From: Brian Warner Date: Tue, 26 Mar 2013 00:57:01 +0000 (-0700) Subject: resurrect provisioning.py X-Git-Tag: allmydata-tahoe-1.10a2~2 X-Git-Url: https://git.rkrishnan.org/listings/reliability?a=commitdiff_plain;h=80b43b409fbbfec2193391983bb076104cc45205;p=tahoe-lafs%2Ftahoe-lafs.git resurrect provisioning.py It looks like commit 916d26e7103208fa207259d62ce453a5a8b9acd0, in addition to making a one-line fix for #1681, also deleted misc/operations_helpers/provisioning/provisioning.py entirely. This brings it back. --- diff --git a/misc/operations_helpers/provisioning/provisioning.py b/misc/operations_helpers/provisioning/provisioning.py new file mode 100644 index 00000000..37acd16d --- /dev/null +++ b/misc/operations_helpers/provisioning/provisioning.py @@ -0,0 +1,776 @@ + +from nevow import inevow, rend, loaders, tags as T +import math +import util + +# factorial and binomial copied from +# http://mail.python.org/pipermail/python-list/2007-April/435718.html + +def div_ceil(n, d): + """ + The smallest integer k such that k*d >= n. + """ + return (n/d) + (n%d != 0) + +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 = loaders.xmlfile(util.sibling("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) + 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 for each user? (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"), + (2e12, "2TB"), + (5e12, "5TB"), + ] + # Estimate ~5gb per user as a more realistic case + space_per_user, i_space_per_user = get_and_set("space_per_user", + space_per_user_sizes, + 5e9) + add_input("Users", + "How much data for each user? (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"), + (1000, "1000 GB"), + (2000, "2000 GB"), + (3000, "3000 GB"), + ] + drive_size, i_drive_size = \ + get_and_set("drive_size", drive_size_choices, 3000, 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 + + 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(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 = div_ceil(total_drives, num_servers) + add_output("Servers", + T.div["Drives per server: ", drives_per_server]) + + # costs + if drive_size == 3000 * 1e9: + add_output("Servers", T.div["3000GB 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 = 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, + ] + + try: + from allmydata import reliability + # we import this just to test to see if the page is available + _hush_pyflakes = reliability + del _hush_pyflakes + f = [T.div[T.a(href="../reliability")["Reliability Math"]], f] + except ImportError: + pass + + 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