1. INTRODUCTION
Because of the continuous growth in container transport volumes, container ports strive to improve their terminal capacities (Drewry, 2007a). While the most traditional solution to this capacity problem is the construction of new berths, most container terminals require large amounts of artificial ground and concrete; additionally, construction is timeconsuming and costly. Drewry (2007b) has stated that the increase in container transport volumes will cause an increase in the size of containerships. However, as mentioned in Kim and Morrison (2012), few existing ports can handle ships with capacities larger than 6,500 TEU (Twentyfoot Equivalent Unit is based on the volume of a 20footlong intermodal container. A common 20footlong container is 20 feet long, 8 feet wide, and 8 feet 6 inches tall.). Because of shallow water depths and lack of berth space, it is difficult to serve such ships in traditional container terminals. Therefore, a port system concept based on offshore container operation is proposed. In such a system, the container (un)loading operations of a ship take place on the sea and are not affected by sea depth or the width of berth. To establish an efficient port system, performance measures for the system should be evaluated in the design stage. However, traditional queueing formulae are difficult to apply to the system because of the additional containerhandling step caused by transportation units and the finite number of units available. The goal of this paper is to assess performance measures for the system by conducting a queueing analysis. The developed techniques can be applied to the resource optimization problem posed by the system.
Midstream operation, an offshore container service concept, has run at the Hong Kong container port for years. In the offshore container service concept, barges equipped with a crance process containers from an anchored container ship in the open sea. The port employs both traditional terminal operations and midstream operationsabout 10% of the total container volume at Hong Kong port is handled by midstream operations. Thus, the capacity of a port system can be improved with midstream operations (Nam and Lee, 2013). However, as mentioned in Nam and Lee (2013), such systems are said to have safety and quality problems due to the difficulty of maintaining good crane balancing during operations. To overcome weaknesses in midstream operations, a mobile harbor system has been proposed by Korea Advanced Institute of Science and Technology (KAIST) (Suh, 2008). Various studies of this system have been conducted. Figure 1(a) depicts the mobile harbor system, and Figure 1(b) shows an open sea test of the mobile harbor system. Similar to midstream operations, the mobile harbor system processes containers from a containership in the open sea; however, the system strived to resolve various problems involved with midstream operations. For detailed information on these efforts, refer to Jung and Kwak (2009), Han and Lee (2009), Kim et al. (2009), Lee et al. (2009), and Shin et al. (2009). As a result, a more efficient and safer system was developed, and open sea tests and economic studies were conducted to prove its feasibility and efficiency (Kim and Morrison, 2012).
Despite some technical successes, there are still important questions to be answered: How can we design an efficient mobile harbor system? Can we evaluate various performance measures for the system in the design stage? Several works have aimed to answer these qauestions. Lee and Park (2010) employed an axiomatic design theory to manage the design process of the system. Sung et al. (2013) and Nam and Lee (2013) studied a scheduling problem for mobile harbor operations in order to enhance its productivity. Feasibility and capacity planning studies for the system were conducted by Kim and Morrison (2012). While several studied have investigated efficient system operations, few studies have been done on analytic performance evaluations for such a system. For a traditional port container service concept, Edmond and Maggs (1978) applied M/M/m and D/M/m queueing models to evaluate the cost incurred by the delay of a ship. While these models can be useful for investment decisions, they provide inaccurate information in many cases. The paper stated: “these models can be useful … when care is taken in the choice of model, their parameters and evaluating their results.”
At system design stage, the system is theoretically similar to the cargo terminal transportation system. A number of studies have veen conducted to analyze the cargo terminal transportation system. Cimpeanu et al. (2015) developed a discrete event simulation model to analyze the refinery in Ireland. The paper considers the unique characteristic of the refinery, such as the flow of the tide, flexible maintenance of the area and other port characteristics. The results of the paper were extended in Cimpeanu et al. (2017). The impact of several maintenance policies on key performance indicators was assessed in the paper. While the results on the performance measures obtained through the simulation models provide more accurate information than the analytical solutions for the performance measures, the analytical solutions can provide the insight necessary to understand the system and explain the relationship between the system parameters. Bugaric et al. (2011) suggested that the analytical solutions for performance measures obtained by queueing models can be used at the beginning of a design process, since they can provide useful and intuitive information faster and more conveniently than approaches using simulations. Bugaric et al. (2011) used queueing theory to study the capacity of unloading a bulk cargo terminal. The results can be used when rough estimations of system behavior are needed. However, the analytic performance evaluation of a mobile harbor system is more complicated by both the additional containerhandling step and the finite number of transportation units available. Recently, in Mishra et al. (2017), a semiopen queueing network model for the interterminal transportation problem is proposed. The theoretical bounds on the throughput time estimates of the model are obtained, and the results are applied to the port of Rotterdam. However, there are few results for other key performance measures such as cycle time or blocking probability. Kim et al. (2010) developed several roughcut evaluation techniques for use with mobile harbor systems; however, they only employed traditional mathematical models. The models did not consider the detailed operations of cranes or movement of transportation units. Therefore, new techniques are required for assessing system performance. Inspired by this gap in the research, our goal is to assess system performance measures for a mobile harbor system using queueing analysis.
Here we study queueing evaluation techniques for a mobile harbor system in order to derive performance measures. Our performance measures of interest are the mean waiting time of a container ship in the system and the probability that a containership will be denied service due to lack of resources (blocking probability). The choice of the performance measures is based on the ship’s response when the containership arrives in the system (or when a request for port use arrives) and all resources are busy. In Tierney et al. (2014), it is stated that “… focus on minimizing the overall delay experienced by containers, an important consideration for port planners, as the costs of delaying outgoing shipments are usually very high.” Because of this high cost, the arriving ship who cannot receive the service immediately will choose one of two options. The first option is to wait. In this case, the average delay experienced by the vessel in the system can be used as an important performance measure. Kim et al. (2010) and Tierney et al. (2014) have studied the measure. Another option is to avoid the system immediately without receiving service. Since the service delay of a container vessel arriving in port is costly, vessels expected to wait excessively for their services will not enter the syssystem and use the other port. This does not mean that the ship leaves the system at the time it arrives. Rather, it means that when the demand for port use occurs, all the resources of the port are busy and the demand cannot be met. In fact, the schedule of the port is strictly controlled and if a demand for the port has occurred in a certain period and the port is not available on the schedule (all resources are already booked), the containership will be notified. In such cases, the containership sometimes uses other ports and this is a loss of demand from the perspective of the port. In order to assess whether it is possible to process the demand immediately, blocking probability, also called Erlang loss formula, can be used.
To obtain the solutions for both performance measures, we modify two queueing models: M/G/m and M/G/m/m queueing models. Both models assume the Poisson arrival process, which is suitable for modeling the ship’s arrival process. In UNCTAD (1985), it is mentioned that “UNCTAD research has confirmed the widely accepted view that the arrival pattern of breakbulk ships is best approximated by a Poisson distribution. This is equivalent to the distribution of the interval between arrivals being approximated by a negative exponential or Erlang 1 distribution.” While the arrival process can be easily modeled by Poisson distribution, obtaining the service time distribution is not an easy task. In the mobile harbor system, the ship service time must consider not only the time to unload the container on the ship, but also the time to travel between the ship and the land berth. We derive a distribution for ship service time considering the behavior of mobile unit fleets. From the distribution, the mean and variance of service time for a ship in the system can be obtained. Modeling the system as a modified M/G/m and M/G/m/m queueing system, we develop analytic performance measures for the mean waiting time and blocking probability of a ship in the system. Both models can be applied to system resource optimization problems.
This remainder of the paper is organized as follows. In section 2, we recall formulae for M/G/m and M/G/m/m queues and describe the mobile harbor system. Our intent in this section is to provide a brief survey of the existing results and explain how the mobile harbor system can be modeled by the queues. We also describe how the mobile harbor system operates and define the variables used in this paper. In section 3, we conduct a queueing analysis. We derive the ship service time and incorporate the results into existing results to modify M/G/m/m and M/G/m queueing systems, thereby obtaining analytic performance measures. In section 4, we apply the developed methodology to actual port systems. Using actual information about the Surabaya port in Indonesia, we show that the developed methodology can be applied to resource allocation problems. Concluding remarks are presented in section 5.
2. PRELIMINARIES
In this section, we introduce relevant queueing models and describe an operational scenario for an unloading operation in a mobile harbor system. First, the results of M/G/m/m and M/G/m queueing models will be introduced and relevant notations will be defined.
We consider a queue with Poisson arrival process (having rate λ) and with m (1 ≤ m ≤ ∞) parallel servers having general service time distribution, each with rate μ. The queue discipline is FCFS: the ships which arrive earliest get served or treated before the ships which arrive later. We assume that the servers do not fail. If there are n ships in the system, and n is less than m, then, in all, n servers are busy and (mn) servers are idle. If there are n (≥m) in the system, then all m servers are busy and (nm) should join the queue and wait for the service completion of the previous customers. As soon as the service completion, the first ship in the queue enter the available server. Figure 2 depicts the system.
Depending on the buffer capacity of the system, different queueing models can be applied. If the buffer capacity is infinite (a system allows the infinite number of ships to wait in the system), then the major performance measure will be the average waiting time of a ship in the system. This measure can be calculated by a cycle time approximation formula of G/G/m queue as follows (Hopp and Spearman, 2011):
The system loading, ρ, is equal to λ/mμ, and is assumed to be less than one. The coefficient of variation of the interarrival time is represented by c_{A} and is equal to 1 in our system (Poisson arrival process). c_{S} is the coefficient of variation for service time (c_{S} = σ_{S}/(1/μ), and σ_{S} is the standard deviation for service time.
As we mentioned above, ships do not wait in some harbor system. In such a system, when an arriving ship sees that all servers are busy, it does not enter, and leaves the system immediately. This case is relevant especially when the cost incurred by the delay of a ship is relatively high. In this type of system, rather than waiting in the queue, the ship travels to other ports or schedules another time to receive service. One important measure for such a system is the probability that all servers are busy when a ship arrives. This probability, also called loss probability, can be calculated using M/G/m/m queueing models. The probability, P_{loss}, is given as follows (Cooper, 1981):(2)
This formula is also called the Erlang loss formula. Because of the invariance of the system, the probability is affected only by the mean interarrival time and mean service time, and not by the distribution of the service time. To calculate the above two measures, the equation parameters must be determined. While the arrival rate of the system can be derived from a demand analysis, obtaining the service rate is complicated. The crucial issue is that the service time for the queueing model must consider the behavior of the mobile harbors. We will study the issue in the next section.
Next, we describe an operational scenario for an unloading operation in a mobile harbor system and explain how the system can be modeled by queueing system. In a mobile harbor system, an arriving container ship anchors at an offshore operation location located in the open sea, rather than berthing at a terminal. Then, transportation units referred to as mobile harbor units approach the ship for unloading operations. Each mobile harbor unit possesses its own container crane and container storage capacity. After the unit arrives at the offshore operation location, the mobile harbor docks with the containership. Once the docking operation is finished, the mobile harbor begins unloading containers from the containership. The unit continues to unload containers until either its maximum storage capacity is reached or there are no remaining containers to be unloaded from the ship. Then, the unit returns to the terminal and releases its containers after docking at a berth. If there are containers that remain to be unloaded on the ship, the unit travels back to the ship for an additional unloading operation. When all unloading operations are complete, the containership departs the system and the mobile harbor units return to the terminal. While mobile harbors can conduct both loading and unloading operations, we omit the case of loading. The results for this case can be easily obtained by modifying the results in this paper. Hereafter, we only consider a case in which mobile harbors unload containers from a containership. Figure 3 depicts this system.
To obtain tractable models, we extract system characteristics from the actual system and define variables appropriately. These are summarized as follows.

There are two locations: an offshore operation location and a land berth.

A ship arrives at the offshore operation location with containers, and departs the system once all containers in the ship have been transported to the land berth.

The arrival process of a ship follows a Poisson process with a rate of λ.

The number of containers in a ship follows a general distribution represented by the random variable nTU.

Initially, all mobile harbors are located in the operation location and are ready to provide services.

The maximum number of units that can simultaneously be docked and operated with a containership is nD.

We assume that the units are grouped in collections of nD units each operating as a fleet. Each group of units moves and conducts unloading operations simultaneously.

The number of fleets serving one ship is defined as k. For example, if k = 2, two fleets, with nD units in each fleet, alternate to conduct unloading operations for one ship. From the standpoint of queueing theory, k·nD units act as a server.

The total number of mobile harbor units in the system is nU. We assume that nU is a multiple of k·nD.

The capacity of a transportation unit is deterministic with respect to cU. This means that the cU is the maximum number of containers transferred by one mobile harbor at a time.

The unloading operation time of a transportation unit is a constant t_{p} at offshore operation locations and land berths, irrespective of containerhandling amount. Thus, in one trip, a mobile harbor unit spends t_{p} time units serving a containership. For example, to serve a ship with 1,000 TEU, one mobile unit with a 250 TEU capacity must make four operations; thus, it spends 4 t_{p} time units finishing its work.

The travel distance between an offshore operation location and a land berth is d. The travel speed of a mobile harbor is a constant: v_{t}. Thus, the time required for a unit to travel between an offshore operation location and a land berth is deterministic with respect to t_{d} (t_{d} = d/v_{t}).

We assume that an arriving ship only enters the offshore operation location when all fleets have returned to the operation location and are ready for service.

The number of land berths for mobile harbors is sufficiently large to avoid mobile harbor waiting time at the land side.
For example, let nU = 12, nD = 2, k = 2, nTU = 500 TEU, cU = 150 TEU t_{p} = 30 min, and t_{d} = 20 min. The total number of mobile harbor units is 12, and two mobile harbors can be docked and operated with a containership simultaneously. Since k = 2, two fleets (four mobile harbors) act as a server to serve one containership. First, two harbor units (the first fleet) dock with a containership and begin to unload containers from the ship. It takes 30 min to finish one unloading operation. After the units are full (150 TEU for each unit), the units begin their travel to the land berth to release their containers. At the land berth, another 30 min are required to release the containers. The two additional units (the second fleet) dock with the containership and begin operations immediately after the previous units begin traveling to the land berth. Although the remaining amount of containers in the ship is 200 TEU, the operation time is assumed to be the same (30 min). After the second fleet unloads the remaining containers, the ship departs the system. Figure 4 depicts this system. In the previous example, we can model the system with a queueing model consisting of three servers. Because two fleets will serve a containership (k = 2), a server will consist of four mobile harbor units. While this system can be modeled by existing multiserver queueing models, the additional handling steps incurred by the mobile harbor units complicate the analysis. We analyze the exact mean service time of a ship in the system in order to derive the measures in the next section.
3. QUEUEING ANALYSIS
Here, we conduct a queueing analysis to derive performance measures for the mobile harbor system. To obtain performance measure results, we first derive the time required for the vessel to be serviced by the system.
When a ship arrives at an empty offshore operation location (server), the first fleet, nD mobile harbors units, docks with the ship and unloads the ship’s containers. Each fleet in the server provides service to the ship in sequential order. After the units in the ith fleet finish their operation, the units begin their travel to the land berth, and the (i + 1)th fleet begins operations (for i = 1, … , k − 1). The first fleet returns to the location and serves the ship after the kth fleet finishes its unloading operation. This process continues until all containers are unloaded. For example, Figure 5 depicts a Gantt chart for a case in which nD = 4, cU = 50 TEU, k = 3, and nTU = 700 TEU. Note that p_{1} and p_{3} represent, respectively, the unloading operation at the operation location, and the releasing operation at the land berth. p_{2} represents the travel of the fleet from the operation location to the land berth, and p4 represents travel from the land berth to the operation location. Each fleet consists of four mobile harbors, and three fleets (that is, a total of 12 mobile harbors) will be assigned to serve one ship. Given these assumptions, four operations are needed to unload 700 TEU. Initially, all fleets are in the operation location. When a ship arrives, the first fleet begins unloading operations. After completing the unloading operation, the first fleet travels to the land berth and the second fleet begins serving a ship. At the land berth, the first fleet releases its containers. After releasing the containers, the fleet returns to the operation location, and provides another service to unload the remaining containers.
Let Y denote a random variable that represents the number of times the mobile harbor fleets operate until all containers are unloaded from any ship. Note that the value of Y can only be a positive integer. The probability mass function of Y, p_{Y}(y), can be expressed by the random variable nTU as follows:
While assigning sufficient fleets to serve one ship guarantees a higher service rate, an excessive number will increase the idle time of the fleets. Thus, we consider two cases separately: 2(t_{p} + t_{d}) < k·t_{p} and 2(t_{p} + t_{d}) ≥ k·t_{p}.
First, when 2(t_{p} + t_{d}) < k·t_{p}, a ship can be serviced consistently. There are sufficient numbers of fleets to serve one ship, so any fleet returning from releasing operations sees the other fleet working with the ship. Idle time may occur for the mobile harbor unit, but the system reaches a maximum service rate (Figure 5). In this case, when a ship with the number of containers between (x − 1) ∙ nD ∙ cU and x ∙ nD ∙ cU (with probability pY(x)) arrives at the empty system, (x+1) ∙ t_{p} + 2t_{d} time units are required to be ready for the next ship because all mobile harbor units should return to the offshore operation location. Let S denote the random variable that represents the time from the start time of a service for a ship to the time that all resources return and are ready for the next ship. Then, the mean can be expressed by Y as follows: E[S] = (E[Y] + 1)t_{p} + 2t_{d}. In addition, the variance of S is t_{p}^{2}Var(Y). Thus, the mean cycle time of a ship in the system can be obtained by modifying equation (1) as follows:
The first term in equation (4) is E[Y] ∙ t_{p} because when the vessel is continuously serviced, unloading all containers takes x ∙ t_{p} time units when the container quantity is between (x − 1) ∙ nD ∙ cU and x ∙ nD ∙ cU. As soon as all the containers are unloaded from the ship, the ship leaves the system. The first term represents the average amount of service the ship receives. However, the next ship must wait until all resources return and be ready, even when the previous ship from the server has already departed. For this reason, the first part in the second multiplication above is (E[Y] + 1)t_{p} + 2t_{d}. The number of servers in the system is nU/(k ∙ nD).
Given the assumption that ships that arrive during a state in which all resources are not available leave the system, the loss probability can be obtained by assigning the mean service time as follows:(5)
For example, suppose that nTU follows a uniform distribution with an interval of [500, 4000], and the system consists of 60 mobile harbor units (nU = 60) with a capacity of 250 TEU each. Let nD = 2, t_{p} = 30, and t_{d} = 10. Different performance metrics are then obtained depending on how the k value and number of fleets assigned to serve one ship are determined. When three fleets are assigned to serve a ship (k = 3), we can think of a queue model consisting of ten servers. Each server has six mobile harbor units. Similarly, the system has six and three servers when k = 5 and k = 10, respectively. Figure 6 shows how performance measures respond to loading increases. As we can see in Figure 6, when the value satisfies the inequality 2(t_{p} + t_{d}) < k · t_{p}, the smaller the k value of the system, the better the performance measure. This is because the system has already achieved its best service rate for a condition of 2(t_{p} + t_{d}) < k · t_{p}. Therefore, allocating more fleets to servers will not only increase the service rate, but will also result in a reduction in the total number of servers in the system.
When 2(t_{p} + t_{d}) ≥ k·t_{p}, the fleet can immediately start its next service when it unloads the vessel’s containers to the land berth and returns. Figure 7 depicts this case. When the first fleet returns from the land berth for the second operation, all other fleets are working on other operations (moving or releasing operation). Upon returning from the land berth, the first fleet begins the second operation. The ship cannot receive service consistently due to the insufficient number of mobile harbor units.
Since the next ship enters the server when all the mobile units are ready and able to service the next ship, the distribution of the service time is the time from when the vessel begins receiving service to the time when all units return after transporting all containers. Recall that the random variable Y is the number of unloading operations that the fleet must perform to unload all containers on the ship, and is determined by nTU, nD, and cU. In Figure 4, Y = 4 means that a total of four operations must be performed to unload all containers on the ship. Because three fleets provide service to the ship in sequential order, the first fleet must work twice, and the second and third fleets work once.
Note that all possible values of y in equation (3) can be uniquely written as y = mk + n for m = 0, 1, 2, … and n = 1, 2, …, k. n can have the same value as k. For example, when y = k, m = 0 and n = k; m = 1 and n = 1 for y = k + 1. Under this definition, m number of unloading operations should be performed by each of the k fleets, and the first n fleets should conduct an additional operation. The nth fleet is the fleet that unloads the last container on the ship. If this fleet returns to the operation location, the next ship can enter the server. Thus, the completion time of the service of the queueing model is determined by the time when the nth fleet returns. For example, the last operation of the previous example is conducted by the first fleet. Thus, the distribution of the service time for a ship, S, can be derived as follows:
For $\text{y}=mk+n,\hspace{0.17em}\left(m=0,1,2,\cdots ,\hspace{0.17em}\text{and}\hspace{0.17em}n=1,2,3,\cdots ,k\right)$ and the service time of a ship is $2\left(m+1\right)\left({t}_{p}+{t}_{d}\right)+\left(n1\right){t}_{p}$ with probability ${p}_{Y}\left(mk+n\right)$.
The mean and variance of the random variable S are
Substituting the above results into the results for the existing M/G/c/c queues yields the following results for loss probability.(8)
For an approximate solution to the mean cycle time, we consider that the ship leaves the system as soon as the last container is unloaded to the mobile harbor. Considering this, equations (6) and (7) are applied to the approximate solution, and developed as follows.(9)
The above equations provide useful information for predicting system performance metrics at the system design stage. In the next section, we apply our results to actual port cases to present design problems associated with resource allocation.
4. APPLICATION TO SURABAYA PORT
Here, we apply our models to the Surabaya Port in Indonesia. The port is considering expanding its capacity to meet everincreasing demand (JICA, 2007). Currently, container ships with a capacity of 2,500 TEU or more cannot be serviced at the port due to the depth of the berth. An alternative to traditional servicing is the mobile harbor system. When the mobile harbor system is constructed, it is expected that the mobile harbor system will service ships with a capacity of 1,000 TEU or more. The remaining ships (with capacities less than 1,000 TEU) will be serviced at the existing terminal.
To design a mobile harbor system, we must determine the number of required resources, and also decide how to configure the system with those resources. To do this, performance measures for each system must be evaluated at the design stage. Therefore, we apply the results of this study to derive and compare the results of performance measures according to system design. According to JICA (2007), the number and size of vessels expected to visit the port in one year are as follows:
According to the literature, there are five types of vessels categorized by capacity. For example, a class 1 ship can hold up to 500 TEU containers. The anticipated number of visiting ship is 39 per year. For class 2 ships, the predicted number of visiting ships per year is 108, and the maximum number of containers they can handle is 1000 TEU. As we can see in Table 1, it is expected that the port will serve ships with capacities greater than 2,500 TEU in future. To obtain analytical models, we make the following assumptions.

The arrival process of the ships in the ith class follows a Poisson process. We assume that the rate of the arrival process for each type of ship is equal to the corresponding forecast in the last column. For example, the mean arrival rate of a class 3 ship is 1,080 ships per year.

When a class i ship arrives at the offshore operation location, the number of containers in the ship is assumed to follow a uniform distribution over the interval in the ith row and second column in Table 1. For example, when a class 2 ship arrives, the number of containers in the ship follows a uniform distribution over the interval [500, 1,000].

Since the container ships with capacities less than 1,000 TEU are assumed to use the existing terminal, we do not consider those ships in the mobile harbor system.
Table 2 presents information from the 2011 mobile harbor project report by the KAIST mobile harbor team. Depending on the capacity of the mobile harbor units, two types of mobile harbors were designed. In the project, three types of mobile harbor, 250 TEU, 600 TEU and 1,200 TEU, were investigated to assess economic viability in comparison to traditional harbor expansion in Surabaya port using discreteevent simulation. The number of mobile harbor of each type was obtained to meet required service level of the port with the expected throughput. As a result, 250 TEU and 1,200 TEU dominated the 600 TEU mobile harbor. However, no results were provided for performance measures such as loss probability of the systems. Therefore, we employ our results to evaluate the two types of mobile harbor systems that have been proposed in the project.
The assumptions for the mobile harbor system can be obtained from the above information as follows

The value of t_{p} for each design is obtained by cU divided by the operation speed. We assume that t_{p} = 20/3 for type 1, and t_{p} = 16 for type 2.

We assume that nD = 4 for type 1 and nD = 2 for type 2.

Let c denote the number of servers in our queueing system. c is equal to nU/(k∙nD). c, k, and nU are decision variables, and we only consider the positive integer values of c, k, and nU that satisfy nU = c∙k∙nD.
The design of an efficient system requires the evaluation of that system. We derive performance measures for various cases by employing our results. Table 3 presents the performance measures for various systems using type 1 mobile harbors. Table 4 presents the results for systems using type 2 mobile harbors.
The results provide us with useful information for designing a mobile harbor system. When a system is configured with a fixed mobile harbor, allocating more fleets to service one ship can provide higher service rates but it results in fewer servers in the system. On the other hands, by choosing a smaller k, we can retain a sufficient number of servers. While this results in low system loading, the service time of a ship will be longer. For example, while the loading is 0.7824 for a case where k = 1 and c = 15, it is higher when k = 2 (see the case where k = 2 and c = 8) for a similar number of mobile harbor units. Because of this higher loading, the cycle time for the case where k = 2 and c = 8 is much longer than the case where k = 1 and c = 15. However, regardless of the value of k, we could reduce loading by increasing the total number of mobile harbors in the system. As loading decreases, the mean cycle time approaches the mean service time; a system with a lower mean service time is more favorable in terms of the mean cycle time. When we compare the case where k = 1 and c = 30 with the case where k = 2 and c = 15, we can verify that the second case is preferable to the first case in terms of mean cycle time. Moreover, we can observe that the minimum integer value of k that satisfies 2(t_{p} + t_{d}) ≥ k·t_{p} should be chosen in order to maximmaximize the service rate of the system. After the inequality 2(t_{p} + t_{d}) ≥ k·t_{p} is satisfied, performance measures for the system do not improve for increasing values of k. The performance measures of a system where k = 3 and k = 4 are the same when those systems have the same number of servers.
Table 4 presents the performance measure results for a system consisting of type 2 mobile harbors. We can use our results to evaluate different system designs.
In order to find the cost efficient design associated with resource allocation, let C_{1} and C_{2} denote total expected costs per year as follow.(10)
We assume that the cost of sojourn per ship per year is A_{1} and that the cost incurred when a ship leaves without entering the system is A_{2}. The variable B represents the operating cost per mobile harbor unit per year. When the expected cycle time is selected as the performance measure, the cost can be calculated by using C_{1}, and when the loss probability is selected, C_{2} represents the cost. The results for the costs calculated using the information in tables 3 and 4 are presented in tables 5 and 6, respectively. For example, the numbers of the first result in table 5 show the values of C_{1} for 15 servers using 60 mobile harbor units. The cost results are shown by the ratio of A_{1} and B. When A_{1} and B are the same (A_{1}/B = 1), the total expected cost is 633863.3.
5. CONCLUDING REMARKS
In this paper, we have suggested techniques based on queueing models to evaluate a mobile harbor system. In a mobile harbor system, transportation units called mobile harbors sail back and forth between offshore operations location and land berths in order to transfer containers from ships to land berths. Because of the complexity of mobile harbor movements, traditional queueing models do not fit well. By incorporating the features of a mobile harbor system, we extend existing queueing models and obtain analytical solutions for addressing the performance measures of the system. These results can provide useful information for such a system, particularly in the design stage. We applied our techniques to the Surabaya Port in Indonesia to design an efficient mobile system.
Many opportunities remain. Here, we only consider unloading operations; however, in many cases, transportation units conduct both loading and unloading operations. Incorporating both operations will complicate our analysis and could be a future direction of our research. A more detailed control algorithm for the system could be developed instead of the FCFS assumptions we used for the queueing models. In addition, this methodology can also be applied to other transportation problems that require cargo unloading, such as those including airplanes, trains and trucks. The results are also applicable to airport railway systems and manufacturing processes. At the airport, a large number of customers arrive at the same time as the plane arrives. When the airport is large, buses and trains operate and transport customers to terminals. The results of this study can be applied to system design problems that must balance customer satisfaction with operational costs. In manufacturing systems, there are units that transport workinprocess inventories between stations. The results can be applied to the problem of estimating the number of transport units in order to efficiently transport inventories produced in various stations. By extracting the features of such systems and collecting data from actual systems, we expect to be able to extend our work.