Network calculus

Modelling flow and server

In network calculus, a flow is modelled as cumulative functions A, where A(t) represents the amount of data (number of bits for example) send by the flow in the interval [0,t). Such functions are non-negative and non-decreasing. The time domain is often the set of non negative reals.

A : R + → R +

∀ u , t ∈ R + : u < t ⟹ A ( u ) ≤ A ( t )

A server can be a link, a scheduler, a traffic shaper, or a whole network. It is simply modelled as a relation between some arrival cumulative curve A and some departure cumulative curve D. It is required that A ≥ D, to model the fact that the departure of some data can not occur before its arrival.

Modelling backlog and delay

Given some arrival and departure curve A and D, the backlog at any instant t, denoted b(A,D,t) can be defined as the difference between A and D. The delay at t, d(A,D,t) is defined as the minimal amount of time such that the departure function reached the arrival function. When considering the whole flows, the supremum of these values is used.

b ( A , D , t ) := A ( t ) − D ( t )

d ( A , D , t ) := inf { d ∈ R +   s . t .   D ( t + d ) ≥ A ( t ) }

b ( A , D ) := sup t ≥ 0 { A ( t ) − D ( t ) }

d ( A , D ) := sup t ≥ 0 { inf { d ∈ R +   s . t .   D ( t + d ) ≥ A ( t ) } }

In general, the flows are not exactly known, and only some constraints on flows and servers are known (like the maximal number of packet sent on some period, the maximal size of packets, the minimal link bandwidth). The aim of network calculus is to compute upper bounds on delay and backlog, based on these constraints. To do so, network calculus uses the min-plus algebra.

Min-plus algebra

In filter theory and linear systems theory the convolution of two functions f and g is defined as

( f ∗ g ) ( t ) := ∫ 0 τ f ( τ ) ⋅ g ( t − τ ) d τ

In min-plus algebra the sum is replaced by the minimum respectively infimum operator and the product is replaced by the sum. So the min-plus convolution of two functions f and g becomes

( f ⊗ g ) ( t ) := inf 0 ≤ τ ≤ t { f ( τ ) + g ( t − τ ) }

e.g. see the definition of service curves. Convolution and min-plus convolution share many algebraic properties. In particular both are commutative and associative.

A so-called min-plus de-convolution operation is defined as

( f ⊘ g ) ( t ) := sup τ ≥ 0 { f ( t + τ ) − g ( τ ) }

e.g. as used in the definition of traffic envelopes.

The vertical and horizontal deviations can be expressed in terms of min-plus operators.

b ( f , g ) = ( f ⊘ g ) ( 0 )

d ( f , g ) = inf { w : ( f ⊘ g ) ( − w ) ≤ 0 }

Traffic envelopes

Cumulative curves are real behaviours, unknown at design time. What is known is some constraint. Network calculus uses the notion of traffic envelope, also known as arrival curves.

A cumulative function A is said to conform to an envelope (or arrival curve) E, iff for all t it holds that

E ( t ) ≥ sup τ ≥ 0 { A ( t + τ ) − A ( τ ) } = ( A ⊘ A ) ( t ) .

Two equivalent definitions can be given

A ≤ A ⊗ E

Thus, E places an upper constraint on flow A. Such function E can be seen as an envelope that specifies an upper bound on the number of bits of flow seen in any interval of length t starting at an arbitrary τ, cf. eq. (1).

Service curves

In order to provide performance guarantees to traffic flows it is necessary to specify some minimal performance of the server (depending on reservations in the network, or scheduling policy, etc.). Service curves provide a means of expressing resource availability. Several kinds of service curves exists, like weakly strict, variable capacity node, etc. See for an overview.

Minimal service

Let A be an arrival flow, arriving at the ingress of a server, and D be the flow departing at the egress. The system is said to provide a simple minimal service curve S to the pair (A,B), if for all t it holds that D ( t ) ≥ ( A ⊗ S ) ( t ) .

Strict minimal service

Let A be an arrival flow, arriving at the ingress of a server, and D be the flow departing at the egress. A backlog period is an interval I such that, on any t ∈ I, A(t)>D(t).

The system is said to provide a strict minimal service curve S to the pair (A,B) iff, ∀ s , t ∈ R + , such that s ≤ t , if ( s , t ] is a backlog period, then D ( t ) − D ( s ) ≥ S ( t − s ) .

If a server offers a strict minimal service of curve S, it also offers a simple minimal service of curve S.

Basic results: Performance bounds and envelope propagation

From traffic envelope and service curves, some bounds on the delay and backlog, and an envelope on the departure flow can be computed.

Let A be an arrival flow, arriving at the ingress of a server, and D be the flow departing at the egress. If the flow as a traffic envelope E, and the server provides a minimal service of curve S, then the backlog and delay can be bounded:

b ( A , D ) ≤ b ( E , S )

d ( A , D ) ≤ d ( E , S )

Moreover, the departure curve has envelope E ′ = E ⊘ S .

Moreover, these bounds are tight i.e. given some E, and S, one may build an arrival and departure such that b(A,D) = b(E,S) and v(A,D)=v(E,S).

Concatenation / PBOO

Consider a sequence of two servers, when the output of the first one is the input of the second one. This sequence can be seen as a new server, built as the concatenation of the two other ones.

Then, if the first (resp. second) server offers a simple minimal service S 1 (resp. S 2 ), then, the concatenation of both offers a simple minimal service S e 2 e = S 1 ⊗ S 2 .

The proof does iterative application of the definition of service curves X ≥ A ⊗ S 1 , D ≥ X ⊗ S 2 and some properties of convolution, isotonicity ( D ≥ ( X ⊗ S 2 ) ⊗ S 1 ), and associativity ( D ≥ X ⊗ ( S 2 ⊗ S 1 ) ).

The interest of this result is that the end-to-end delay bound is not greater than the sum of local delays: d ( E , S 2 ⊗ S 1 ) ≤ d ( E , S 1 ) + d ( E ⊘ S 1 , S 2 ) .

This result is known as Pay burst only once (PBOO).

Tools

There are several tools based on network calculus.