## Bug #15653

### crush: low weight devices get too many objects for num_rep > 1

80%

**Description**

### discussion¶

### description, with example¶

CRUSH will correctly choose items with relative weights with the right probabilities for each independent choice. However, when choosing multiple replicas, each choice is **not** indepent, since it

has to be unique. The result is that low-weighted devices get too many items.

Simple example:

maetl:src (master) 03:20 PM $ cat cm.txt # begin crush map # devices device 0 device0 device 1 device1 device 2 device2 device 3 device3 device 4 device4 # types type 0 osd type 1 domain type 2 pool # buckets domain root { id -1 # do not change unnecessarily # weight 5.000 alg straw2 hash 0 # rjenkins1 item device0 weight 10.00 item device1 weight 10.0 item device2 weight 10.0 item device3 weight 10.0 item device4 weight 1.000 } # rules rule data { ruleset 0 type replicated min_size 1 max_size 10 step take root step choose firstn 0 type osd step emit } # end crush map maetl:src (master) 03:20 PM $ ./crushtool -c cm.txt -o cm maetl:src (master) 03:20 PM $ ./crushtool -i cm --test --show-utilization --num-rep 1 --min-x 1 --max-x 1000000 --num-rep 1 rule 0 (data), x = 1..1000000, numrep = 1..1 rule 0 (data) num_rep 1 result size == 1: 1000000/1000000 device 0: stored : 243456 expected : 200000 device 1: stored : 243624 expected : 200000 device 2: stored : 244486 expected : 200000 device 3: stored : 243881 expected : 200000 device 4: stored : 24553 expected : 200000 maetl:src (master) 03:20 PM $ ./crushtool -i cm --test --show-utilization --num-rep 1 --min-x 1 --max-x 1000000 --num-rep 3 rule 0 (data), x = 1..1000000, numrep = 3..3 rule 0 (data) num_rep 3 result size == 3: 1000000/1000000 device 0: stored : 723984 expected : 600000 device 1: stored : 722923 expected : 600000 device 2: stored : 723153 expected : 600000 device 3: stored : 723394 expected : 600000 device 4: stored : 106546 expected : 600000

Note that in the 1x case, we get 1/10th the items on device 4, as expected. For 3x, it grows to 1/7th. For lower weights the amplification is more pronounced.

### detailed explanation¶

The chances of getting a particular device during the first draw is the weight of the device divided by the sum of the weight of all devices. For example let say there are 5 devices in a bucket, with the following weights a = 10, b = 10, c = 10, d = 10, e = 1. The chances of getting e is 1/41 and the chances of getting a is 10/41.

Things get more complicated for the second draw because we have to account for a first draw that does not include a given device: it is the sum of the weight of all devices except the one we're interested in, divided by the weight of all devices. So, if we want to know the chances of e showing up in the second draw, the first draw must not include it and this has a 40/41 chance of happening. Also, during the second draw, the chance of getting e is increased because there is one less device to chose from (the one that was picked during the first draw): 1/31 (i.e. 41 - the weight of the device that was chosen). Because the second draw depends on the first draw, the probability must be multiplied: 40/41 * 1/31.

Since the chance of getting the device e in a first draw or getting the device e in a second draw are independent, the chances of getting the device e in both situations is the sum of their probability: 1/41 (first draw) + (40/41 * 1/31) (second draw).

This is a special case because all devices have the same weight except for e. If we are to calculate the probability of a being selected in the second draw, we have to sum the case where e is selected and the case where b, c or d is selected during the first draw, because they do not have the same weight. If e is selected during the first draw, a will be selected during the second draw with a probability of (1/41 * 10/40). If b, c, or d is selected during the first draw, a will be selected during the second draw with a probability of (30/41 * 10/31).

The chances of getting the device a in the first draw and the second draw is therefore: 10/41+(30/41 * 10/31)+(1/41 * 10/40)

To summarize:

- probability of getting e : 1/41 + (40/41 * 1/31) = .05586
- probability of getting a : 10/41+(30/41 * 10/31)+(1/41 * 10/40) = .48603

We are therefore 8.7 ( 0.48603/0.05586 ) more likely to get e than to get a.

From the point of view of the users, this is counter intuitive because they expect that the weight reflects the probability, which is only true for a single draw. With just one draw a is (10/41)/(1/41) = 10 times more likely to be selected than e. With two draws, a is only 8.7 times more likely to be selected than e, as shown above.

### History

#### #1 Updated by Sage Weil almost 3 years ago

**Description**updated (diff)

#### #2 Updated by Sage Weil almost 3 years ago

**Description**updated (diff)

#### #3 Updated by Adam Emerson over 2 years ago

**Category**set to*10***Assignee**set to*Adam Emerson***% Done**changed from*0*to*50*

I just need to put in a special case for LIST, (I don't know if I should just not bother with TREE) test it some, and then everything should be set.

#### #4 Updated by Adam Emerson over 2 years ago

**Status**changed from*Verified*to*Testing***% Done**changed from*50*to*80*

#### #5 Updated by Dan van der Ster about 2 years ago

Does this issue explain our uneven distribution? We have four racks, with 7, 8, 8, 4 hosts in each, respectively. The rack with 4 hosts (RA13) is getting noticeably more data than the others (as is the 7-host rack, RA01):

ID WEIGHT REWEIGHT SIZE USE AVAIL %USE VAR TYPE NAME -2 3532.61450 - 3530T 1528T 2002T 43.29 1.19 room 0513-R-0050 -72 911.81860 - 911T 399T 511T 43.85 1.20 rack RA01 -4 1048.31836 - 1047T 424T 622T 40.54 1.11 rack RA05 -6 1048.31836 - 1047T 421T 626T 40.21 1.10 rack RA09 -9 524.15918 - 523T 282T 241T 53.94 1.48 rack RA13

I suppose the workaround for this is to decrease the weight of the smaller rack?

#### #6 Updated by Loic Dachary about 2 years ago

See https://github.com/ceph/ceph/pull/10218 for a discussion and a tentative fix.

#### #7 Updated by Loic Dachary about 2 years ago

The test Adam wrote to demonstrate the problem, made into a pull request: https://github.com/ceph/ceph/pull/13083

#### #8 Updated by Loic Dachary about 2 years ago

**Description**updated (diff)**Status**changed from*Testing*to*In Progress*

#### #9 Updated by Loic Dachary about 2 years ago

**Description**updated (diff)

#### #10 Updated by Loic Dachary about 2 years ago

**Description**updated (diff)

#### #11 Updated by Loic Dachary about 2 years ago

**Description**updated (diff)

#### #12 Updated by Greg Farnum over 1 year ago

**Project**changed from*Ceph*to*RADOS***Category**deleted ()*10*

#### #13 Updated by Greg Farnum over 1 year ago

**Category**set to*Performance/Resource Usage***Component(RADOS)***CRUSH*added

#### #14 Updated by Sage Weil over 1 year ago

**Status**changed from*In Progress*to*Verified*

#### #15 Updated by Sage Weil about 1 year ago

**Assignee**deleted ()*Adam Emerson*

The new weight-set capability in crush gives us the tool to fix this, but balancer module does not try to do fix it yet via crush. the 'upmap' mode will work around it, though.