Arrival time of the whole group of hikers on a narrow trail I am learning about arrival times with constraints. I've constructed a statistical problem based on a metaphor I heard about business processes. I think better understanding how to solve the problem below will help me tackle similar problems.
Imagine a finite collection of $n$ hikers start up a narrow trail of length $l$ (in meters please) that does not allow for passing, thus the order of the hikers does not change. Each hiker's speed if they were alone would be a folded normal distribution with their own mean $\mu_i$ and standard deviation $\sigma_i$, but their speed is truncated when a hiker is ahead of them due to the non-passing rule.
Question: What is distribution of the arrival time of the group in the sense of the elapsed time from when the first hiker starts up the trail to the time that the last hiker arrives at the end of the trail?

My thoughts on approaching the solution.
The first hiker is unconstrained while every other hiker's arrival time depends on the hikers in front of them. So it makes sense to first consider the first hiker's arrival time which I think will be distributed by an inverse folded normal distribution. Since a speed of zero is sometimes possible, the case $\frac{l}{0}$ will have to be considered. Perhaps by arguing it has measure zero.
Every hiker's speed after the first will be dependent on those hikers ahead of them. This dependence aside, they'll each have an arrival time distribution related to the inverse folded normal distribution. Perhaps the best way to tackle this is with the chain rule of probability where we consider the speeds that are compatible with the "no passing" constraint. Since we are imposing a total order on the hikers' positions, we can focus on keeping the speed of a given hiker consistent with only the hiker directly ahead of them while recursively applying the chain rule.
If my approach is valid, then it is just a matter of figuring out how to set up the appropriate formalisms.

Update
Assume that each hiker starts at an arbitrary time at the start of the trail with an arbitrary speed. For distribution purposes, these initial conditions have a degenerate distribution.
 A: Let the speed of hiker $i$ be modeled by a random variable $S_i$ and let their start position be a distance $P_i$ from the end of the hike, where the hikers form a set $\mathcal H$ and $i\in\mathcal H.$   Their position at time $t$ would therefore be $P_i - t S_i,$ assuming they were unimpeded, and the time needed to arrive at the goal unimpeded would be $A_i=P_i / S_i.$
By time $t\ge 0,$ a hiker will have encountered anyone whose initial position was ahead of them but whose computed unimpeded position is now behind them.  So, define
$$\mathcal{B}(t; \mathcal H, i) = \{j \in \mathcal H \mid P_j \le P_i\  \&\ P_j - tS_j \ge P_i - tS_i\}.$$
This is the subset of hikers whom $i$ has encountered through time $t.$  This bunch of hikers will reach the goal together at the last arrival time of everyone in it.  That is, hiker $i$'s arrival time now is projected to be
$$A_i(t;\mathcal H) = \max\left(A_j \mid j \in \mathcal{B}(t;\mathcal H, i)\right).$$
Obviously none of these exceeds the largest of the $A_j.$  However, it is equally obvious that at least one of them always equals the largest of the $A_j:$ namely, the hiker with the latest unimpeded arrival time.  Thus,

The problem is to find the distribution of the maximum of the $A_i, i\in\mathcal H.$

That has a standard solution derivable from basic concepts.  In the simplest case, where the $A_i$ are $n$ independent and identically distributed variables with common distribution $F,$ the distribution function of the maximum is $F^n.$  In general, though, this distribution is messy and requires numerical integration.
To see this result intuitively, ponder a space-time diagram of a hike, where position along the trail is plotted vertically (with the goal at top) and time is indicated horizontally, flowing to the right.  In this example five hikers $\mathcal H = \{a,b,c,d,e\}$ begin at the positions indicated by the triangles at the left at speeds indicated by the slopes of the rays emanating to their right, arriving at the goal at times $A_i$ indicated by the circles on the goal line at the top.  Their unimpeded progress is indicated by the solid and dotted continuations, one color per hiker.

The true progress of the hikers is indicated by the solid line segments, colored according to the speed of the slowest hiker in each group.  So, $e$ starts behind all hikers but quickly encounters the slower $d,$ forming a bunch of two hikers moving at $d$'s pace.  A little while later the fast-moving $b$ runs into slower $a,$ forming another pair of hikers moving at $a$'s pace.  This latter pair reaches the goal a little before hiker $c,$ who encountered nobody along the way.  Finally, the $e,d$ pair cross the line exactly at the time $A_d$ that was anticipated for $d$ at the outset.
The analysis with which this answer began simply amounts to pointing out that $d$ cannot arrive any later than originally anticipated.  After all, if they are ever delayed en route, then they will arrive with one or more hikers whose original arrival time was even later then $A_d,$ contradicting the characterization of $d$ as the hiker with the largest value of $A_d.$
R code to create the example.
f <- function(arrivals, speeds) {
  #
  # Compute an impeded hiker's position at time `time`.
  # (Negative positions have yet to reach the goal.)
  #
  path <- Vectorize(function(time, arrival, speed) {
    x <- speeds * (time - arrivals)   # Positions of all hikers
    x.0 <- speed * (time - arrival)   # This hiker's position
    crossed <- which(x <= x.0 & arrivals * speeds <= arrival * speed)
    min(x[crossed])                   # *Furthest* from the goal
  }, "time")
  #
  # Plot the unimpeded and impeded paths, respectively.
  #
  plot.path. <- function(arrival, speed, ...) curve(speed * (x - arrival), add = TRUE, ...)
  plot.path <- function(arrival, speed, ...) curve(path(x, arrival, speed), add = TRUE, ...)
  #
  # Plot and decorate the space-time diagram for these hikers.
  #
  colors <- hsv(seq(0, 3/4, length.out=length(arrivals)), 1, .8, alpha=2/3)
  plot(c(0, 1.1) * max(arrivals), c(-1, 0.1) * max(arrivals * speeds), type="n", bty="n",
       xlab="Time", ylab="Position", xaxt="n", yaxt="n",
       main="Space-Time Diagram of the Hikers")
  abline(h=0)                                                              # Goal line
  mtext("Goal", side=2, at=0, line=1)                                      # "Goal" label
  invisible(mapply(plot.path., arrivals, speeds, col=colors, lty=3, lwd=2))# Dotted paths
  invisible(mapply(plot.path, arrivals, speeds, col=colors, lty=1, lwd=2)) # Solid paths
  points(arrivals, rep(0, length(arrivals)), pch=21, bg=colors)            # Arrival points
  points(rep(0,length(arrivals)), -arrivals*speeds, pch=24, bg=colors)     # Departure points
  mtext(names(arrivals), side=2, at=-arrivals*speeds, line=0)              # Path labels
}
#
# Create sample data.
#
# set.seed(17)
arrivals <- signif(sort(rgamma(5, 20)), 3)
speeds <- signif(abs(rnorm(length(arrivals), 1, 3/4)) + 0.25, 3)
# # Put the last arriver near the middle at the start.
# i <- which.max(arrivals)
# speeds[i] <- 1/mean(1/speeds[-i])
names(arrivals) <- names(speeds) <- head(letters, length(arrivals))[rank(arrivals*speeds)]
f(arrivals, speeds)

