# Choosing the right pace for your lessons (thanks to queueing theory)

Hi everyone! I often see on the forums questions about how many lessons one should do every day, or people swamped by reviews and wondering how to get on top of it, etc.

I present to you the ultimate guide to choosing your lessons rate, based on the awesome power of queueing theory.

Final edit: the post below is correct, but only assuming a few things that are not true (same accuracy at all levels, and number of of level drops when answering wrong). I now built a webtool that uses your history of reviews to gather all your accuracies by level and compute all of the math below for you: `https://castux.github.io/wanikani-stats/?key=your_key_here`. This post is left here for those curious about how it works.

# The results

The first thing youâll need to know is your accuracy, that is your average percentage of correct reviews. You can find that for instance with the WKStats website.

Then, consider the following table. All the values (except time to burn) are based on a reference lesson rate of 1 lesson per day, which means you can either:

• Multiply all the values by your actual average lesson rate, and it will give you: average number of reviews per day and average queue size (plus the detail for apprentice items specifically).
• OR: set a target for yourself (either in reviews per day, or queue size), and deduce the corresponding lesson rate.

Examples:

1. My accuracy is about 94%. Iâll worry only about the 94% line. I set myself to do about 20 lessons per day. That means Iâll have, on average: about 9.3 * 20 = 186 reviews per day (including 94 for apprentice items), and my average number of non-burned items will be 3824 (including 84 apprentice items).
2. My accuracy is 85%. I follow the rule of thumb I saw on the forums of âno more than 100 apprentice items at onceâ. At the reference rate of 1 lesson per day, the apprentice queue size is 6.1, so I deduce that my lesson rate should be no more than 100 / 6.1 = 16.4 lessons per day. Note that reading the rest of the line, I realize that this will actually mean ~203 reviews per day!
3. Maybe Iâll base my pacing on reviews instead. Still at 85% percent accuracy, I see that the reviews per day is 12.4 (for the reference 1 lesson per day). If I want to limit myself to 80 reviews per day, I deduce that my lesson rate should be 80 / 12.4 = 6.4 lessons per day.

The average time to burn does not depend on your lesson rate, only on your accuracy, so donât multiply it, you can just read it directly from the table. Notice that if your accuracy is 100% (no mistakes ever), time to burn is simply the sum of the SRS delay times: 4 month + 1 month + 2 weeks + 1 week + 2 days + 1 day + 8 hours + 4 hours = 174.5 days.

Accuracy Total reviews per day Apprentice reviews per day Total queue size Apprentice queue size Time to burn (days)
50% 178.0 150.0 679.3 109.3 679
51% 154.9 128.6 642.7 96.2 643
52% 135.4 110.8 609.6 85.0 610
53% 119.0 95.9 579.5 75.3 580
54% 105.2 83.3 552.2 66.9 552
55% 93.3 72.8 527.3 59.7 527
56% 83.2 63.8 504.4 53.4 504
57% 74.5 56.2 483.4 47.9 483
58% 67.1 49.7 464.0 43.0 464
59% 60.6 44.1 446.1 38.8 446
60% 55.0 39.3 429.5 35.1 429
61% 50.1 35.2 414.1 31.8 414
62% 45.8 31.6 399.7 29.0 400
63% 42.0 28.5 386.4 26.4 386
64% 38.7 25.8 373.8 24.1 374
65% 35.8 23.5 362.1 22.1 362
66% 33.2 21.4 351.1 20.3 351
67% 30.9 19.6 340.8 18.7 341
68% 28.8 18.0 331.0 17.3 331
69% 26.9 16.6 321.8 16.0 322
70% 25.3 15.3 313.1 14.8 313
71% 23.8 14.2 304.9 13.7 305
72% 22.4 13.2 297.1 12.8 297
73% 21.2 12.3 289.7 11.9 290
74% 20.1 11.5 282.7 11.2 283
75% 19.0 10.8 276.0 10.5 276
76% 18.1 10.2 269.6 9.8 270
77% 17.2 9.6 263.6 9.2 264
78% 16.5 9.1 257.8 8.7 258
79% 15.7 8.6 252.2 8.2 252
80% 15.1 8.2 246.9 7.8 247
81% 14.4 7.8 241.8 7.4 242
82% 13.9 7.4 237.0 7.0 237
83% 13.3 7.1 232.3 6.7 232
84% 12.8 6.8 227.8 6.3 228
85% 12.4 6.5 223.5 6.1 224
86% 12.0 6.2 219.4 5.8 219
87% 11.5 6.0 215.4 5.5 215
88% 11.2 5.8 211.5 5.3 212
89% 10.8 5.6 207.8 5.1 208
90% 10.5 5.4 204.3 4.9 204
91% 10.2 5.2 200.8 4.7 201
92% 9.9 5.0 197.5 4.5 198
93% 9.6 4.9 194.3 4.4 194
94% 9.3 4.7 191.2 4.2 191
95% 9.1 4.6 188.2 4.1 188
96% 8.8 4.4 185.3 4.0 185
97% 8.6 4.3 182.4 3.8 182
98% 8.4 4.2 179.7 3.7 180
99% 8.2 4.1 177.1 3.6 177
100% 8.0 4.0 174.5 3.5 175

Here it is in graph form (again, for the reference lesson rate of 1 lesson per day):

We can notice that review rate blows up dramatically when accuracy decreases, but that at any accuracy, most of the reviews are apprentice. That makes sense, since bad accuracy means a lot of items fall back down (or stay) in apprentice levels.

For items in the system, itâs less dramatic, but we also notice that only a small portion are apprentice items. That also makes sense, since the higher levels have so much bigger SRS delays: most items are patiently waiting in the Master or Enlightened levels.

Finally, here is a Google Docs spreadsheet you can copy and toy with: simply change the lesson rate value in the yellow box to update the table and graphs.

Notes:

1. You can, if you like, make separate computations for each type of item (radical, kanji, vocab), using the accuracy and lesson rate specifically for each item type, and then add the results together. It will be more precise.
2. I assumed that accuracy is the same at every SRS level, which might not be true, but I donât know if itâs possibly to get your accuracy split by level, and it would also make the formulas much bigger, without much benefit.
3. This also assumes that you have studied long enough, so that you already started burning items. Before your first burn, the system is âramping upâ to its full capacity, and all the real numbers will be smaller than those indicated here.
4. Last assumption: you do reviews immediately when they become available. In practice, there is usually a small delay (for instance if you review once per day only), but that would only increase the numbers by a little bit. We do assume however that when reviewing, you review all available items: if you do let items accumulate, either the system wonât be stable (see below), or it will add longer delays, which would need to be taken in account in the model.

# The math

Assumptions

In order to apply a mathematical model, I made the assumption that the system is stable, that is: no level will accumulate items indefinitely: every item that comes in will come out eventually. This has the interesting consequence that your burn rate is exactly your lesson rate.

The entire Wanikani process is modeled as series of queues (one per SRS level), between which flow items (radicals, kanjis, vocabs):

The lesson rate is denoted Î».

All the flows are Poisson processes, and all the queues are M/D/â (which roughly means that items arrive in queues are random times, but once in the queue, they start being processed immediately, and stay there for a determined amount of time: the SRS delay for that level).

I denote the queuesâ departure rates Î»1 to Î»8.

When an item exits a queue (ie. a level), it is reviewed immediately. That means that each queueâs departure rate is also that levelâs review rate. After being reviewed, the item has a probability `s` of moving to the next level (correct review), and a probability `(1-s)` of moving down max two levels (incorrect review).

Because every queue is stable (first assumption), the total arrival rate equals the departure rate, for every queue. This gives this system of equations, solved using Maxima (Î»âs are written as `l`):

The total review rate `Lt` is simply the sum of review rates of all levels. `Lt` is the review rate for apprentice items.

We notice that all these are proportional to Î», the chosen lesson rate.

For the queue sizes, we use Littleâs Formula: the average size of a stable queue is the arrival rate (same as departure rate) multiplied by the average time spent in the queue by an item. Q = Î» * T.

We know exactly the amount of time spent in a queue by an item: itâs the SRS delay for that level (written here in days), so we can solve for the average queue sizes:

``````q1 = l1 * 1/6,
q2 = l2 * 1/3,
q3 = l3 * 1,
q4 = l4 * 2,
q5 = l5 * 7,
q6 = l6 * 14,
q7 = l7 * 30,
q8 = l8 * 120
``````

For the average number of apprentice items, we simply take the sum of the first four levels, and for the total number, the sum of all levels:

``````Qa = q1 + q2 + q3 + q4,
Qt = Qa + q5 + q6 + q7 + q8
``````

These results are also proportional to Î», which is why we can simply give the values for Î» = 1 item/day, and multiply everything according to the real rate.

Finally, for the average burn time, we can apply Littleâs Formula to the entire system (which was assumed to be stable). The average amount of items in the whole system Qt equals the arrival rate Î» times the average time spent in the system T. Qt = Î» * T gives T = Qt / Î»:

``````T = Qt / l
``````

We see that this doesnât depend on Î» anymore, only on the accuracy `s`. A final note: because we assume that the system is stable, its departure rate (which is the burn rate) must be equal to the arrival rate (the lesson rate). Once the system has ramped up to capacity, youâll burn items as fast as you learn them!

Thatâs it, folks! The ultimate power of maths! Now to do these reviews

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Noice

Also

Donât judge me

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Ah, my bad, I must not be using that term correctly.

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I was just joking

I just woke up so my brain isnât processing things well enough for me to read this properly yetâŠ

But it looks awesome and thank you for posting it. ^ _ ^

Thank you for this.
I am going to spend a little time with the math later. I have been an SRS nerd for several decades, but I donât think that I have seen SRS analyzed this way. I saw some of this math (but donât remember it), back in a discreet mathematics class. I saw more of it, and remember more of it, in advanced probability.
Good stuff.

I didnât read through your whole post, so sorry if you covered this.

I donât think using the accuracy percentage from the stats site is useful for this type of calculation. The stats site tells you how many reviews you got wrong (reading or meaning separately), not how many items dropped in the SRS. The latter is much more important for determining how many reviews youâll have to do.

Iâm not sure if your calculation balances for that already, so maybe itâs not an issue. But I figured Iâd mention it.

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Aaah, you are absolutely right! The model needs the probability of an item going up a level, which is not what the site gives The numbers are correct, but not my instruction on how to get that percentageâŠ Might be able to deduce it though. I will think about this. Thanks for the tip!

OK, so itâs actually not very difficult, but it does make a slight difference.

For radicals, the radical-specific accuracy indicated on WKStats is fine, since itâs meaning only. For kanji and vocab, there are reviews for both meaning and reading, so:

The probability of a kanji/vocab to go up a level is the probability of answer both meaning and reading right, which is equal to the product of the individual probabilities. For instance my WKStats look like this:

Kanji 91.04% 98.97%
Vocab 89.01% 97.02%

So my accuracy (as defined in this post) for kanjis is 91.04 * 98.97 / 100 = 90.10%, and my accuracy for vocab is 89.01 * 97.02 / 100 = 86.36%

Then these can be used to compute the results separated by item. Or I suppose they can be combined using the relative amounts of each item type, which gives a final number.

Iâll update the post. Thanks again for noticing this!

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Tagging you because youâll love this @Naphthalene.

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No problem. By the way, youâll get the most accurate and fine-grained user data from the `/results` endpoint of the new WaniKani API, if youâre interested.

Also, you only drop 1 SRS level if you answer wrong on an Apprentice item. If Iâm reading your model right, youâve got all levels dropping by 2 SRS levels, which is only correct for Guru1 and above.

Also, you might find [this] interesting. Itâs a code snippet for gathering your accuracy at each SRS level using the `/reviews` endpoint line @seanblue mentioned. And it uses WKâs paired accuracy, rather than separate reading/meaning accuracy like wkstats. (Iâm going to add this to wkstats eventually)

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Indeed.
Iâm glad @Tenoch actually sat down and did the math, 'cause ahem some of us were too lazy to do it. (With the caveat that the queues should go back one SRS back for apprentice rather than two, as mentioned by @rfindley, and that the uniform probability s can be replaced by a queue-specific one, but I donât think it would change much in the grand scheme of things)

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Oh wow, I totally missed that aspect! Not that I donât believe you, but do you have an official source for that (or just observation?) Thanks for that tip, I can update the equations and results.

And that SRS specific accuracy script is great! Iâm starting to believe this whole thing should be come a webtool, given all the parameters that come into play

I donât think thereâs an official source because the FAQ and Guide are way out of date. But thatâs definitely how it works. (The best official source is the code.)

It goes down by:
`ceil(WA/2) * M`
Where `WA` is the number of wrong answers given for that item (meaning or reading) during the same review session and `M` is 1 if the item started in Apprentice and 2 if the item started at Guru or higher.

For example:

# of Wrong Answers Decrease in SRS if Apprentice Decrease in SRS if Guru+
1 1 2
2 1 2
3 2 4
4 2 4
5 3 6

Oh boy I guess for now I could assume 1 and 2 for simplicity. If the complete statistics are available in API v2, one might also integrate them into this (which lower level is reached, with what probability).

(This smells of a âsmall articleâ turning into a âlarge projectâ :D)

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My brain canât process that too much information. Can some great folk sum up what great ideas can I apply from this great post?

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As @seanblue mentioned, I got it from the source code:

r = (t>=5 ? 2*Math.round(o/2) : 1*Math.round(o/2))

ârâ is the number of SRS levels youâll drop.
âtâ is the current SRS level, and 5 is Guru1.
âoâ is the number of wrong answers (sum of reading and meaning).

If youâre not familiar with the ternary operator (a ? b : c), it means if a is true, then return b, otherwise return c

So, if Guru1 or higher, use:
2*Math.round(o/2)
otherwise:
1*Math.round(o/2)

Alright, I updated the post assuming 1 drop in apprentice, and 2 drops for guru+. Of course, thatâs not always true, but the accuracy parameter is already an approximation. Might be enough for getting an order of magnitude.

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If it gets turned into a web tool or something, youâll be able to input your accuracy and desired max number of reviews per day, and it will tell you approximately how many lessons to do per day to achieve that pace. Or give you a target accuracy to try to achieve in order to reduce your workload.

Without a web tool, youâll have to understand how to plug your numbers into the forumae to achieve the same result.

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