Add notes describing Synapse's streams (#16015)

Co-authored-by: Patrick Cloke <clokep@users.noreply.github.com>

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diff --git a/changelog.d/16015.doc b/changelog.d/16015.doc
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+Add a internal documentation page describing the ["streams" used within Synapse](https://matrix-org.github.io/synapse/v1.90/development/synapse_architecture/streams.html).
diff --git a/docs/SUMMARY.md b/docs/SUMMARY.md
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--- a/docs/SUMMARY.md
+++ b/docs/SUMMARY.md
@@ -97,6 +97,7 @@
     - [Cancellation](development/synapse_architecture/cancellation.md)
     - [Log Contexts](log_contexts.md)
     - [Replication](replication.md)
+    - [Streams](development/synapse_architecture/streams.md)
     - [TCP Replication](tcp_replication.md)
     - [Faster remote joins](development/synapse_architecture/faster_joins.md)
   - [Internal Documentation](development/internal_documentation/README.md)
diff --git a/docs/development/synapse_architecture/streams.md b/docs/development/synapse_architecture/streams.md
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+## Streams
+
+Synapse has a concept of "streams", which are roughly described in [`id_generators.py`](
+    https://github.com/matrix-org/synapse/blob/develop/synapse/storage/util/id_generators.py
+).
+Generally speaking, streams are a series of notifications that something in Synapse's database has changed that the application might need to respond to.
+For example:
+
+- The events stream reports new events (PDUs) that Synapse creates, or that Synapse accepts from another homeserver.
+- The account data stream reports changes to users' [account data](https://spec.matrix.org/v1.7/client-server-api/#client-config).
+- The to-device stream reports when a device has a new [to-device message](https://spec.matrix.org/v1.7/client-server-api/#send-to-device-messaging).
+
+See [`synapse.replication.tcp.streams`](
+    https://github.com/matrix-org/synapse/blob/develop/synapse/replication/tcp/streams/__init__.py
+) for the full list of streams.
+
+It is very helpful to understand the streams mechanism when working on any part of Synapse that needs to respond to changes—especially if those changes are made by different workers.
+To that end, let's describe streams formally, paraphrasing from the docstring of [`AbstractStreamIdGenerator`](
+    https://github.com/matrix-org/synapse/blob/a719b703d9bd0dade2565ddcad0e2f3a7a9d4c37/synapse/storage/util/id_generators.py#L96
+).
+
+### Definition
+
+A stream is an append-only log `T1, T2, ..., Tn, ...` of facts[^1] which grows over time.
+Only "writers" can add facts to a stream, and there may be multiple writers.
+
+Each fact has an ID, called its "stream ID".
+Readers should only process facts in ascending stream ID order.
+
+Roughly speaking, each stream is backed by a database table.
+It should have a `stream_id` (or similar) bigint column holding stream IDs, plus additional columns as necessary to describe the fact.
+Typically, a fact is expressed with a single row in its backing table.[^2]
+Within a stream, no two facts may have the same stream_id.
+
+> _Aside_. Some additional notes on streams' backing tables.
+>
+> 1. Rich would like to [ditch the backing tables](https://github.com/matrix-org/synapse/issues/13456).
+> 2. The backing tables may have other uses.
+     >    For example, the events table serves backs the events stream, and is read when processing new events.
+     >    But old rows are read from the table all the time, whenever Synapse needs to lookup some facts about an event.
+> 3. Rich suspects that sometimes the stream is backed by multiple tables, so the stream proper is the union of those tables.
+
+Stream writers can "reserve" a stream ID, and then later mark it as having being completed.
+Stream writers need to track the completion of each stream fact.
+In the happy case, completion means a fact has been written to the stream table.
+But unhappy cases (e.g. transaction rollback due to an error) also count as completion.
+Once completed, the rows written with that stream ID are fixed, and no new rows
+will be inserted with that ID.
+
+### Current stream ID
+
+For any given stream reader (including writers themselves), we may define a per-writer current stream ID:
+
+> The current stream ID _for a writer W_ is the largest stream ID such that
+> all transactions added by W with equal or smaller ID have completed.
+
+Similarly, there is a "linear" notion of current stream ID:
+
+> The "linear" current stream ID is the largest stream ID such that
+> all facts (added by any writer) with equal or smaller ID have completed.
+
+Because different stream readers A and B learn about new facts at different times, A and B may disagree about current stream IDs.
+Put differently: we should think of stream readers as being independent of each other, proceeding through a stream of facts at different rates.
+
+**NB.** For both senses of "current", that if a writer opens a transaction that never completes, the current stream ID will never advance beyond that writer's last written stream ID.
+
+For single-writer streams, the per-writer current ID and the linear current ID are the same.
+Both senses of current ID are monotonic, but they may "skip" or jump over IDs because facts complete out of order.
+
+
+_Example_.
+Consider a single-writer stream which is initially at ID 1.
+
+| Action     | Current stream ID | Notes                                           |
+|------------|-------------------|-------------------------------------------------|
+|            | 1                 |                                                 |
+| Reserve 2  | 1                 |                                                 |
+| Reserve 3  | 1                 |                                                 |
+| Complete 3 | 1                 | current ID unchanged, waiting for 2 to complete |
+| Complete 2 | 3                 | current ID jumps from 1 -> 3                    |
+| Reserve 4  | 3                 |                                                 |
+| Reserve 5  | 3                 |                                                 |
+| Reserve 6  | 3                 |                                                 |
+| Complete 5 | 3                 |                                                 |
+| Complete 4 | 5                 | current ID jumps 3->5, even though 6 is pending |
+| Complete 6 | 6                 |                                                 |
+
+
+### Multi-writer streams
+
+There are two ways to view a multi-writer stream.
+
+1. Treat it as a collection of distinct single-writer streams, one
+   for each writer.
+2. Treat it as a single stream.
+
+The single stream (option 2) is conceptually simpler, and easier to represent (a single stream id).
+However, it requires each reader to know about the entire set of writers, to ensures that readers don't erroneously advance their current stream position too early and miss a fact from an unknown writer.
+In contrast, multiple parallel streams (option 1) are more complex, requiring more state to represent (map from writer to stream id).
+The payoff for doing so is that readers can "peek" ahead to facts that completed on one writer no matter the state of the others, reducing latency.
+
+Note that a multi-writer stream can be viewed in both ways.
+For example, the events stream is treated as multiple single-writer streams (option 1) by the sync handler, so that events are sent to clients as soon as possible.
+But the background process that works through events treats them as a single linear stream.
+
+Another useful example is the cache invalidation stream.
+The facts this stream holds are instructions to "you should now invalidate these cache entries".
+We only ever treat this as a multiple single-writer streams as there is no important ordering between cache invalidations.
+(Invalidations are self-contained facts; and the invalidations commute/are idempotent).
+
+### Writing to streams
+
+Writers need to track:
+ - track their current position (i.e. its own per-writer stream ID).
+ - their facts currently awaiting completion.
+
+At startup, 
+ - the current position of that writer can be found by querying the database (which suggests that facts need to be written to the database atomically, in a transaction); and
+ - there are no facts awaiting completion.
+
+To reserve a stream ID, call [`nextval`](https://www.postgresql.org/docs/current/functions-sequence.html) on the appropriate postgres sequence.
+
+To write a fact to the stream: insert the appropriate rows to the appropriate backing table.
+
+To complete a fact, first remove it from your map of facts currently awaiting completion.
+Then, if no earlier fact is awaiting completion, the writer can advance its current position in that stream.
+Upon doing so it should emit an `RDATA` message[^3], once for every fact between the old and the new stream ID.
+
+### Subscribing to streams
+
+Readers need to track the current position of every writer.
+
+At startup, they can find this by contacting each writer with a `REPLICATE` message,
+requesting that all writers reply describing their current position in their streams.
+Writers reply with a `POSITION` message.
+
+To learn about new facts, readers should listen for `RDATA` messages and process them to respond to the new fact.
+The `RDATA` itself is not a self-contained representation of the fact;
+readers will have to query the stream tables for the full details.
+Readers must also advance their record of the writer's current position for that stream.
+
+# Summary
+
+In a nutshell: we have an append-only log with a "buffer/scratchpad" at the end where we have to wait for the sequence to be linear and contiguous.
+
+
+---
+
+[^1]: we use the word _fact_ here for two reasons.
+Firstly, the word "event" is already heavily overloaded (PDUs, EDUs, account data, ...) and we don't need to make that worse.
+Secondly, "fact" emphasises that the things we append to a stream cannot change after the fact.
+
+[^2]: A fact might be expressed with 0 rows, e.g. if we opened a transaction to persist an event, but failed and rolled the transaction back before marking the fact as completed.
+In principle a fact might be expressed with 2 or more rows; if so, each of those rows should share the fact's stream ID.
+
+[^3]: This communication used to happen directly with the writers [over TCP](../../tcp_replication.md);
+nowadays it's done via Redis's Pubsub.