tahoe-lafs/src/allmydata/provisioning.py
robk-tahoe 9ed5533760 added tweaked sibpath implementation
use of twisted.python.util.sibpath to find files relative to modules doesn't
work when those modules are bundled into a library by py2exe.  this provides
an alternative implementation (in allmydata.util.sibpath) which checks for
the existence of the file, and if it is not found, attempts to find it relative
to sys.executable instead.
2008-01-10 15:23:41 -07:00

763 lines
35 KiB
Python

from nevow import inevow, loaders, rend, tags as T
import math
from allmydata.util import mathutil, sibpath
def getxmlfile(name):
return loaders.xmlfile(sibpath.sibpath(__file__, "web/%s" % name))
# factorial and binomial copied from
# http://mail.python.org/pipermail/python-list/2007-April/435718.html
def factorial(n):
"""factorial(n): return the factorial of the integer n.
factorial(0) = 1
factorial(n) with n<0 is -factorial(abs(n))
"""
result = 1
for i in xrange(1, abs(n)+1):
result *= i
assert n >= 0
return result
def binomial(n, k):
assert 0 <= k <= n
if k == 0 or k == n:
return 1
# calculate n!/k! as one product, avoiding factors that
# just get canceled
P = k+1
for i in xrange(k+2, n+1):
P *= i
# if you are paranoid:
# C, rem = divmod(P, factorial(n-k))
# assert rem == 0
# return C
return P//factorial(n-k)
class ProvisioningTool(rend.Page):
addSlash = True
docFactory = getxmlfile("provisioning.xhtml")
def render_forms(self, ctx, data):
req = inevow.IRequest(ctx)
def getarg(name, astype=int):
if req.method != "POST":
return None
if name in req.fields:
return astype(req.fields[name].value)
return None
return self.do_forms(getarg)
def do_forms(self, getarg):
filled = getarg("filled", bool)
def get_and_set(name, options, default=None, astype=int):
current_value = getarg(name, astype)
i_select = T.select(name=name)
for (count, description) in options:
count = astype(count)
selected = False
if ((current_value is not None and count == current_value) or
(current_value is None and count == default)):
o = T.option(value=str(count), selected="true")[description]
else:
o = T.option(value=str(count))[description]
i_select = i_select[o]
if current_value is None:
current_value = default
return current_value, i_select
sections = {}
def add_input(section, text, entry):
if section not in sections:
sections[section] = []
sections[section].extend([T.div[text, ": ", entry], "\n"])
def add_output(section, entry):
if section not in sections:
sections[section] = []
sections[section].extend([entry, "\n"])
def build_section(section):
return T.fieldset[T.legend[section], sections[section]]
def number(value, suffix=""):
scaling = 1
if value < 1:
fmt = "%1.2g%s"
elif value < 100:
fmt = "%.1f%s"
elif value < 1000:
fmt = "%d%s"
elif value < 1e6:
fmt = "%.2fk%s"; scaling = 1e3
elif value < 1e9:
fmt = "%.2fM%s"; scaling = 1e6
elif value < 1e12:
fmt = "%.2fG%s"; scaling = 1e9
elif value < 1e15:
fmt = "%.2fT%s"; scaling = 1e12
elif value < 1e18:
fmt = "%.2fP%s"; scaling = 1e15
else:
fmt = "huge! %g%s"
return fmt % (value / scaling, suffix)
user_counts = [(5, "5 users"),
(50, "50 users"),
(200, "200 users"),
(1000, "1k users"),
(10000, "10k users"),
(50000, "50k users"),
(100000, "100k users"),
(500000, "500k users"),
(1000000, "1M users"),
]
num_users, i_num_users = get_and_set("num_users", user_counts, 50000)
add_input("Users",
"How many users are on this network?", i_num_users)
files_per_user_counts = [(100, "100 files"),
(1000, "1k files"),
(10000, "10k files"),
(100000, "100k files"),
(1e6, "1M files"),
]
files_per_user, i_files_per_user = get_and_set("files_per_user",
files_per_user_counts,
1000)
add_input("Users",
"How many files in each user's vdrive? (avg)",
i_files_per_user)
space_per_user_sizes = [(1e6, "1MB"),
(10e6, "10MB"),
(100e6, "100MB"),
(200e6, "200MB"),
(1e9, "1GB"),
(2e9, "2GB"),
(5e9, "5GB"),
(10e9, "10GB"),
(100e9, "100GB"),
(1e12, "1TB"),
]
# current allmydata average utilization 127MB per user
space_per_user, i_space_per_user = get_and_set("space_per_user",
space_per_user_sizes,
200e6)
add_input("Users",
"How much data is in each user's vdrive? (avg)",
i_space_per_user)
sharing_ratios = [(1.0, "1.0x"),
(1.1, "1.1x"),
(2.0, "2.0x"),
]
sharing_ratio, i_sharing_ratio = get_and_set("sharing_ratio",
sharing_ratios, 1.0,
float)
add_input("Users",
"What is the sharing ratio? (1.0x is no-sharing and"
" no convergence)", i_sharing_ratio)
# Encoding parameters
encoding_choices = [("3-of-10-5", "3.3x (3-of-10, repair below 5)"),
("3-of-10-8", "3.3x (3-of-10, repair below 8)"),
("5-of-10-7", "2x (5-of-10, repair below 7)"),
("8-of-10-9", "1.25x (8-of-10, repair below 9)"),
("27-of-30-28", "1.1x (27-of-30, repair below 28"),
("25-of-100-50", "4x (25-of-100, repair below 50)"),
]
encoding_parameters, i_encoding_parameters = \
get_and_set("encoding_parameters",
encoding_choices, "3-of-10-5", str)
encoding_pieces = encoding_parameters.split("-")
k = int(encoding_pieces[0])
assert encoding_pieces[1] == "of"
n = int(encoding_pieces[2])
# we repair the file when the number of available shares drops below
# this value
repair_threshold = int(encoding_pieces[3])
add_input("Servers",
"What are the default encoding parameters?",
i_encoding_parameters)
# Server info
num_server_choices = [ (5, "5 servers"),
(10, "10 servers"),
(15, "15 servers"),
(30, "30 servers"),
(50, "50 servers"),
(100, "100 servers"),
(200, "200 servers"),
(300, "300 servers"),
(500, "500 servers"),
(1000, "1k servers"),
(2000, "2k servers"),
(5000, "5k servers"),
(10e3, "10k servers"),
(100e3, "100k servers"),
(1e6, "1M servers"),
]
num_servers, i_num_servers = \
get_and_set("num_servers", num_server_choices, 30, int)
add_input("Servers",
"How many servers are there?", i_num_servers)
# availability is measured in dBA = -dBF, where 0dBF is 100% failure,
# 10dBF is 10% failure, 20dBF is 1% failure, etc
server_dBA_choices = [ (10, "90% [10dBA] (2.4hr/day)"),
(13, "95% [13dBA] (1.2hr/day)"),
(20, "99% [20dBA] (14min/day or 3.5days/year)"),
(23, "99.5% [23dBA] (7min/day or 1.75days/year)"),
(30, "99.9% [30dBA] (87sec/day or 9hours/year)"),
(40, "99.99% [40dBA] (60sec/week or 53min/year)"),
(50, "99.999% [50dBA] (5min per year)"),
]
server_dBA, i_server_availability = \
get_and_set("server_availability",
server_dBA_choices,
20, int)
add_input("Servers",
"What is the server availability?", i_server_availability)
drive_MTBF_choices = [ (40, "40,000 Hours"),
]
drive_MTBF, i_drive_MTBF = \
get_and_set("drive_MTBF", drive_MTBF_choices, 40, int)
add_input("Drives",
"What is the hard drive MTBF?", i_drive_MTBF)
# http://www.tgdaily.com/content/view/30990/113/
# http://labs.google.com/papers/disk_failures.pdf
# google sees:
# 1.7% of the drives they replaced were 0-1 years old
# 8% of the drives they repalced were 1-2 years old
# 8.6% were 2-3 years old
# 6% were 3-4 years old, about 8% were 4-5 years old
drive_size_choices = [ (100, "100 GB"),
(250, "250 GB"),
(500, "500 GB"),
(750, "750 GB"),
]
drive_size, i_drive_size = \
get_and_set("drive_size", drive_size_choices, 750, int)
drive_size = drive_size * 1e9
add_input("Drives",
"What is the capacity of each hard drive?", i_drive_size)
drive_failure_model_choices = [ ("E", "Exponential"),
("U", "Uniform"),
]
drive_failure_model, i_drive_failure_model = \
get_and_set("drive_failure_model",
drive_failure_model_choices,
"E", str)
add_input("Drives",
"How should we model drive failures?", i_drive_failure_model)
# drive_failure_rate is in failures per second
if drive_failure_model == "E":
drive_failure_rate = 1.0 / (drive_MTBF * 1000 * 3600)
else:
drive_failure_rate = 0.5 / (drive_MTBF * 1000 * 3600)
# deletion/gc/ownership mode
ownership_choices = [ ("A", "no deletion, no gc, no owners"),
("B", "deletion, no gc, no owners"),
("C", "deletion, share timers, no owners"),
("D", "deletion, no gc, yes owners"),
("E", "deletion, owner timers"),
]
ownership_mode, i_ownership_mode = \
get_and_set("ownership_mode", ownership_choices,
"A", str)
add_input("Servers",
"What is the ownership mode?", i_ownership_mode)
# client access behavior
access_rates = [ (1, "one file per day"),
(10, "10 files per day"),
(100, "100 files per day"),
(1000, "1k files per day"),
(10e3, "10k files per day"),
(100e3, "100k files per day"),
]
download_files_per_day, i_download_rate = \
get_and_set("download_rate", access_rates,
100, int)
add_input("Users",
"How many files are downloaded per day?", i_download_rate)
download_rate = 1.0 * download_files_per_day / (24*60*60)
upload_files_per_day, i_upload_rate = \
get_and_set("upload_rate", access_rates,
10, int)
add_input("Users",
"How many files are uploaded per day?", i_upload_rate)
upload_rate = 1.0 * upload_files_per_day / (24*60*60)
delete_files_per_day, i_delete_rate = \
get_and_set("delete_rate", access_rates,
10, int)
add_input("Users",
"How many files are deleted per day?", i_delete_rate)
delete_rate = 1.0 * delete_files_per_day / (24*60*60)
# the value is in days
lease_timers = [ (1, "one refresh per day"),
(7, "one refresh per week"),
]
lease_timer, i_lease = \
get_and_set("lease_timer", lease_timers,
7, int)
add_input("Users",
"How frequently do clients refresh files or accounts? "
"(if necessary)",
i_lease)
seconds_per_lease = 24*60*60*lease_timer
check_timer_choices = [ (1, "every week"),
(4, "every month"),
(8, "every two months"),
(16, "every four months"),
]
check_timer, i_check_timer = \
get_and_set("check_timer", check_timer_choices, 4, int)
add_input("Users",
"How frequently should we check on each file?",
i_check_timer)
file_check_interval = check_timer * 7 * 24 * 3600
if filled:
add_output("Users", T.div["Total users: %s" % number(num_users)])
add_output("Users",
T.div["Files per user: %s" % number(files_per_user)])
file_size = 1.0 * space_per_user / files_per_user
add_output("Users",
T.div["Average file size: ", number(file_size)])
total_files = num_users * files_per_user / sharing_ratio
user_file_check_interval = file_check_interval / files_per_user
add_output("Grid",
T.div["Total number of files in grid: ",
number(total_files)])
total_space = num_users * space_per_user / sharing_ratio
add_output("Grid",
T.div["Total volume of plaintext in grid: ",
number(total_space, "B")])
total_shares = n * total_files
add_output("Grid",
T.div["Total shares in grid: ", number(total_shares)])
expansion = float(n) / float(k)
total_usage = expansion * total_space
add_output("Grid",
T.div["Share data in grid: ", number(total_usage, "B")])
if n > num_servers:
# silly configuration, causes Tahoe2 to wrap and put multiple
# shares on some servers.
add_output("Servers",
T.div["non-ideal: more shares than servers"
" (n=%d, servers=%d)" % (n, num_servers)])
# every file has at least one share on every server
buckets_per_server = total_files
shares_per_server = total_files * ((1.0 * n) / num_servers)
else:
# if nobody is full, then no lease requests will be turned
# down for lack of space, and no two shares for the same file
# will share a server. Therefore the chance that any given
# file has a share on any given server is n/num_servers.
buckets_per_server = total_files * ((1.0 * n) / num_servers)
# since each such represented file only puts one share on a
# server, the total number of shares per server is the same.
shares_per_server = buckets_per_server
add_output("Servers",
T.div["Buckets per server: ",
number(buckets_per_server)])
add_output("Servers",
T.div["Shares per server: ",
number(shares_per_server)])
# how much space is used on the storage servers for the shares?
# the share data itself
share_data_per_server = total_usage / num_servers
add_output("Servers",
T.div["Share data per server: ",
number(share_data_per_server, "B")])
# this is determined empirically. H=hashsize=32, for a one-segment
# file and 3-of-10 encoding
share_validation_per_server = 266 * shares_per_server
# this could be 423*buckets_per_server, if we moved the URI
# extension into a separate file, but that would actually consume
# *more* space (minimum filesize is 4KiB), unless we moved all
# shares for a given bucket into a single file.
share_uri_extension_per_server = 423 * shares_per_server
# ownership mode adds per-bucket data
H = 32 # depends upon the desired security of delete/refresh caps
# bucket_lease_size is the amount of data needed to keep track of
# the delete/refresh caps for each bucket.
bucket_lease_size = 0
client_bucket_refresh_rate = 0
owner_table_size = 0
if ownership_mode in ("B", "C", "D", "E"):
bucket_lease_size = sharing_ratio * 1.0 * H
if ownership_mode in ("B", "C"):
# refreshes per second per client
client_bucket_refresh_rate = (1.0 * n * files_per_user /
seconds_per_lease)
add_output("Users",
T.div["Client share refresh rate (outbound): ",
number(client_bucket_refresh_rate, "Hz")])
server_bucket_refresh_rate = (client_bucket_refresh_rate *
num_users / num_servers)
add_output("Servers",
T.div["Server share refresh rate (inbound): ",
number(server_bucket_refresh_rate, "Hz")])
if ownership_mode in ("D", "E"):
# each server must maintain a bidirectional mapping from
# buckets to owners. One way to implement this would be to
# put a list of four-byte owner numbers into each bucket, and
# a list of four-byte share numbers into each owner (although
# of course we'd really just throw it into a database and let
# the experts take care of the details).
owner_table_size = 2*(buckets_per_server * sharing_ratio * 4)
if ownership_mode in ("E",):
# in this mode, clients must refresh one timer per server
client_account_refresh_rate = (1.0 * num_servers /
seconds_per_lease)
add_output("Users",
T.div["Client account refresh rate (outbound): ",
number(client_account_refresh_rate, "Hz")])
server_account_refresh_rate = (client_account_refresh_rate *
num_users / num_servers)
add_output("Servers",
T.div["Server account refresh rate (inbound): ",
number(server_account_refresh_rate, "Hz")])
# TODO: buckets vs shares here is a bit wonky, but in
# non-wrapping grids it shouldn't matter
share_lease_per_server = bucket_lease_size * buckets_per_server
share_ownertable_per_server = owner_table_size
share_space_per_server = (share_data_per_server +
share_validation_per_server +
share_uri_extension_per_server +
share_lease_per_server +
share_ownertable_per_server)
add_output("Servers",
T.div["Share space per server: ",
number(share_space_per_server, "B"),
" (data ",
number(share_data_per_server, "B"),
", validation ",
number(share_validation_per_server, "B"),
", UEB ",
number(share_uri_extension_per_server, "B"),
", lease ",
number(share_lease_per_server, "B"),
", ownertable ",
number(share_ownertable_per_server, "B"),
")",
])
# rates
client_download_share_rate = download_rate * k
client_download_byte_rate = download_rate * file_size
add_output("Users",
T.div["download rate: shares = ",
number(client_download_share_rate, "Hz"),
" , bytes = ",
number(client_download_byte_rate, "Bps"),
])
total_file_check_rate = 1.0 * total_files / file_check_interval
client_check_share_rate = total_file_check_rate / num_users
add_output("Users",
T.div["file check rate: shares = ",
number(client_check_share_rate, "Hz"),
" (interval = %s)" %
number(1 / client_check_share_rate, "s"),
])
client_upload_share_rate = upload_rate * n
# TODO: doesn't include overhead
client_upload_byte_rate = upload_rate * file_size * expansion
add_output("Users",
T.div["upload rate: shares = ",
number(client_upload_share_rate, "Hz"),
" , bytes = ",
number(client_upload_byte_rate, "Bps"),
])
client_delete_share_rate = delete_rate * n
server_inbound_share_rate = (client_upload_share_rate *
num_users / num_servers)
server_inbound_byte_rate = (client_upload_byte_rate *
num_users / num_servers)
add_output("Servers",
T.div["upload rate (inbound): shares = ",
number(server_inbound_share_rate, "Hz"),
" , bytes = ",
number(server_inbound_byte_rate, "Bps"),
])
add_output("Servers",
T.div["share check rate (inbound): ",
number(total_file_check_rate * n / num_servers,
"Hz"),
])
server_share_modify_rate = ((client_upload_share_rate +
client_delete_share_rate) *
num_users / num_servers)
add_output("Servers",
T.div["share modify rate: shares = ",
number(server_share_modify_rate, "Hz"),
])
server_outbound_share_rate = (client_download_share_rate *
num_users / num_servers)
server_outbound_byte_rate = (client_download_byte_rate *
num_users / num_servers)
add_output("Servers",
T.div["download rate (outbound): shares = ",
number(server_outbound_share_rate, "Hz"),
" , bytes = ",
number(server_outbound_byte_rate, "Bps"),
])
total_share_space = num_servers * share_space_per_server
add_output("Grid",
T.div["Share space consumed: ",
number(total_share_space, "B")])
add_output("Grid",
T.div[" %% validation: %.2f%%" %
(100.0 * share_validation_per_server /
share_space_per_server)])
add_output("Grid",
T.div[" %% uri-extension: %.2f%%" %
(100.0 * share_uri_extension_per_server /
share_space_per_server)])
add_output("Grid",
T.div[" %% lease data: %.2f%%" %
(100.0 * share_lease_per_server /
share_space_per_server)])
add_output("Grid",
T.div[" %% owner data: %.2f%%" %
(100.0 * share_ownertable_per_server /
share_space_per_server)])
add_output("Grid",
T.div[" %% share data: %.2f%%" %
(100.0 * share_data_per_server /
share_space_per_server)])
add_output("Grid",
T.div["file check rate: ",
number(total_file_check_rate,
"Hz")])
total_drives = max(mathutil.div_ceil(int(total_share_space),
int(drive_size)),
num_servers)
add_output("Drives",
T.div["Total drives: ", number(total_drives), " drives"])
drives_per_server = mathutil.div_ceil(total_drives, num_servers)
add_output("Servers",
T.div["Drives per server: ", drives_per_server])
# costs
if drive_size == 750 * 1e9:
add_output("Servers", T.div["750GB drive: $250 each"])
drive_cost = 250
else:
add_output("Servers",
T.div[T.b["unknown cost per drive, assuming $100"]])
drive_cost = 100
if drives_per_server <= 4:
add_output("Servers", T.div["1U box with <= 4 drives: $1500"])
server_cost = 1500 # typical 1U box
elif drives_per_server <= 12:
add_output("Servers", T.div["2U box with <= 12 drives: $2500"])
server_cost = 2500 # 2U box
else:
add_output("Servers",
T.div[T.b["Note: too many drives per server, "
"assuming $3000"]])
server_cost = 3000
server_capital_cost = (server_cost + drives_per_server * drive_cost)
total_server_cost = float(num_servers * server_capital_cost)
add_output("Servers", T.div["Capital cost per server: $",
server_capital_cost])
add_output("Grid", T.div["Capital cost for all servers: $",
number(total_server_cost)])
# $70/Mbps/mo
# $44/server/mo power+space
server_bandwidth = max(server_inbound_byte_rate,
server_outbound_byte_rate)
server_bandwidth_mbps = mathutil.div_ceil(int(server_bandwidth*8),
int(1e6))
server_monthly_cost = 70*server_bandwidth_mbps + 44
add_output("Servers", T.div["Monthly cost per server: $",
server_monthly_cost])
add_output("Users", T.div["Capital cost per user: $",
number(total_server_cost / num_users)])
# reliability
any_drive_failure_rate = total_drives * drive_failure_rate
any_drive_MTBF = 1 // any_drive_failure_rate # in seconds
any_drive_MTBF_days = any_drive_MTBF / 86400
add_output("Drives",
T.div["MTBF (any drive): ",
number(any_drive_MTBF_days), " days"])
drive_replacement_monthly_cost = (float(drive_cost)
* any_drive_failure_rate
*30*86400)
add_output("Grid",
T.div["Monthly cost of replacing drives: $",
number(drive_replacement_monthly_cost)])
total_server_monthly_cost = float(num_servers * server_monthly_cost
+ drive_replacement_monthly_cost)
add_output("Grid", T.div["Monthly cost for all servers: $",
number(total_server_monthly_cost)])
add_output("Users",
T.div["Monthly cost per user: $",
number(total_server_monthly_cost / num_users)])
# availability
file_dBA = self.file_availability(k, n, server_dBA)
user_files_dBA = self.many_files_availability(file_dBA,
files_per_user)
all_files_dBA = self.many_files_availability(file_dBA, total_files)
add_output("Users",
T.div["availability of: ",
"arbitrary file = %d dBA, " % file_dBA,
"all files of user1 = %d dBA, " % user_files_dBA,
"all files in grid = %d dBA" % all_files_dBA,
],
)
time_until_files_lost = (n-k+1) / any_drive_failure_rate
add_output("Grid",
T.div["avg time until files are lost: ",
number(time_until_files_lost, "s"), ", ",
number(time_until_files_lost/86400, " days"),
])
share_data_loss_rate = any_drive_failure_rate * drive_size
add_output("Grid",
T.div["share data loss rate: ",
number(share_data_loss_rate,"Bps")])
# the worst-case survival numbers occur when we do a file check
# and the file is just above the threshold for repair (so we
# decide to not repair it). The question is then: what is the
# chance that the file will decay so badly before the next check
# that we can't recover it? The resulting probability is per
# check interval.
# Note that the chances of us getting into this situation are low.
P_disk_failure_during_interval = (drive_failure_rate *
file_check_interval)
disk_failure_dBF = 10*math.log10(P_disk_failure_during_interval)
disk_failure_dBA = -disk_failure_dBF
file_survives_dBA = self.file_availability(k, repair_threshold,
disk_failure_dBA)
user_files_survives_dBA = self.many_files_availability( \
file_survives_dBA, files_per_user)
all_files_survives_dBA = self.many_files_availability( \
file_survives_dBA, total_files)
add_output("Users",
T.div["survival of: ",
"arbitrary file = %d dBA, " % file_survives_dBA,
"all files of user1 = %d dBA, " %
user_files_survives_dBA,
"all files in grid = %d dBA" %
all_files_survives_dBA,
" (per worst-case check interval)",
])
all_sections = []
all_sections.append(build_section("Users"))
all_sections.append(build_section("Servers"))
all_sections.append(build_section("Drives"))
if "Grid" in sections:
all_sections.append(build_section("Grid"))
f = T.form(action=".", method="post", enctype="multipart/form-data")
if filled:
action = "Recompute"
else:
action = "Compute"
f = f[T.input(type="hidden", name="filled", value="true"),
T.input(type="submit", value=action),
all_sections,
]
return f
def file_availability(self, k, n, server_dBA):
"""
The full formula for the availability of a specific file is::
1 - sum([choose(N,i) * p**i * (1-p)**(N-i)] for i in range(k)])
Where choose(N,i) = N! / ( i! * (N-i)! ) . Note that each term of
this summation is the probability that there are exactly 'i' servers
available, and what we're doing is adding up the cases where i is too
low.
This is a nuisance to calculate at all accurately, especially once N
gets large, and when p is close to unity. So we make an engineering
approximation: if (1-p) is very small, then each [i] term is much
larger than the [i-1] term, and the sum is dominated by the i=k-1
term. This only works for (1-p) < 10%, and when the choose() function
doesn't rise fast enough to compensate. For high-expansion encodings
(3-of-10, 25-of-100), the choose() function is rising at the same
time as the (1-p)**(N-i) term, so that's not an issue. For
low-expansion encodings (7-of-10, 75-of-100) the two values are
moving in opposite directions, so more care must be taken.
Note that the p**i term has only a minor effect as long as (1-p)*N is
small, and even then the effect is attenuated by the 1-p term.
"""
assert server_dBA > 9 # >=90% availability to use the approximation
factor = binomial(n, k-1)
factor_dBA = 10 * math.log10(factor)
exponent = n - k + 1
file_dBA = server_dBA * exponent - factor_dBA
return file_dBA
def many_files_availability(self, file_dBA, num_files):
"""The probability that 'num_files' independent bernoulli trials will
succeed (i.e. we can recover all files in the grid at any given
moment) is p**num_files . Since p is close to unity, we express in p
in dBA instead, so we can get useful precision on q (=1-p), and then
the formula becomes::
P_some_files_unavailable = 1 - (1 - q)**num_files
That (1-q)**n expands with the usual binomial sequence, 1 - nq +
Xq**2 ... + Xq**n . We use the same approximation as before, since we
know q is close to zero, and we get to ignore all the terms past -nq.
"""
many_files_dBA = file_dBA - 10 * math.log10(num_files)
return many_files_dBA