How would you design a real-time chat system?

// Chat delivery: per-conversation sequence numbers, idempotent sends, offline sync, gap repair, group fan-out.
const fmt = (n) => n.toLocaleString("en-US");
console.log("== 1. Capacity (assumptions: 500M DAU, 40 messages/user/day, 300 bytes stored per message, RF 3) ==");
const msgsDay = 500_000_000 * 40;
console.log("messages/day " + fmt(msgsDay) + ", avg " + fmt(Math.floor(msgsDay / 86400)) + "/s, peak (x3) " + fmt(Math.floor(msgsDay / 86400) * 3) + "/s");
console.log("storage/day " + fmt(msgsDay * 300 / 1e12) + " TB raw, " + fmt(msgsDay * 300 * 3 / 1e12) + " TB with 3 replicas");
const online = 500_000_000 * 30 / 100, perGw = 500_000;
console.log("peak concurrent sockets " + fmt(online) + " (30% of DAU) -> " + fmt(online / perGw) + " gateways at " + fmt(perGw) + " sockets each");

console.log("\n== 2. Delivery simulation (store = append-only log per conversation) ==");
const store = {}, seqOf = {}, dedup = {}, registry = {}, pushes = [];
let gatewaySends = 0;
const clients = {};
const client = (u) => (clients[u] ||= { lastSeq: {}, held: {}, shown: [] });
function apply(u, conv, m) { // client side: apply in order, hold anything after a gap
  const c = client(u), want = (c.lastSeq[conv] || 0) + 1;
  if (m.seq < want) return u + " ignores duplicate seq " + m.seq;
  if (m.seq > want) { c.held[m.seq] = m; return u + " holds seq " + m.seq + " (expected " + want + ", gap)"; }
  c.shown.push(m.seq + ":" + m.text); c.lastSeq[conv] = m.seq;
  let note = u + " shows seq " + m.seq;
  while (c.held[c.lastSeq[conv] + 1]) { const h = c.held[c.lastSeq[conv] + 1]; delete c.held[h.seq]; c.shown.push(h.seq + ":" + h.text); c.lastSeq[conv] = h.seq; note += ", then held seq " + h.seq; }
  return note;
}
function send(conv, members, from, clientId, text, opts = {}) {
  const key = conv + "/" + clientId;
  if (dedup[key]) return console.log("  " + from + " retry " + clientId + " -> already stored as seq " + dedup[key] + ", nothing re-sent");
  const seq = (seqOf[conv] = (seqOf[conv] || 0) + 1);
  const m = { seq, text };
  (store[conv] ||= []).push(m); dedup[key] = seq;
  const byGw = {};
  for (const u of members) if (u !== from) {
    if (!registry[u]) { pushes.push(u + ":" + conv); continue; }
    (byGw[registry[u]] ||= []).push(u);
  }
  const notes = [];
  for (const gw of Object.keys(byGw).sort()) {
    gatewaySends++; // one publish per gateway, which then writes to each local socket
    for (const u of byGw[gw]) {
      if (opts.drop === u) notes.push(u + " (dropped in transit)");
      else if (opts.delay === u) notes.push(u + " (delayed)");
      else notes.push(apply(u, conv, m));
    }
  }
  console.log("  " + from + " " + clientId + " '" + text + "' -> seq " + seq + (opts.ackLost ? " (ack lost)" : " acked") + (notes.length ? " | " + notes.join("; ") : " | recipient offline, push queued"));
  return m;
}
const dm = ["alice", "bob"];
registry.alice = "gw1"; registry.bob = "gw2";
send("dm1", dm, "alice", "a-1", "hi");
send("dm1", dm, "alice", "a-2", "lunch?", { ackLost: true });
send("dm1", dm, "alice", "a-2", "lunch?");
console.log("  stored messages in dm1: " + store.dm1.length);
delete registry.bob; console.log("  bob goes offline");
send("dm1", dm, "alice", "a-3", "12:30?");
send("dm1", dm, "alice", "a-4", "or 13:00");
registry.bob = "gw1";
const missed = store.dm1.filter((m) => m.seq > client("bob").lastSeq.dm1);
console.log("  bob reconnects to gw1, sync(after=" + client("bob").lastSeq.dm1 + ") returns seq " + missed.map((m) => m.seq).join(","));
for (const m of missed) console.log("  " + apply("bob", "dm1", m));
const late = send("dm1", dm, "alice", "a-5", "booked", { delay: "bob" });
send("dm1", dm, "alice", "a-6", "see you");
console.log("  " + apply("bob", "dm1", late) + "   (delayed copy arrives)");
send("dm1", dm, "alice", "a-7", "bring id", { drop: "bob" });
send("dm1", dm, "alice", "a-8", "thanks");
const gapFrom = client("bob").lastSeq.dm1;
const repair = store.dm1.filter((m) => m.seq > gapFrom && !client("bob").held[m.seq]);
console.log("  bob gap timer fires: fetch after seq " + gapFrom + " -> seq " + repair.map((m) => m.seq).join(","));
for (const m of repair) console.log("  " + apply("bob", "dm1", m));
console.log("  bob screen: " + client("bob").shown.join(" | "));
console.log("  push notifications queued: " + pushes.length + " (" + pushes.join(", ") + ")");

console.log("\n== 3. Group of 6: fan-out is per gateway, not per member ==");
const group = ["alice", "bob", "carl", "dana", "eve", "fay"];
registry.carl = "gw2"; registry.dana = "gw2"; registry.eve = "gw3"; registry.fay = "gw3";
gatewaySends = 0;
send("g1", group, "alice", "a-1", "standup in 5");
console.log("  recipients 5, gateway publishes " + gatewaySends + ", stored copies " + store.g1.length);