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.

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.

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.

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.

The case of ballistic packet transmission where the S is directly proportional to the packet size.

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.

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.

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.

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.

Make a degenerate distribution.

https://en.wikipedia.org/wiki/Degenerate_distribution

The GSV for both directions with a peer, outbound and inbound.

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