This section analyzes the performance of the LB scheme system and investigates its characteristics. Initially, the size of the EV waiting space is classified as either small, medium, or large sizes, with

$K=5$ for small,

$K=10$ for medium, and

$K=30$ for large sizes. The operating method of the system is classified into three modes according to the battery-supply interval and the EVs’ interarrival time. In Mode 1, the battery-supply interval is

${T}_{1}$ and EVs’ interarrival time is

${\lambda}_{1}$. In Mode 3, these parameters are

${T}_{2}$ and

${\lambda}_{2}$, respectively. Mode 2 varies depending on the policy: The battery-supply interval is

${T}_{1}$ and the interarrival time of the EVs is

${\lambda}_{2}$ for the “AF” policy and

${T}_{2}$ and

${\lambda}_{1}$ for the “SF” policy. Numerical examples were created assuming the low, medium, and high levels, which were determined based upon

${\lambda}_{1}\times {T}_{1}$ for when the system started. However, because of the battery-supply interval and interarrival time of EVs change according to the control policy, the aforementioned offered load was unsuitable for expressing the overall characteristics of the system and was used only for simple classification. In

Section 4.1 the performance of the system is examined under various conditions, and in

Section 4.2 the cost structure is defined, and the economic system operation is investigated using cost analysis.

#### 4.1. System Performance

In this section, we investigate the system performance using varying parameters. Initially,

$({T}_{1},{T}_{2})=(0.125,0.1)$ and

$({\lambda}_{1},{\lambda}_{2})=(3,1)$ were assumed for the low traffic circumstance, that is, the initial offered load is

${\lambda}_{1}\times {T}_{1}=0.375$. In the low loading situation, the congestion control is not necessary because EVs wait in the system.

Table 1 shows the probability that the BSCS system operates in Mode 1 according to the combination of

$({L}_{1},{L}_{2})$ when

$K=10$ and

$M=3$. ‘SF’ and ‘AF’ denote the “Supply First” and “Arrival First” policy, respectively. In the case of low loading, the system is not congested, there are very few examples of exceeding the threshold

${L}_{1}$, and the probability of residing in Mode 1 is very high. The waiting time and EVs’ queue size are negligible because their values are close to 0. Furthermore, in this situation, these values are small and insignificant for a comparison of the two policies proposed in this study. When

M is larger than 3, there is almost no case of an EV waiting and the proportion of time during which the system remains in Modes 2 or 3 is zero. When the loading is low, the number of batteries in the battery queue exceeds 90% of its capacity. For every combination of

$({L}_{1},{L}_{2})$, the average number of batteries in the queue is approximately 2.7 when

$M=3$, 6.7 when

$M=7$, and 9.7 when

$M=10$. While, there are waiting EVs when

$M=3$, the average is less than 0.02 and is negligible.

Next, the mid-size case of

$K=10$ is examined. When

$({T}_{1},{T}_{2})=(0.167,0.125)$ and

$({\lambda}_{1},{\lambda}_{2})=(3,1)$, the initial load

${\lambda}_{1}\times {T}_{1}$ is approximately 0.5. In the case of mid-level traffic, the probability of EV blocking is close to zero as well. This means that the EVs are not blocked and almost all of them can receive service.

Figure 2 shows the mean number of EVs (

${L}_{EV}$) and batteries (

${L}_{B}$) waiting according to the change of

$({L}_{1},{L}_{2})$. In mid traffic, the number of EVs is observed but the value is small. The mean EV queue length value is higher for the “SF” policy when

${L}_{1}=0$. When

${L}_{1}=0$, the “AF” policy operates with

$({T}_{1},{\lambda}_{2})$ and the “SF” policy operates with

$({T}_{2},{\lambda}_{1})$, meaning that controlling the interarrival time with high volatility is more effective than shortening the battery-supply interval.

Figure 3 shows a comparison of EV waiting time between the cases of

$M=3$ and

$M=7$ where

$K=10,({T}_{1},{T}_{2})=(0.125,0.1)$ and

$({\lambda}_{1},{\lambda}_{2})=(5,3)$. It is observed that, with an increase in

M, there is a reduction in the EV waiting time. Evidently, this is because batteries can be immediately replaced if the number of completely charged and waiting batteries increases. The increase of the

M value implies an increase of in the EV processing capability; this is helpful for decreasing the mean EV queue length and the blocking probability. However, in a situation where the EV queue is completely filled, the queue length and the blocking probability cannot be drastically decreased regardless of the increase in the

M value; this is because a battery is installed in the EV as soon as it is supplied. As observed for the mean EV queue length, the “AF” policy shows relatively better performance when

${L}_{1}=0$. In the case of a large system with the mid-level traffic queue size of 30, EV blocking did not occur, and the EV queue size was zero. For the battery queue size, batteries were always added to exceed the capacity setup of 95% of

M. When

$M=10$ and

$M=20$, on average 9.5 and 19.5 batteries were ready in the battery queue. While there were some differences in system performance between the two policies operating with the mid-sized traffic load, they were not significant.

The performance difference between the two policies occurs in a high traffic situation. For high traffic, we assumed cases in which the initial offered load

${\lambda}_{1}\times {T}_{1}$ is 0.8 or higher. Because this study considers a finite system, a situation in which

${\lambda}_{1}\times {T}_{1}>1$ can be assumed. Initially,

Figure 4a shows the mean EV queue length, confirming that this value increases as the

${L}_{2}$ value increases for the same

${L}_{1}$ value. For

Figure 4a,

$M=10,K=3,({T}_{1},{T}_{2})=(0.125,0.1)$, and

$({\lambda}_{1},{\lambda}_{2})=(7,5)$ are assumed. Furthermore, in the case of “SF” policy applied to a BSCS, when the volatility of EV-arrival is relatively high, it is confirmed that the number of waiting EVs is larger than that with the “AF” policy. Large

${L}_{1}$ or

${L}_{2}$ values imply that the congestion control policy is applied when the number of waiting EVs is large. This result, in which the initial load

${\lambda}_{1}\times {T}_{1}$ is 0.875, is similar to the low and medium traffic cases. However, if the initial offered load is increased to 1.25, where

$M=10,K=7,({T}_{1},{T}_{2})=(0.25,0.125)$ and

$({\lambda}_{1},{\lambda}_{2})=(7,5)$ are applied, the mean EV queue length is reversed:

Figure 4b shows that the queue length for the “AF” policy becomes larger than that seen with “SF” policy. This means that the effect of controlling the arrival interval is small if the EVs are severely queued up. Moreover, as a peculiar phenomenon, when the “AF” policy is applied first, the mean queue EV queue length increases more sharply than does that for the “SF” policy.

Figure 5 shows the blocking probability for the case of

$K=30,M=10,({T}_{1},{T}_{2}=0.25,0.125)$, and

$({\lambda}_{1},{\lambda}_{2})=(9,7)$. When

${L}_{1}$ and

${L}_{2}$ increase, it is confirmed that the increased range of blocking probability is larger with the “AF” policy as compared to the case of “SF” (where severe queued-up is observed).

#### 4.2. Cost Analysis of the Operation of a Battery Swapping and Charging Station

The economic system operation method is examined using cost analysis. Initially, the total cost per unit time for each policy is defined in Equation (

23), and detailed terms are listed in

Table 2. Blocking cost

${P}_{Block}$ refers to a cost that occurs when an EV leaves a BSCS without receiving service. While this is not an incurred cost, it is considered as an opportunity cost arising from losing a customer who is dissatisfied with the service. EV and battery holding costs, denoted by

$Cos{t}_{Holding,EV}$ and

$Cos{t}_{Holding,Battery}$, are incurred when there is an EV or a battery in the system and this includes the costs of providing and maintaining the physical space. In particular, EV holding cost also includes the cost for customer dissatisfaction resulting from the EV waiting time, and it is assumed that this cost increases with an increase in waiting time. Note that we assume that the blocking and two types of holding costs are not dependent upon congestion control policies. The operating cost

$Cos{t}_{Operating,\phantom{\rule{3.33333pt}{0ex}}i}$ includes the maintenance cost, e.g., electricity and manpower, which allow the system to operate. A higher cost is likely incurred if a congestion control policy is applied by, for example, diminishing the battery-supply interval; this is because more resource (electrical power) is available. It is assumed that higher costs occur in the Mode 2 of the “SF” policy relative to those of the “AF” policy. This assumes that more resource is available for reducing the supply interval than for supporting bypassing efforts.

The total costs of the two policies are similar when loading is of low or medium level because the performance measures are similar. For a high loading case, we assume that

$K=30,M=10,({T}_{1},{T}_{2})=(0.25,0.125)$ and

$({\lambda}_{1},{\lambda}_{2})=(7,5)$. Total costs are depicted in

Figure 6a, which shows that total cost of the “AF” policy increases in accordance with the increase in the EV blocking probability (

${P}_{EV}$) and is greater than that of “SF” due to the blocking cost. Moreover, the total cost of the “AF” policy increases sharply since the blocking probability increases rapidly as the

${L}_{2}$ value increases for the same

${L}_{1}$ value. It is observed that when medium loading is assumed (

$K=10,M=3,({T}_{1},{T}_{2})=(0.167,0.125)$ and

$({\lambda}_{1},{\lambda}_{2})=(3,1)$), the total costs tend to decrease. This is because the probability that the BSCS is operating in Mode 1 increases. In other words, when the BSCS is not overly crowded, the length of the EV queue rarely exceeds the value

${L}_{1}$ and

${L}_{2}$ given that

${L}_{1}$ and

${L}_{2}$ are large.