Yesterday, I took a day trip to Palisades Tahoe for the fresh powder—along with a lot of other people who had the same idea. Naturally, the lines were longer than usual.

Palisades uses two lanes to fill each lift chair: one for groups, and a separate “singles” lane that moves up to fill any leftover seats. My girlfriend and I usually use the singles lane (even though we’re not actually single) because it’s often faster.

This time it felt like the singles lane was lagging behind the group lane, which got me thinking more carefully about how each line works. To figure out which option might be better in the future, I decided to do a little math.

Setup and assumptions

Let’s establish some of the notation and assumptions:

1. Lift capacity: ( $c$ ).
   • How many people a single chair can seat, for example: $c = 4$ or $c = 6$.
2. Average group size: ( $g$ ).
   • For example: If most groups are 2–3 people, we might estimate $g \approx 2.5$.
   • We assume each “group” that shows up at the front of the group line has on average $g$ people.
3. Two separate queues:
   • Group line: People who wish to ride together.
   • Singles line: Individual riders (or couples/groups who decide to split up and act as singles).
4. How chairs fill:
   • If there is a group waiting, the lift operator takes exactly one group from the group line to fill seats (to not overcomplicate the math at first).
   • If the group has fewer than $c$ people, then any leftover seats are filled from the singles line.
5. Goal: Decide which line (group or singles) minimizes your waiting time if you are standing there in real time, observing:
   • $X_g =$ the number of people ahead of you in the group line.
   • $X_s =$ the number of singles ahead of you in the singles line.

This simplified picture reflects many ski resorts (such as Palisades Tahoe) where each chair is always filled to capacity (because leftover seats are “topped off” with singles).

Approximate Rate Analysis

A key simplification is to imagine that, on average, each chair that arrives takes:

• One group of size $g$ from the group line.
• The remaining $(c - g)$ seats from the singles line.

This is, of course, a simplification. In reality, multiple small groups might share a single chair, or the group line might be temporarily empty, etc. But if both lines are long and groups come in typical sizes near g, this is a decent first approximation.

Service Rates

1. Chairs arrive at a rate $\mu$ (chairs per unit time). For instance, if one chair arrives every 8 seconds, $\mu = 1 / 8\ \text{chairs/sec}$.
2. Effective service rate for the Group line ($\lambda_g$): Each arriving chair loads one “average group” of size $g$. $\lambda_g = g\mu \quad (\text{people per unit time})$
3. Effective service rate for the Singles line ($\lambda_s$): Each arriving chair (after loading a group) has $(c - g)$ leftover seats for singles. $\lambda_s = (c - g)\mu$

Waiting Times

If you see:

• $X_g$ people ahead of you in the Group line,
• $X_s$ people ahead of you in the Singles line,

then a rough formula for expected waiting time in each line is:

Group‐line wait: $W_{G} \approx \frac{X_g}{g\mu}$ Singles‐line wait: $W_{S} \approx \frac{X_s}{(c - g)\mu}$

You (and your companions) should choose whichever lane has a shorter W.

Compare the Two Lines

To decide which line is faster for you, compare $W_G$ and $W_S$.

If $W_S < W_G \quad\Longleftrightarrow\quad \frac{X_s}{(c-g)\mu} < \frac{X_g}{g\mu}\quad\Longleftrightarrow\quad X_s g < X_g (c - g)$,

then Singles is faster. Otherwise, the Group line is faster.

Hence a quick rule of thumb, if $g$ (avg group size) and $c$ (chair capacity) are known:

Review of the Simplified “One Group per Chair” Model

Previously, we assumed that one group loads per chair. In many resorts (including Palisades Tahoe), the lift operator will often combine multiple smaller groups on the same chair until there are no more seats—or until the next group in line does not fit. That can significantly increase the effective throughput of the group line.

How Chairs Actually Fill

In reality, with a capacity‐c chair and a group queue of many small groups, the operator (or a “merge” attendant) loads as many entire groups from the front of the group line as can fit on the remaining seats:

• Start with $c$ empty seats.
• Take the first group in line if it fits; reduce seat count accordingly.
• Take the next group if it also fits, and so on, until you cannot fit the next group without splitting them.
• Any leftover seats (that the next group would not fit into) are made available to the singles line.

For example, if $c = 5$ and the first two groups in line are size 2 and 3, they both ride in one chair (2 + 3). No singles get on that chair. Or if the line were 2, 2, and 1, then all three groups (2 + 2 + 1) share one chair (filling exactly 5 seats).

When multiple small groups can pack themselves efficiently into the same chair, the group line ends up occupying more than just “g seats per chair” on average. In other words, the group line’s throughput is higher than the simple “one group per chair” formula of $g\mu$.

Effective Group Throughput in the “Packing” Model

To see why the group line throughput rises, imagine loading many chairs in a row. Let:

• $L =$ the (random) number of “leftover” seats wasted on a typical chair because the next group in line is too large to fit. Those leftover seats go to singles.
• $\mathbb{E}[L] =$ the average leftover seats per chair.

Then, on average, the group line is using $c - \mathbb{E}[L]$ seats per chair. Since chairs arrive at rate $\mu$, the effective people‐per‐unit‐time flowing out of the group line is approximately $(c - \mathbb{E}[L])\mu$

Meanwhile, the singles line (on average) is getting $\mathbb{E}[L]$ seats per chair (leftover seats). So their throughput is roughly $\mathbb{E}[L]\mu$.

Hence, in steady‐state, we can think of there being some fraction $\alpha$ (between 0 and 1) of each chair allocated to the group line, with the remaining fraction $1-\alpha$ going to singles:

$\text{Let } \alpha = \frac{c - \mathbb{E}[L]}{c}$. Then:

• Group‐line service rate $\approx \alpha c \mu$.
• Singles‐line service rate $\approx (1 - \alpha) c \mu$.

Once you know $\alpha$, the approximate waiting‐time formulas become:

$W_G \approx \frac{X_g}{\alpha c \mu} \qquad W_S \approx \frac{X_s}{(1-\alpha) c \mu}$.

And you choose Singles if and only if $\frac{X_s}{(1-\alpha) c} < \frac{X_g}{\alpha c} \quad\Longleftrightarrow\quad X_s \alpha < X_g (1-\alpha)$.

Estimating $\alpha$

Unfortunately, $\mathbb{E}[L]$ (the average leftover) depends on the distribution of group sizes, not just the mean $g$. If almost all groups are of size 2 and $c=6$, then typically you can squeeze 2–2–2 into each chair, leaving very little leftover. That leads to a high $\alpha$. On the other hand, if many groups are big (3–5 people each) relative to $c=6$, it becomes more common that you cannot fit a second (or third) group, so you leave more leftover seats to singles, lowering $\alpha$.

In practice, if you see that the operator is often combining multiple small groups on the same chair, you can expect the group line’s throughput to be significantly better than the naive “one group per chair” assumption. Conversely, if the line is dominated by large groups near capacity c, you may be closer to the “one group per chair” scenario.

Practical interpretation

When both queues are long, here’s a rule of thumb in real time:

1. Estimate how often multiple small groups can be seated on the same chair. If you commonly see 2–2–1 or 3–2 combos loading on a 6‐seater, then the group line is “packing efficiently,” so its effective throughput is better than the naive “g seats per chair” guess.
2. Look at the actual queue lengths:
   • $X_g$: people in the group line.
   • $X_s$: people in singles line.
3. Apply whichever version of the formula (one‐group approximation vs. multi‐group “packing” approximation) seems closer to what the lift operator is actually doing. That tells you whether your personal wait time would be shorter in the group line or in singles.

In short, if your group is willing to split up, you should still often use the Singles lane—especially if many large groups are arriving (leading to big leftover seat counts). But on certain powder days (where a ton of pairs/trios fill each chair perfectly), the group line can move faster than expected, and you might do just as well or better by staying as a group.