The escalator of the station [b]champion butcher[/b] has this property that if $m$ persons are on it, then it's speed is $m^{-\alpha}$ where $\alpha$ is a fixed positive real number.
Suppose that $n$ persons want to go up by the escalator and the width of the stairs is such that all the persons can stand on a stair. If the length of the escalator is $l$, what's the least time that is needed for these persons to go up? Why?

[i]proposed by Mohammad Ghiasi[/i]
1. **Understanding the Problem:**
   - The speed of the escalator is given by \( m^{-\alpha} \) where \( m \) is the number of people on it and \( \alpha \) is a fixed positive real number.
   - We need to find the least time required for \( n \) persons to go up an escalator of length \( l \).

2. **Case Analysis:**
   - We need to consider two cases based on the value of \( \alpha \):
     - Case 1: \( \alpha \ge 1 \)
     - Case 2: \( \alpha < 1 \)

3. **Case 1: \( \alpha \ge 1 \)**
   - If \( \alpha \ge 1 \), the speed of the escalator decreases rapidly as more people get on it.
   - To minimize the time, it is optimal to have people go up one by one.
   - When one person is on the escalator, the speed is \( 1^{-\alpha} = 1 \).
   - The time taken for one person to travel the length \( l \) is \( l \).
   - Since there are \( n \) people, the total time is \( n \times l \).

4. **Case 2: \( \alpha < 1 \)**
   - If \( \alpha < 1 \), the speed of the escalator does not decrease as rapidly with more people.
   - To minimize the time, it is optimal to have all \( n \) people go up simultaneously.
   - When \( n \) people are on the escalator, the speed is \( n^{-\alpha} \).
   - The time taken for all \( n \) people to travel the length \( l \) is \( \frac{l}{n^{-\alpha}} = l \cdot n^{\alpha} \).

5. **Combining the Results:**
   - For \( \alpha \ge 1 \), the total time is \( n \times l \).
   - For \( \alpha < 1 \), the total time is \( l \cdot n^{\alpha} \).
   - The least time needed is given by \( l \cdot n^{\min(\alpha, 1)} \).

The final answer is \( \boxed{l \cdot n^{\min(\alpha, 1)}} \).