Modelling study of a container distribution system for high rise factories

Using cranes to transport the containers is a tried-and-tested efficient distribution method and the crane design can be incorporated in the vertical hoisting of the containers to variou

CHAPTER 1

INTRODUCTION

Singapore is currently the world’s largest transshipment hub and is connected to

600 ports in 123 countries In the year 2002, the Port of Singapore Authority (PSA), handled 17 million TEUs (Twenty-foot Equivalent Units) [1] More than 80% of this volume is transshipment cargo destined for countries other than Singapore This has resulted in 8% annualized growth in container traffic over the past two decades and rapid economic growth in many developing countries To handle the ever increasing container traffic, an entire logistics system is developed comprising of the warehouses and freight stations, road and rail transport, ground handling equipment like straddle carriers and gantry cranes in the port terminals, and larger container ships (in excess of 10,000 TEUs) to benefit from the economies of scale

To meet the increased container capacity as well as logistics warehouse space, it

is inevitable that warehouses and freight stations will increase in size and more efficient designs and container handling systems are being proposed to increase the throughput and efficiencies Single-storey factories and warehouses are the norm in many countries The container trucks arrive at the docks of the warehouses, where forklifts are driven into the containers for the cargo loading and unloading activities Often, the trailer and the container will be left behind as the loading of the cargo may take up to a day This is inefficient, as this also requires a large tract of land for building the docks and wide roads for the container trucks to be driven to the doorsteps of the warehouses

Continuously rising land prices and limited availability of building space are some of the reasons to build factories and warehouses skywards Multi-storey industrial buildings increase the usage of land resources, and this is particularly more significant for countries such as Hong Kong and Singapore, where land is scarce The future trend

of factories will be in the form of high-rise buildings Material handling and transportation economies also favor this new trend Vertical movement in multi-storey buildings eliminates longer, slower and costlier horizontal handling of containers in sprawling single-storey factories Compact multi-level design also offers advantage in factory design construction and operation Excavation, foundation, building and maintenance costs are often less per square metre of usable floor space in high-rise buildings compared to single-storey factories

The Jurong Town Corporation (JTC) was established in 1968 to plan, develop and manage industrial estates in Singapore Over the last 30 years, JTC has been conscious of the importance of optimizing the use of its industrial land It has done so through intensification of land use, making more productive use of land, and improving the planning and development of the supporting infrastructure It has also been constantly reviewing the allocation policies for industrial and ready-built factories, and revising the planning development of industrial estates and factories

Singapore has reached a point in economic growth where it can no longer rely solely on increases in labour and capital investments to fuel further growth It has to focus on productivity gains and innovation for greater output per unit of input [2] Industries that operate in a multi-storey environment are better able to achieve higher land productivity levels than those that do not Hence, JTC has been building its

factories to higher plot ratios (from 1.8 to 2.5) to increase the efficiency of land use and

to encourage industrialists to go high-rise whenever possible

These high-rise flatted factories, such as the nine-storey flatted factories located

at Woodlands East, are designed to integrate marketing, management, production, storage and other industrial activities They are served by cargo/passenger lifts and loading bays, and some latest factories are designed with ramps that go all the way to the higher floors of these buildings to enable container trucks to transport the cargo to the sheltered loading bays at the entrance of the factory units on every level Hong Kong also solves land scarcity by building industrial facilities that go as high as twenty storeys and ramps that allow vehicular access up to thirteen storeys

Although going skywards is the solution to solving land scarcity, building such a large vehicular ramp requires an extensive land area and large capital cost A better alternative system is one that delivers the containers to the various floors of a high-rise factory by other means, instead of the existing vehicular ramp The envisioned factories

of the future would be high-rise buildings that incorporate both the office and manufacturing plant in a “single building” Containers will be lifted to the various floors and placed in the container lobby of the factory unit The company would then unload or load the cargo into the container This is more efficient and economical since less land and building cost is incurred to construct the docking areas and the vehicular ramp Furthermore, the empty container can be transported one day prior to loading of the goods and no trailer will be left idling Having the container in the container lobby also provides the extra security compared to leaving the container in the open docks in existing factories

The main concern is on the delivery of containers to the various floors of the high-rise factories The final proposed distribution system must allow for the smooth and problem-free movement of containers and reduce operating and maintenance costs Less land and lower capital outlay will be the foremost criteria in proposing a new container distribution system

This thesis is organized as follows Chapter 2 first reviews the existing methods

of delivering containers to various floors and later proposes a new container distribution system Analytical and simulation studies of the proposed system are presented in Chapters 3 and 4 respectively Chapter 5 discusses the simulation results and a cost analysis is performed to select the optimal crane configuration for uncertain truck arrivals Conclusions are drawn in Chapter 6, together with some recommendations for further study

CHAPTER 2

CONTAINER DISTRIBUTION SYSTEMS

2.1 CURRENT CONTAINER DISTRIBUTION SYSTEMS

2.1.1 Vehicular Ramp

Incorporated in 1981, ATL Logistics Centre Hong Kong Ltd (a subsidiary of CSX World Terminals) owns and operates ATL Logistics Centre - the world's first and largest intelligent multi-storey drive-in cargo logistics centre (Figure 2.1) [3]

(c) Cargo Being Loaded/Unloaded at Docks

(d) Wide Access Roads Inside Buildings

Conveniently located in the heart of Kwai Chung Container Terminals and within easy reach of Hong Kong's commercial and population centres, airport and the Mainland border, ATL Logistics Centre offers warehouse and office leasing with a full range of cargo handling, container freight station and distribution services ATL Logistics Centre is made up of two multi-storey warehouses: Centre A and Centre B, comprising of seven and thirteen storeys respectively It consists of a three-lane (two lanes up and one lane down) vehicular ramp to provide direct drive-in access to all levels of the buildings The ramp and internal loading bays are accessible for all vehicular types including 40-ft container trucks The Centre comprises a total floor space of 9.4 million square feet, provides over 1,730 loading bays and handles an average of 8,000 vehicles daily

Similar logistics warehouses are also found in Singapore As part of Industrial Land Plan 21 (IP21) [4], JTC has also learnt from Hong Kong by building multi-storey warehouses to solve land scarcity Jurong Port (a subsidiary of JTC) not only is a key bulk and conventional cargo gateway in Singapore, with 23 berths serving over 7,000 vessels every year, it also owns the Jurong Logistics Hub, which is Singapore’s largest multi-storey drive-up warehouse (Figure 2.2) [5]

The Hub is a multi-storey drive-up warehouse, which allows 40-ft containers to

be trucked to every level, right to the doorsteps of customers and under all weather conditions Strategically located from the Port, Jurong Island, Jurong Industrial Estate and Tuas industrial zone, the ultra-modern warehouse comprises of 118,000 square metres of warehouse space and 6,200 square metres of office space Jurong Logistics

Hub's customers include multi-national corporations and logistics providers such as Sony, Volvo, Translink, L’Oreal and Dell Computers

Figure 2.2: Jurong Logistics Hub

(a) The Main Complex (b) Vehicular Ramp

(c) Large Turning Radius (d) Container Left at Dock

JTC has also adopted a similar drive-in concept for its stack-up factories to optimize land usage Named Woodlands Spectrum (Figure 2.3) and located in Woodlands East Industrial Park, these high-rise facilities offer ground-floor convenience (through a large ramp for container trucks), private loading areas and car parks The ramp has to be wide enough to accommodate the turning radius of 40-ft trucks and large access roads have to be constructed for easy manoeuvere to reach the units at different levels [6]

Figure 2.3: Woodlands Spectrum

(a) Overview of One Unit (b) Interior of Vehicular Ramp

(c) Wide Access Roads

(d) Unloading/Loading Dock

Large capital cost is required to construct such a container delivery system and also a large plot of land is required for the building of the vehicular ramps and wide roads Figures from JTC show that the ramp amounts for one-third ($600 million) of the total construction cost (for Woodlands Spectrum) It is ironical that what causes factories to move skywards (to optimize land space and cost) in the beginning ends up with a design using a large plot of land for the vehicular ramps and wide roads, and incurs a large capital cost A solution to this would be a new system of delivering the containers to the various floors of the multi-storey factories without using a large vehicular ramp

2.1.2 Container Hoisting Crane

One of the key objectives of this research is to propose a new and innovative method of delivering the containers to each and every unit’s doorsteps without utilizing the direct drive-in model of the vehicular ramp Since the item being transported is the 20-ft and 40-ft ISO containers, the handling methods at maritime terminals may provide suitable alternatives Using cranes to transport the containers is a tried-and-tested efficient distribution method and the crane design can be incorporated in the vertical hoisting of the containers to various floors of the buildings The small ground area required, the low dead weight and the resulting low load on the building are some

of the advantages that illustrate the expediency and economy of using container cranes

In fact, container cranes had already been implemented in many high-rise factories and warehouses around the world A customer list (Table 2.1) obtained from Mannesmann Demag (a company that manufactures and installs container hoists) shows that Hong Kong has many high-rise factories and warehouses that utilize

container cranes to deliver the containers to the various floors

Table 2.1: Customer List from Mannesmann Demag

Reference List (Container Hoist Installations) Customer Country

No of storeys served (plus ground floor)

Year of Construction

Tai Sang Land

Singapore

Singapore

Tai Sang Land

Southwinds Land

& Investments Hong Kong 15 1982

The machine room with the hoist and electrical equipment, the hoist shaft outside

or inside the building with horizontal travel tracks into the lobbies, the vertical

guidance system, the spreader and the various container positions in the lobby, all form

a single unified system Each floor has a control panel from which the operator starts

and monitors all functions for the particular lobby Display panels provide information

on the operations currently being carried out and on those that have been completed

The container truck delivering the container will first position itself underneath

the hoisting crane The crane, guided by vertical beams, will be lowered and the

self-adjusting spreader will then lift up the container to the pre-selected level Selection of

the optimum lifting speed is dependent on the number of floors and the number of

containers to be handled per hour Once it has been lifted to the level, the spreader will

be transferred horizontally via tracks and deposit the container at the lobby Thereafter, the loading or unloading of cargo can be carried out with forklifts Such a system offers better security than leaving the container in the docking area Figure 2.4 illustrates the sequence of a container being delivered to a unit in a high-rise warehouse in Taikoo, Hong Kong

Figure 2.4: Sequence of Delivering Container to a Unit

(a) Truck Positioned Underneath Spreader (b) Container Being Lifted to Pre-selected Level (c) Container Being Transferred Horizontally Into Lobby

(d) Loading and Unloading of Cargo via Forklift

A high-rise factory at River Valley Road employs the same distribution system as that of Taikoo, Hong Kong However, the factory utilizes two container hoists, each serving one face of the building Similarly, the truck will be positioned underneath the spreader, and once in position, the spreader will be lowered to hoist up the container to the selected level, which is then transferred horizontally into the lobby Each hoist can only handle containers meant for factory units located on the same side of the building

No crossover of containers is possible This creates a problem during the breakdown of the hoisting cranes, as this will affect the whole container distribution system for that building The hoisting service for that building is virtually down whenever repair or maintenance work is required Figure 2.5 shows the crane hoists used in the factory

Figure 2.5: Crane Hoists Used to Lift Containers

(a) Double Cranes (b) Close-up View of Crane

Although container hoisting cranes have already been implemented in many multi-storey factories around the world, they still have deficiencies Although the system of using hoisting cranes has freed up land and capital that would have been used for the vehicular ramp, the vertical distribution system still contains some deficiencies, such as the disruption to the factory when the cranes fail

2.1.3 Automated Storage and Retrieval System (AS/RS)

In recent years, the AS/RS has had an important impact on storage and warehousing operations These high-rise storage modules are becoming increasingly popular and have been successfully integrated in many manufacturing and distribution processes and warehousing enterprises around the world dealing with numerous items

in large volumes AS/RS is an attractive solution to limited storage space, high labour costs, shorter as well as reliability in cycle times, random access requirements and real-time material identification and tracking capability

Recently, much research has also been reported on the feasibility of implementing AS/RS for maritime container terminals [7-9] Faced with substantial increases in container traffic, limited land availability, larger vessels and the need to become cost competitive, this high density storage system will play an important role in the future success of many container terminal activities It is believed that the current container handling processes result in the misallocation of expensive and scarce land resources at terminal sites, wastage of capital in inventory, longer waiting time of trucks and ships, and a larger fleet size of yard-trucks Implementing AS/RS in a container terminal would improve terminal operations

Ioannou et al [9] published an extensive report for the Center for Advanced Transportation Technologies of the University of Southern California, which provides

an engineering evaluation and quantitative assessment of the performance of existing, emerging and conceptual cargo handling technologies for terminal operations, and proposes three automated container terminal concepts employing advanced technologies One of the proposed designs is an automated container yard using AS/RS Three high-rise storage and retrieval systems are proposed, namely the Seaport Container Storage Systems (in association with Transact), Earl’s Computainer and Krupp’s Fast Handling System

Figure 2.6: Physical Model of Container Storage System by Seaport (From [9])

The Seaport Container Storage System (Figure 2.6) is based on the Transact design of automated air cargo handling system employed in some airports It adopts a proprietary design of double deep racks, where each rack is ten levels high, with four storage positions on each level An automatic stacker crane called Elevating Transfer Vehicle (ETV) interfaces horizontally and vertically with the storage cells A shuttle mounted on the ETV stores and retrieves containers from the storage cells on each side

of the ETV aisle An automatic overhead crane with a 40-ft spreader provides the means to receive and deliver containers to and from the horizontal material handling systems (trucks, transfer cars or AGVs)

Figure 2.7: Earl’s Computainer (From [9])

Earl’s, a leading manufacturer of container spreader bars, has built a full-scale prototype of an AS/RS called Computainer (Figure 2.7) The Computainer is a multi-storey steel structure with a small number of storage cells Less than four acres of land

is required for 2,000 40-ft containers and related access and truck queueing areas

Multiple access bays are provided for rapid truck turn around The Computainer includes an integrated hoist transfer system based on proven technology Its mechanical design and operational simplicity account for its attractiveness as a viable storing solution for container terminals

Figure 2.8: Prototype of Krupp Fast Handling System (From [9])

Krupp, a German manufacturer of marine cranes and mining material handling, has developed an automated system design specifically for intermodal rail terminals (but can be adapted for marine terminals) known as “Krupp Fast Handling System” A prototype (Figure 2.8) of this concept has been installed at the Duisburg-Rheinhausen terminal Each module comprises a set of end and middle pickup/deliver stands, a high-rack handling device and channeling vehicles The high-rack handling device moves along the transverse aisle on guide rails and mainly serves to transport the loading units vertically to the storage levels

By the year 2020, it is projected that the amount of cargo transferred between container terminals will be doubled The scarcity of land in many areas makes it almost impossible for many terminals to respond to this increasing demand by expanding their yard facilities The high-density storage AS/RS can be built on a small piece of land and capacity is increased by adding more floors The high productivity of the AS/RS lies in its capability to access any container within the storage structure The high productivity and high storage capacity on a small piece of land, make it attractive to employ the AS/RS concept for the proposed container distribution system for high-rise factories

2.2 PROPOSED CONTAINER DISTRIBUTION SYSTEM

2.2.1 Description of the Proposed Container Distribution System

The design of the proposed container distribution system is based on the concept

of utilizing overhead cranes currently employed in port operations Instead of using the hoisting cranes in one fixed location, the proposed design uses automated overhead cranes that are able to travel to different column sections (different factory units) This design provides backup options in the event a crane malfunctions, in which case, it can

be pushed to a free space between the columns The other remaining cranes can then be deployed to service the factory

The overhead crane runs on top of two buildings that are about 15 metres (three vehicle lanes) apart and this allows the trucks to have easy manoveure The spreader is able to travel across the span of the buildings as well as to cross over to either building

to handle the containers meant for different columns Figure 2.9 shows the proposed container distribution system on a factory consisting of two columns and three storeys

Figure 2.9: Proposed Container Distribution System

(a) Overview of the Factory (b) Innerview of the Factory

The trucks will first be positioned in different column sections depending on where the factory units are located For safety reasons, the containers can only be hoisted or lowered vertically This means that the overhead crane will first travel to the designated column section before hoisting up the container vertically Wind gusts can impose a considerable external load on the hanging container, causing it to sway and knock against the building walls To eliminate swaying, vertical guides are installed for the spreader

Figure 2.10: Container Lobby

Once the crane has latched onto the container and hoisted to the selected level, the lobby platform will be extended out to receive the container The extensible platform moves on steel wheels and is operated by hydraulic pistons The platform uses flangeless track wheels and travels on tracks with flat head while guided by lateral guide wheels This is to prevent skewness of the platform and ensure it moves in a straight path The travel tracks for the platform are spaced 13 metres apart to allow the crane to move a 40-ft container (or two 20-ft containers) vertically between the tracks Figure 2.10 shows the design of the container lobby

The design of the container lobby is for housing one 40-ft or two 20-ft containers The self-adjusting spreader allows the crane to pick up different types of containers The crane can be equipped with a twin-lift spreader to hoist up two 20-ft containers together The width of the lobby can also be increased to accommodate two 40-ft containers side by side (lengthwise) if the factory requires this There are doors on both sides of the lobby to solve the container orientation problem, so that the cargo can be loaded or unloaded in either direction, depending on the position of the container

The advantage of this proposed container distribution system is the availability of more loading and unloading bays compared to the single hoisting system at high-rise factories in Taikoo, Hong Kong and River Valley, Singapore An integrated computerized system performs the following functions:

• The identification of the arriving truck

• The positioning of the overhead crane in advance

• Latching on and lifting of the container to the various floors

• Lowering it onto the extended platform

The factory has a ground control station, from which an operator is able to initiate and monitor all operations The interested reader is referred to [10] for an extensive discussion on the operations of the various systems (such as automated overhead travelling crane, smart spreader, container positioning system) employed in the factory

2.2.2 Operation of the Proposed Container Distribution System

The proposed factory consists of two five-storey buildings, each with nine columns, giving a total of ninety units The two buildings are referred to as Block A and Block B and the nine columns are numbered Column 1 to Column 9 There is a

common waiting area for trucks queueing for service when the cranes are not available

A number of cranes travel on top of the buildings to handle the containers meant for each column Figure 2.11 shows a schematic diagram of the proposed factory

The operation of the Factory consists of inflow and outflow processes For the inflow process, the loaded truck arriving at the factory is identified for its destination, with details such as the Block Type, Column No., Unit No and the Type of Operation (loading or unloading) If the crane is available, the truck will proceed to the column; otherwise it will be directed to a waiting area The crane assigned to that column then starts unloading the container After the container has been transferred to the designated lobby, the truck leaves the factory and the next truck in queue proceeds to be serviced

by the crane For the outflow process, a similar sequence of events is applied In this instance, the empty trucks will arrive to be loaded with the containers from the various units Figure 2.12 shows the flowchart of the operation of the factory

Block A

Columns of factory units

9

8 7

6 5

4

3 2

1

9

8 7

6 5

4

3 2

1

Block B Legend

Crane Movements

Figure 2.11: Schematic Picture of Factory (Top View)

Truck identified to obtain information on Arrival Time, Block Type, Column No., Unit No., Type of Operation (unload or load)

Proceeds to column to begin service OR remains in waiting area if crane is busy

Figure 2.12: Flowchart of Operation of Factory

The overall aim of the study is to investigate whether the proposed factory, its individual components and operating logic would interact efficiently to produce an optimal performance For this purpose, performance parameters such as the number of cranes required, the assignment of cranes, the average delay encountered by the trucks

in queue and the size of the waiting area for the trucks must be determined The inverse relationship between the number of resources and queueing time requires the

“optimization” of the number of cranes to be used in the factory More cranes may result in reducing waiting times, but may increase the overall cost of the proposed factory Estimates for the queueing times and crane utilization will help in the decision

of identifying the appropriate number of cranes to be used and how they are assigned to service the trucks

For the factory in consideration (Figure 2.11), the number of cranes can range from one to a maximum of nine However, the extreme values are not desirable because

a single crane does not allow for any backup during breakdowns and using nine cranes

is a waste of resources as the cranes would be idle most of the time Another important research issue is crane assignment Since the number of cranes is less than the columns,

it is essential to assign the cranes effectively to the columns In this research, configurations of using two, three and four cranes are examined The main concern is to choose the appropriate number of cranes in order to achieve acceptable queueing times for service and crane utilizations A minimum of two cranes is selected to ensure that the Factory’s operation is not hindered when one of the cranes breaks down Figure 2.13 shows the crane configurations using two, three and four cranes

Denotes crane Denotes columns Denotes allocation flexibility

Figure 2.13: Various Crane Configurations (a) Two-crane Configuration

(b) Three-crane Configuration Type I

(c) Three-crane Configuration Type II

(d) Four-crane Configuration Type I

(e) Four-crane Configuration Type II

Crane configurations are classified into groups based on the different number of cranes employed, each of which consists of a few configurations depending on the flexibility of assigning the columns to the cranes With full allocation flexibility, the average waiting time may be reduced However, in the factory, the cranes are constrained such that they may not cross each other In the context of machine allocation [11,12], it has been shown that the performance of a system with slight flexibility is almost equal to that with full routing flexibility In the next chapter, the performance of the factory under different crane configurations and truck arrival rates

is analyzed

to determine the customer’s waiting time in the queue and the number of customers waiting in queue

Queueing theory was developed to provide models that give insights to how systems behave when attempting to provide service for randomly arising demands Many applications of the theory have been well documented in the literature of probability, operations research, management science, and industrial engineering Some examples are traffic flow (vehicles, aircraft, people, communications), scheduling (patients in hospitals, jobs on machines, programs on a computer), and facility design (banks, post offices, amusement parks, fast-food restaurants) [13-16]

The purpose of the models is to develop mathematical equations of the performance measures (average waiting time as a function of customer arrival rate) for different configurations (different arrival and service patterns, number of servers and server configuration and queue organization) The performance measures provide an important indicator on how well the alternative configurations meet the system objectives There are three types of system responses of interest They are the average waiting time a typical customer has to endure, the average number of customers waiting and the idle time of the servers The task of a queueing analyst is generally one of two

issues: to determine the values of appropriate performance measures for a given process, or to design an “optimal” (according to some criterion) system To do the former, one must relate waiting delays, queue lengths, and other measures to the performance specifications of the queueing operations

On the other hand, for the “optimal” design of a system, the analyst balances customer-waiting time with the idle time of servers based on the inherent cost structure

of the system If the costs of waiting and idle service can be obtained directly, they can

be used to determine the optimum number of servers to employ and the service rates at which to operate these servers It is necessary to be able to obtain an estimate of the size of the queue to plan for the waiting room There may also be a space cost which should be considered along with customer-waiting and idle-server costs to obtain the optimal system design In any case, the analyst will strive to solve this problem by analytical means

Queueing theory results usually are not used directly in making decisions regarding the design of the system Rather, these results are used to construct decision models that seek the minimization of an appropriate cost function The objective of the cost models is to determine the optimum operation parameters such as service rate, number and size of service facilities The cost decision model normally includes the two basic cost quantities, namely the cost of offering the service and cost of waiting for service As the level of service increases, the first cost increases and the second simultaneously decreases The optimum level of service corresponds to the minimum sum of these two conflicting costs

However, most real systems comprise of complicated operating procedures and may not be accurately represented by queueing models, due to the elaborate input or service mechanisms, the complexity of the system design, the nature of the queue discipline, or combinations of the above Queueing analysis is often restricted as it is based on exponential assumptions for arrival and service distributions and is highly limited in system complexity Furthermore, most of the models provide only steady-state results, and if one were interested in transient effects or if the possibility distributions (e.g arrival rates) were to change with time, it might not be possible to develop analytical solutions or efficient numerical schemes in these cases For such problems, it may be necessary to resort to analysis by simulation

It should be emphasized, however, that if analytical models are available, they should be used, and that simulation should be resorted for cases where either analytical models are not available and approximations are not acceptable or they are so complex that the solution time is prohibitive A hybrid approach combining analytical queueing analysis and simulation can be used to analyze the queueing process of complex systems Queueing theory provides the conceptual framework and limits the number of variants to be examined, while simulation is used to compare and evaluate the variants [17,18] Analytical queueing results help in verifying the simulation model for specific cases and accordingly, help to build confidence In the remaining of this chapter, the performance of the proposed factory is studied by using analytical queueing theory to model simple crane configurations

The main objective of this study is to determine an effective crane configuration for uncertain truck arrivals In order to construct a cost model to compare various crane configurations, performance measures that describe the operations of the proposed factory accurately are evaluated These performance measures are as follows:

1 Average number of trucks waiting in queue (Lq)

2 Average number of trucks in system (Ls)

3 Average waiting time per truck in queue (Wq)

4 Average time per truck spent in system (Ws)

5 Average crane utilization (ρ)

One of the most powerful relationships in queueing theory is known as Little’s Law and was developed by John D.C Little [19] in the early 1960s It states that the average number of customers in a queueing system (Ls or Lq) is equal to the product

of the average arrival rate of customers (λ) to that system and the average time spent in

that system (Ws or Wq) Lq provides an estimate of the required size of the waiting

area Wq helps in computing the waiting cost for service and crane utilization helps to

evaluate the operating cost of the cranes

The most common stochastic queueing models assume that interarrival times and service times obey the exponential distribution, or equivalently, the arrival and service rates follow a Poisson distribution Assuming the number of occurrences in some time interval to be a Poisson random variable is equivalent to assuming that the time between successive occurrences to be an exponentially distributed random variable An important property of the exponential distribution is the Markovian or memoryless

property, which states that the probability distribution of remaining time until the next arrival or service completion is always the same, regardless of how much time has passed In effect, the process “forgets” its history For interarrival time, this property describes the situation where the time of the next arrival is completely uninfluenced by the last arrival

Most queueing models assume that inputs (arrivals of customers) and outputs (departures of served customers) of the queueing system occur according to the birth and death processes respectively Individual births and deaths occur randomly, where their mean occurrence rates depend upon the current state of the system, and is based

on the assumption that only one birth or death can occur at a time This assumption is valid, because in a Poisson process, the probability of two simultaneous arrivals is zero The Poisson process is usually assumed in many queueing models, not only because of the many mathematically agreeable properties of the Poisson-exponential process, but also due to the fact that occurrences of events in many real-life situations do obey the postulates of the Poisson process, and thus its use in probability modelling is considered realistic

Arrival processes in several situations can be modelled as Poisson processes, such as telephone calls received at a call center, arrival of customers in a bank, and breakdown of machines in manufacturing facility For a detailed discussion on the Poisson process, refer to [20-22]

On further examination of the crane configurations illustrated in Figure 2.13, it may be observed that the configurations may be decomposed into a number of independent subsystems as shown in Figure 3.1 The basis for analyzing the configurations as separate independent subsystems is due to the partitioning of the cranes to service the columns If there were no such partitioning, the configurations are analyzed as single systems, for example, the configuration presented in Figure 2.13(e)

1 2 3 4

1 2 3 4 5 6 7 8 9

1 2 3 4

1 2 3 4 5 6 7 8 9

Subsystem A Subsystem B Subsystem A

Figure 3.1: Subsystems of Four-crane Configuration Type I

There are actually nine service locations (represented by the nine columns), which the cranes need to travel to in order to service the columns on either building With the additive property of the Poisson process, each service location consists of the sum of two independent Poisson processes (represented by the columns on each building) Hence, Subsystem A consists of a single server servicing two separate queues Similarly, Subsystem B consists of two servers servicing three queues, as

arrivals in Columns 3 and 4 and Columns 6 and 7 are summed as a single queue respectively The average waiting time of a truck in the system is obtained from the weighted sum of the average waiting times in the subsystems

Much research has been reported on the improvement of the efficiency of a production facility by the optimal control and allocation of arrivals to workstations to reduce the average waiting times of the work-in-progress Stidham [23] studied the problem of routing customers to one or two queues without the option of accepting or rejecting an arriving customer Dyer and Proll [24] and Rolfe [25] studied the optimal allocation of servers in multiserver queues in a multiple facility system that minimizes the total expected waiting time Tavana and Rappaport [26] presented an aggregate optimization model for a collection of individual queueing systems in order to find the optimal allocation of arrival rate that minimizes various objectives (e.g Wq) in a group

of systems These studies provide justification that the average waiting time of an arbitrary customer in the system is given by the weighted sum of the average waiting times of customers in the individual subsystems (Refer to Appendix A for illustration)

The individual subsystems are studied and tractable queueing models that describe their operations are obtained Subsystem A is commonly known as the single server multiple queues (SSMQ) system The system consists of a number of parallel queues, being serviced by a single server Subsystem B is known as multiple servers multiple queues (MSMQ) system Some SSMQ models are first analyzed to provide performance statistics on Subsystem A, and subsequently, MSMQ models are studied for Subsystem B

3.1 SINGLE SERVER MULTIPLE QUEUES (SSMQ) MODELS

3.1.1 Priority Queueing Model

Queueing models can have different queue disciplines such as served (FCFS), last-come-first-served (LCFS), service in random order (SIRO), and selection by priority [27,28] In queueing models with priority, it is assumed that several parallel queues are formed at the facility with each queue accounting for customers belonging to certain order of priority In priority schemes, customers with the highest priorities are selected for service ahead of those with lower priorities, independent of their arrival times into the system Customers within the same priority are served FCFS All high priority customers are served before any of those from lower priority classes Only equilibrium stable behaviour is considered, i.e the fraction of time the server is busy (server utilization, ρ) is less than one

first-come-first-The service of a lower ranking priority customer can be interrupted when a customer of higher priority enters the system This is known as preemptive priority service In addition, a decision is made as to whether to continue the preempted customer’s service from the point of preemption when resumed (preemptive-resume) or

to start anew (preemptive-repeat) On the other hand, a priority discipline is said to be non-preemptive if there is no interruption and the customer, once in service, will leave the facility only after the service is completed and regardless of the priority of newly arriving customers

Consider the priority queueing model shown in Figure 3.2 There are a number of queues, each of infinite capacity, attended by a single server Each class of customers arrives at the queues with independent Poisson process of rate λi (i = 1,2,…,m) The

highest priority class is indexed with value 1 (lowest priority class is indexed with value m) Each class of customers has service time distribution of mean E[ti] and second moment E[ti2]

m classes

Figure 3.2: Priority Queueing Model

Mean service time is [ ] [ ]

1

i m

i

iE t t

=

= λ

λ

(3.2)

Second moment of overall service time distribution is

] [ ]

1

2

i m

i

i

t E t

=

= λ

where [ ] (3.5)

1

i m

i i

=

ρ

The interpretation of ρ is the fraction of time the server is busy and ρi is the fraction of

time the server is busy with customers from class i

=

= k

i i k

Average waiting time of a class k customer in queue is

2

1 1

2

] [

m

i

i i

k

t E Wq

α α

λ

(3.7) Average number of customers in queue is

∑

=

= m

i i

Lq Lq

1 (3.8)

Average waiting time of an arbitrary customer in queue regardless of priority is

Lq

Lq Wq

1

1

λ λ

λ ρ

i m

i i

t E

Wq (3.10)

* 1

i

iWq = E Lq E t

ρ (3.12) the left-hand side of (3.11) is often called the amount of work in the system Independent of how the queues are visited, this amount is always equal to the steady-state work in a model in which the service order is FCFS (right-hand side of (3.11))

The conservation law states that the sum of the average waiting times per class, weighted by their utilizations, remains the same, independent of the priority assignment

to the various classes [27] In the special case where E[ti] = E[t] for all classes, then the conservation law gives

λ λ

i m

i i

t E

If no priority is assigned, customers will be serviced based on the FCFS discipline and the system is equivalent to a M/G/1 system with second moment of overall mean service time as described by (3.3) The average waiting time of each individual class of customers is different but the sum of the weighted average is equivalent to that of a M/G/1 system This is because the change in queue discipline does not change the overall dynamics of the system The probability that say, n units are in the system is the same; the mean number present, regardless of priority, is the same, and so on Only the individual history of particular units are affected and not the average behavior The average waiting time of an arbitrary customer is given by the Pollaczek-Khintchine (P-K) Formula (3.14)

λ

−

= 1 2

] [ t2E

The case of no priority between the two classes is considered This model is analyzed in [20,29] and the average number of each class of customers in queue is as follows:

2

2 1 1

2

2 2 1

1

1 1

1

1 1

µ

λ µ λ

µ

λ µ

µ ρ

2

1 2

1 2

2

2 1

1

2 2

1

1

µ

λ µ λ

µ

λ µ

µ µ

µ ρ µ

(3.16), Lq1 =2.4 customers and Lq2 =0.8 customers The average waiting time of an

arbitrary customer in queue is evaluated to be four minutes using (3.9)

Consider now a single queue model with an arrival rate of 0.8 customers/min (λ=λ1+λ2) This is known as the M/M/1 queueing model The average waiting time of a

customer in queue (Wq) is evaluated from formulas in [20] and found to be exactly four

minutes Hence, for a number of separate queues serviced by a single server with equal

service rates for all customers, the performance statistics (Lq,Ls,Wq,Ws) of an

arbitrary customer in the system is the same as that of a single queue system even though each individual queue performs differently This is in agreement to the Kleinrock’s conservation law Hence, a SSMQ system can be analyzed with a single server single queue model

For completeness, a model is next analyzed with a general service time distribution rather than the exponential example discussed previously Avi-Itzhak et al [30] studied a queueing system where a single server serves two classes of customers under the alternating discipline, i.e if the server is attending to customers of class 1, it cannot switch to customers of class 2 until the system is empty of class 1 customers, and vice versa If the server is idle, then the first arriving customer that enters service will acquire priority rights for customers of its class Customers within a class are served FCFS A generalization of this model to m queues was analyzed in [31]

Referring to Figure 3.3, the stable condition is achieved when λ1/µ1 + λ2/µ2 = ρ1 +

ρ2 = ρ < 1 The mean and second moment of the service requirement of each class of customers are given by E[ti] = 1/µi and E[ti2] respectively for i = 1,2 The average waiting time of each class of customers in queue is as follows:

( ) ( 1)( )( 1 2)

2 2 2 1 1 2 1 2

2 2

1

2 1 1 1

21

11

2

][1

][)

1(2

][

ρ ρ ρ ρ ρ

ρ λ

ρ

λ ρ

1 1

2

2 2 2 2

21

11

2

][1

][)

1(2

][

ρ ρ ρ ρ ρ

ρ λ

ρ

λ ρ

is studied from the perspective of another queueing model

3.1.2 Polling Model

A basic polling model is a system of multiple queues accessed by a single server

in some predefined order Customers arrive at the queues following some arrival process Upon visiting a particular queue, customers are served based on a scheduling strategy After which, the server leaves the queue and visits the next queue Moving from one queue to another involves some time, usually known as the switchover time Examples of server visit orders include cyclic polling, Markovian polling and tabular polling The scheduling strategy defines how long or how many customers are served

by the server once it visits a particular queue Among the well-known scheduling disciplines are: exhaustive, gated, limited and decrementing service (see [32-36] for a more complete survey)

Consider a cyclic polling model with N stations, modelled by queues Q1 through

QN (Figure 3.4) Queues are indexed by i (i = 1,2,…,N) At queue i, customers arrive according to a Poisson process with rate λi The mean and second moment of the service time at queue i are denoted by E[ti] and E[ti2] respectively The total offered load is as follows:

The mean and variance of the time required by the server to switch from queue i to queue (i mod N+1) are denoted by δi and δi2 respectively The main performance

measure of interest is the average customer waiting time for queue i, i.e Wq i

The cyclic polling models are generally not work conserving because the server does no useful work when it switches from one queue to another If the switching times are zero, the polling model would have been work conserving and Kleinrock’s conservation law would apply (3.10) If there were only one station (N=1) and zero switchover times, a normal (work conserving) M/G/1 model would be obtained, and the right-hand side of (3.10) is just the average waiting time in the M/G/1 model

When the model is not work conserving (i.e nonzero switchover times), Kleinrock’s conservation law does not hold However, it has been shown by Boxma et

al [37] that a pseudo-conservation law expresses the sum of the waiting times in queues, weighted by their relative utilizations to be a constant The law does not give explicit expressions for the individual mean waiting times but it provides insight into the system operation and the efficiency of scheduling strategies It can also be used as a basis for approximation schemes or to verify simulation results

Chang [38] studied a FCFS polling system operating under a central controlled or distributed system In a central controlled system, a central controller has information

of the arrival times of the customers The server follows the order of arrival times to serve customers on the FCFS basis The server switches frequently among queues if the next customer on the list is located at a different queue Chang defined the augmented service time (AST) of a customer as the sum of the time spent corresponding to the

switchover (if any) and the following service time He showed that if Poisson arrival processes are assumed, and if all service and switchover times are independent, then the ASTs are independent and the system is modelled as a M/G/1 system with the service times being identical to the corresponding ASTs in the polling system

The arrival process of queue i is a Poisson process with rate λi (i = 1,2,…,N) and mean and second moment of the service time of any customer is E[ti] and E[ti2] respectively The switchover time from one queue to another is assumed to have mean

ν and second moment ν(2)

respectively The average waiting time of an arbitrary customer in queue as derived by Chang [38] is

1 2

] [ 1

] [ 2

1 2

2 1

2 )

2 ( )

2 (

t E

t E t

E Wq

N

i i

N

i i

ρ ν

λ

ρ ν

ν λ

(3.20)

When switchover times are zero, i.e ν = 0, ν (2)

= 0, (3.20) reduces to the P-K Formula for a M/G/1 model (3.14) Therefore, regardless of which models (priority queuing model or polling model with zero switchover times) are used, a SSMQ system reduces

to a M/G/1 model and the results for Subsystem A can be easily evaluated using the

P-K Formula