Consider an electric vehicle charging station with a single charger installed and let us observe the number of vehicles at the station at regular time intervals. Arriving vehicles at the station are lined up in the queue in the order of arrival, and only the first vehicle in the queue can be charged. In the interval between one observation and the next observation, assume that one new vehicle arrives with probability $p ( 0 < p < 1 )$, and that the vehicle charging at the head of the queue completes charging with probability $q ( 0 < q < 1 )$. Here, assume that $p$ and $q$ are constants and $p + q < 1$. The queue can accommodate $N ( N \geq 2 )$ vehicles, including the vehicle being charged at the head of the queue, and the $( N + 1 )$-th vehicle shall give up and leave the station without queuing up. The vehicle which completes charging leaves the station immediately. In the interval between one observation and the next observation, either only one or no vehicles arrive at the station and either only one or no vehicles complete charging. Moreover, assume that both arrival of new vehicle and completion of charging for the first vehicle do not occur together in any one interval. I. When there are $i ( 0 < i < N )$ vehicles in the queue, find the probability for the following condition: no new vehicle arrives and the first vehicle does not leave in the interval between one observation and the next observation. Let $\pi _ { i } ( 0 \leq i \leq N )$ be the probability that $i$ vehicles are in the queue in the steady state. II. Express the relationship between $\pi _ { i }$ and $\pi _ { i + 1 }$. Here, $i \leq N - 1$. III. Express $\pi _ { i }$ using $p , q$ and $N$. IV. Find the expected value of the number of vehicles at the station in the steady state using $p , q$ and $N$. Here, $p < q$.
\section*{Problem 6}
Consider an electric vehicle charging station with a single charger installed and let us observe the number of vehicles at the station at regular time intervals.
Arriving vehicles at the station are lined up in the queue in the order of arrival, and only the first vehicle in the queue can be charged. In the interval between one observation and the next observation, assume that one new vehicle arrives with probability $p ( 0 < p < 1 )$, and that the vehicle charging at the head of the queue completes charging with probability $q ( 0 < q < 1 )$. Here, assume that $p$ and $q$ are constants and $p + q < 1$.
The queue can accommodate $N ( N \geq 2 )$ vehicles, including the vehicle being charged at the head of the queue, and the $( N + 1 )$-th vehicle shall give up and leave the station without queuing up. The vehicle which completes charging leaves the station immediately.
In the interval between one observation and the next observation, either only one or no vehicles arrive at the station and either only one or no vehicles complete charging. Moreover, assume that both arrival of new vehicle and completion of charging for the first vehicle do not occur together in any one interval.
I. When there are $i ( 0 < i < N )$ vehicles in the queue, find the probability for the following condition: no new vehicle arrives and the first vehicle does not leave in the interval between one observation and the next observation.
Let $\pi _ { i } ( 0 \leq i \leq N )$ be the probability that $i$ vehicles are in the queue in the steady state.
II. Express the relationship between $\pi _ { i }$ and $\pi _ { i + 1 }$. Here, $i \leq N - 1$.
III. Express $\pi _ { i }$ using $p , q$ and $N$.
IV. Find the expected value of the number of vehicles at the station in the steady state using $p , q$ and $N$. Here, $p < q$.