From: Shawn Willden <shawn-tahoe@willden.org>
Date: Wed, 14 Jan 2009 03:12:35 +0000 (-0700)
Subject: Statistics module
X-Git-Tag: allmydata-tahoe-1.4.0~228
X-Git-Url: https://git.rkrishnan.org/listings/vdrive/quickstart.html?a=commitdiff_plain;h=21832280da410f8e7b2d5a055b32ca1e4e996c67;p=tahoe-lafs%2Ftahoe-lafs.git

Statistics module

Added a statistics module for calculating various facets of
share survival statistics.
---

diff --git a/docs/lossmodel.lyx b/docs/lossmodel.lyx
new file mode 100644
index 00000000..d2d852bf
--- /dev/null
+++ b/docs/lossmodel.lyx
@@ -0,0 +1,688 @@
+#LyX 1.6.1 created this file. For more info see http://www.lyx.org/
+\lyxformat 345
+\begin_document
+\begin_header
+\textclass amsart
+\use_default_options true
+\begin_modules
+theorems-ams
+theorems-ams-extended
+\end_modules
+\language english
+\inputencoding auto
+\font_roman default
+\font_sans default
+\font_typewriter default
+\font_default_family default
+\font_sc false
+\font_osf false
+\font_sf_scale 100
+\font_tt_scale 100
+
+\graphics default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_amsmath 1
+\use_esint 1
+\cite_engine basic
+\use_bibtopic false
+\paperorientation portrait
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\defskip medskip
+\quotes_language english
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\author "" 
+\author "" 
+\end_header
+
+\begin_body
+
+\begin_layout Title
+Tahoe Distributed Filesharing System Loss Model
+\end_layout
+
+\begin_layout Author
+Shawn Willden
+\end_layout
+
+\begin_layout Email
+shawn@willden.org
+\end_layout
+
+\begin_layout Abstract
+The abstract goes here
+\end_layout
+
+\begin_layout Section
+Problem Statement
+\end_layout
+
+\begin_layout Standard
+The allmydata Tahoe distributed file system uses Reed-Solomon erasure coding
+ to split files into 
+\begin_inset Formula $N$
+\end_inset
+
+ shares, each of which is then delivered to a randomly-selected peer in
+ a distributed network.
+ The file can later be reassembled from any 
+\begin_inset Formula $k\leq N$
+\end_inset
+
+ of the shares, if they are available.
+\end_layout
+
+\begin_layout Standard
+Over time shares are lost for a variety of reasons.
+ Storage servers may crash, be destroyed or simply be removed from the network.
+ To mitigate such losses, Tahoe network clients employ a repair agent which
+ scans the peers once per time period 
+\begin_inset Formula $A$
+\end_inset
+
+ and determines how many of the shares remain.
+ If less than 
+\begin_inset Formula $R$
+\end_inset
+
+ (
+\begin_inset Formula $k\leq R\leq N$
+\end_inset
+
+) shares remain, then the repairer reconstructs the file shares and redistribute
+s the missing ones, bringing the availability back up to full.
+\end_layout
+
+\begin_layout Standard
+The question we're trying to answer is "What's the probability that we'll
+ be able to reassemble the file at some later time 
+\begin_inset Formula $T$
+\end_inset
+
+?".
+ We'd also like to be able to determine what values we should choose for
+ 
+\begin_inset Formula $k$
+\end_inset
+
+, 
+\begin_inset Formula $N$
+\end_inset
+
+, 
+\begin_inset Formula $A$
+\end_inset
+
+, and 
+\begin_inset Formula $R$
+\end_inset
+
+ in order to ensure 
+\begin_inset Formula $Pr[loss]\leq t$
+\end_inset
+
+ for some threshold probability 
+\begin_inset Formula $t$
+\end_inset
+
+.
+ This is an optimization problem because although we could obtain very low
+ 
+\begin_inset Formula $Pr[loss]$
+\end_inset
+
+ by choosing small 
+\begin_inset Formula $k,$
+\end_inset
+
+ large 
+\begin_inset Formula $N$
+\end_inset
+
+, small 
+\begin_inset Formula $A$
+\end_inset
+
+, and setting 
+\begin_inset Formula $R=N$
+\end_inset
+
+, these choices have costs.
+ The peer storage and bandwidth consumed by the share distribution process
+ are approximately 
+\begin_inset Formula $\nicefrac{N}{k}$
+\end_inset
+
+ times the size of the original file, so we would like to reduce this ratio
+ as far as possible consistent with 
+\begin_inset Formula $Pr[loss]\leq t$
+\end_inset
+
+.
+ Likewise, frequent and aggressive repair process can be used to ensure
+ that the number of shares available at any time is very close to 
+\begin_inset Formula $N,$
+\end_inset
+
+ but at a cost in bandwidth.
+\end_layout
+
+\begin_layout Section
+Reliability
+\end_layout
+
+\begin_layout Standard
+The probability that the file becomes unrecoverable is dependent upon the
+ probability that the peers to whom we send shares are able to return those
+ copies on demand.
+ Shares that are returned in corrupted form can be detected and discarded,
+ so there is no need to distinguish between corruption and loss.
+\end_layout
+
+\begin_layout Standard
+There are a large number of factors that affect share availability.
+ Availability can be temporarily interrupted by peer unavailability, due
+ to network outages, power failures or administrative shutdown, among other
+ reasons.
+ Availability can be permanently lost due to failure or corruption of storage
+ media, catastrophic damage to the peer system, administrative error, withdrawal
+ from the network, malicious corruption, etc.
+\end_layout
+
+\begin_layout Standard
+The existence of intermittent failure modes motivates the introduction of
+ a distinction between 
+\noun on
+availability
+\noun default
+ and 
+\noun on
+reliability
+\noun default
+.
+ Reliability is the probability that a share is retrievable assuming intermitten
+t failures can be waited out, so reliability considers only permanent failures.
+ Availability considers all failures, and is focused on the probability
+ of retrieval within some defined time frame.
+\end_layout
+
+\begin_layout Standard
+Another consideration is that some failures affect multiple shares.
+ If multiple shares of a file are stored on a single hard drive, for example,
+ failure of that drive may lose them all.
+ Catastrophic damage to a data center may destroy all shares on all peers
+ in that data center.
+\end_layout
+
+\begin_layout Standard
+While the types of failures that may occur are pretty consistent across
+ even very different peers, their probabilities differ dramatically.
+ A professionally-administered blade server with redundant storage, power
+ and Internet located in a carefully-monitored data center with automatic
+ fire suppression systems is much less likely to become either temporarily
+ or permanently unavailable than the typical virus and malware-ridden home
+ computer on a single cable modem connection.
+ A variety of situations in between exist as well, such as the case of the
+ author's home file server, which is administered by an IT professional
+ and uses RAID level 6 redundant storage, but runs on old, cobbled-together
+ equipment, and has a consumer-grade Internet connection.
+\end_layout
+
+\begin_layout Standard
+To begin with, let's use a simple definition of reliability:
+\end_layout
+
+\begin_layout Definition
+
+\noun on
+Reliability
+\noun default
+ is the probability 
+\begin_inset Formula $p_{i}$
+\end_inset
+
+ that a share 
+\begin_inset Formula $s_{i}$
+\end_inset
+
+ will surve to (be retrievable at) time 
+\begin_inset Formula $T=A$
+\end_inset
+
+, ignoring intermittent failures.
+ That is, the probability that the share will be retrievable at the end
+ of the current repair cycle, and therefore usable by the repairer to regenerate
+ any lost shares.
+\end_layout
+
+\begin_layout Definition
+Reliability is clearly dependent on 
+\begin_inset Formula $A$
+\end_inset
+
+.
+ Short repair cycles offer less time for shares to 
+\begin_inset Quotes eld
+\end_inset
+
+decay
+\begin_inset Quotes erd
+\end_inset
+
+ into unavailability.
+\end_layout
+
+\begin_layout Subsection
+Fixed Reliability
+\end_layout
+
+\begin_layout Standard
+In the simplest case, the peers holding the file shares all have the same
+ reliability 
+\begin_inset Formula $p$
+\end_inset
+
+, and are all independent from one another.
+ Let 
+\begin_inset Formula $K$
+\end_inset
+
+ be a random variable that represents the number of shares that survive
+ 
+\begin_inset Formula $A$
+\end_inset
+
+.
+ Each share's survival can be viewed as an indepedent Bernoulli trial with
+ a succes probability of 
+\begin_inset Formula $p$
+\end_inset
+
+, which means that 
+\begin_inset Formula $K$
+\end_inset
+
+ follows the binomial distribution with paramaters 
+\begin_inset Formula $N$
+\end_inset
+
+ and 
+\begin_inset Formula $p$
+\end_inset
+
+ (
+\begin_inset Formula $K\sim B(N,p)$
+\end_inset
+
+).
+ The probability mass function (PMF) of the binomial distribution is:
+\begin_inset Formula \begin{equation}
+Pr(K=i)=f(i;N,p)=\binom{n}{i}p^{i}(1-p)^{n-i}\label{eq:binomial-pdf}\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+A file survives if at least 
+\begin_inset Formula $k$
+\end_inset
+
+ of the 
+\begin_inset Formula $N$
+\end_inset
+
+ shares survive.
+ Equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:binomial-pdf"
+
+\end_inset
+
+ gives the probability that exactly 
+\begin_inset Formula $i$
+\end_inset
+
+ shares survive, so the probability that fewer than 
+\begin_inset Formula $k$
+\end_inset
+
+ survive is the sum of the probabilities that 
+\begin_inset Formula $0,1,2,\ldots,k-1$
+\end_inset
+
+ shares survive.
+ That is:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{equation}
+Pr[failure]=\sum_{i=0}^{k-1}\binom{n}{i}p^{i}(1-p)^{n-i}\label{eq:simple-failure}\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Independent Reliability
+\end_layout
+
+\begin_layout Standard
+Equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:simple-failure"
+
+\end_inset
+
+ assumes that each share has the same probability of survival, but as explained
+ above, this is not typically true.
+ A more accurate model allows each share 
+\begin_inset Formula $s_{i}$
+\end_inset
+
+ an independent probability of survival 
+\begin_inset Formula $p_{i}$
+\end_inset
+
+.
+ Each share's survival can still be treated as an independent Bernoulli
+ trial, but with success probability 
+\begin_inset Formula $p_{i}$
+\end_inset
+
+.
+ Under this assumption, 
+\begin_inset Formula $K$
+\end_inset
+
+ follows a generalized distribution with parameters 
+\begin_inset Formula $N$
+\end_inset
+
+ and 
+\begin_inset Formula $p_{i},1\leq i\leq N$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+The PMF for this generalized 
+\begin_inset Formula $K$
+\end_inset
+
+ does not have a simple closed-form representation.
+ However, the PMFs for random variables representing individual share survival
+ do.
+ Let 
+\begin_inset Formula $S_{i}$
+\end_inset
+
+ be a random variable such that:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+S_{i}=\begin{cases}
+1 & \textnormal{if }s_{i}\textnormal{ survives}\\
+0 & \textnormal{if }s_{i}\textnormal{ fails}\end{cases}\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+The PMF for 
+\begin_inset Formula $Si$
+\end_inset
+
+ is very simple, 
+\begin_inset Formula $Pr(S_{i}=1)=p_{i}$
+\end_inset
+
+ and 
+\begin_inset Formula $Pr(S_{i}=0)=p_{i}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Observe that 
+\begin_inset Formula $\sum_{i=1}^{N}S_{i}=K$
+\end_inset
+
+.
+ Effectively, 
+\begin_inset Formula $K$
+\end_inset
+
+ has just been separated into the series of Bernoulli trials that make it
+ up.
+\end_layout
+
+\begin_layout Standard
+The discrete convolution theorem states that given random variables 
+\begin_inset Formula $X$
+\end_inset
+
+ and 
+\begin_inset Formula $Y$
+\end_inset
+
+ and their sum 
+\begin_inset Formula $Z=X+Y$
+\end_inset
+
+, if 
+\begin_inset Formula $Pr[X=x]=f(x)$
+\end_inset
+
+ and 
+\begin_inset Formula $Pr[Y=y]=f(y)$
+\end_inset
+
+ then 
+\begin_inset Formula $Pr[Z=z]=(f\star g)(z)$
+\end_inset
+
+ where 
+\begin_inset Formula $\star$
+\end_inset
+
+ denotes the convolution operation.
+ Stated in English, the probability mass function of the sum of two random
+ variables is the convolution of the probability mass functions of the two
+ random variables.
+\end_layout
+
+\begin_layout Standard
+Discrete convolution is defined as
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+(f\star g)(n)=\sum_{m=-\infty}^{\infty}f(m)\cdot g(n-m)\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+The infinite summation is no problem because the probability mass functions
+ we need to convolve are zero outside of a small range.
+\end_layout
+
+\begin_layout Standard
+According to the discrete convolution theorem, then, if 
+\begin_inset Formula $Pr[K=i]=f(i)$
+\end_inset
+
+ and 
+\begin_inset Formula $Pr[S_{i}=j]=g_{i}(j)$
+\end_inset
+
+, then 
+\begin_inset Formula $ $
+\end_inset
+
+
+\begin_inset Formula $f=g_{1}\star g_{2}\star g_{3}\star\ldots\star g_{N}$
+\end_inset
+
+.
+ Since convolution is associative, this can also be written as 
+\begin_inset Formula $ $
+\end_inset
+
+
+\begin_inset Formula \begin{equation}
+f=(g_{1}\star g_{2})\star g_{3})\star\ldots)\star g_{N})\label{eq:convolution}\end{equation}
+
+\end_inset
+
+which enables 
+\begin_inset Formula $f$
+\end_inset
+
+ to be implemented as a sequence of convolution operations on the simple
+ PMFs of the random variables 
+\begin_inset Formula $S_{i}$
+\end_inset
+
+.
+ In fact, as values of 
+\begin_inset Formula $N$
+\end_inset
+
+ get large, equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:convolution"
+
+\end_inset
+
+ turns out to be a more effective means of computing the PMF of 
+\begin_inset Formula $K$
+\end_inset
+
+ even in the case of the standard bernoulli distribution, primarily because
+ the binomial calculation in equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:binomial-pdf"
+
+\end_inset
+
+ produces very large values that overflow unless arbitrary precision numeric
+ representations are used, or unless the binomial calculation is very cleverly
+ interleaved with the powers of 
+\begin_inset Formula $p$
+\end_inset
+
+ and 
+\begin_inset Formula $1-p$
+\end_inset
+
+ to keep the values manageable.
+\end_layout
+
+\begin_layout Standard
+Note also that it is not necessary to have very simple PMFs like those of
+ the 
+\begin_inset Formula $S_{i}$
+\end_inset
+
+.
+ Any share or set of shares that has a known PMF can be combined with any
+ other set with a known PMF by convolution, as long as the two share sets
+ are independent.
+ Since PMFs are easily represented as simple lists of probabilities, where
+ the 
+\begin_inset Formula $i$
+\end_inset
+
+th element in the list corresponds to 
+\begin_inset Formula $Pr[K=i]$
+\end_inset
+
+, these functions are easily managed in software, and computing the convolution
+ is both simple and efficient.
+\end_layout
+
+\begin_layout Subsection
+Multiple Failure Modes
+\end_layout
+
+\begin_layout Standard
+In modeling share survival probabilities, it's useful to be able to analyze
+ separately each of the various failure modes.
+ If reliable statistics for disk failure can be obtained, then a probability
+ mass function for that form of failure can be generated.
+ Similarly, statistics on other hardware failures, administrative errors,
+ network losses, etc., can all be estimated independently.
+ If those estimates can then be combined into a single PMF for that server,
+ then we can use it to predict failures for that server.
+\end_layout
+
+\begin_layout Standard
+In the case of independent failure modes for a single server, this is very
+ simple to do.
+ If 
+\begin_inset Formula $p_{i,j}$
+\end_inset
+
+ is the probability of survival of the 
+\begin_inset Formula $j$
+\end_inset
+
+th failure mode of server 
+\begin_inset Formula $i$
+\end_inset
+
+, and there are 
+\begin_inset Formula $m$
+\end_inset
+
+ failure modes then
+\begin_inset Formula \[
+p_{i}=\prod_{j=1}^{m}p_{i,j}\]
+
+\end_inset
+
+ is the probability of server 
+\begin_inset Formula $i$
+\end_inset
+
+'s survival and
+\begin_inset Formula \[
+Pr[S_{i}=k]=f(k)=\begin{cases}
+1-p_{i} & k=0\\
+p_{i} & k=1\end{cases}\]
+
+\end_inset
+
+ is the full survival PMF.
+\end_layout
+
+\begin_layout Standard
+
+\end_layout
+
+\end_body
+\end_document
diff --git a/src/allmydata/test/test_util.py b/src/allmydata/test/test_util.py
index 32abb7ef..973c798a 100644
--- a/src/allmydata/test/test_util.py
+++ b/src/allmydata/test/test_util.py
@@ -1,7 +1,7 @@
 
 def foo(): pass # keep the line number constant
 
-import os, time
+import os, time, random
 from twisted.trial import unittest
 from twisted.internet import defer, reactor
 from twisted.python import failure
@@ -9,6 +9,7 @@ from twisted.python import failure
 from allmydata.util import base32, idlib, humanreadable, mathutil, hashutil
 from allmydata.util import assertutil, fileutil, deferredutil, abbreviate
 from allmydata.util import limiter, time_format, pollmixin, cachedir
+from allmydata.util import statistics
 
 class Base32(unittest.TestCase):
     def test_b2a_matches_Pythons(self):
@@ -163,6 +164,108 @@ class Math(unittest.TestCase):
         self.failUnlessEqual(f([0,0,0,4]), 1)
         self.failUnlessAlmostEqual(f([0.0, 1.0, 1.0]), .666666666666)
 
+    def test_round_sigfigs(self):
+        f = mathutil.round_sigfigs
+        self.failUnlessEqual(f(22.0/3, 4), 7.3330000000000002)
+
+class Statistics(unittest.TestCase):
+    def should_assert(self, msg, func, *args, **kwargs):
+        try:
+            func(*args, **kwargs)
+            self.fail(msg)
+        except AssertionError, e:
+            pass
+
+    def failUnlessListEqual(self, a, b, msg = None):
+        self.failUnlessEqual(len(a), len(b))
+        for i in range(len(a)):
+            self.failUnlessEqual(a[i], b[i], msg)
+
+    def failUnlessListAlmostEqual(self, a, b, places = 7, msg = None):
+        self.failUnlessEqual(len(a), len(b))
+        for i in range(len(a)):
+            self.failUnlessAlmostEqual(a[i], b[i], places, msg)
+
+    def test_binomial_coeff(self):
+        f = statistics.binomial_coeff
+        self.failUnlessEqual(f(20, 0), 1)
+        self.failUnlessEqual(f(20, 1), 20)
+        self.failUnlessEqual(f(20, 2), 190)
+        self.failUnlessEqual(f(20, 8), f(20, 12))
+        self.should_assert("Should assert if n < k", f, 2, 3)
+
+    def test_binomial_distribution_pmf(self):
+        f = statistics.binomial_distribution_pmf
+
+        pmf_comp = f(2, .1)
+        pmf_stat = [0.81, 0.18, 0.01]
+        self.failUnlessListAlmostEqual(pmf_comp, pmf_stat)
+        
+        # Summing across a PMF should give the total probability 1
+        self.failUnlessAlmostEqual(sum(pmf_comp), 1)
+        self.should_assert("Should assert if not 0<=p<=1", f, 1, -1)
+        self.should_assert("Should assert if n < 1", f, 0, .1)
+
+    def test_survival_pmf(self):
+        f = statistics.survival_pmf
+        # Cross-check binomial-distribution method against convolution
+        # method.
+        p_list = [.9999] * 100 + [.99] * 50 + [.8] * 20
+        pmf1 = statistics.survival_pmf_via_conv(p_list)
+        pmf2 = statistics.survival_pmf_via_bd(p_list)
+        self.failUnlessListAlmostEqual(pmf1, pmf2)
+        self.failUnlessTrue(statistics.valid_pmf(pmf1))
+        self.should_assert("Should assert if p_i > 1", f, [1.1]);
+        self.should_assert("Should assert if p_i < 0", f, [-.1]);
+        
+
+    def test_convolve(self):
+        f = statistics.convolve
+        v1 = [ 1, 2, 3 ]
+        v2 = [ 4, 5, 6 ]
+        v3 = [ 7, 8 ]
+        v1v2result = [ 4, 13, 28, 27, 18 ]
+        # Convolution is commutative
+        r1 = f(v1, v2)
+        r2 = f(v2, v1)
+        self.failUnlessListEqual(r1, r2, "Convolution should be commutative")
+        self.failUnlessListEqual(r1, v1v2result, "Didn't match known result")
+        # Convolution is associative
+        r1 = f(f(v1, v2), v3)
+        r2 = f(v1, f(v2, v3))
+        self.failUnlessListEqual(r1, r2, "Convolution should be associative")
+        # Convolution is distributive
+        r1 = f(v3, [ a + b for a, b in zip(v1, v2) ])
+        tmp1 = f(v3, v1)
+        tmp2 = f(v3, v2)
+        r2 = [ a + b for a, b in zip(tmp1, tmp2) ]
+        self.failUnlessListEqual(r1, r2, "Convolution should be distributive")
+        # Convolution is scalar multiplication associative
+        tmp1 = f(v1, v2)
+        r1 = [ a * 4 for a in tmp1 ]
+        tmp2 = [ a * 4 for a in v1 ]
+        r2 = f(tmp2, v2)
+        self.failUnlessListEqual(r1, r2, "Convolution should be scalar multiplication associative")
+
+    def test_find_k(self):
+        f = statistics.find_k
+        g = statistics.pr_file_loss
+        plist = [.9] * 10 + [.8] * 10
+        t = .0001
+        k = f(plist, t)
+        self.failUnlessEqual(k, 10)
+        self.failUnless(g(plist, k) < t)
+
+    def test_pr_file_loss(self):
+        f = statistics.pr_file_loss
+        plist = [.5] * 10
+        self.failUnlessEqual(f(plist, 3), .0546875)
+
+    def test_pr_backup_file_loss(self):
+        f = statistics.pr_backup_file_loss
+        plist = [.5] * 10
+        self.failUnlessEqual(f(plist, .5, 3), .02734375)
+
 
 class Asserts(unittest.TestCase):
     def should_assert(self, func, *args, **kwargs):
diff --git a/src/allmydata/util/mathutil.py b/src/allmydata/util/mathutil.py
index 1d83e28c..81b12cfb 100644
--- a/src/allmydata/util/mathutil.py
+++ b/src/allmydata/util/mathutil.py
@@ -73,3 +73,7 @@ def log_floor(n, b):
         p *= b
         k += 1
     return k - 1
+
+def round_sigfigs(f, n):
+    fmt = "%." + str(n-1) + "e"
+    return float(fmt % f)
diff --git a/src/allmydata/util/statistics.py b/src/allmydata/util/statistics.py
new file mode 100644
index 00000000..5315e95e
--- /dev/null
+++ b/src/allmydata/util/statistics.py
@@ -0,0 +1,225 @@
+# Copyright (c) 2009 Shawn Willden
+# mailto:shawn@willden.org
+
+from __future__ import division
+from mathutil import round_sigfigs
+import math
+import array
+
+def pr_file_loss(p_list, k):
+    """
+    Probability of single-file loss for shares with reliabilities in
+    p_list.
+
+    Computes the probability that a single file will become
+    unrecoverable, based on the individual share survival
+    probabilities and and k (number of shares needed for recovery).
+
+    Example: pr_file_loss([.9] * 5 + [.99] * 5, 3) returns the
+    probability that a file with k=3, N=10 and stored on five servers
+    with reliability .9 and five servers with reliability .99 is lost.
+
+    See survival_pmf docstring for important statistical assumptions.
+
+    """
+    assert 0 < k <= len(p_list)
+    assert valid_probability_list(p_list)
+
+    # Sum elements 0 through k-1 of the share set PMF to get the
+    # probability that less than k shares survived.
+    return sum(survival_pmf(p_list)[0:k])
+
+def survival_pmf(p_list):
+    """
+    Return the collective PMF of share survival count for a set of
+    shares with the individual survival probabilities in p_list.
+
+    Example: survival_pmf([.99] * 10 + [.8] * 6) returns the
+    probability mass function for the number of shares that will
+    survive from an initial set of 16, 10 with p=0.99 and 6 with
+    p=0.8.  The ith element of the resulting list is the probability
+    that exactly i shares will survive.
+
+    This calculation makes the following assumptions:
+
+    1.  p_list[i] is the probability that any individual share will
+    will survive during the time period in question (whatever that may
+    be).
+
+    2.  The share failures are "independent", in the statistical
+    sense.  Note that if a group of shares are stored on the same
+    machine or even in the same data center, they are NOT independent
+    and this calculation is therefore wrong.
+    """
+    assert valid_probability_list(p_list)
+
+    pmf = survival_pmf_via_conv(p_list)
+
+    assert valid_pmf(pmf)
+    return pmf
+
+def survival_pmf_via_bd(p_list):
+    """
+    Compute share survival PMF using the binomial distribution PMF as
+    much as possible.
+
+    This is more efficient than the convolution method below, but
+    doesn't work for large numbers of shares because the
+    binomial_coeff calculation blows up.  Since the efficiency gains
+    only matter in the case of large numbers of shares, it's pretty
+    much useless except for testing the convolution methond.
+
+    Note that this function does little to no error checking and is
+    intended for internal use and testing only.
+    """
+    pmf_list = [ binomial_distribution_pmf(p_list.count(p), p) 
+                 for p in set(p_list) ]
+    return reduce(convolve, pmf_list)
+
+def survival_pmf_via_conv(p_list):
+    """
+    Compute share survival PMF using iterated convolution of trivial
+    PMFs.
+
+    Note that this function does little to no error checking and is
+    intended for internal use and testing only.
+    """
+    pmf_list = [ [1 - p, p] for p in p_list ];
+    return reduce(convolve, pmf_list)
+
+def print_pmf(pmf, n):
+    """
+    Print a PMF in a readable form, with values rounded to n
+    significant digits. 
+    """
+    for k, p in enumerate(pmf):
+        print "i=" + str(k) + ":", round_sigfigs(p, n)
+
+def pr_backup_file_loss(p_list, backup_p, k):
+    """
+    Probability of single-file loss in a backup context
+
+    Same as pr_file_loss, except it factors in the probability of
+    survival of the original source, specified as backup_p.  Because
+    that's a precondition to caring about the availability of the
+    backup, it's an independent event.
+    """
+    assert valid_probability_list(p_list)
+    assert 0 < backup_p <= 1
+    assert 0 < k <= len(p_list)
+
+    return pr_file_loss(p_list, k) * (1 - backup_p)
+
+
+def find_k(p_list, target_loss_prob):
+    """
+    Find the highest k value that achieves the targeted loss
+    probability, given the share reliabilities given in p_list.
+    """
+    assert valid_probability_list(p_list)
+    assert 0 < target_loss_prob < 1
+
+    pmf = survival_pmf(p_list)
+    return find_k_from_pmf(pmf, target_loss_prob)
+
+def find_k_from_pmf(pmf, target_loss_prob):
+    """
+    Find the highest k value that achieves the targeted loss 
+    probability, given the share survival PMF given in pmf.
+    """
+    assert valid_pmf(pmf)
+    assert 0 < target_loss_prob < 1
+
+    loss_prob = 0.0
+    for k, p_k in enumerate(pmf):
+        loss_prob += p_k
+        if loss_prob > target_loss_prob:
+            return k
+
+    k = len(pmf) - 1
+    return k
+
+def valid_pmf(pmf):
+    """
+    Validate that pmf looks like a proper discrete probability mass
+    function in list form.
+
+    Returns true if the elements of pmf sum to 1.
+    """
+    return round(sum(pmf),5) == 1.0
+
+def valid_probability_list(p_list):
+    """
+    Validate that p_list is a list of probibilities
+    """
+    for p in p_list:
+        if p < 0 or p > 1:
+            return False
+
+    return True
+
+def convolve(list_a, list_b):
+    """
+    Returns the discrete convolution of two lists.
+
+    Given two random variables X and Y, the convolution of their
+    probability mass functions Pr(X) and Pr(Y) is equal to the
+    Pr(X+Y).
+    """
+    n = len(list_a)
+    m = len(list_b)
+    
+    result = []
+    for i in range(n + m - 1):
+        sum = 0.0
+
+        lower = max(0, i - n + 1)
+        upper = min(m - 1, i)
+        
+        for j in range(lower, upper+1):
+            sum += list_a[i-j] * list_b[j]
+
+        result.append(sum)
+
+    return result
+
+def binomial_distribution_pmf(n, p):
+    """
+    Returns Pr(K), where K ~ B(n,p), as a list of values.
+
+    Returns the full probability mass function of a B(n, p) as a list
+    of values, where the kth element is Pr(K=k), or, in the Tahoe
+    context, the probability that exactly k copies of a file share
+    survive, when placed on n independent servers with survival
+    probability p.
+    """
+    assert p >= 0 and p <= 1, 'p=%s must be in the range [0,1]'%p
+    assert n > 0
+
+    result = []
+    for k in range(n+1):
+        result.append(math.pow(p    , k    ) * 
+                      math.pow(1 - p, n - k) * 
+                      binomial_coeff(n, k))
+
+    assert valid_pmf(result)
+    return result;
+
+def binomial_coeff(n, k):
+    """
+    Returns the number of ways that k items can be chosen from a set
+    of n.
+    """
+    assert n >= k
+
+    if k > n:
+        return 0
+
+    if k > n/2:
+        k = n - k
+
+    accum = 1.0
+    for i in range(1, k+1):
+        accum = accum * (n - k + i) // i;
+
+    return int(accum + 0.5)