Capacity and Queueing Evaluation of Port Systems with Offshore Container Unloading
Woo-sung Kim1, Jong Hoe Kim2, Hyoyoung Kim3, Hyeyon Kwon4 and James R. Morrison5*
Department of Industrial and Systems Engineering
KAIST
335 Gwahangno, Yuseong-gu
Daejeon 305-701, Korea
e-mail:{kws0924,windf,hyhouse5,hyeyonnie,james.morrison}@kaist.ac.kr
Abstract: Analytic performance evaluation of port systems that include offshore unloading of containerships, as in Hong Kong’s midstream operations and recent mobile harbor concepts, is complicated by the additional container handling step and the finite number of resources available. Commonly used decoupling approximations, that consider each resource as an independent queueing system, may not be sufficient to obtain adequately accurate estimates of system performance. In this paper, we develop several rough cut evaluation techniques for port systems with offshore operations; there are two thrusts. First we focus on container ship service time and capacity analysis. Based on system primitives such as the container capacity of the offshore vessels, the ship unloading speed and the land berth unloading rate, we calculate a lower bound on the minimum number of offshore vessels to serve a given demand for port services. Based on this throughput approach, we in turn obtain an approximation for the ship service. We also develop a Petri net model to obtain more exact results for ship service time. Our second thrust is to develop approximations for queueing time to enter service. Here we use a modified M/G/c queue to model the system. This model is then employed to develop an approximation for the queuing and service time of container ships. With the goal of assessing the accuracy of our approaches, we develop simulations of port systems with offshore components. We discuss the results and highlight which methods are of practical value.
1. INTRODUCTION
As the volume of trade between nations has grown, worldwide maritime container transport has followed. In fact, it was projected in Drewry(2006), that by 2011, the container transportation market demand will exceed capacity. While this projected date has no doubt been delayed by the worldwide economic slow down, the need for additional port capacity will continue. To address this problem, various countries have increased their container terminal capacity in hopes of attracting the global container market. One of the traditional solutions to increase capacity is to construct new berths. However, this approach may not be ideal because it requires substantial time, is very costly, and is not matched with green development or ecological technology. Most container terminal construction uses large amounts of artificial ground and concrete and consumes large tracts of seashore. An alternative is the mobile harbor system. This system is detailed in Suh N.P (2008) and is intended to address the existing problems of traditional container terminals, such as a lack of berth space and water depth. To efficiently design and operate a mobile harbor system, it is important to evaluate the measures of system performance such as throughput and ship queueing time. However, commonly used queueing approximations do not well fit the mobile harbor system. Therefore, new techniques are required to assess the system performance.
In this paper, we study evaluation techniques for the mobile harbor system. First, conducting throughput analysis, we calculate the minimum number of mobile harbors to serve a ship. With this, we can calculate ship service time under ideal service assumptions. After relaxing the assumptions, we propose another approximation for ship service time. Second, we conduct queueing analysis. Employing an M/M/c queueing model, we can obtain the number of MHs and berths to achieve a required service level. Waiting time to service time ratio (W/S ratio) is used as service level metric. We also develop approximations for the cycle time of a ship. Modeling the system as a modified M/G/c queueing system, we develop approximations for queueing and service time of container ships. To verify our approximations, we develop simulation, and compare with our approximations.
This remainder of the paper is organized as follows. In section 2, we describe the mobile harbor system adopted in this paper. We introduce how the mobile harbor system operates and define variables which we use in this paper. In Figure 1. Mobile Harbor (MH) system
section 3, throughput analysis is conducted. With different assumptions, several techniques to evaluate system performance are introduced. Two techniques which calculate a minimum number of offshore vessels to serve ships are introduced under ideal assumptions. Using the techniques, we can estimate the ship service time without waiting time. Then, based on a petri-net model, we conduct throughput analysis. We can obtain throughput and cycle time assuming no queueing. In section 4, approximations for the queueing and service time of container ships are developed. We model the mobile harbor system as a modified M/G/c queueing system. We compare the approximation with simulation results. Concluding remarks will be presented in section 5.
2. SYSTEM DESCRIPTION
In this section, the concept of the mobile harbor system is introduced. In traditional harbor systems, ships dock at land berths to unload containers. There are challenges associated with the traditional system. Due to container transportation market trends, the size of ships has increased (Drewry(2006)). However, most ports do not have the capability to serve the new containerships because of water depth. Insufficiency of berth space is another problem. The mobile harbor system addresses these problems. In the mobile harbor system, mobile ships/barges, each possessing its own container crane and storage capacity for hundreds of containers or more, travel on the ocean to vessels anchored offshore. There it employs its onboard crane to conduct loading and unloading of cargo at sea. We call such mobile crane and container storage ships as mobile harbors (MH). When a container ship arrives to the offshore operation location, mobile harbors provide service to the ship. The mobile harbor next transports the containers to a dedicated land berth. Once arriving to the dedicated land berth, the mobile harbor releases the containers to the land berth. Because the (un)loading operations between a ship and a MH is conducted on the ocean, it can more readily guarantee water depth. At the inland berth, an Ultra Fast Interface (UFI) with high container transfer speed moves containers between the berth and the MHs. Therefore, the MH system can minimize the dependency between the port capacity and the length of the seashore. Figure 1 depicts a MH system.
The following notation is used in this paper:
λArrival rate of a ships (ships/day),
cMHCapacity of a mobile harbor (TEU/MH),
nMHNumber of mobile harbors,
nBNumber of berths,
nTUNumber of containers in a ship,
nDMaximum number of MHs that can simultaneously be docked with a containership,
VtTravel speed of a Mobile harbor (Km/hour),
VsOperation speed of a Mobile harbor at ship (Containers/hour),
VbOperation speed of a Mobile harbor at berth (Containers/hour),
dDistance between offshore operation location and land berth (Km).
We also use MH fleet concept here. That is, MHs are grouped in each collections of nD MHs and operates as a fleet. Each such group of MHs moves, travels and conducts (un) loading simultaneously. We assume that the number of MHs is a multiple of nD in this paper.
3. THROUGHPUT ANALYSIS
In this section, several techniques to calculate the MH system capacity and ship service time are introduced. Here, we assume that containership inter-arrival times are deterministic. When a ship arrives, all resources in the system are devoted to serve the ship. We obtain the ship service time using three techniques; there are differences between them. In section 3.1, assuming sufficient berth resources, we calculate a bound on the number of MH fleets required to provide ideal ship service. In section 3.2, we relax the assumption on sufficient berth resources. An approximation for service time term will be proposed. In section 3.3, our Petri-net model will be introduced. If we assume there are no containers to load, we can obtain the exact ship service time. Based on the Petri-net model, we can obtain the system throughput capacity.
3.1 Bound on the number of mobile harbor fleets: Ideal ship service
In this section, we obtain a lower bound on the number of mobile harbor fleets (MH fleets) required to provide uninterrupted service to incoming ships. There are several assumptions. First, loading begins after the unloading is complete. Second, all unloading and loading operations must finish before the next container ship arrives (We relax this assumption later.). Third, each group of nD MHs operates as fleet. Fourth, ship interarrival times are deterministic and there is never contention for a land berth. Fifth, number of containers to unload and load is equal (nTU); the ship enters and leaves with the same amount of containers. Sixth, the total loading and total unloading time are equal. When the first MH fleets is full, it leaves the container ship to release the containers at the port. As soon as the first MH fleet leaves, the next MH fleets starts unloading. The deterministic variables U, L, Tr, R, , and indicate unloading time (), loading time (, traveling time (, releasing time (), number of containers to unload, and number of containers to load, respectively. For simplicity, we assume that = = nTU. When the total number of containers (both load and unload) is less than or equal to the MH fleet capacity, the number of mobile harbor fleets is equal to the container amount divided by the container capacity of a MH fleet, rounded up to the nearest integer. When the container amount is larger than a MH fleet capacity, if unloading time is longer than traveling time and releasing time (U≥2Tr+R), we need only two MH fleets, because they can alternate to provide uninterrupted services. When the unloading time is shorter than the traveling and releasing time (U<2Tr+R), we need more than two MH fleets. There are three cases for this condition. First is when the initial MH fleets returns before the entire unloading operations end. Second is when the initial MH fleet returns after all the unloading operations are complete, but before all the unloading and loading operations end. Last is when the initial MH fleet returns after all the unloading and loading operations are complete. The bound on the number of MHs required to provide continuous service is
2, if U ≥ 2TR+R,
U+2Tr+R
U+2Tr+R,
(1)
We will explain the third case in (1) to help understand the bound. Under the condition that the initial MH fleet returns after the unloading is complete but before the entire unloading and loading operations end (U+2Tr+R). Let’s assume that cMH = 200TEU, nTU = 5000 TEU, nD = 6, = 50 TEU/hr, Tr = 9, and R = 4. Then, = 16.67 U+2TR+R = 26 = 33.34. This means that the berth is located farther, so that the MH fleet returns to the ship after the unloading, but before the loading operations are complete. If we insert the numbers to the equation, = 5, = 7, and minimum of these is 5. So we need 5 MH fleets to unload and an extra 2 MH fleets for loading (), because as soon as the last unloading MH fleet leaves to the berth, we need alternate MH fleet for loading. This is obtained by subtracting total time to unload ) from time a fleet takes to come back to the ship (U+2Tr+R) and divide by loading time (L). Therefore, at least 7 MH fleets are needed to complete the service.
In addition to these equations, we consider the situation where another ship arrives before the service is complete for the previous ship. There are two situations. One is when the next ship arrives before the entire unloading operations of the previous ship are complete and the other is when the next ship arrives after the entire unloading operations are done but before all the loading operations of the previous ship are complete. For the first situation, additional MH fleets are needed to conduct service for the next ship. To calculate the multiple of MH fleets, we need to divide total unloading time by inter-arrival time. In case of second situation, only one additional of MH fleets is required, because the MH fleets used in the previous ship to load the containers to ship can be used again.
Using these relationships, we can obtain an approximation for the service time as (nMH x ). In this section, the number of mobile harbors is calculated with an assumption that all the unloading and loading operations are completed within the inter-arrival time and therefore, it is certain that there will be no waiting time for the ships. This method clearly shows the relationship between size of the mobile harbor fleet, distance between the ship and berth, and the number of mobile harbor fleets. If the size of the mobile harbor fleets and distance between the ship and berth are given, the lower bound of the number of MH fleets can be easily obtained. However, there exist several restrictions. First, there may not be sufficient berths to guarantee no MH contention at the land berths. Second, mooring time is not considered. In the next section, a method to calculate the number of MHs considering the restrictions stated above will be proposed.
3.2 Ship service time approximation: MH delays at the land berth and ships
Relaxing the assumptions of section 3.1, we develop a ship service time (SST) approximation with finite berth resources. There are three container flows in the mobile harbor system. First, containers are transferred from the containership to the MH at sea via the MH container crane. Second, containers travel from the offshore operation location to the land berth on the MHs. Third, containers are transferred from the MH to the land at the MH-land berth interface. The maximum container throughput is less than the minimum capacity at each of three flows. That is the system capacity is less than (or equal) to the system bottleneck capacity. Let λ1, λ2, λ3 denote the cargo flow rate between ship and offshore operation location, cargo flow rate between offshore operation location and MH berth and cargo flow rate between MH berth and land berth.
A first order approximation for the SST is
(2)
However, MHs may suffer queueing at the land berths and ships if their number is greater than the number of docking points or berths. The queueing also depends on the arrival pattern of MHs to the berths and docking points. As a gross approximation to this queueing, we assume an arrival pattern of MHs to the berths and docking points as in figure 2.
Figure 2. The assumed arrived pattern of MHs to their servers
All MHs come to the server (which can be berth or docking point) simultaneously. The server immediate begins service of as many MHs as possible. In figure 2, MHs 1 and 2 enter service with the two servers immediately. The remaining MHs in the fleet 3,4,5,6 in figure 2 are queued until MHs 1 and 2 complete services. MHs 3,4 and 5,6 enter service immediately as they are able. When MHs 5 and 6 have received service the process repeats with a new MH fleet. In this case, the average waiting time for MHs to enter service is
(3)
where, Nc is the number of MHs in a fleet, Ns is the number of servers and T is the service time. Thus, the capacity of the second flow between the ocean location and the land berth location, becomes in which Twb and Tws are the average waiting time of MHs at the berth and ship respectively. Using (3) and the approximate SST becomes
(4)
The result of this equation was compared with exact simulation under the same assumptions. We compared 154 systems and the average absolute gap between the SST approximation and the simulation was 6.49%. The approximation is intuitive and generates reasonable results. Also, one can easily obtain the service time distribution of the system from the distribution of the cargo volume, since the denominator of (4) is constant given the system parameters.
3.3 Petri net model
Figure 3. Petri-net model
Petri net is a graphical and mathematical modeling tool that can be used for many systems, see Tadao Murata (1989). In C.V. Ramamoorthy and Gary S. Ho(1980) deterministic timed nets and their minimum cycle time are discussed. A Petri net consists of circles that represent conditions and bars that represent events. The circles are called places and the bars are called transitions. If each input place of a transition has at least one token (dot), the transition fires and a token is removed from each of the input places and a token appears in each of the output places. We assume that transitions can fire again before the previous firing is complete, so long as there are sufficient tokens available. That is, the firings of a transition can overlap in time. Here we develop a Petri net model of a port with mobile harbors. The model is depicted in Figure 3; we describe its transitions and places next. A mobile harbor attaches to a ship docking point during transition. Then, it loads cargo during transition and detaches from the ship during . A token indicating docking location returns to its place when the mobile harbor is detached. Travel to the land berth occurs during transition . It docks to a land berth during and unloads the cargo. After undocking (, the token arriving at place represents a land berth being ready for another mobile harbor full of cargo. The unloaded and undocked mobile harbor travels back to the ship and repeats the cycle. Transitions and model the cyclic arrival of subsequent ships – that is, another ship enters the service after all cargo has been removed from the previous ship. In Fig. 3, the tokens indicate that there are 8 mobile harbors, 7 full mobile harbors worth of cargo on each ship, 4 berths, and 4 docking points on a ship. Note that in Fig. 3 requires 7 tokens to fire. As a consequence, when every single token arc fires 7 times, fires once.