# Resource Capacity Requirement for Multi-Terminal Cooperation in Container Ports
## 2 Resource Profiles for Representing Workloads
### Workload Distributions  
The random variables \(C_{a}\), \(C_{b}\), \(C_{c}\), \(C_{d}\), \(C_{a}^{f},C_{b}^{f}\), and \(C_{c}^{f}\) change the workload of the resources over time-shifts. The set of randomness of the random variables, described in the above subsection, also generate probability distributions for requesting workload for storage and retrieval operations as conceptually depicted in the Figure 3.  
Figure 2: Simplified illustration of resource profiles for unloading (**left**) and loading (**right**) operations for a feeder vessel. The cumulative conditional workload distribution for TEU arrivals before the mainline vessel arrival, associated with the random variable \(C_{a}\), is  
\[W_{arrival}\big{(}t_{open}\leq t\leq t_{close}\big{|}\ T_{arrival}=t_{arrival} \big{)}=\sum\
olimits_{i=t_{open}}^{t}E[C_{a}]P[C_{a}|T\leq i], \tag{1}\]  
where \(T_{arrival}\) is a random variable with mild variation and \(P[C_{a}|T\leq i]\) is the cumulative distribution of outbound TEU arrivals from \(t_{open}\). The cumulative workload distribution is conditioned by \(T_{arrival}\) as \(t_{open}\) and \(t_{close}\) are determined by the realization of \(T_{arrival}\). The cumulative conditional workload distribution of the unloading operation is  
\[ \begin{array}{l}W_{unload}(T_{switch}-(T_{arrival}+t_{unload})\leq t\ |\ T_{arrival}=t_{arrival})=\\ \sum\
olimits_{i=T_{arrival}+t_{unload}}^{t}E[C_{b}]P[C_{b}|T\leq i]=min\Big{(} (t-i)H_{Q}^{U},E[C_{b}]\Big{)},\end{array}  \tag{2}\]  
where \(H_{Q}^{U}\) is the random variable for the QC gross productivity of a vessel during the unloading operation measured by the amount of TEUs handled per time unit. \(P[C_{b}|T\leq i]\) is replaced with the time-independent unloading rate. The gross productivity is dependent on the QC intensity, which in turn depends on the vessel size and the TEU volume. Hence, \(H_{Q}^{U}=H_{Q}^{U}I_{Q}\), where \(h_{Q}^{U}\) is the gross productivity distribution of a QC for a vessel for the  
Figure 3: Illustrated TEU workload distributions for storage space requirement. unloading operation, which can be realized by the number of moves per time unit. For S being a random variable with a small deviation for the vessel category, \(I_{Q}=F_{(C_{k}+C_{s},S)}\) is the QC intensity distribution for the vessel operation that varies with the amount of TEUs and the vessel size as summarized in Appendix A. Accordingly, the cumulative conditional workload distribution for the loading operation is given as,  
\[ \begin{split} W_{load}(T_{end}-T_{load}\leq t\mid T_{load}=t_{ load})=\sum_{i=T_{load}}^{t}E[C_{c}]P[C_{c}|T\leq i]=\\ min\Big{(}(t-i)H_{Q^{\prime}}^{L},E[C_{c}]\Big{)},\end{split}  \tag{3}\]  
where \(H_{Q}^{L}=h_{Q}^{L}I_{Q}\). \(h_{Q}^{L}\) is the gross productivity distribution for the loading operations. Note that the workload for the loading operation take a positive effect on reducing the storage space requirement but it requires handling effort to be performed. As for the transferring operations from a terminal to another, the cumulative conditional workload distribution is  
\[ \begin{split} W_{transfer}\Big{(}T_{finish}-T_{transfer}\leq t \mid T_{transfer}=t_{transfer}\Big{)}=\\ \sum_{i=T_{transfer}}^{t}E[C_{d}]P[C_{d}|T\leq i],\end{split}  \tag{4}\]  
where \(P[C_{d}|T\leq i]\) represents the transferring productivity of trucks that transport TEUs out to other terminals including the traffic congestion and the gate process. \(T_{transfer}\) should correspond to the \(t_{open}\) for another vessel incoming to another terminal (Terminal 2 in Figure 1). The same goes for the workload distributions \(C_{s}^{f}\), \(C_{b}^{f}\) and \(C_{c}^{f}\) of the feeder vessels in Figure 2. The cumulative conditional workload distribution for the unloading operation of a feeder vessel is  
\[