Scenario:
Suppose we have infinite cache memory size. Caching is just limited by timeout, value of this timeout is half an hour. Cache is initially empty.
Problem:
We have 50,000 distinct request. Our system is querying, randomly, at the rate of 15 request/second i.e. 27,000 request in half an hour . What kind of curve or average value of cache hit rate could we expect for first 5 hours?
Note: This scenario is fixed. I need an approach to find out hit rate. If you think tag is wrong, please suggest appropriate tag.
I think you're right and this is a math question (certainly not a programming
problem).
One approach is to consider the extremes -- what is the hit rate for the
first query when the the system starts running? For the second query?
After one second? After 10? After a minute? And what is the likelyhood
that any random query will be found in the cache once the system has been
running a long time?
These are few specific values, and together they give you a curve.
I don't think great numeric precision is necessary; the long-term average
and the shape of the curve is more interesting.
Related
Certain sensors are to trigger a signal based on the rate of change of the value rather than a threshold.
For instance, heat detectors in fire alarms are supposed to trigger an alarm quicker if the rate of temperature rise is higher: A temperature rise of 1K/min should trigger an alarm after 30 minutes, a rise of 5K/min after 5 minutes and a rise of 30K/min after 30 seconds.
I am wondering how this is implemented in embedded systems, where resources are scares. Is there a clever data structure to minimize the data stored?
The naive approach would be to measure the temperature every 5 seconds or so and keep the data for 30 minutes. On these data one can calculate change rates over arbitrary time windows. But this requires a lot of memory.
I thought about small windows (e.g. 10 seconds) for which min and max are stored, but this would not save much memory.
From a mathematical point of view, the examples you have described can be greatly simplified:
1K/min for 30 mins equals a total change of 30K
5K/min for 5 mins equals a total change of 25K
Obviously there is some adjustment to be made because you have picked round numbers for the example, but it sounds like what you care about is having a single threshold for the total change. This makes sense because taking the integral of a differential results in just a delta.
However, if we disregard the numeric example and just focus on your original question then here are some answers:
First, it has already been mentioned in the comments that one byte every five seconds for half an hour is really not very much memory at all for almost any modern microcontroller, as long as you are able to keep your main RAM turned on between samples, which you usually can.
If however you need to discard the contents of RAM between samples to preserve battery life, then a simpler method is just to calculate one differential at a time.
In your example you want to have a much higher sample rate (every 5 seconds) than the time you wish to calculate the delta over (eg: 30 mins). You can reduce your storage needs to a single data point if you make your sample rate equal to your delta period. The single previous value could be stored in a small battery retained memory (eg: backup registers on STM32).
Obviously if you choose this approach you will have to compromise between accuracy and latency, but maybe 30 seconds would be a suitable timebase for your temperature alarm example.
You can also set several thresholds of K/sec, and then allocate counters to count how many consecutive times the each threshold has been exceeded. This requires only one extra integer per threshold.
In signal processing terms, the procedure you want to perform is:
Apply a low-pass filter to smooth quick variations in the temperature
Take the derivative of its output
The cut-off frequency of the filter would be set according to the time frame. There are 2 ways to do this.
You could apply a FIR (finite impulse response) filter, which is a weighted moving average over the time frame of interest. Naively, this requires a lot of memory, but it's not bad if you do a multi-stage decimation first to reduce your sample rate. It ends up being a little complicated, but you have fine control over the response.
You could apply in IIR (Infinite impulse response) filter, which utilizes feedback of the output. The exponential moving average is the simplest example of this. These filters require far less memory -- only a few samples' worth, but your control over the precise shape of the response is limited. A classic example like the Butterworth filter would probably be great for your application, though.
If i do a benchmark, and for example i found the following:
With 1 concurrent user, The api give 150 req/s. (9000 req/minute)
With more than 300 concurrent user, The api start throwing exception.
An app is doing request 1 every 30 minute.
Is it correct if I say:
the best cases is that the api could handle (30 * 9000 = 270.000 user). That is under 30 minute, there would be 270.000 sequential request and each are coming from different user
The worst cases would be when there is 300 user posting request at the same time.
And if it's true, would there any way to calculate the average case ?
Is is the same as calculating worst case, average case complexity of an algorithm ?
One theoretical tool to answer these questions is http://en.wikipedia.org/wiki/Queueing_theory. It says that you are very unlikely to get the level of performance that you are assuming, because the load applied to the system fluctuates, so that there are busy periods and quiet periods. If the system has nothing to do in quiet periods it is forced into idleness that you haven't accounted for. In busy periods, on the other hand, it will typically build up long queues of pending work, until the queues get so long that customers walk away, or the queues become longer than the system can support and it collapses, or both.
The graph at figure 1 page 3 of http://pages.cs.wisc.edu/~dsmyers/cs547/lecture_12_mm1_queue.pdf shows a graph of response time vs applied load for what is probably the most optimistic even vaguely realistic situation. You can see that response time gets very large as you approach maximum load.
By far the most sensible thing to do is to run tests which apply a realistic load to your application - this is important enough for people to build things like http://jmeter.apache.org/. If you want a rule of thumb I'd say don't plan to stress the system at more than 50% of theoretical capacity as you originally calculated.
I am using codahale metrics for monitoring purposes. Lets say there is a spike in latency at some point and later there are no values reported due to attribute that there are no traffic, the value in the graph stays as is(I am using a histogram). At times it gives a notion that the spike remains and we might need to address it, but it actually means that no values are reported after that and hence the graph doesn't decay. Am I missing any config parameter in this case or is the behaviour expected?
The way we update the metrics is
metrics.processingTime.update(processingTime);
So, when there is no traffic, we don't update this metric.
I know that the histogram takes into consideration datapoints from the past (for an irregular period of time) in order to display a statistical image of the data.
When there are no new datapoints, only the outlier is taken into consideration and averaged on and on.
The meters have the same behavior, displaying the data through moving averages of 1,5,15 minutes.
The solution in the histogram case is to use HDRhistogram and flush it periodically.
Background
Echo Nest have a rate limited API. A given application (identified in requests using an API key) can make up to 120 REST calls a minute. The service response includes an estimate of the total number of calls made in the last minute; repeated abuse of the API (exceeding the limit) may cause the API key to be revoked.
When used from a single machine (a web server providing a service to clients) it is easy to control access - the server has full knowledge of the history of requests and can regulate itself correctly.
But I am working on a program where distributed, independent clients make requests in parallel.
In such a case it is much less clear what an optimal solution would be. And in general the problem appears to be undecidable - if over 120 clients, all with no previous history, make an initial request at the same time, then the rate will be exceeded.
But since this is a personal project, and client use is expected to be sporadic (bursty), and my projects have never been hugely successful, that is not expected to be a huge problem. A more likely problem is that there are times when a smaller number of clients want to make many requests as quickly as possible (for example, a client may need, exceptionally, to make several thousand requests when starting for the first time - it is possible two clients would start at around the same time, so they must cooperate to share the available bandwidth).
Given all the above, what are suitable algorithms for the clients so that they rate-limit appropriately? Note that limited cooperation is possible because the API returns the total number of requests in the last minute for all clients.
Current Solution
My current solution (when the question was written - a better approach is given as an answer) is quite simple. Each client has a record of the time the last call was made and the number of calls made in the last minute, as reported by the API, on that call.
If the number of calls is less than 60 (half the limit) the client does not throttle. This allows for fast bursts of small numbers of requests.
Otherwise (ie when there are more previous requests) the client calculates the limiting rate it would need to work at (ie period = 60 / (120 - number of previous requests)) and then waits until the gap between the previous call and the current time exceeds that period (in seconds; 60 seconds in a minute; 120 max requests per minute). This effectively throttles the rate so that, if it were acting alone, it would not exceed the limit.
But the above has problems. If you think it through carefully you'll see that for large numbers of requests a single client oscillates and does not reach maximum throughput (this is partly because of the "initial burst" which will suddenly "fall outside the window" and partly because the algorithm does not make full use of its history). And multiple clients will cooperate to an extent, but I doubt that it is optimal.
Better Solutions
I can imagine a better solution that uses the full local history of the client and models other clients with, say, a Hidden Markov Model. So each client would use the API report to model the other (unknown) clients and adjust its rate accordingly.
I can also imagine an algorithm for a single client that progressively transitions from unlimited behaviour for small bursts to optimal, limited behaviour for many requests without introducing oscillations.
Do such approaches exist? Can anyone provide an implementation or reference? Can anyone think of better heuristics?
I imagine this is a known problem somewhere. In what field? Queuing theory?
I also guess (see comments earlier) that there is no optimal solution and that there may be some lore / tradition / accepted heuristic that works well in practice. I would love to know what... At the moment I am struggling to identify a similar problem in known network protocols (I imagine Perlman would have some beautiful solution if so).
I am also interested (to a lesser degree, for future reference if the program becomes popular) in a solution that requires a central server to aid collaboration.
Disclaimer
This question is not intended to be criticism of Echo Nest at all; their service and conditions of use are great. But the more I think about how best to use this, the more complex/interesting it becomes...
Also, each client has a local cache used to avoid repeating calls.
Updates
Possibly relevant paper.
The above worked, but was very noisy, and the code was a mess. I am now using a simpler approach:
Make a call
From the response, note the limit and count
Calculate
barrier = now() + 60 / max(1, (limit - count))**greedy
On the next call, wait until barrier
The idea is quite simple: that you should wait some length of time proportional to how few requests are left in that minute. For example, if count is 39 and limit is 40 then you wait an entire minute. But if count is zero then you can make a request soon. The greedy parameter is a trade-off - when greater than 1 the "first" calls are made more quickly, but you are more likely hit the limit and end up waiting for 60s.
The performance of this is similar to the approach above, and it's much more robust. It is particularly good when clients are "bursty" as the approach above gets confused trying to estimate linear rates, while this will happily let a client "steal" a few rapid requests when demand is low.
Code here.
After some experimenting, it seems that the most important thing is getting as good an estimate as possible for the upper limit of the current connection rates.
Each client can track their own (local) connection rate using a queue of timestamps. A timestamp is added to the queue on each connection and timestamps older than a minute are discarded. The "long term" (over a minute) average rate is then found from the first and last timestamps and the number of entries (minus one). The "short term" (instantaneous) rate can be found from the times of the last two requests. The upper limit is the maximum of these two values.
Each client can also estimate the external connection rate (from the other clients). The "long term" rate can be found from the number of "used" connections in the last minute, as reported by the server, corrected by the number of local connections (from the queue mentioned above). The "short term" rate can be estimated from the "used" number since the previous request (minus one, for the local connection), scaled by the time difference. Again, the upper limit (maximum of these two values) is used.
Each client computes these two rates (local and external) and then adds them to estimate the upper limit to the total rate of connections to the server. This value is compared with the target rate band, which is currently set to between 80% and 90% of the maximum (0.8 to 0.9 * 120 per minute).
From the difference between the estimated and target rates, each client modifies their own connection rate. This is done by taking the previous delta (time between the last connection and the one before) and scaling it by 1.1 (if the rate exceeds the target) or 0.9 (if the rate is lower than the target). The client then refuses to make a new connection until that scaled delta has passed (by sleeping if a new connected is requested).
Finally, nothing above forces all clients to equally share the bandwidth. So I add an additional 10% to the local rate estimate. This has the effect of preferentially over-estimating the rate for clients that have high rates, which makes them more likely to reduce their rate. In this way the "greedy" clients have a slightly stronger pressure to reduce consumption which, over the long term, appears to be sufficient to keep the distribution of resources balanced.
The important insights are:
By taking the maximum of "long term" and "short term" estimates the system is conservative (and more stable) when additional clients start up.
No client knows the total number of clients (unless it is zero or one), but all clients run the same code so can "trust" each other.
Given the above, you can't make "exact" calculations about what rate to use, but you can make a "constant" correction (in this case, +/- 10% factor) depending on the global rate.
The adjustment to the client connection frequency is made to the delta between the last two connection (adjusting based on the average over the whole minute is too slow and leads to oscillations).
Balanced consumption can be achieved by penalising the greedy clients slightly.
In (limited) experiments this works fairly well (even in the worst case of multiple clients starting at once). The main drawbacks are: (1) it doesn't allow for an initial "burst" (which would improve throughput if the server has few clients and a client has only a few requests); (2) the system does still oscillate over ~ a minute (see below); (3) handling a larger number of clients (in the worst case, eg if they all start at once) requires a larger gain (eg 20% correction instead of 10%) which tends to make the system less stable.
The "used" amount reported by the (test) server, plotted against time (Unix epoch). This is for four clients (coloured), all trying to consume as much data as possible.
The oscillations come from the usual source - corrections lag signal. They are damped by (1) using the upper limit of the rates (predicting long term rate from instantaneous value) and (2) using a target band. This is why an answer informed by someone who understand control theory would be appreciated...
It's not clear to me that estimating local and external rates separately is important (they may help if the short term rate for one is high while the long-term rate for the other is high), but I doubt removing it will improve things.
In conclusion: this is all pretty much as I expected, for this kind of approach. It kind-of works, but because it's a simple feedback-based approach it's only stable within a limited range of parameters. I don't know what alternatives might be possible.
Since you're using the Echonest API, why don't you take advantage of the rate limit headers that are returned with every API call?
In general you get 120 requests per minute. There are three headers that can help you self-regulate your API consumption:
X-Ratelimit-Used
X-Ratelimit-Remaining
X-Ratelimit-Limit
**(Notice the lower-case 'ell' in 'Ratelimit'--the documentation makes you think it should be capitalized, but in practice it is lower case.)
These counts account for calls made by other processes using your API key.
Pretty neat, huh? Well, I'm afraid there is a rub...
That 120-request-per-minute is really an upper bound. You can't count on it. The documentation states that value can fluctuate according to system load. I've seen it as low as 40ish in some calls I've made, and have in some cases seen it go below zero (I really hope that was a bug in the echonest API!)
One approach you can take is to slow things down once utilization (used divided by limit) reaches a certain threshold. Keep in mind though, that on the next call your limit may have been adjusted download significantly enough that 'used' is greater than 'limit'.
This works well up until a point. Since the Echonest doesn't adjust the limit in a predictable mannar, it is hard to avoid 400s in practice.
Here are some links that I've found helpful:
http://blog.echonest.com/post/15242456852/managing-your-api-rate-limit
http://developer.echonest.com/docs/v4/#rate-limits
I am trying to spread out data that is received in bursts. This means I have data that is received by some other application in large bursts. For each data entry I need to do some additional requests on some server, at which I should limit the traffic. Hence I try to spread up the requests in the time that I have until the next data burst arrives.
Currently I am using a token-bucket to spread out the data. However because the data I receive is already badly shaped I am still either filling up the queue of pending request, or I get spikes whenever a bursts comes in. So this algorithm does not seem to do the kind of shaping I need.
What other algorithms are there available to limit the requests? I know I have times of high load and times of low load, so both should be handled well by the application.
I am not sure if I was really able to explain the problem I am currently having. If you need any clarifications, just let me know.
EDIT:
I'll try to clarify the problem some more and explain, why a simple rate limiter does not work.
The problem lies in the bursty nature of the traffic and the fact, that burst have a different size at different times. What is mostly constant is the delay between each burst. Thus we get a bunch of data records for processing and we need to spread them out as evenly as possible before the next bunch comes in. However we are not 100% sure when the next bunch will come in, just aproximately, so a simple divide time by number of records does not work as it should.
A rate limiting does not work, because the spread of the data is not sufficient this way. If we are close to saturation of the rate, everything is fine, and we spread out evenly (although this should not happen to frequently). If we are below the threshold, the spreading gets much worse though.
I'll make an example to make this problem more clear:
Let's say we limit our traffic to 10 requests per seconds and new data comes in about every 10 seconds.
When we get 100 records at the beginning of a time frame, we will query 10 records each second and we have a perfect even spread. However if we get only 15 records we'll have one second where we query 10 records, one second where we query 5 records and 8 seconds where we query 0 records, so we have very unequal levels of traffic over time. Instead it would be better if we just queried 1.5 records each second. However setting this rate would also make problems, since new data might arrive earlier, so we do not have the full 10 seconds and 1.5 queries would not be enough. If we use a token bucket, the problem actually gets even worse, because token-buckets allow bursts to get through at the beginning of the time-frame.
However this example over simplifies, because actually we cannot fully tell the number of pending requests at any given moment, but just an upper limit. So we would have to throttle each time based on this number.
This sounds like a problem within the domain of control theory. Specifically, I'm thinking a PID controller might work.
A first crack at the problem might be dividing the number of records by the estimated time until next batch. This would be like a P controller - proportional only. But then you run the risk of overestimating the time, and building up some unsent records. So try adding in an I term - integral - to account for built up error.
I'm not sure you even need a derivative term, if the variation in batch size is random. So try using a PI loop - you might build up some backlog between bursts, but it will be handled by the I term.
If it's unacceptable to have a backlog, then the solution might be more complicated...
If there are no other constraints, what you should do is figure out the maximum data rate that you are comfortable with sending additional requests, and limit your processing speed according to that. Then monitor what is happening. If that gets through all of your requests quickly, then there is no harm . If its sustained level of processing is not fast enough, then you need more capacity.