

on the weather; on labor disputes, and so on, and varies greatly fromship to ship.

45. However, the average service time per ship can be reducedby improving the overall efficiency of service through factors suchas better warehouse facilities, improved cargo handling equipment andmethods, better labor relations.

Containerization

46. One very significant way of reducing the average servicetime per ship is containerization, which might well develop rapidlyin the fairly near future. Containerization has two main implicationsin the context of this paper. First, containerization of cargo reducessubstantially (twfo to tenfold) the service time needed by ships,therebysubstc.ntially increasing the capacity of a given number of berths.

Depending on the speed with which containerization will become generallyadopted, the problem of port expansion by adding berths might very well

become less relevant in the foreseeable future. If this is so, moreattention should be given to the issue of how to adapt existing port

facilities to handling containerized cargo.

Lighterage

47. Another alternative to additional berths is additionallighterage capacity. Because of the different cost structure oflighterage compared to that of berths (higher variable but lowercapital costs), lighterage in some situations may be a viable alter-native. This is illustrated schematically in Chart V-1.

48. Chart V-1 shows both the cost of an additional berth (OF)and the costs of additional lighterage (DH).1/ The cost of lighterage

1/ The "capital costs" of berth and lighterage involved here are theinvestment costs wfhich could be saved by postponing the investmantstill the next period. See also paragraph 31.

- 21 -

Chart V -1: Berth Versus Lithterage

Total LighterageCosts

G (1) Reduction in WaitingTime Costs (1)

F / e .Capital CostsF ' ' Additional Berth

0 E (2) _ ' Reduction in Waiting

(2) Time Costs-(2)

/ ~~~~~~~~~~~~~~ ;

D -_ -_ Capital CostsAdditionalLighterage

.~~~~~~~~~~ S

0 A B C

Number of Days

- 22 -

depends on the number of days lighterage is used during the periodconsidered, say, one year. To keep the argument simple, we assumethat the (additional) lighterage capacity equals the capacity of oneberth (e.g., three lighters equal one berth). Moreover, because ofthe variable cost element in lighterage services, we make the realisticassumption that lighterage capacity will function as "lender of lastresort", L,.e., will only be used if all berths are occupied and a queueis waiting. Finally, the reduction in waiting time costs resultingfrom the additional berth or lighterage capacity should be considered.Chart V-1 considers two cases: (1) a reduction OG; and (2) a reductionOE.

49. If the waiting time cost reduction from additional berthor lighterage capacity equals OG, an additional berth is the bestsolution provided the lighterage capacity would have to be utilizedmore than OB days. Otherwise, lighterage is preferable. In casethe waiting time cost reduction is OE, an extra berth will not beconsidered. Additional lighterage capacity would then be indicatedprovided it would not have to be used more than OA days, i.e.,provided the total lighterage costs did not exceed the reduction whichan additional berth would provide.

50. In summary,

Days of Waiting Time Cost ReductionLighterage Alternative (1); Alternative OERequired Berth Lighterage Berth Lighterage

<;OA 2nd best Yes No YesOA - OB 2nd best Yes No NoOB - OC Yes 2nd best No No

;>Oc Yes No No No

C. Distribution of Benefits

51. The criterion used in this paper in determining the optimuminvestment strategy has been maximization of net benefits irrespectiveof who :bhe beneficiaries are. From a national economic point of viewthis would seem to be the right approach provided that all benefitsaccrue to the national economy. This will happen if the shippingindustry is operating under fullycompetitive conditions, forcing it topass on cost reductions in the form of rate reductions. If less thanfully competitive conditions exist, rate reductions might tend to beless than cost reductions.

- 23 -

52. The policy implications are that under fully competitiveconditions in the shipping industry the Port Authority does not have totake any action: cost reductions will then be reflected in ratereductions and all benefits will accrue to the economy. Under mono-polistic conditions in the shipping industry, the Port Authoritycould bring pressure to bear on the shipping industry to pass on costreductions, or if this line of action is unsuccessful, it could intro-duce an additional port charge. It is generally felt, however, thatthe shipping industry is fairly competitive and that after some delay,cost reductions will be reflected in rate reductions.

VI. POSSIBLE FURTHER MODEL-TESTING AND RESEARCH

53. The basic model presented in this paper has already beentested by Jones and Blunder, as described in their article on thePort of Bangkok.l/ They tested the assumptions of Poisson andErlang distributions. In addition, they compared the theoreticalwaiting time from the model and from two related models with thewaiting time actually observed. It would be interesting now to tack'ea few concrete, more complicated cases such as those proposed inparagraphs 36-41, and try out the Monte-Carlo approach to the solutionof the problem.

54. We have already indicated two subjects for further research:the future of containerization and the value of a ship's time. Annex Ito this paper already supplies some information on this last subject,but further study might be desirable especially in view of its impor-tance in the cost-benefit analysis.

1/ Jones, H. H. and Blunder, W. R., "Ship Turn-around Time at thePort of Bangkok", Journal of the Waterways and Harbors Division,Proceedings of the American Society of Civil Engineers, May 1968.

ANNEX IPage 1

SHIP OPERATING COST DATA

1. Because the value of a ship's time is important in the problemdealt with in this paper, this Annex presents data on the cost of

operating ships, as well as possible sources for further research on the

topic. At this stage we have not tried to analyze the data in detail

or check the underlying assumptions, nor have we undertaken an intensive

literature search. This should be done in a separate paper (see PreZace).

2. Amnex Table T-1 and corresponding chart present data from the

Westinform Service, the U. S. Department of the Army, and unidentifiedconsultants to the U. S. Department of the Army. The U. S. data forforeign ships are consistently higher than the Westinform data.

3. Sources which might usefully be consulted for additional costestimates are: Godd, R. D., "Tlhe Turnaround of Cargo Liners and itsEffects upon Sea Transport Cost", Journal of Transport Economics andPolicy, Vol. I, No. 1, January 1967 (London). Sources mentioned in thisarticWe (page 76, footnote) are: Lane, P. A., "An Aspect of the Costof Port Delays", Yorkshire Bulletin of Economics and Social Research,

November 1957; OECD: Maritime Transport, 1963, Paris, page 37.

ANNEX IPage 2

Annex Table I-1l Ship Size and Operating Costs(in US dollars)

Kind Deadweight Daily Operating Cost Daily Operating Cost perof Tonnage 1,000 ton of Cargo 1/Ship At Sea In Port At Sea In Port

(1) ~~(2) (37) 74) 7(5)

A. Westinform Data

Tramp 7,445 1,148 907 166.68 131.7113,600 1,831 1,489 145.54 118.3816,070 2,022 1,610 136.00 108.30

Ore Carrier 15,341 2,007 1,596 143.19 113.8218,655 2,072 1,643 121.38 96.2921,588 2,268 1,756 113.06 87.5326,707 2,458 1,946 98.70 78.1234,997 2,750 2,150 87.89 68.7448,564 3,288 2,516 74.12 56.7353,285 3,748 2,808 76.05 56.9866,662 4,379 3,181 71.01 51.5880,815 5,134 3,763 68.68 50.3489,67h 5,209 3,665 62.80 44.18

B. Department of the Army Data

Ore Carrier 20,000 2,880 2,b2b 156.52 131.5225,000 3,120 2,592 135.65 112.7930,000 3,408 2,832 123.48 102.6135,000 3,864 2,000 120.00 93.1740,000 4,128 3,216 112.17 87.391,5,000 LU,34 3,360 104.93 81.1650,000 4,,608 3,504 100.17 77.7455,000 4,8b8 3,7bL 95.81 73.9960,000 5,088 3,960 92.17 71.7b70,000 5,808 b,368 90.19 67.8280,000 6,264 4,68o 85.11 63.59

C. Consultants' Data

General Cargo 13,300 5,035 3,847 L11.47 3114.38

21,600 5,775 4,210 290.59 211.84Dry Bulk Carrier 25,000 3,386 2,757 147.21 119.86

35,000 3,787 2,989 117.60 92.8250,000 L,3W2 3,3b1 94.39 72.6355,000 4,201 3,435 83.02 67.8870,000 4,893 3,706 75.97 57.5480,000 5,185 3,909 70.44 53.11

1/ On the basis of the relevant Westinform information the volume of cargo per shipis assumed to amount to 92% of the ship's deadweight tonnage (column 1 of the table).

Sources: The Westinform Service, Westinform Standard Vessel TvDes, London, June 1967.Westinform also provides cost data on dry bulk carriers. These are, however,not significantly different from the cost data for ore carriers for comparableship sizes.

U. S. Department of the Army, Chief of Engineers, Staff of the Board of &igi-neers for Rivers and Harbors, Preliminary Economic Analysis for FurtherDeepening of Delaware River. Delaware. New Jersey and Pennsylvania, April1967, Table 12, indicating operating costs of non-American ore carriers.

Consultants (unidentified) to the U.S. Department of the Army, Tables:Estimated Averaee Hourly Oneratine Cost of Ocean-eoine Genera) Careo Vessels -

Foreign Flag, July 1967, and Estimated Operatine Costs of Bulk CargoVessels - Foreign Flae, July 1967.

Annex Chart I-1: Daily QOerating Costs of Ships in Port

Ore Carrier: Bulk Carrier: . .

Tramp : ----- General Cargo:

5000

. (Consultants)

.OOO (Consultants)

g *S

A ~~~~~~~(Department of_-h3000 the Army) _

X ~~~~~~~~~~~* (~~~~~~~~~Westinform)

S. L2000

U)

1000

||>4

0 10 20 30 LO 50 60 70 80

Deadweight Tonnage (1CCC)

ANNEX IIPage 1

THE TWO BASIC ASSUMPTIONS: POISSON AID ERLANG DISTRIBUTIONS

1. The two basic assumptions made in this paper (see paragraphs18 and 19) refer to the form of the frequency distribution of the dailyship arrivals (Poisson distribution) and the form of the distributionof the service time (Erlang distribution, k = 1). In each specificcase under consideration the analyst will have to find out whetherthese assumptions adequately reflect the actual situation. If they donot, the conclusions arrived at on the basis of these assumptions donot apply. More realistic assumptions should then be adopted 1/ andthe conclusions revised.

2. To facilitate testing the two assumptions in specific casesthe following two tables present the Poisson and Erlang (k = 1) distri-butions for a range of ship arrivals and average service times. AnnexTable II-1 indicates, e.g., that if 200 ships arrive annually, and thearrival pattern is Poisson, during 57.81% of the days (Part A of the table)or 211 days (Part B of the table) no ships will arrive; during 31.68%of the days, or 116 days, one ship will arrive. And so on. Similarly,if 40O instead of 200 ships arrive the corresponding figures will be33.42% or 122 days, and 30.63% or 134 days.

3. Table II-2 indicates,e.g., that if the average service time isi day, and the service time distribution is Erlang (k = 1), 39.35% ofall ships will have a service time between zero and ¼ day; 23.87% ofall ships a service time between -4 and ½z da, etc. Similarly, if theaverage service time is one day instead of - day, the correspondingiigures will be 22.12% and 17.23%.

4. Annex Chart II-1 finally presents some of the data from thetables in the form of curves. The upper part of the chart shows thatas the number of ship arrivals increases, the resulting Poisson distri-bution approaches a normal distribution.

1/ See also Annex III.

ANNEX IIPage 2

Annex Table II-1: Poisson Distribution for Selected Numbers of Ships Arriving

Number of Ships Number of Ships Arriving AnnuallyArriving Daily 200 boo 600 oO0 1,000 1,500

A. In Percentages

0 57.81 33.42 19.32 11.17 6.46 i.6b

1 31.68 36.63 31.76 24.48 17.69 6.74

2 8.68 20.07 26.11 26.83 2L.2L 13.86

3 1.59 7.33 l4.31 19.61 22.14 18.99

4. 0.22 2.01 5.88 10.74 15.16 19.51

5 0.02 0.44 1.93 4.71 8.31 16.04

6 0 0.08 0.53 1.72 3.79 10.98

7 0.01 0.12 0.54 i.49 6.45

8 0 0.03 0.15 0.51 3.31

9 0 0.O0 0.15 1.51

10 0.01 0.04 0.62

11 0 0.01 0.23

12 0 0.08

13 0.03

lb 0.01

15 0

B. In Days per year'-/

0 211 122 71 Li 24 6

1 116 134 116 89 65 25

2 32 73 95 98 88 51

3 6 27 52 72 81 69

b 1 7 21 39 59 71

5 0 2 7 17 30 59

6 0 2 6 14 LO

7 0 2 5 2b

a I 12

9 0 1 6

10 0 2

11 1

12 0

1/ Due to rounding, the totalo may not add up to precisely 365 days.

Annex Table II-2. Erlang (k = 1) Disiribution for Selected Average Service Time

Service Time Average Service Time in Daysin Days ii 1 2 3 5

In Percentages of the Total Number of Ships

o - 1/4 39.35 22.12 11.75 8.CC 6.C6 4.88

1/b - 1/2 23.87 17.23 1C.37 7.36 5.69 h.6b

1/2 - 3/b 14.47 13.b2 9.15 6.77 5.35 4.L'1

3/A - 1 8.78 10.45 8.C8 6.23 5.02 4.20

1 - 1 1/2 8.55 14.47 13.42 11.00 9.15 7.79

1 1/2 - 2 3.15 8.78 1C.b5 9.31 8.C8 7.C5

2 - 2 1/2 1.16 5.33 8.14 7.88 7.13 6.38

2 1/2 - 3 0.3 3.23 6.34 6.67 6.29 5.77

3 - --- 3 1/2 0.16 1.96 4.9b 5.65 5.55 5.22

3 1/2 - 4 0.06 1.19 3.84 4.78 4.90 4.73

b ---- 5 C.03 1.16 5.33 7.47 8.1b 8.14

5 ---- 6 C 0.43 3.23 5.35 6.3b 6.67

6 ---- 7 C.i6 1.96 3.84 4.9b 5.46

7 ---- 8 C.C6 1.19 2.75 3.8b b4.7

8 ---- 9 0.02 C.72 1.97 2.99 3.66

9 --- 1C C.01 O.bb 1.41 2.33 3.CC

1C ---- Co C.67 3.57 8.21 13.53

ANNEX II

Annex Chart II-1: Page 4 200

50\ Assumed Arrival and Service Time Distribution

\ - < 200 ships per yearX \ ~~~~~~~~~~~~~(Poisson)X

r. 30.

C.) 30\100c < 800 ships per year

> 20 /

t ~ ~ ~ ~ ~ ~ ~ ~ ~~~4 10 / 0\0 ships per year fi5 50

0).

0' 50 0) -

10~~~~~~~~~~~~~~~~~~~~~

° 1 2 3 8 5 6 7 8 9 10

Number of Daily Ship Arrivals

100.

(Erlang, k = 1; cumulative)

75. \\ \ . Average service time: 5 days

0)

0)-

*,-Average ServiceCT, tiXme: 2 days,~25.

- Average\ service ~-_

tifme: i day

0 1 2 3 b 5 6 7) 8 9 10 v 12

Service Time in Days

ANNEX IIIPage 1

THE TWO BASIC ASSUMPTIONS TESTED: THE PORT OF CAILAO

1. On the basis of data referring to the port of Callao (Peru)we tested the two assumptions that (a) the actual arrival pattern ofships resembles a Poisson distribution; and (b) the actual service timedistribution resembles an Erlang (k = 1) distribution.

2. The data refer to six months in total: two months in 1965,two in 1966, and two in 1967. The two months in each year are thebusiest and the least busy monthsduring the year. 1/

Number ofMonth Ship Arrivals

1965 - February 87- November 136

1966 - February 124November 140

1967 - April 143- September 133

All six months 763

3. The data indicate for each individual ship, inter alia, theday and time of (a) arrival; (b) entering a berth; and Tc leaving theberth. On the basis of these data we calculated the actual distributionsof ship arrival and service time. We then compared the actual arrivaldistribution with the Poisson distribution and the actual service timedistribution with three Erlang distributions: k = 1; k = 2; and k = 3.

1/ I.e., with the highest and the lowest number of ship arrivals.

ANNEX IIIPage 2

4. To determine how well the theoretical distributions fit the

actual distributions we applied the Chi Square test. 1/ The results

of this test are as follows:2/

Arrival Distribution Service Time Distribution(Poisson) (Erlang)

k = 1 k = 2 k = 3

x2 _ test: 12.87 11.75 5.77 16.93

Degrees ofFreedom: 10 13 13 13

5. The test results indicate that all four tested theoretical

distributions offer an acceptable approximation of the corresponding

actual distributions. Among the three Erlang distributions, Erlang

k = 2 offers the best fit, followed by Erlang k = 1.

6. The queuing model presented in this paper refers to a servicetime distribution Erlang k = 1. As compared with Erlang k = 2 (or in

general: k : 1), the k = 1 distribution reflects more variabilityamong the individual service times. For k = oo, the individual servicetimes are uniform.

1/ See Cramer, H., Mathematical MIethods of Statistics, Princeton,

1946, Chapter 30, Test of Goodness of Fit and Allied Tests.

2/ Jones and Blunder, op. cit., computed the following values forthe port of Bangkok.

Arrival Distribution Service Time Distribution(P sons 7Eriang, k = 3)

x2 _ test: 14.87 13.27

Degrees ofFreedom: 42 19

AMNEX IlIPage 3

7. Since the actual service time distribution resembles mostclosely Erlang (k = 2), using the queuing model based on Erlang (k = 1)will result in overestimating the waiting time. But this overestimationwill be small because the difference between the Erlang (k = 1) and

Erlang (k = 2) distribution is small. Moreover, the larger the number

of berths involved, the smaller the overestimation will be.

8. For a multiple channel queuing model with an unlimited queue,a Poisson arrival pattern, and an Erlang service time distributionX a

mathematical solution is highly inconvenient unless the Erlang distri-

bution is characterized by k = 1 or k = oo. If the actual service timedistribution resembles Erlang k = 2 (Port of Callao) or k = 3 (Portof Bangkok), we will have to choose between approximating the waiting

time on the basis of the model and approximating it on the basis of

computer simulation (see also paragraph 39).

ANNEX IVPage 1

MATHEMATICS OF WAITING TIME COMPUTATION

1. The model used in this paper takes into account the followingassumptions:

(a) The time between successive arrivals is a randomvariable and has a negative exponential distri-bution. As a consequence of this assumption,wehave the probability that n arrivals occurwithin an interval of time of duration t, Un (t)

given byU (t) = t) e which is an t

Poisson distribution. > is the mean (or expected)arrival rate in terms of our model.

(b) The service time per ship is a random variable andhas a negative exponential (Erlang k = 1) distri-bution, with the mean service rate,, , as theparameter. The service rate is the inverse ofthe service time.

(c) All berths have the same mean service rate.

(d) There is no limit to the length of the queue.

(e) The order of servicing is on the basis of "first-come-first served".

2. On the basis of these assumptions, the mean length of the queueand the mean waiting time per ship can be calculated. If M = number ofberths;

a = k; and/4C-

b = > ; then in the steady state, the mean length of theq .M

queue is given by:

ANNEX IVPage 2

Mb a

M !L=

M-12 nrg f M

(1 -b) > a + (1 -b) an=O n M

and the mean waiting time by:

W = L

Note to paragraph 1, (d).

Assumption (d) of paragraph 1, might be replaced by an assumption limit-ing the length of the queue at any time to, say, K ships. Or formu-

lated alternatively, we might assume an upper bound for the maximumwaiting time any ship arriving at the port will tolerate. In the one

berth case this would be K , i.e., the number of ships in the queue divi-

ded by the mean service rate.

Imposing the restriction that the queue length will never exceed K ships,

gives the following formula for the mean length of the queue in themultiple berth case.

e (a) K + 1 KL = b X Kb - (K +1'b +

1 -b D L b _

where, a and b are as defined above;K+1

D = (1 -b) EM-1 (a) + eM (a) (1 -b )

EM_l (a) = e (a); and

M -aeM (a)= a e

M !

ANNEX VPage 1

QUANTIFICATION OF WAITING TIME

1. The data in the two following tables have been computed onthe basis of the formula arrived at in the foregoing annex.

2. Annex Table V-1 presents the total annual waiting time ofships in days for different numbers of berths (1-15), average service

times (1-5 days) and annual ship arrivals (100-2,OOO).

3. Annex Table V-2 presents the total annual waiting time of shipsin days for different numbers of berths and products of annual ship

arrivals and average service time. For a graphical presentation of the

data in this table, see Chart III-2 (page 31 )o

ANNEX VPage 2

Annex Table V-1: Total Arnnual Waiting Time (in days)

Number of Average Service Time (days)Berths

2 3 4 5

A. Annual Ship Arrivals: 10C

1 38 243 1386 n.f. n.f.2 2 16 61 162 3803 0 2 8 22 45L o 0 0 3 75 C C 0 0 16-15 0 0 0 0 0

B. Annual Ship Arrivals: 2aC

1 243 n.f. n.f. n.f. n.f.2 16 162 1072 n.f. n.f.3 2 22 88 323 22b84 0 3 lb 39 965 0 0 2 6 116 0 C 0 1 17-15 C C 0 0 C

C. Annual Ship Arrivals: 300

1 1386 n.f. n.f. n.f. n.f.2 61 1C72 n.f. n.f. n.f.3 8 88 700 n.f. n.f.4 0 1l 6C 298 n.f5 0 2 8 21 696 0 C- 1 2 47-15 0 C 0 O C

D. Annual Ship Arrivals: LOG

1 n.f. n.f. n.f. n.f. n.f.2 162 n.f. n.f. n.f. n.f.3 22 323 n.f. n.f. n.f.b 3 39 298 n.f. n.f.5 0 6 21 13C n.f.6 0 1 2 5 317 0 0 0 0 18-15 0 0 0 C C

.. contd.

ANNEX VPage 3

Annex Table V-1 contd.

Number of Average Service Time (days)3erths

2 3

E. Annual Ship Arrivals: 500

1 n.f. n.f. n.f. n.f. n.f.

2 380 n.f. n.f. n.f. n.f.

3 L5 22L8 n.f. n.f. n.f.

14 7 96 n.f. n.f. n.f.

5 1 11 69 n.f. n.f.

6 0 1 4 31 n.f.

7 0 C 0 1 36

8-15 0 0 0 C C

F. Annual Ship Arrivals: 600

1 n.f. n.f. n.f. n.f. n.f.

2 1072 n.f. n.f. n.f. n;f.

3 88 n.f. n.f. n.f. n.f.

14 1Lb 298 n.f. n.f. n.f.

5 2 21 6894 n.f. n.f.

6 C 2 1C n.f. n.f.

7 C 0 0 5 n.f.

8-15 zero or not>feasible

G. Annual Ship Arrivals: 7CC

1 n.f. n.f. n.f.2 7t591 n.f. n.f.3 170 n.f. n.f.

4 25 3671 n.f.5 4 43 n.f. n.f.

6 1 3 121

7 0 0 1*

8-15 zero or not feasible

H. Annual Ship Arrivals: 800

1 n.f. n.f.2 n.f. n.f.3 323 n.f.4 39 n.f. n.f.

5 6 1306 1 57-15 zero or not feasible

..contd.

ANNEX VPage h

Annex Table V-1 : contd.

Number of Average Service Time (days)Bertls

1 2 3 5

I. Annual Ship Arrivals: 9C0

1 n.f. n.f.2 n.f. n.f.3 67C n.f.L 60 n.f. n.f.5 8 689b6 1 107 0 18-15 zero or not feasible

J. Annual Ship Arrivals: 10CC

1 n.f. n.f.2 n.f. n.f.3 22)18 n.f.4 96 n.f. n.f.5 11 n.f.6 1 317-15 zero or not feasible

K. Annual Ship Arrivals: 1500

1-b not feasible5 69 n.f.6 b n.f.7-15 zero or not feasible

L. Annual Ship Arrivals: 2CCO

1-5 not feasible6 31 n.f.7 1 n.f.8-15 zero or not feasible

Annex Table V-2: Total Annual Waiting Time and 'Droauct of Annual Ship Arrivals and Average Service Time(in days)

Product of Annual N u m b e r o f B e r t h sShip Arrival andAverage Service

Time 1 2 3 5 6 7 8-15

1CC 38 2 0 C 0 0 0 02CC 243 16 2 C 0 0 0 03CC 1386 61 8 1 0 0 0 0350 8410 105 1l 2 0 0 0 0boo n.f. 162 22 3 0 0 0 05CC t 38C 45 7 1 0 0 06cc " 1072 88 1l 2 0 0 0700 if 7591 170 25 b 1 0 0800 Bt n.f. 323 39 6 1 0 0900 700 60 8 1 0 0

1000 2248 96 11 1 0 015CO n.f. n.f. 69 b 0 0200C if if n.f. 31 1 C 1/3000 " " if n.f. n.f. 0 or n.f.-4CCO "i " " " 0 or n.f.2/50CC "I II If II It if if 0 or n.f.3/lGOCO " " " " " " " n.f.

1/ 9-15 berths and 8 berths respectively.

2/ 11-15 and 8-10 berths respectively.

3/ lb-15 and 8-13 berths respectively.

NOTE: n.f. indicates not feasible, i.e., an endless queue would be building up.