I have updated the algorithm to handle an arbitrary number of replicas and arbitrary constraints. Notebook: https://github.com/plafl/notebooks/blob/master/replication.ipynb PDF: https://github.com/plafl/notebooks/blob/master/converted/replication.pdf (Note: GitHub's renderization of the notebook and the PDF is quite deficient, I recommend downloading/cloning) In the following by policy I mean the concrete set of probabilities of selecting the first replica, the second replica, etc... In practical terms there are several problems: - It's not practical for a high number of disks or replicas. Possible solution: approximate summation over all possible disk selections with a Monte Carlo method. the algorithm would be: we start with a candidate solution, we run a simulation and based on the results we update the probabilities. Repeat until we are happy with the result. Other solution: cluster similar disks together. - Since it's a non-linear optimization problem I'm not sure right now about it's convergence properties. Does it converge to a global optimum? How fast does it converge? Possible solution: the algorithm always converges, but it can converge to a locally optimum policy. I see no escape except by carefully designing the policy. All solutions to the problem are going to be non linear since we must condition current probabilities on previous disk selections. - Although it can handle arbitrary constraints it does so by rejecting disks selections that violate at least one constraint. This means that for bad policies it can spend all the time rejecting invalid disks selection candidates. Possible solution: the policy cannot be designed independently of the constraints. I don't know what constraints are typical use cases but having a look should be the first step. The constraints must be an input to the policy. I hope it's of some use. Quite frankly I'm not a ceph user, I just found the problem an interesting puzzle. Anyway I will try to have a look at the CRUSH paper this weekend. 2017-02-13 15:21 GMT+01:00 Sage Weil <sweil@xxxxxxxxxx>: > On Mon, 13 Feb 2017, Loic Dachary wrote: >> Hi, >> >> Dan van der Ster reached out to colleagues and friends and Pedro >> López-Adeva Fernández-Layos came up with a well written analysis of the >> problem and a tentative solution which he described at : >> https://github.com/plafl/notebooks/blob/master/replication.ipynb >> >> Unless I'm reading the document incorrectly (very possible ;) it also >> means that the probability of each disk needs to take in account the >> weight of all disks. Which means that whenever a disk is added / removed >> or its weight is changed, this has an impact on the probability of all >> disks in the cluster and objects are likely to move everywhere. Am I >> mistaken ? > > Maybe (I haven't looked closely at the above yet). But for comparison, in > the normal straw2 case, adding or removing a disk also changes the > probabilities for everything else (e.g., removing one out of 10 identical > disks changes the probability from 1/10 to 1/9). The key property that > straw2 *is* able to handle is that as long as the relative probabilities > between two unmodified disks does not change, then straw2 will avoid > moving any objects between them (i.e., all data movement is to or from > the disk that is reweighted). > > sage > > >> >> Cheers >> >> On 01/26/2017 04:05 AM, Sage Weil wrote: >> > This is a longstanding bug, >> > >> > http://tracker.ceph.com/issues/15653 >> > >> > that causes low-weighted devices to get more data than they should. Loic's >> > recent activity resurrected discussion on the original PR >> > >> > https://github.com/ceph/ceph/pull/10218 >> > >> > but since it's closed and almost nobody will see it I'm moving the >> > discussion here. >> > >> > The main news is that I have a simple adjustment for the weights that >> > works (almost perfectly) for the 2nd round of placements. The solution is >> > pretty simple, although as with most probabilities it tends to make my >> > brain hurt. >> > >> > The idea is that, on the second round, the original weight for the small >> > OSD (call it P(pick small)) isn't what we should use. Instead, we want >> > P(pick small | first pick not small). Since P(a|b) (the probability of a >> > given b) is P(a && b) / P(b), >> > >> > P(pick small | first pick not small) >> > = P(pick small && first pick not small) / P(first pick not small) >> > >> > The last term is easy to calculate, >> > >> > P(first pick not small) = (total_weight - small_weight) / total_weight >> > >> > and the && term is the distribution we're trying to produce. For exmaple, >> > if small has 1/10 the weight, then we should see 1/10th of the PGs have >> > their second replica be the small OSD. So >> > >> > P(pick small && first pick not small) = small_weight / total_weight >> > >> > Putting those together, >> > >> > P(pick small | first pick not small) >> > = P(pick small && first pick not small) / P(first pick not small) >> > = (small_weight / total_weight) / ((total_weight - small_weight) / total_weight) >> > = small_weight / (total_weight - small_weight) >> > >> > This is, on the second round, we should adjust the weights by the above so >> > that we get the right distribution of second choices. It turns out it >> > works to adjust *all* weights like this to get hte conditional probability >> > that they weren't already chosen. >> > >> > I have a branch that hacks this into straw2 and it appears to work >> > properly for num_rep = 2. With a test bucket of [99 99 99 99 4], and the >> > current code, you get >> > >> > $ bin/crushtool -c cm.txt --test --show-utilization --min-x 0 --max-x 40000000 --num-rep 2 >> > rule 0 (data), x = 0..40000000, numrep = 2..2 >> > rule 0 (data) num_rep 2 result size == 2: 40000001/40000001 >> > device 0: 19765965 [9899364,9866601] >> > device 1: 19768033 [9899444,9868589] >> > device 2: 19769938 [9901770,9868168] >> > device 3: 19766918 [9898851,9868067] >> > device 6: 929148 [400572,528576] >> > >> > which is very close for the first replica (primary), but way off for the >> > second. With my hacky change, >> > >> > rule 0 (data), x = 0..40000000, numrep = 2..2 >> > rule 0 (data) num_rep 2 result size == 2: 40000001/40000001 >> > device 0: 19797315 [9899364,9897951] >> > device 1: 19799199 [9899444,9899755] >> > device 2: 19801016 [9901770,9899246] >> > device 3: 19797906 [9898851,9899055] >> > device 6: 804566 [400572,403994] >> > >> > which is quite close, but still skewing slightly high (by a big less than >> > 1%). >> > >> > Next steps: >> > >> > 1- generalize this for >2 replicas >> > 2- figure out why it skews high >> > 3- make this work for multi-level hierarchical descent >> > >> > sage >> > >> > -- >> > To unsubscribe from this list: send the line "unsubscribe ceph-devel" in >> > the body of a message to majordomo@xxxxxxxxxxxxxxx >> > More majordomo info at http://vger.kernel.org/majordomo-info.html >> > >> >> -- >> Loïc Dachary, Artisan Logiciel Libre >> -- >> To unsubscribe from this list: send the line "unsubscribe ceph-devel" in >> the body of a message to majordomo@xxxxxxxxxxxxxxx >> More majordomo info at http://vger.kernel.org/majordomo-info.html >> >> -- To unsubscribe from this list: send the line "unsubscribe ceph-devel" in the body of a message to majordomo@xxxxxxxxxxxxxxx More majordomo info at http://vger.kernel.org/majordomo-info.html
