{-# LANGUAGE GeneralizedNewtypeDeriving #-}
{-# LANGUAGE NamedFieldPuns             #-}
{-# LANGUAGE StandaloneDeriving         #-}

-- | 𝚫Q representation and primitives.
--
-- See also <https://www.ofcom.org.uk/__data/assets/pdf_file/0024/71682/traffic-management-detection.pdf>
-- appendix A.
--
module Ouroboros.Network.DeltaQ
  ( -- * DeltaQ
    DeltaQ (..)
  , deltaqQ99thPercentile
  , deltaqQ50thPercentile
  , deltaqProbabilityMassBeforeDeadline
    -- * GSV
  , GSV (..)
  , SizeInBytes
  , ballisticGSV
  , gsvLeadingEdgeArrive
  , gsvTrailingEdgeDepart
  , gsvTrailingEdgeArrive
    -- * Distribution
    -- | Needed to construct a 'DeltaQ' or 'GSV'
  , Distribution
  , degenerateDistribution
    -- * Bi-directional GSV.
    -- | 'gsvRequestResponseDuration' is provided as an example of the 'GSV'
    -- and 'DeltaQ' primitives.
  , PeerGSV (..)
  , gsvRequestResponseDuration
  , defaultGSV
  , fromSample
  ) where

import           Control.Monad.Class.MonadTime (DiffTime, Time (..), diffTime)
import           Data.Word (Word32)


--
-- Sketch of API for DeltaQ
--

-- | A \"𝚫Q\" is a probability distribution on the duration between two events.
-- It is an \"improper\" probability distribution in that it may not integrate
-- to 1. The \"missing\" probability mass represents failure. This allows both
-- timing and failure to be represented in one mathematical object.
--
-- In the case of networks a 𝚫Q can be used for example distributions such as
-- the time for a leading edge or trailing edge of a packet to traverse a
-- network (or failing to do so), and many others besides.
--
newtype DeltaQ = DeltaQ (Distribution DiffTime)

-- | DeltaQ distributions (as independent random variables) are semi-groups
-- by convolution.
--
deriving instance Semigroup DeltaQ

-- | The point in time in the distribution for which 99% of the probability
-- mass is before that time.
--
-- This tells us how long we would have to wait to have a 99% chance of the
-- end event having happened. Of course 99% is somewhat arbitrary and other
-- points in the distribution could be of interest.
--
-- TODO: this needs to be specified better for improper distributions.
--
deltaqQ99thPercentile :: DeltaQ -> DiffTime
deltaqQ99thPercentile :: DeltaQ -> DiffTime
deltaqQ99thPercentile (DeltaQ (DegenerateDistribution DiffTime
t)) = DiffTime
t

deltaqQ50thPercentile :: DeltaQ -> DiffTime
deltaqQ50thPercentile :: DeltaQ -> DiffTime
deltaqQ50thPercentile (DeltaQ (DegenerateDistribution DiffTime
t)) = DiffTime
t

-- | This is another way of looking at a 𝚫Q distribution. Instead of giving
-- a fraction of the probability mass (like 99%) and asking how long we would
-- have to wait, we can say how long we are prepared to wait and ask what
-- fraction of the probability mass is before that time.
--
-- So this tells us the chance of the end event happening within the time we
-- are prepared to wait. This is useful for evaluating different options for
-- which has the greatest probability of success within a deadline.
--
deltaqProbabilityMassBeforeDeadline :: DiffTime
                                    -> DeltaQ
                                    -> Double
deltaqProbabilityMassBeforeDeadline :: DiffTime -> DeltaQ -> Double
deltaqProbabilityMassBeforeDeadline DiffTime
deadline (DeltaQ (DegenerateDistribution DiffTime
t))
  | DiffTime
t DiffTime -> DiffTime -> Bool
forall a. Ord a => a -> a -> Bool
< DiffTime
deadline = Double
1
  | Bool
otherwise    = Double
0


--
-- Sketch of API for Distribution
--

-- | An improper probability distribution over some underlying type (such as
-- time durations).
--
-- The current representation only covers the case of degenerate distributions,
-- that take a single value with probability 1. This is just a proof of concept
-- to illustrate the API.
--
data Distribution n = DegenerateDistribution n

-- | Distributions are semi-groups by convolution.
--
instance Num n => Semigroup (Distribution n) where
  <> :: Distribution n -> Distribution n -> Distribution n
(<>) = Distribution n -> Distribution n -> Distribution n
forall n.
Num n =>
Distribution n -> Distribution n -> Distribution n
convolveDistribution


-- | Make a degenerate distribution.
--
-- <https://en.wikipedia.org/wiki/Degenerate_distribution>
--
degenerateDistribution :: n -> Distribution n
degenerateDistribution :: n -> Distribution n
degenerateDistribution = n -> Distribution n
forall n. n -> Distribution n
DegenerateDistribution


-- | The convolution of two (improper) probability distributions.
--
-- <https://en.wikipedia.org/wiki/Convolution>
--
-- For now the implementation only covers degenerate distributions.
--
convolveDistribution :: Num n
                     => Distribution n -> Distribution n -> Distribution n
convolveDistribution :: Distribution n -> Distribution n -> Distribution n
convolveDistribution (DegenerateDistribution n
d)
                     (DegenerateDistribution n
d') =
    n -> Distribution n
forall n. n -> Distribution n
DegenerateDistribution (n
dn -> n -> n
forall a. Num a => a -> a -> a
+n
d')


-- | Shift a distribution by a constant. This is equivalent to convolution
-- with a degenerate distribution.
--
-- > shiftDistribution n d = degenerateDistribution n `convolveDistribution` d
--
shiftDistribution :: Num n => n -> Distribution n -> Distribution n
shiftDistribution :: n -> Distribution n -> Distribution n
shiftDistribution n
n (DegenerateDistribution n
d) = n -> Distribution n
forall n. n -> Distribution n
DegenerateDistribution (n
nn -> n -> n
forall a. Num a => a -> a -> a
+n
d)


--
-- Sketch of API for GSV
--

-- | A \"GSV\" corresponds to a 𝚫Q that is a function of the size of a data
-- unit to be transmitted over a network. That is, it gives the 𝚫Q of the
-- transmission time for different sizes of data in 'SizeInBytes'.
--
-- The 𝚫Q is broken out into three separate 𝚫Q distributions, 𝚫Q∣G, 𝚫Q∣S and
-- 𝚫Q∣V, with the overall 𝚫Q being the convolution of the three components.
-- The G and S components captures the /structural/ aspects of networks, while
-- the V captures the /variable/ aspects:
--
-- [/G/]: the /geographical/ component of network delay. This is the minimum
--        time to transmit a hypothetical zero-sized data unit. This component
--        of the distribution does not depend on the data unit size. It is
--        a degenerate distribution, taking only one value.
--
-- [/S/]: the /serialisation/ component of network delay. This is time to
--        serialise a data unit as it is being transmitted. This is of course
--        a function of the data unit size. For each size it is a degenerate
--        distribution, taking only one value.
--
-- [/V/]: the /variable/ aspect of network delay. This captures the variability
--        in network delay due to issues such as congestion. This does not
--        depend on the data unit size, and is /not/ a degenerate disruption.
--
-- For ballistic transmission of packets, /S/ is typically directly proportional
-- to the size. Thus the combination of /G/ and /S/ is simply a linear function
-- of the size.
--
data GSV = GSV !DiffTime                  -- G as seconds
               !(SizeInBytes -> DiffTime) -- S as seconds for size
               !(Distribution DiffTime)   -- V as distribution

-- | GSVs are semi-groups by convolution on the three individual components.
--
instance Semigroup GSV where
  GSV DiffTime
g1 SizeInBytes -> DiffTime
s1 Distribution DiffTime
v1 <> :: GSV -> GSV -> GSV
<> GSV DiffTime
g2 SizeInBytes -> DiffTime
s2 Distribution DiffTime
v2 = DiffTime
-> (SizeInBytes -> DiffTime) -> Distribution DiffTime -> GSV
GSV (DiffTime
g1DiffTime -> DiffTime -> DiffTime
forall a. Num a => a -> a -> a
+DiffTime
g2) (\SizeInBytes
sz -> SizeInBytes -> DiffTime
s1 SizeInBytes
sz DiffTime -> DiffTime -> DiffTime
forall a. Num a => a -> a -> a
+ SizeInBytes -> DiffTime
s2 SizeInBytes
sz) (Distribution DiffTime
v1 Distribution DiffTime
-> Distribution DiffTime -> Distribution DiffTime
forall a. Semigroup a => a -> a -> a
<> Distribution DiffTime
v2)


instance Show GSV where
    show :: GSV -> String
show (GSV DiffTime
g SizeInBytes -> DiffTime
s (DegenerateDistribution DiffTime
v)) =
        String
"GSV g " String -> ShowS
forall a. [a] -> [a] -> [a]
++ DiffTime -> String
forall a. Show a => a -> String
show DiffTime
g String -> ShowS
forall a. [a] -> [a] -> [a]
++ String
" s " String -> ShowS
forall a. [a] -> [a] -> [a]
++ DiffTime -> String
forall a. Show a => a -> String
show (SizeInBytes -> DiffTime
s SizeInBytes
1) String -> ShowS
forall a. [a] -> [a] -> [a]
++ String
" v " String -> ShowS
forall a. [a] -> [a] -> [a]
++ DiffTime -> String
forall a. Show a => a -> String
show DiffTime
v

-- | The case of ballistic packet transmission where the /S/ is directly
-- proportional to the packet size.
--
ballisticGSV :: DiffTime               -- ^ /G/
             -> DiffTime               -- ^ /S/ as time per byte.
             -> Distribution DiffTime  -- ^ /V/ distribution
             -> GSV
ballisticGSV :: DiffTime -> DiffTime -> Distribution DiffTime -> GSV
ballisticGSV DiffTime
g DiffTime
s Distribution DiffTime
v = DiffTime
-> (SizeInBytes -> DiffTime) -> Distribution DiffTime -> GSV
GSV DiffTime
g (\SizeInBytes
sz -> DiffTime
s DiffTime -> DiffTime -> DiffTime
forall a. Num a => a -> a -> a
* SizeInBytes -> DiffTime
forall a b. (Integral a, Num b) => a -> b
fromIntegral SizeInBytes
sz) Distribution DiffTime
v


--
-- Basic calculations based on GSV
--

type SizeInBytes = Word32

-- | The 𝚫Q for when the leading edge of a transmission unit arrives at the
-- destination. This is just the convolution of the /G/ and /V/ components.
--
gsvLeadingEdgeArrive  :: GSV ->                DeltaQ

-- | The 𝚫Q for when the trailing edge of a transmission unit departs the
-- sending end. This is just the convolution of the /S/ and /V/ components.
--
-- Since it involves /S/ then it depends on the 'SizeInBytes' of the
-- transmission unit.
--
gsvTrailingEdgeDepart :: GSV -> SizeInBytes -> DeltaQ

-- | The 𝚫Q for when the trailing edge of a transmission unit arrives at the
-- destination. This is the convolution of the /G/, /S/ and /V/ components.
--
-- Since it involves /S/ then it depends on the 'SizeInBytes' of the
-- transmission unit.
--
gsvTrailingEdgeArrive :: GSV -> SizeInBytes -> DeltaQ

gsvLeadingEdgeArrive :: GSV -> DeltaQ
gsvLeadingEdgeArrive (GSV DiffTime
g SizeInBytes -> DiffTime
_s Distribution DiffTime
v) =
  Distribution DiffTime -> DeltaQ
DeltaQ (DiffTime -> Distribution DiffTime -> Distribution DiffTime
forall n. Num n => n -> Distribution n -> Distribution n
shiftDistribution DiffTime
g Distribution DiffTime
v) -- dubious for anything other than ballistic

gsvTrailingEdgeDepart :: GSV -> SizeInBytes -> DeltaQ
gsvTrailingEdgeDepart (GSV DiffTime
_g SizeInBytes -> DiffTime
s Distribution DiffTime
v) SizeInBytes
bytes =
  Distribution DiffTime -> DeltaQ
DeltaQ (DiffTime -> Distribution DiffTime -> Distribution DiffTime
forall n. Num n => n -> Distribution n -> Distribution n
shiftDistribution (SizeInBytes -> DiffTime
s SizeInBytes
bytes) Distribution DiffTime
v)

gsvTrailingEdgeArrive :: GSV -> SizeInBytes -> DeltaQ
gsvTrailingEdgeArrive (GSV DiffTime
g SizeInBytes -> DiffTime
s Distribution DiffTime
v) SizeInBytes
bytes =
  Distribution DiffTime -> DeltaQ
DeltaQ (DiffTime -> Distribution DiffTime -> Distribution DiffTime
forall n. Num n => n -> Distribution n -> Distribution n
shiftDistribution (DiffTime
g DiffTime -> DiffTime -> DiffTime
forall a. Num a => a -> a -> a
+ SizeInBytes -> DiffTime
s SizeInBytes
bytes) Distribution DiffTime
v)


--
-- Derived calculations for bi-directional GSV
--

-- | The 'GSV' for both directions with a peer, outbound and inbound.
--
data PeerGSV = PeerGSV {
                 PeerGSV -> Time
sampleTime  :: !Time,
                 PeerGSV -> GSV
outboundGSV :: !GSV,
                 PeerGSV -> GSV
inboundGSV  :: !GSV
               }
               deriving Int -> PeerGSV -> ShowS
[PeerGSV] -> ShowS
PeerGSV -> String
(Int -> PeerGSV -> ShowS)
-> (PeerGSV -> String) -> ([PeerGSV] -> ShowS) -> Show PeerGSV
forall a.
(Int -> a -> ShowS) -> (a -> String) -> ([a] -> ShowS) -> Show a
showList :: [PeerGSV] -> ShowS
$cshowList :: [PeerGSV] -> ShowS
show :: PeerGSV -> String
$cshow :: PeerGSV -> String
showsPrec :: Int -> PeerGSV -> ShowS
$cshowsPrec :: Int -> PeerGSV -> ShowS
Show

-- | The current tracking model is based on an EWMA
--  (https://en.wikipedia.org/wiki/Moving_average#Exponential_moving_average).
--  Typically implementations of EWMA assume a regular update, but EWMA is based
--  on Exponential Smoothing
--  (https://en.wikipedia.org/wiki/Exponential_smoothing). Such smoothing has a
--  time constant, which captures the time for a unit impulse to decay to 1 -
--  1/e (~ 63.2%), the &#x1D6FC (smoothing factor) is a function of relative
--  frequency of the sample interval and this time constant.
--
-- The approach being taken here is one that does not assume a fixed sample
-- interval (and hence a fixed &#x1D6FC), instead we calculate, given the
-- interval from when the last sample was taken, the &#x1D6FC needed to ensure
-- that the old value has sufficiently decayed.
--
-- The exact calcuation involves exponentiation, however where the number of
-- samples within the time constant is sufficiently large a simple ratio of the
-- sample's interval over the time constant will suffice. The relative error of
-- this numerical approximation is, for our use case, small. Eg 1/50 (20s
-- between samples with a 1000s time constant) has a relative error of 1%. The
-- expected typical range of this relative error is between 5% (ratio of 1/10),
-- to 0.5% (1/100).
--
-- Given the inherent measurement noise in this measurement, the use of the
-- approximation is well justified. We choose (reaonably aribtarily) 1000s as
-- the time constant, it is unclear if this should be a configuration variable
-- or not.
-- Note that this semigroup is  is non-commutative. The new value must come first.
instance Semigroup PeerGSV where
  <> :: PeerGSV -> PeerGSV -> PeerGSV
(<>) PeerGSV
a PeerGSV
b = let timeConstant :: DiffTime
timeConstant = DiffTime
1000 :: DiffTime
                 sampleInterval :: DiffTime
sampleInterval = PeerGSV -> Time
sampleTime PeerGSV
a Time -> Time -> DiffTime
`diffTime` PeerGSV -> Time
sampleTime PeerGSV
b
                 alpha :: DiffTime
alpha = (DiffTime
sampleInterval DiffTime -> DiffTime -> DiffTime
forall a. Fractional a => a -> a -> a
/ DiffTime
timeConstant) DiffTime -> DiffTime -> DiffTime
forall a. Ord a => a -> a -> a
`min` DiffTime
1
                 updateG :: GSV -> GSV -> GSV
updateG (GSV DiffTime
g1 SizeInBytes -> DiffTime
s Distribution DiffTime
v) (GSV DiffTime
g0 SizeInBytes -> DiffTime
_ Distribution DiffTime
_)
                   = DiffTime
-> (SizeInBytes -> DiffTime) -> Distribution DiffTime -> GSV
GSV (DiffTime
g0 DiffTime -> DiffTime -> DiffTime
forall a. Num a => a -> a -> a
+ DiffTime
alpha DiffTime -> DiffTime -> DiffTime
forall a. Num a => a -> a -> a
* (DiffTime
g1 DiffTime -> DiffTime -> DiffTime
forall a. Num a => a -> a -> a
- DiffTime
g0)) SizeInBytes -> DiffTime
s Distribution DiffTime
v
             in PeerGSV :: Time -> GSV -> GSV -> PeerGSV
PeerGSV { sampleTime :: Time
sampleTime  = PeerGSV -> Time
sampleTime PeerGSV
a
                        , outboundGSV :: GSV
outboundGSV = GSV -> GSV -> GSV
updateG (PeerGSV -> GSV
outboundGSV PeerGSV
a) (PeerGSV -> GSV
outboundGSV PeerGSV
b)
                        , inboundGSV :: GSV
inboundGSV  = GSV -> GSV -> GSV
updateG (PeerGSV -> GSV
inboundGSV  PeerGSV
a) (PeerGSV -> GSV
inboundGSV  PeerGSV
b)
                        }

-- | This is an example derived operation using the other 'GSV' and 'DeltaQ'
-- primitives.
--
-- It calculates the 𝚫Q for the time to send a request of a certain size and
-- receive a reply of an expected size. It then takes the 99% percentile as an
-- approximation of the maximum time we might be prepared to wait.
--
-- > deltaqQ99thPercentile $
-- >     gsvTrailingEdgeArrive outboundGSV reqSize
-- >  <> gsvTrailingEdgeArrive inboundGSV respSize
--
--
-- This is not realistic in that it omits processing time, but that could be
-- added as yet another 'DeltaQ' value, if there's any estimate for it:
--
-- > deltaqQ99thPercentile $
-- >     gsvTrailingEdgeArrive outboundGSV reqSize
-- >  <> gsvTrailingEdgeArrive inboundGSV respSize
-- >  <> processingDeltaQ
--
gsvRequestResponseDuration :: PeerGSV
                           -> SizeInBytes -- ^ Request size
                           -> SizeInBytes -- ^ Expected response size
                           -> DiffTime
gsvRequestResponseDuration :: PeerGSV -> SizeInBytes -> SizeInBytes -> DiffTime
gsvRequestResponseDuration PeerGSV{GSV
outboundGSV :: GSV
outboundGSV :: PeerGSV -> GSV
outboundGSV, GSV
inboundGSV :: GSV
inboundGSV :: PeerGSV -> GSV
inboundGSV}
                           SizeInBytes
reqSize SizeInBytes
respSize =
    DeltaQ -> DiffTime
deltaqQ99thPercentile (DeltaQ -> DiffTime) -> DeltaQ -> DiffTime
forall a b. (a -> b) -> a -> b
$
        GSV -> SizeInBytes -> DeltaQ
gsvTrailingEdgeArrive GSV
outboundGSV SizeInBytes
reqSize
     DeltaQ -> DeltaQ -> DeltaQ
forall a. Semigroup a => a -> a -> a
<> GSV -> SizeInBytes -> DeltaQ
gsvTrailingEdgeArrive GSV
inboundGSV SizeInBytes
respSize


defaultGSV :: PeerGSV
defaultGSV :: PeerGSV
defaultGSV = PeerGSV :: Time -> GSV -> GSV -> PeerGSV
PeerGSV {Time
sampleTime :: Time
sampleTime :: Time
sampleTime, GSV
outboundGSV :: GSV
outboundGSV :: GSV
outboundGSV, GSV
inboundGSV :: GSV
inboundGSV :: GSV
inboundGSV }
  where
    default_g :: DiffTime
default_g = DiffTime
500e-3 -- Old hardcoded default value. Only available value when running without KeepAlive.
    default_s :: DiffTime
default_s = DiffTime
2e-6 -- 4Mbps.
    inboundGSV :: GSV
inboundGSV  = DiffTime -> DiffTime -> Distribution DiffTime -> GSV
ballisticGSV DiffTime
default_g DiffTime
default_s (DiffTime -> Distribution DiffTime
forall n. n -> Distribution n
degenerateDistribution DiffTime
0)
    outboundGSV :: GSV
outboundGSV = GSV
inboundGSV
    sampleTime :: Time
sampleTime  = DiffTime -> Time
Time DiffTime
0

fromSample :: Time -> Time -> SizeInBytes -> PeerGSV
fromSample :: Time -> Time -> SizeInBytes -> PeerGSV
fromSample t :: Time
t@(Time DiffTime
start) (Time DiffTime
end) SizeInBytes
_size =
    PeerGSV :: Time -> GSV -> GSV -> PeerGSV
PeerGSV  {Time
sampleTime :: Time
sampleTime :: Time
sampleTime, GSV
outboundGSV :: GSV
outboundGSV :: GSV
outboundGSV, GSV
inboundGSV :: GSV
inboundGSV :: GSV
inboundGSV }
  where
    g :: DiffTime
g =  (DiffTime
end DiffTime -> DiffTime -> DiffTime
forall a. Num a => a -> a -> a
- DiffTime
start) DiffTime -> DiffTime -> DiffTime
forall a. Fractional a => a -> a -> a
/ DiffTime
2

    sampleTime :: Time
sampleTime = Time
t
    inboundGSV :: GSV
inboundGSV = DiffTime -> DiffTime -> Distribution DiffTime -> GSV
ballisticGSV DiffTime
g DiffTime
2e-6 (DiffTime -> Distribution DiffTime
forall n. n -> Distribution n
degenerateDistribution DiffTime
0)
    outboundGSV :: GSV
outboundGSV  = GSV
inboundGSV