Network management and QoS provisioning - Network Calculus

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1 Network Calculus Network calculus is a metodology to study in a deterministic approach theory of queues. First a linear modelization is needed: it means that, for example, a system like: ρ can be modelized like: σ(t) x(t) ρ y(t) where:. x(t) is the arrival function;. y(t) is the departure function;. σ(t) is a sort of transmission function. The purpose is to define: y(t) = x(t) σ(t) because in that way the system is linear. From this assumption it is possible deduce that the following system: x(t) σ 1 (t) σ 2 (t) y(t) can be reduced as: x(t) σ T OT (t) y(t) where: y(t) = x(t) (σ 1 (t) σ 2 (t)) = y(t) = x(t) σ T OT (t) Infact, if the system is linear, the following relation is always true: σ T OT (t) = σ 1 (t) σ 2 (t) (1) 1

2 Algebra In order to define compute easily convolutions a new algebra is introduced, the min-plus-algebra. The standard algebra is defined as: A (R, +, ) while the min-plus-algebra has this definition: MPA (R, min{}, +) With the two operators, min{} and +, all properties are satisfy in order to define a new algebra. Let us look at the following example. Example Compute (3 + 4) 5 in algebra A. One way to solve it is: (3 5) + (4 5) = = 35 Now the same problem it is solved with min-plus-algebra: min{3, 4} + 5 = min{(3 + 5), (4 + 5)} = min{8, 9} = 8 Convolution Given the min-plus-algebra, how would be the convolution? In algebra A the convolution is defined as: (x h)(t) = and the following relation holds: (x h)(t) = t t x(t s) h(s) ds h(t s) x(s) ds The integral operator is simply a sum for all terms form to t so it is replaced with min{} while the multiplication is replaced with +: (x σ)(t) = min{x(t s) + σ(s)} <s<t Proof and: Putting: (t s) = θ = s = t θ s = = θ = t s = t = θ = At the end: (x σ)(t) = min{x(θ) + σ(t θ)} <θ<t which is again a min-plus-convolution. Notice that means A convolution while min-plus-convolution. 2

3 Linear property If in the min-plus-algebra it is possible to realize a minplus-convolution it is possible too characterize linear systems by mean of the equivalence of: [(x 1 + x 2 ) h](t) = [x 1 h](t) + [x 2 h](t) Infact, this is the outcome of a system which has 2 inputs added together. In the min-plus-algebra this operation will be: [(x 1 + x 2 ) σ](t) = min{(x 1 σ), (x 2 σ)} Exercise Consider x(t) defined for t, an increasing and continous function and consider δ T (t) defined as: δ T (t) T t where T >. Compute: [x δ T ](t) Solution It is possible find two cases: [x δ T ](t) = min{x(s) + δ T (t s)} s t. (t s) T = δ T (t s) = ;. (t s) > T = δ T (t s) =. The second one does not influence at all the minimum operation, so only the case one is considered. Re-writing the condition: (t s) T = s t T There is another condition that we have to take in account: s t Putting them together: { s t T we are able to find two cases: s t 3

4 . case 1: t T = (t T ) s t; it means that the operation who have to be compute is: min{x(s)} (t T ) s t = x(t T ). case 2: t T < = s (t T ); it means that the operation who have to be compute is: Final, the two solutions are: [x δ T ](t) = min{x(s)} s (t T ) = x() { x(t T ) t T x() t T < The first one it s the same initial function tralsated by T. Queuing systems Considering queuing systems as: R(t) S R (t) we have to characterize:. the input R(t);. the system S;. the output R (t). Input Curve R(t) is the cumulative amount of data received up to time t. Those assuptions are assumed:. R() = ;. t. Properties. R(t) is an increasing function;. the arrival rate r(t) is defined as: r(t) = R(t) t 4

5 Output Curve R (t) is the cumulative amount of data leaving the server up to time t. As before, the following assuptions are assumed:. R () = ;. t. Properties. R (t) is an increasing function;. the istantaneous departure rate r (t) is defined as: Relation r (t) = R (t) t There is a very important relation between input and output curve. Look at the following graph: Byte R (t) R(t) t This situation is impossible: it means that there is more data leaving the server than data arriving into the server. So: Considerations about system R (t) R(t) Looking at a graph in where the two curves are represented it is possible to make considerations about the system. For example: Byte R(t) R (t) d(t) t x t 5

6 In the previous graph we can identify:. BACKLOG: t x is the time in which a certain amount of data is stored into the system; in general: BACKLOG(t) = R(t) R (t) this parameter can be visualized looking the horizontal double arrow;. VIRTUAL DELAY: d(t) is the period of time in which input and output have the same amount of data; in general it is true for: R(t) = R (t + d(t)) It is possible to characterize the term d(t) as: d(t) = inf {R(t) R (t + τ)} where τ. Infact, τ = is a particular case in which: R(t) = R (t) The virtual delay is really a delay if FIFO discipline in provided by the system. The standard definitio is reported: the virtual delay is the delay that would be expected by a bit arriving at time t if alla bits received before it were served in FIFO order. Bounds bounds: Over the two parameters already characterized is useful introduce. MAX BACKLOG: max{r(t) R (t)};. MAX VIRTUAL DELAY: max{d(t)}. Observe that compute the maximum of the curves on the graph it is not practical so input and output of the system have to be describe in another way in order to compute easily bounds. Arrival curve An arrival curve is described as:. α(θ): increasing function with θ > ;. flow R(t) is constrained by α (arrival classes) in the following way: R(t + θ) R(t) α(θ) θ, t 6

7 The term R(t + θ) R(t) represents the total amount of data arrived into the period [t, t + θ]. As before there is a condition that have to be always respected: R(t) R(s) α(t s) where t and s are two istants of time. To understand it, consider for α(θ) the following function: α(θ) θ In order to not violate the constraint, taken two instants s and t, the graph will be: α(θ) s t θ Examples Some examples of α classes are reported below. CBR source: the bit rate is constant so data grows up constantly; the graph is: α(θ) r θ it can be defined as γ r, ; 7

8 source that emits b bits then stops; the graph is: α(θ) b θ token bucket that emits an initial burstiness, then it is constant; the graph is: α(θ) r b θ it can be defined as γ r,b, where:. r is the rate of the bucket, also called ρ;. b is the bucket size. Theorem R(t) is constrained by α if and only if: R (R α) Proof Using the constraint of arrival curve: R(t) R(s) α(t s) t, s t so: R(t) R(s) + α(t s) t, s t if it is true for s t, the previous expression is also true for the infimum: R(t) inf R(s) + α(t s) but that expression is: R(t) (R α) 8

9 Service Curve Till now only the input has been defined. In order to define the output let us start with two examples. GPS Consider a GPS system in which each flow is served at rate r (it is also the minimum rate). R(t) GPS(r) R (t) The departure rate should satisfy: R (t) R (t ) r(t t ) (2) because r(t t ) it is the minimum service rate in the period [t, t]: it is simply γ r,. Consider now as an hypothesis, the fact that at the beginning input and output have the same value: and by appling 2: R(t ) = R (t ) (3) R (t) R (t ) + r(t t ) t, t t R (t) inf {R (t ) + r(t t )} but, for 3, at time t queues inside the system are empty, so: R (t) inf {R(t ) + r(t t )} and this is nothing else that: R (t) [R r(t t )] R (t) (R γ r, ) Notice that this relation is in the form: output=input trans. function. 9

10 Max Delay System In this example a max delayed system in which T is the maximum delay is taken in account: R(t) MD(T ) R (t) As an hypothesis a FIFO order is considered; it means that: If FIFO is implemented: So: d(t) T t R(t) = R (t + d(t)) (t + d(t)) < (t + T ) R (t + d(t)) < R (t + T ) R(t) R (t + T ) = R R(t T ) But the last expression is nothing else that: Definition R R δ T (t) Now it is possible define the service curve which describe completly the behavior of the following system: R(t) β R (t) β is a service curve if and only if: R R β 1

11 Resume Type of curve Formula α arrival curve R (R α) β service curve R (R β) Considerations about system As before, it is possible to define bounds looking at the two curves together on a graph; for example: Byte(θ) α(θ) β(θ) θ The maximum BACKLOG is the maximum vertical gap while the maximum d(t) is the maximum horizontal gap. Theorem The difference R(t) R (t) is: Deconvolution R(t) R (t) sup{α(θ) β(θ)} It is possible consider that R (t) can be defined as an arrival curve of another system: in this case it is called α. R(t) α S β R (t) α Using the max-plus-algebra: α = α β The max-plus-convolution is defined as: (α β)(t) = sup{α(t + s) β(t)} 11

12 Example Consider a token bucket constrained source and a latency rate server (Weighted Fair Queue). 12