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+/*
+ * Copyright (C) 2012 The Guava Authors
+ *
+ * Licensed under the Apache License, Version 2.0 (the "License");
+ * you may not use this file except in compliance with the License.
+ * You may obtain a copy of the License at
+ *
+ * http://www.apache.org/licenses/LICENSE-2.0
+ *
+ * Unless required by applicable law or agreed to in writing, software
+ * distributed under the License is distributed on an "AS IS" BASIS,
+ * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+ * See the License for the specific language governing permissions and
+ * limitations under the License.
+ */
+
+package com.google.common.util.concurrent;
+
+import com.google.common.annotations.Beta;
+import com.google.common.annotations.VisibleForTesting;
+import com.google.common.base.Preconditions;
+import com.google.common.base.Ticker;
+
+import java.util.concurrent.TimeUnit;
+
+import javax.annotation.concurrent.ThreadSafe;
+
+/**
+ * A rate limiter. Conceptually, a rate limiter distributes permits at a
+ * configurable rate. Each {@link #acquire()} blocks if necessary until a permit is
+ * available, and then takes it. Once acquired, permits need not be released.
+ *
+ * <p>Rate limiters are often used to restrict the rate at which some
+ * physical or logical resource is accessed. This is in contrast to {@link
+ * java.util.concurrent.Semaphore} which restricts the number of concurrent
+ * accesses instead of the rate (note though that concurrency and rate are closely related,
+ * e.g. see <a href="http://en.wikipedia.org/wiki/Little's_law">Little's Law</a>).
+ *
+ * <p>A {@code RateLimiter} is defined primarily by the rate at which permits
+ * are issued. Absent additional configuration, permits will be distributed at a
+ * fixed rate, defined in terms of permits per second. Permits will be distributed
+ * smoothly, with the delay between individual permits being adjusted to ensure
+ * that the configured rate is maintained.
+ *
+ * <p>It is possible to configure a {@code RateLimiter} to have a warmup
+ * period during which time the permits issued each second steadily increases until
+ * it hits the stable rate.
+ *
+ * <p>As an example, imagine that we have a list of tasks to execute, but we don't want to
+ * submit more than 2 per second:
+ *<pre>  {@code
+ *  final RateLimiter rateLimiter = RateLimiter.create(2.0); // rate is "2 permits per second"
+ *  void submitTasks(List<Runnable> tasks, Executor executor) {
+ *    for (Runnable task : tasks) {
+ *      rateLimiter.acquire(); // may wait
+ *      executor.execute(task);
+ *    }
+ *  }
+ *}</pre>
+ *
+ * <p>As another example, imagine that we produce a stream of data, and we want to cap it
+ * at 5kb per second. This could be accomplished by requiring a permit per byte, and specifying
+ * a rate of 5000 permits per second:
+ *<pre>  {@code
+ *  final RateLimiter rateLimiter = RateLimiter.create(5000.0); // rate = 5000 permits per second
+ *  void submitPacket(byte[] packet) {
+ *    rateLimiter.acquire(packet.length);
+ *    networkService.send(packet);
+ *  }
+ *}</pre>
+ *
+ * <p>It is important to note that the number of permits requested <i>never</i>
+ * affect the throttling of the request itself (an invocation to {@code acquire(1)}
+ * and an invocation to {@code acquire(1000)} will result in exactly the same throttling, if any),
+ * but it affects the throttling of the <i>next</i> request. I.e., if an expensive task
+ * arrives at an idle RateLimiter, it will be granted immediately, but it is the <i>next</i>
+ * request that will experience extra throttling, thus paying for the cost of the expensive
+ * task.
+ *
+ * <p>Note: {@code RateLimiter} does not provide fairness guarantees.
+ *
+ * @author Dimitris Andreou
+ * @since 13.0
+ */
+// TODO(user): switch to nano precision. A natural unit of cost is "bytes", and a micro precision
+//     would mean a maximum rate of "1MB/s", which might be small in some cases.
+@ThreadSafe
+@Beta
+public abstract class RateLimiter {
+  /*
+   * How is the RateLimiter designed, and why?
+   *
+   * The primary feature of a RateLimiter is its "stable rate", the maximum rate that
+   * is should allow at normal conditions. This is enforced by "throttling" incoming
+   * requests as needed, i.e. compute, for an incoming request, the appropriate throttle time,
+   * and make the calling thread wait as much.
+   *
+   * The simplest way to maintain a rate of QPS is to keep the timestamp of the last
+   * granted request, and ensure that (1/QPS) seconds have elapsed since then. For example,
+   * for a rate of QPS=5 (5 tokens per second), if we ensure that a request isn't granted
+   * earlier than 200ms after the the last one, then we achieve the intended rate.
+   * If a request comes and the last request was granted only 100ms ago, then we wait for
+   * another 100ms. At this rate, serving 15 fresh permits (i.e. for an acquire(15) request)
+   * naturally takes 3 seconds.
+   *
+   * It is important to realize that such a RateLimiter has a very superficial memory
+   * of the past: it only remembers the last request. What if the RateLimiter was unused for
+   * a long period of time, then a request arrived and was immediately granted?
+   * This RateLimiter would immediately forget about that past underutilization. This may
+   * result in either underutilization or overflow, depending on the real world consequences
+   * of not using the expected rate.
+   *
+   * Past underutilization could mean that excess resources are available. Then, the RateLimiter
+   * should speed up for a while, to take advantage of these resources. This is important
+   * when the rate is applied to networking (limiting bandwidth), where past underutilization
+   * typically translates to "almost empty buffers", which can be filled immediately.
+   *
+   * On the other hand, past underutilization could mean that "the server responsible for
+   * handling the request has become less ready for future requests", i.e. its caches become
+   * stale, and requests become more likely to trigger expensive operations (a more extreme
+   * case of this example is when a server has just booted, and it is mostly busy with getting
+   * itself up to speed).
+   *
+   * To deal with such scenarios, we add an extra dimension, that of "past underutilization",
+   * modeled by "storedPermits" variable. This variable is zero when there is no
+   * underutilization, and it can grow up to maxStoredPermits, for sufficiently large
+   * underutilization. So, the requested permits, by an invocation acquire(permits),
+   * are served from:
+   * - stored permits (if available)
+   * - fresh permits (for any remaining permits)
+   *
+   * How this works is best explained with an example:
+   *
+   * For a RateLimiter that produces 1 token per second, every second
+   * that goes by with the RateLimiter being unused, we increase storedPermits by 1.
+   * Say we leave the RateLimiter unused for 10 seconds (i.e., we expected a request at time
+   * X, but we are at time X + 10 seconds before a request actually arrives; this is
+   * also related to the point made in the last paragraph), thus storedPermits
+   * becomes 10.0 (assuming maxStoredPermits >= 10.0). At that point, a request of acquire(3)
+   * arrives. We serve this request out of storedPermits, and reduce that to 7.0 (how this is
+   * translated to throttling time is discussed later). Immediately after, assume that an
+   * acquire(10) request arriving. We serve the request partly from storedPermits,
+   * using all the remaining 7.0 permits, and the remaining 3.0, we serve them by fresh permits
+   * produced by the rate limiter.
+   *
+   * We already know how much time it takes to serve 3 fresh permits: if the rate is
+   * "1 token per second", then this will take 3 seconds. But what does it mean to serve 7
+   * stored permits? As explained above, there is no unique answer. If we are primarily
+   * interested to deal with underutilization, then we want stored permits to be given out
+   * /faster/ than fresh ones, because underutilization = free resources for the taking.
+   * If we are primarily interested to deal with overflow, then stored permits could
+   * be given out /slower/ than fresh ones. Thus, we require a (different in each case)
+   * function that translates storedPermits to throtting time.
+   *
+   * This role is played by storedPermitsToWaitTime(double storedPermits, double permitsToTake).
+   * The underlying model is a continuous function mapping storedPermits
+   * (from 0.0 to maxStoredPermits) onto the 1/rate (i.e. intervals) that is effective at the given
+   * storedPermits. "storedPermits" essentially measure unused time; we spend unused time
+   * buying/storing permits. Rate is "permits / time", thus "1 / rate = time / permits".
+   * Thus, "1/rate" (time / permits) times "permits" gives time, i.e., integrals on this
+   * function (which is what storedPermitsToWaitTime() computes) correspond to minimum intervals
+   * between subsequent requests, for the specified number of requested permits.
+   *
+   * Here is an example of storedPermitsToWaitTime:
+   * If storedPermits == 10.0, and we want 3 permits, we take them from storedPermits,
+   * reducing them to 7.0, and compute the throttling for these as a call to
+   * storedPermitsToWaitTime(storedPermits = 10.0, permitsToTake = 3.0), which will
+   * evaluate the integral of the function from 7.0 to 10.0.
+   *
+   * Using integrals guarantees that the effect of a single acquire(3) is equivalent
+   * to { acquire(1); acquire(1); acquire(1); }, or { acquire(2); acquire(1); }, etc,
+   * since the integral of the function in [7.0, 10.0] is equivalent to the sum of the
+   * integrals of [7.0, 8.0], [8.0, 9.0], [9.0, 10.0] (and so on), no matter
+   * what the function is. This guarantees that we handle correctly requests of varying weight
+   * (permits), /no matter/ what the actual function is - so we can tweak the latter freely.
+   * (The only requirement, obviously, is that we can compute its integrals).
+   *
+   * Note well that if, for this function, we chose a horizontal line, at height of exactly
+   * (1/QPS), then the effect of the function is non-existent: we serve storedPermits at
+   * exactly the same cost as fresh ones (1/QPS is the cost for each). We use this trick later.
+   *
+   * If we pick a function that goes /below/ that horizontal line, it means that we reduce
+   * the area of the function, thus time. Thus, the RateLimiter becomes /faster/ after a
+   * period of underutilization. If, on the other hand, we pick a function that
+   * goes /above/ that horizontal line, then it means that the area (time) is increased,
+   * thus storedPermits are more costly than fresh permits, thus the RateLimiter becomes
+   * /slower/ after a period of underutilization.
+   *
+   * Last, but not least: consider a RateLimiter with rate of 1 permit per second, currently
+   * completely unused, and an expensive acquire(100) request comes. It would be nonsensical
+   * to just wait for 100 seconds, and /then/ start the actual task. Why wait without doing
+   * anything? A much better approach is to /allow/ the request right away (as if it was an
+   * acquire(1) request instead), and postpone /subsequent/ requests as needed. In this version,
+   * we allow starting the task immediately, and postpone by 100 seconds future requests,
+   * thus we allow for work to get done in the meantime instead of waiting idly.
+   *
+   * This has important consequences: it means that the RateLimiter doesn't remember the time
+   * of the _last_ request, but it remembers the (expected) time of the _next_ request. This
+   * also enables us to tell immediately (see tryAcquire(timeout)) whether a particular
+   * timeout is enough to get us to the point of the next scheduling time, since we always
+   * maintain that. And what we mean by "an unused RateLimiter" is also defined by that
+   * notion: when we observe that the "expected arrival time of the next request" is actually
+   * in the past, then the difference (now - past) is the amount of time that the RateLimiter
+   * was formally unused, and it is that amount of time which we translate to storedPermits.
+   * (We increase storedPermits with the amount of permits that would have been produced
+   * in that idle time). So, if rate == 1 permit per second, and arrivals come exactly
+   * one second after the previous, then storedPermits is _never_ increased -- we would only
+   * increase it for arrivals _later_ than the expected one second.
+   */
+
+  /**
+   * Creates a {@code RateLimiter} with the specified stable throughput, given as
+   * "permits per second" (commonly referred to as <i>QPS</i>, queries per second).
+   *
+   * <p>The returned {@code RateLimiter} ensures that on average no more than {@code
+   * permitsPerSecond} are issued during any given second, with sustained requests
+   * being smoothly spread over each second. When the incoming request rate exceeds
+   * {@code permitsPerSecond} the rate limiter will release one permit every {@code
+   * (1.0 / permitsPerSecond)} seconds. When the rate limiter is unused,
+   * bursts of up to {@code permitsPerSecond} permits will be allowed, with subsequent
+   * requests being smoothly limited at the stable rate of {@code permitsPerSecond}.
+   *
+   * @param permitsPerSecond the rate of the returned {@code RateLimiter}, measured in
+   *        how many permits become available per second.
+   */
+  public static RateLimiter create(double permitsPerSecond) {
+    return create(SleepingTicker.SYSTEM_TICKER, permitsPerSecond);
+  }
+
+  @VisibleForTesting
+  static RateLimiter create(SleepingTicker ticker, double permitsPerSecond) {
+    RateLimiter rateLimiter = new Bursty(ticker);
+    rateLimiter.setRate(permitsPerSecond);
+    return rateLimiter;
+  }
+
+  /**
+   * Creates a {@code RateLimiter} with the specified stable throughput, given as
+   * "permits per second" (commonly referred to as <i>QPS</i>, queries per second), and a
+   * <i>warmup period</i>, during which the {@code RateLimiter} smoothly ramps up its rate,
+   * until it reaches its maximum rate at the end of the period (as long as there are enough
+   * requests to saturate it). Similarly, if the {@code RateLimiter} is left <i>unused</i> for
+   * a duration of {@code warmupPeriod}, it will gradually return to its "cold" state,
+   * i.e. it will go through the same warming up process as when it was first created.
+   *
+   * <p>The returned {@code RateLimiter} is intended for cases where the resource that actually
+   * fulfils the requests (e.g., a remote server) needs "warmup" time, rather than
+   * being immediately accessed at the stable (maximum) rate.
+   *
+   * <p>The returned {@code RateLimiter} starts in a "cold" state (i.e. the warmup period
+   * will follow), and if it is left unused for long enough, it will return to that state.
+   *
+   * @param permitsPerSecond the rate of the returned {@code RateLimiter}, measured in
+   *        how many permits become available per second
+   * @param warmupPeriod the duration of the period where the {@code RateLimiter} ramps up its
+   *        rate, before reaching its stable (maximum) rate
+   * @param unit the time unit of the warmupPeriod argument
+   */
+  // TODO(user): add a burst size of 1-second-worth of permits, as in the metronome?
+  public static RateLimiter create(double permitsPerSecond, long warmupPeriod, TimeUnit unit) {
+    return create(SleepingTicker.SYSTEM_TICKER, permitsPerSecond, warmupPeriod, unit);
+  }
+
+  @VisibleForTesting
+  static RateLimiter create(
+      SleepingTicker ticker, double permitsPerSecond, long warmupPeriod, TimeUnit timeUnit) {
+    RateLimiter rateLimiter = new WarmingUp(ticker, warmupPeriod, timeUnit);
+    rateLimiter.setRate(permitsPerSecond);
+    return rateLimiter;
+  }
+
+  @VisibleForTesting
+  static RateLimiter createBursty(
+      SleepingTicker ticker, double permitsPerSecond, int maxBurstSize) {
+    Bursty rateLimiter = new Bursty(ticker);
+    rateLimiter.setRate(permitsPerSecond);
+    rateLimiter.maxPermits = maxBurstSize;
+    return rateLimiter;
+  }
+
+  /**
+   * The underlying timer; used both to measure elapsed time and sleep as necessary. A separate
+   * object to facilitate testing.
+   */
+  private final SleepingTicker ticker;
+
+  /**
+   * The timestamp when the RateLimiter was created; used to avoid possible overflow/time-wrapping
+   * errors.
+   */
+  private final long offsetNanos;
+
+  /**
+   * The currently stored permits.
+   */
+  double storedPermits;
+
+  /**
+   * The maximum number of stored permits.
+   */
+  double maxPermits;
+
+  /**
+   * The interval between two unit requests, at our stable rate. E.g., a stable rate of 5 permits
+   * per second has a stable interval of 200ms.
+   */
+  double stableIntervalMicros;
+
+  /**
+   * The time when the next request (no matter its size) will be granted. After granting a request,
+   * this is pushed further in the future. Large requests push this further than small requests.
+   */
+  private long nextFreeTicketMicros = 0L; // could be either in the past or future
+
+  private RateLimiter(SleepingTicker ticker) {
+    this.ticker = ticker;
+    this.offsetNanos = ticker.read();
+  }
+
+  /**
+   * Updates the stable rate of this {@code RateLimiter}, that is, the
+   * {@code permitsPerSecond} argument provided in the factory method that
+   * constructed the {@code RateLimiter}. Currently throttled threads will <b>not</b>
+   * be awakened as a result of this invocation, thus they do not observe the new rate;
+   * only subsequent requests will.
+   *
+   * <p>Note though that, since each request repays (by waiting, if necessary) the cost
+   * of the <i>previous</i> request, this means that the very next request
+   * after an invocation to {@code setRate} will not be affected by the new rate;
+   * it will pay the cost of the previous request, which is in terms of the previous rate.
+   *
+   * <p>The behavior of the {@code RateLimiter} is not modified in any other way,
+   * e.g. if the {@code RateLimiter} was configured with a warmup period of 20 seconds,
+   * it still has a warmup period of 20 seconds after this method invocation.
+   *
+   * @param permitsPerSecond the new stable rate of this {@code RateLimiter}.
+   */
+  public final synchronized void setRate(double permitsPerSecond) {
+    Preconditions.checkArgument(permitsPerSecond > 0.0
+        && !Double.isNaN(permitsPerSecond), "rate must be positive");
+    resync(readSafeMicros());
+    double stableIntervalMicros = TimeUnit.SECONDS.toMicros(1L) / permitsPerSecond;
+    this.stableIntervalMicros =  stableIntervalMicros;
+    doSetRate(permitsPerSecond, stableIntervalMicros);
+  }
+
+  abstract void doSetRate(double permitsPerSecond, double stableIntervalMicros);
+
+  /**
+   * Returns the stable rate (as {@code permits per seconds}) with which this
+   * {@code RateLimiter} is configured with. The initial value of this is the same as
+   * the {@code permitsPerSecond} argument passed in the factory method that produced
+   * this {@code RateLimiter}, and it is only updated after invocations
+   * to {@linkplain #setRate}.
+   */
+  public final synchronized double getRate() {
+    return TimeUnit.SECONDS.toMicros(1L) / stableIntervalMicros;
+  }
+
+  /**
+   * Acquires a permit from this {@code RateLimiter}, blocking until the request can be granted.
+   *
+   * <p>This method is equivalent to {@code acquire(1)}.
+   */
+  public void acquire() {
+    acquire(1);
+  }
+
+  /**
+   * Acquires the given number of permits from this {@code RateLimiter}, blocking until the
+   * request be granted.
+   *
+   * @param permits the number of permits to acquire
+   */
+  public void acquire(int permits) {
+    checkPermits(permits);
+    long microsToWait;
+    synchronized (this) {
+      microsToWait = reserveNextTicket(permits, readSafeMicros());
+    }
+    ticker.sleepMicrosUninterruptibly(microsToWait);
+  }
+
+  /**
+   * Acquires a permit from this {@code RateLimiter} if it can be obtained
+   * without exceeding the specified {@code timeout}, or returns {@code false}
+   * immediately (without waiting) if the permit would not have been granted
+   * before the timeout expired.
+   *
+   * <p>This method is equivalent to {@code tryAcquire(1, timeout, unit)}.
+   *
+   * @param timeout the maximum time to wait for the permit
+   * @param unit the time unit of the timeout argument
+   * @return {@code true} if the permit was acquired, {@code false} otherwise
+   */
+  public boolean tryAcquire(long timeout, TimeUnit unit) {
+    return tryAcquire(1, timeout, unit);
+  }
+
+  /**
+   * Acquires the given number of permits from this {@code RateLimiter} if it can be obtained
+   * without exceeding the specified {@code timeout}, or returns {@code false}
+   * immediately (without waiting) if the permits would not have been granted
+   * before the timeout expired.
+   *
+   * @param permits the number of permits to acquire
+   * @param timeout the maximum time to wait for the permits
+   * @param unit the time unit of the timeout argument
+   * @return {@code true} if the permits were acquired, {@code false} otherwise
+   */
+  public boolean tryAcquire(int permits, long timeout, TimeUnit unit) {
+    checkPermits(permits);
+    long timeoutMicros = unit.toMicros(timeout);
+    long microsToWait;
+    synchronized (this) {
+      long nowMicros = readSafeMicros();
+      if (nextFreeTicketMicros > nowMicros + timeoutMicros) {
+        return false;
+      } else {
+        microsToWait = reserveNextTicket(permits, nowMicros);
+      }
+    }
+    ticker.sleepMicrosUninterruptibly(microsToWait);
+    return true;
+  }
+
+  private static void checkPermits(int permits) {
+    Preconditions.checkArgument(permits > 0, "Requested permits must be positive");
+  }
+
+  /**
+   * Reserves next ticket and returns the wait time that the caller must wait for.
+   */
+  private long reserveNextTicket(double requiredPermits, long nowMicros) {
+    resync(nowMicros);
+    long microsToNextFreeTicket = nextFreeTicketMicros - nowMicros;
+    double storedPermitsToSpend = Math.min(requiredPermits, this.storedPermits);
+    double freshPermits = requiredPermits - storedPermitsToSpend;
+
+    long waitMicros = storedPermitsToWaitTime(this.storedPermits, storedPermitsToSpend)
+        + (long) (freshPermits * stableIntervalMicros);
+
+    this.nextFreeTicketMicros = nextFreeTicketMicros + waitMicros;
+    this.storedPermits -= storedPermitsToSpend;
+    return microsToNextFreeTicket;
+  }
+
+  /**
+   * Translates a specified portion of our currently stored permits which we want to
+   * spend/acquire, into a throttling time. Conceptually, this evaluates the integral
+   * of the underlying function we use, for the range of
+   * [(storedPermits - permitsToTake), storedPermits].
+   *
+   * This always holds: {@code 0 <= permitsToTake <= storedPermits}
+   */
+  abstract long storedPermitsToWaitTime(double storedPermits, double permitsToTake);
+
+  private void resync(long nowMicros) {
+    // if nextFreeTicket is in the past, resync to now
+    if (nowMicros > nextFreeTicketMicros) {
+      storedPermits = Math.min(maxPermits,
+          storedPermits + (nowMicros - nextFreeTicketMicros) / stableIntervalMicros);
+      nextFreeTicketMicros = nowMicros;
+    }
+  }
+
+  private long readSafeMicros() {
+    return TimeUnit.NANOSECONDS.toMicros(ticker.read() - offsetNanos);
+  }
+
+  @Override
+  public String toString() {
+    return String.format("RateLimiter[stableRate=%3.1fqps]", 1000000.0 / stableIntervalMicros);
+  }
+
+  /**
+   * This implements the following function:
+   *
+   *          ^ throttling
+   *          |
+   * 3*stable +                  /
+   * interval |                 /.
+   *  (cold)  |                / .
+   *          |               /  .   <-- "warmup period" is the area of the trapezoid between
+   * 2*stable +              /   .       halfPermits and maxPermits
+   * interval |             /    .
+   *          |            /     .
+   *          |           /      .
+   *   stable +----------/  WARM . }
+   * interval |          .   UP  . } <-- this rectangle (from 0 to maxPermits, and
+   *          |          . PERIOD. }     height == stableInterval) defines the cooldown period,
+   *          |          .       . }     and we want cooldownPeriod == warmupPeriod
+   *          |---------------------------------> storedPermits
+   *              (halfPermits) (maxPermits)
+   *
+   * Before going into the details of this particular function, let's keep in mind the basics:
+   * 1) The state of the RateLimiter (storedPermits) is a vertical line in this figure.
+   * 2) When the RateLimiter is not used, this goes right (up to maxPermits)
+   * 3) When the RateLimiter is used, this goes left (down to zero), since if we have storedPermits,
+   *    we serve from those first
+   * 4) When _unused_, we go right at the same speed (rate)! I.e., if our rate is
+   *    2 permits per second, and 3 unused seconds pass, we will always save 6 permits
+   *    (no matter what our initial position was), up to maxPermits.
+   *    If we invert the rate, we get the "stableInterval" (interval between two requests
+   *    in a perfectly spaced out sequence of requests of the given rate). Thus, if you
+   *    want to see "how much time it will take to go from X storedPermits to X+K storedPermits?",
+   *    the answer is always stableInterval * K. In the same example, for 2 permits per second,
+   *    stableInterval is 500ms. Thus to go from X storedPermits to X+6 storedPermits, we
+   *    require 6 * 500ms = 3 seconds.
+   *
+   *    In short, the time it takes to move to the right (save K permits) is equal to the
+   *    rectangle of width == K and height == stableInterval.
+   * 4) When _used_, the time it takes, as explained in the introductory class note, is
+   *    equal to the integral of our function, between X permits and X-K permits, assuming
+   *    we want to spend K saved permits.
+   *
+   *    In summary, the time it takes to move to the left (spend K permits), is equal to the
+   *    area of the function of width == K.
+   *
+   * Let's dive into this function now:
+   *
+   * When we have storedPermits <= halfPermits (the left portion of the function), then
+   * we spend them at the exact same rate that
+   * fresh permits would be generated anyway (that rate is 1/stableInterval). We size
+   * this area to be equal to _half_ the specified warmup period. Why we need this?
+   * And why half? We'll explain shortly below (after explaining the second part).
+   *
+   * Stored permits that are beyond halfPermits, are mapped to an ascending line, that goes
+   * from stableInterval to 3 * stableInterval. The average height for that part is
+   * 2 * stableInterval, and is sized appropriately to have an area _equal_ to the
+   * specified warmup period. Thus, by point (4) above, it takes "warmupPeriod" amount of time
+   * to go from maxPermits to halfPermits.
+   *
+   * BUT, by point (3) above, it only takes "warmupPeriod / 2" amount of time to return back
+   * to maxPermits, from halfPermits! (Because the trapezoid has double the area of the rectangle
+   * of height stableInterval and equivalent width). We decided that the "cooldown period"
+   * time should be equivalent to "warmup period", thus a fully saturated RateLimiter
+   * (with zero stored permits, serving only fresh ones) can go to a fully unsaturated
+   * (with storedPermits == maxPermits) in the same amount of time it takes for a fully
+   * unsaturated RateLimiter to return to the stableInterval -- which happens in halfPermits,
+   * since beyond that point, we use a horizontal line of "stableInterval" height, simulating
+   * the regular rate.
+   *
+   * Thus, we have figured all dimensions of this shape, to give all the desired
+   * properties:
+   * - the width is warmupPeriod / stableInterval, to make cooldownPeriod == warmupPeriod
+   * - the slope starts at the middle, and goes from stableInterval to 3*stableInterval so
+   *   to have halfPermits being spend in double the usual time (half the rate), while their
+   *   respective rate is steadily ramping up
+   */
+  private static class WarmingUp extends RateLimiter {
+
+    final long warmupPeriodMicros;
+    /**
+     * The slope of the line from the stable interval (when permits == 0), to the cold interval
+     * (when permits == maxPermits)
+     */
+    private double slope;
+    private double halfPermits;
+
+    WarmingUp(SleepingTicker ticker, long warmupPeriod, TimeUnit timeUnit) {
+      super(ticker);
+      this.warmupPeriodMicros = timeUnit.toMicros(warmupPeriod);
+    }
+
+    @Override
+    void doSetRate(double permitsPerSecond, double stableIntervalMicros) {
+      double oldMaxPermits = maxPermits;
+      maxPermits = warmupPeriodMicros / stableIntervalMicros;
+      halfPermits = maxPermits / 2.0;
+      // Stable interval is x, cold is 3x, so on average it's 2x. Double the time -> halve the rate
+      double coldIntervalMicros = stableIntervalMicros * 3.0;
+      slope = (coldIntervalMicros - stableIntervalMicros) / halfPermits;
+      if (oldMaxPermits == Double.POSITIVE_INFINITY) {
+        // if we don't special-case this, we would get storedPermits == NaN, below
+        storedPermits = 0.0;
+      } else {
+        storedPermits = (oldMaxPermits == 0.0)
+            ? maxPermits // initial state is cold
+            : storedPermits * maxPermits / oldMaxPermits;
+      }
+    }
+
+    @Override
+    long storedPermitsToWaitTime(double storedPermits, double permitsToTake) {
+      double availablePermitsAboveHalf = storedPermits - halfPermits;
+      long micros = 0;
+      // measuring the integral on the right part of the function (the climbing line)
+      if (availablePermitsAboveHalf > 0.0) {
+        double permitsAboveHalfToTake = Math.min(availablePermitsAboveHalf, permitsToTake);
+        micros = (long) (permitsAboveHalfToTake * (permitsToTime(availablePermitsAboveHalf)
+            + permitsToTime(availablePermitsAboveHalf - permitsAboveHalfToTake)) / 2.0);
+        permitsToTake -= permitsAboveHalfToTake;
+      }
+      // measuring the integral on the left part of the function (the horizontal line)
+      micros += (stableIntervalMicros * permitsToTake);
+      return micros;
+    }
+
+    private double permitsToTime(double permits) {
+      return stableIntervalMicros + permits * slope;
+    }
+  }
+
+  /**
+   * This implements a trivial function, where storedPermits are translated to
+   * zero throttling - thus, a client gets an infinite speedup for permits acquired out
+   * of the storedPermits pool. This is also used for the special case of the "metronome",
+   * where the width of the function is also zero; maxStoredPermits is zero, thus
+   * storedPermits and permitsToTake are always zero as well. Such a RateLimiter can
+   * not save permits when unused, thus all permits it serves are fresh, using the
+   * designated rate.
+   */
+  private static class Bursty extends RateLimiter {
+    Bursty(SleepingTicker ticker) {
+      super(ticker);
+    }
+
+    @Override
+    void doSetRate(double permitsPerSecond, double stableIntervalMicros) {
+      double oldMaxPermits = this.maxPermits;
+      /*
+       * We allow the equivalent work of up to one second to be granted with zero waiting, if the
+       * rate limiter has been unused for as much. This is to avoid potentially producing tiny
+       * wait interval between subsequent requests for sufficiently large rates, which would
+       * unnecessarily overconstrain the thread scheduler.
+       */
+      maxPermits = permitsPerSecond; // one second worth of permits
+      storedPermits = (oldMaxPermits == 0.0)
+          ? 0.0 // initial state
+          : storedPermits * maxPermits / oldMaxPermits;
+    }
+
+    @Override
+    long storedPermitsToWaitTime(double storedPermits, double permitsToTake) {
+      return 0L;
+    }
+  }
+
+  @VisibleForTesting
+  static abstract class SleepingTicker extends Ticker {
+    abstract void sleepMicrosUninterruptibly(long micros);
+
+    static final SleepingTicker SYSTEM_TICKER = new SleepingTicker() {
+      @Override
+      public long read() {
+        return systemTicker().read();
+      }
+
+      @Override
+      public void sleepMicrosUninterruptibly(long micros) {
+        if (micros > 0) {
+          Uninterruptibles.sleepUninterruptibly(micros, TimeUnit.MICROSECONDS);
+        }
+      }
+    };
+  }
+}