ABSTRACT

The Apapa port is part of the ports operated by the Nigerian Ports Authority which was established in 1955 to oversee the activities and operations of all Nigerian sea ports. Over the years the ports have witnessed tremendous increase in volume of trade causing heavy congestion as ships queue up waiting to discharge importers‟ cargoes. The study demonstrates the applicability of queuing theory models in addressing the problem of congestion in Nigerian Ports, by dealing with the application of multi-queue multi-server queuing model with infinite capacity, First come-First Served model in tackling the congestion problem at the Apapa port, Lagos, Nigeria. The performance indicators of the existing single-queue multi-server model at the general goods cargo terminal of the Apapa port were computed while the results for the performance indicators of the multi-queue multi-server queuing model were also computed and examined. The results obtained were found to be effective in improving port efficiency as it shows that average number of ships in the system (queue and at berth), average queue length, average waiting time of ships in the queue and in the system are reduced in the proposed model by up to 73%, 93%, 78% and 93% respectively. Consequently the study recommended that the multi-queue multi-server queuing model should be the model of choice to solve the ship congestion problem at the Nigerian Ports. This is more so as its implementation is at very minimal cost which is attractive to the port owners.

TABLE OF CONTENTS

Contents
Declaration …………………………………………………………………………………………………………………… iii
Certification ………………………………………………………………………………………………………………….. iv
Dedication ……………………………………………………………………………………………………………………..v
Acknowledgements ………………………………………………………………………………………………………… vi
Table of Contents ………………………………………………………………………………………………………….. vii
List of Tables ………………………………………………………………………………………………………………… ix
Abstract ………………………………………………………………………………………………………………………… x
CHAPTER ONE ……………………………………………………………………………………………………………. 1
INTRODUCTION …………………………………………………………………………………………………………. 1
1.1 Background of the Study ………………………………………………………………………………………… 1
1.2 Statement of the Problem………………………………………………………………………………………… 4
1.3 Aim and Objectives of the Study ……………………………………………………………………………… 6
1.4 Significance of the Study ………………………………………………………………………………………… 6
1.6 Limitations of the Study …………………………………………………………………………………………. 7
1.7 Assumptions of the Study ……………………………………………………………………………………….. 8
1.8 Definition of Terms ……………………………………………………………………………………………….. 8
CHAPTER TWO …………………………………………………………………………………………………………. 10
LITERATURE REVIEW ………………………………………………………………………………………………. 10
2.1 Review of Literature …………………………………………………………………………………………….. 10
CHAPTER THREE ……………………………………………………………………………………………………… 18
METHODOLOGY ………………………………………………………………………………………………………. 18
3.1 Chi-Square Goodness-of-fit-tests ……………………………………………………………………………. 18
3.1.1 Inter-arrivals are exponentially distributed …………………………………………………………. 18
3.2 The Port Basic Parameters and Utilization Coefficient ……………………………………………….. 19
3.3 The Single Server, Unlimited Queue Model(M/M/I): (/FCFS) ………………………………….. 20
3.3.1 Expected Number of Ships in the System …………………………………………………………… 25
3.3.2 Expected Number of Ships in the Queue ……………………………………………………………. 25
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3.3.3 Expected Time a Ship Spends in the System ………………………………………………………. 27
3.3.4 Expected Time a Ship Spends in the Queue ……………………………………………………….. 27
3.4 Little‟s Law ………………………………………………………………………………………………………… 28
3.5 Multi-Berth Queuing Model (M/M/c): (/FCFS) ………………………………………………………. 29
3.6 Multiple Queue – Multiple Berth Queuing Model ………………………………………………………. 29
3.7 Performance Indicators of the Multiple Server Queuing model ……………………………………. 30
CHAPTER FOUR………………………………………………………………………………………………………… 34
RESULTS AND DISCUSSION ……………………………………………………………………………………… 34
4.1 Goodness of fit test ………………………………………………………………………………………………. 34
4.11 Arrivals of ships into the queue at Apapa Port ……………………………………………………… 34
4.12 Service distribution …………………………………………………………………………………………. 36
4.2 The Port Basic Parameters …………………………………………………………………………………….. 38
4.4 Multi-Queue Multi-Berth Model …………………………………………………………………………….. 40
CHAPTER FIVE …………………………………………………………………………………………………………. 45
SUMMARY, CONCLUSION AND RECOMENDATIONS ……………………………………………….. 45
5.1 Summary ……………………………………………………………………………………………………………. 45
5.2 Conclusion …………………………………………………………………………………………………………. 45
5.3 Recommendations ……………………………………………………………………………………………….. 46
REFERENCES ……………………………………………………………………………………………………………. 47
APPENDICES …………………………………………………………………………………………………………….. 50
Notations Used ……………………………………………………………………………………………………………. 61

CHAPTER ONE

NTRODUCTION
1.1 Background of the Study
Queuing theory deals with the study of waiting for services at service point of any kind (Sztrik, 2012). It is always desirable by all parties to stay on the queue for as short a time as possible. However, reduction of the waiting time usually requires extra and most times huge capital outlays. To decide whether or not to use any port by merchants they always consider the efficiency of the port operations and for port owners (Federal Government of Nigeria) to invest, it is important to know if their investment reduces the waiting time. Queuing models are able to analyze such situations. In this study, attention is paid to application of multiple queues to a multi-server situation to improve on the efficiency of operations in Nigerian Ports. Data collected from Apapa Ports for a period of five months, August to December 2016 were analyzed.Shipping is the cheapest method of all other common modes of transportation. Shipping carries large volume of cargo, which is almost four times more than rail and four hundred times higher than air transportation in total (Martin and Stopford, 2009). Most of goods shipped through Apapa ports are in containers.
Containerizationcomes with opportunities for safe and secure shipping and handling of cargo. Challenges come into consideration because containerization significantly changes the requirements for terminal facilities (Islam and Olsen, 2011). Important application areas of queuing models are production systems, transportation and stocking systems, communication systems and information processing systems. Queuing models are particularly useful for the design of these systems in terms of layout, capacities and control. In this work, our attention is restricted to a model with multiple queues (i.e. in tackling the problem of port congestion at Apapa Port, Lagos, Nigeria). Queuing theory deals with one
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of the most unpleasant experiences of life, such as waiting. Erlang (1909)was the first to treat congestion problems in the beginning of the 20th century using the principle of queuing theory. (Sztrik, 2012) One of the characteristics of the queuing model is the arrival process of ships. Here the ships arrive according to a Poisson stream (i.e. exponential inter arrival times). Furthermore the behavior of arriving ships at the fairway buoy is such that they cannot leave after arrival irrespective of the state of the queue and how long they may have to wait. The service times are independent and identically distributed, and are independent of the inter-arrival times. The service time were exponentially distributed. The service discipline was observed to be on a one by one and a first-come-first-served and there are multiple numbers of berths that offer services. Hence the situation was a single queue multiple server type. The fairway buoy where ships first weigh anchor and wait to be served was assumed also to have an infinite capacity. Kendall introduced the notations used to characterize queuing models.It is a three-part code denoted as M/M/c. The first letter specifies the inter-arrival time distribution and the second one the service time distribution. The letter M is for the exponential distribution and stands for Memory less property. The third and last letter specifies the number of berths. In this case, the M/M/c model was applied. The notation can be extended with an extra letter to cover port waiting area capacity. For example, a system with exponential inter- arrival and service times, c number of berths and having a waiting room only for N customers (including the one in service) is abbreviated by the four letter code M/M/c/N. (Adan and Resing,2015)
In the case of the Nigerian Ports Authority, the ships arrivals are independent of each other and are suspected to be Poisson distributed. Once a ship arrives at the fair way buoy it
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cannot leave irrespective of the length or state of the queue. It has to wait until served. Hence the importers have to have a fair idea of the anticipated waiting time at each port to decide on which port will serve as final destination economically. The queue systems at Apapa Port are being served by parallel service berth channels where each berth has an independently and identically distributed exponential service time distribution. The mechanism of a queue process is such that ships arrive at fairway buoy to join the queue and are called to berth as soon as a service point (berth) is free. The ships are served upon arrival at the berth and then leave the system thereafter. Thus the queuing system may be described as composed of ships arriving for service, waiting for service if all berths are occupied (busy) and leaving the system after being served. Ships arrive randomly and independently and in accordance to a Poisson process i.e. the number of ships arriving until any specific time has a Poisson distribution. The pattern of arrival in a time period t follows Poisson distribution which assumes that arrival is completely independent of other arrivals i.e. arrivals are completely random. Obviously if the arrival process follows the Poisson distribution, an associated random variable defined as the time T (inter-arrival time) follows the exponential distribution.
Finally, arrivals of ships are said to be in tandem with Poisson process are random since the probability of arrival of a ship in a small interval of time of length ℎ is proportional to the length ℎ, and is independent of the amount of lapsed time from the arrival period of the last ship.
The service discipline at the Nigerian Ports is first come first served (FCFS), no reneging, balking, jockeying or collusion is allowed or possible. If the service is assumed continuous, then the mean service time ? is also assumed to be constant over long time and is
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independent of the number of units already serviced, queue length or any other random property of the system. Therefore, probability of complete services impliesthat the density function of inter-service time is exponential. However in the case under study it is not possible for service to be continuous within a long time interval as the possibility of a service berth to be idle for sometimes exists, therefore Poisson distribution cannot be applied to servicing. The exponential distribution is the only continuous distribution with the important property of forgetfulness or lack of memory suitable for the service process(Cooper, 1972).
1.2 Statement of the Problem
Capacity shortage increases port costs (i.e. higher surcharge or demurrage for port users in a competitive open market). This implies that if costs are kept low then other cheaper ports or less congested ports (e.g. Cotonou and Lome) become more noticeable (Dekker,2005). The ocean transport industry is growing at a faster rate than seaports can build facilities (Pallis and de Langen, 2010) and it takes from two to over ten years from decision to completion of changes in the infrastructure to increase capacity (Henesey, 2006). As many ports are exceeding their capacities and finding it difficult to find the capital outlay, ports worldwide are increasingly faced with the problem of congestion as time goes on.Hence the significance of falling on the use of queuing theory to address the problem becomes pertinent. The case of Nigeria is of great concern to port and economic planners as was re-emphasized in a Patrick Doyle Nigeria Television Authority (NTA) international production for Nigeria Maritime Safety Agency (NIMASA) on August 31, 2017 that maritime trade in Nigeria will quadruple by the year 2035.
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Congestion brings delays for port users and increases the cost to many stakeholders; for example, to shipping lines, it brings about increase in cost of charter, while to terminals it brings yard congest and increases cost of re-handling(Mabs, 2009). The Nigerian Ports Authority (NPA) was established in 1955 to manage all Nigerian Sea and Inland Container Terminals or Ports. Apapa Ports, Lagos is one of such ports. The volume of international trade in and out of Nigeria has been on a steady increase, hence the demand for the services of the various ports have been stretched causing delay and ships have to wait for their turn to discharge / load cargo into/out of the various ports. In a shipping business, a wait of a single day is of tremendous cost and should be avoided as much as possible. The importer/exporter would rather not wait, while the port owners cannot afford to provide a dedicated berth for every ship that calls as the cost will be unbearable, so a balance must be stroke. The importers, on the one hand, would not want to pay heavy demurrage and ship charter on the one hand while the Nigerian Ports would not want to lose business (if importers re-route their cargo through neighboring country‟s ports which may be cheaper or better managed) as a result of its high charges or having to run the ports at a loss on the other hand. If the number of berths were to be infinite then all the ships will be served instantaneously on arrival and there will be no queue. However,, this is not practically possible.In view of these problems, we intend to examine the existing queuing system in the Nigerian Ports with a view to proposing a better model that would improve overall Ports Operations in Nigeria.
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1.3 Aim and Objectives of the Study
The aim of this work is to examine the existing queuing system at Apapa Port, Lagos with a view to applying a model that would substantially improve the port‟s operations and the objectives are:
i. to establish the arrival and service distributions of ships at Apapa Port by conducting a chi-square goodness-of-fit test respectively
ii. to identify and applythe existingqueuing model forthe port‟s operations;
iii. to apply the appropriate queuing model in order to reduce the length of the queues and length of stay bythe ships.
iv. to comparethe multiple queue model with the existing model
1.4Significance of the Study
The problem of congestion is a source of concern to the Federal Government in that different solutions proffered were either capital intensive to implement or unattractive to importers. This study is therefore significant in that it seeks to provide another method of solving or reducing the congestion problem to a mutually agreeable level. The study is significant in providing better services with tolerable waiting. The Nigerian Ports Authority (NPA)‟s poor performance restricts national growth, adds to the cost of all goods that transit in and out of the country. Considering that through the years 2000 to 2003, the NPA cost the Federal Government N86.7billion naira (Bureau for Public Enterprise, BPE, 2005) it is therefore of great significance to examine ways of improving the efficiency of NPA at minimum costs. Furthermore the study is significant in evaluating the efficiency of the port operations, port planning, capacity assessment and building.
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1.5 Contribution to Knowledge The dissertation‟s contribution to knowledge is in the application of this variation of multiple queue multiple berth model to port ship arrival and services process which have hitherto not been seen in literature. The model applied is a multiple queue multiple berth model which is found to shorten the number of ships in the system and on the queue. Similarly the applied model gave results indicating shorter waiting time in the system and on the queue without having to resort to infrastructural development like building more service berths.
1.6 Limitations of the Study
The study was limited by time and financial resources to go round all the ports and collect real time data from the several seaports and inland container terminals spread in the country. Furthermore, this study did not cover all the differentaspects of the port operations which could also cause queuing or congestion problems. For example the efficiency of the port bureaucratic processes, port union activities, Nigerian Customs, clearing agents, stevedores, adequacy of space to store the cargo after offloading, efficiency of other material handling equipments like forklifts, trucks, cranes etc.All other activities that could cause queuing were assumed to be at optimal performance levels.It was not possible to obtain actual cost in monetary value of the different required operating indices like cost of ship wait, berth un-occupancy cost etc. per unit time. Finally, queuing models that could possibly be better but require increasing the number of berths and hence heavy capital outlay and considerable number of years to put in place were not considered.
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1.7 Assumptions of the Study
The assumptions of the study are as follows:
i. Ships arrivals are independent, random and are served on a first-come-first-served basis.
ii. The fairway buoy has infinite capacity to accommodate arriving ships.
iii. The arrival of ships is Poisson distributed with a mean arrival rate of λ and the inter-arrival rate is exponentially distributed.
iv. Balking or jockeying is not allowed as all arriving ships are held until served.
v. The service times are assumed to be independent and identically distributed, and are independent of the inter-arrival times. Service times are exponentially distributed.
vi. The mean service rate  is the same for each server.
1.8 Definition of Terms
a) Queue:This is a line or sequence of ships waiting to be served.
b) Berths: This refers to the service point where offloading/loading services are being provided to ships calling.
c) M/M/cModel: Is a queuing model with queues which are being served by parallel service channels c, in which each server has an independently and identically distributed exponential service time distribution.
d) Demurrage: Those are charges that a charterer or importer pays to the ship or port owner for extra time or over usage of the vessel or space at the port.
e) Inter-Arrival time:This is the time between successive arrivals and is exponentially distributed.
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f) Steady State: On the long run, the probability distribution of arrivals, waiting time and service times are independent of time and are hence said to be in steady state since its operating characteristics are independent of time.
g) Saddle Point or equilibrium point:This is when the average arrival rate equals average service rate.
h) Service Discipline: This refers to the manner, in which those in a queue are chosen for service. The different types of service discipline are First come, first served (FCFS);Last come, first served (LCFS);Service in Random Order (SIRO); andGeneral Service Discipline (GSD)
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