Packetization Guaranteed Rate Nodes · Greedy Shaper and Packetizer Consider a -packetizer and a “good” func-tion . Call the greedy shaper with shap-ing curve . Assume that there

Packetization

&

Guaranteed Rate NodesK-G Stenborg

[email protected]

Packetization & Guaranteed Rate Nodes – p.1/49

Variable Length Packets

A sequence

�

of cumulative packet lengths is awide sense increasing sequence such that��

� � ��

is finite.We interpret

� � � � � ��

as the length of the

th packet.For any real number � define� � � � � ��

� � ��

. We have thecharacterization� � � � � � � � � ��

A

�

-packetizer is the system transforming the in-

put

��

into� � � � � �

.


Packetizer

We say that a flow is

�

-packetized if� � � � � � � � � �

for all

�

.


Greedy Shaper and Packetizer

Consider a

�

-packetizer

�

and a “good” func-

tion �. Call � the greedy shaper with shap-

ing curve �. Assume that there is a sub-additive

function �� and a number� � � � � such that

�� . Then for any input the out-

put of the composition

��

�

is �-smooth.


Packetized Greedy Shapers (P. G. S.)

Consider an input sequence of packets,represented by the function

��

�� .

Call

�

the cumulative packet lengths.

Packetized Shaper (with shaping curve ): Asystem that forces its output to have as anarrival curve and be -packetized.Packetized Greedy Shaper: A packetized shaperthat delays the input packets in a buffer,whenever sending a packet would violate theconstraint , but outputs them as soon aspossible.




��

�� .

Call

�

the cumulative packet lengths.Packetized Shaper (with shaping curve �): Asystem that forces its output to have � as anarrival curve and be

�

-packetized.

Packetized Greedy Shaper: A packetized shaperthat delays the input packets in a buffer,whenever sending a packet would violate theconstraint , but outputs them as soon aspossible.




��

�� .

Call

�

the cumulative packet lengths.Packetized Shaper (with shaping curve �): Asystem that forces its output to have � as anarrival curve and be

�

-packetized.Packetized Greedy Shaper: A packetized shaperthat delays the input packets in a buffer,whenever sending a packet would violate theconstraint �, but outputs them as soon aspossible.


Proposition

If ��

�� then the packetizedgreedy shaper blocks all packets for ever(namely,

� � � � �

).Thus in this seminar, we assume that

��

�� for

� � �

.

Thus the arrival curve � has a discontinuity at the

origin (at least as large as one maximum packet

size).


Realization of the P. G. S.

Consider a sequence

�

of cumulative packet

lengths and a “good” function � that satisfies

�� there �� is sub-additive and

� �� . Let the inputs be

�packetized. Then the

packetized greedy shaper for � and

�

can be re-

alized as the concatenation of the greedy shaper

with shaping curve � and the

�

-packetizer.


Proof

� � �

packetized input.

� � �

packetized greedyshaper output. .

The definition of the packetized greedy shaperimplies that .

=>


Proof

� � �

packetized input.

� � �


��

��

��

� � � � � � � � � �

The definition of the packetized greedy shaperimplies that .

=>


Proof

� � �

packetized input.

� � �


��

��

��

� � � � � � � � � �

The definition of the packetized greedy shaperimplies that

�� .

=>


Proof

� � �

packetized input.

� � �


��

��

��

� � � � � � � � � �

The definition of the packetized greedy shaperimplies that

�� .

=> � �� Packetization & Guaranteed Rate Nodes – p.8/49

Corollary

For

�

-packetized inputs, the implementations of

buffered leaky bucket controllers based on bucket

replenishment and virtual finish times are equiva-

lent.


I/O characterisation of P. G. S.

Consider a packetized greedy shaper withshaping curve � and cumulative packet length

�

.Assume that � is a “good” function. The output� � �

of the packetized greedy shaper is given by

� � � ��

� � ��

� � ��

�

with

��

� ��

and

� � � � � � � � � ��

� ��

� � � � � � �

for

� �

.


Lemma

Consider a sequence

�

of cumulative packetlengths and a “good” function �. Among all flows

� � � �

such that(1) �

(2) � is

�

-packetized(3) � has � as an arrival curve

there is one flow� � �

that upper-bounds all. It is

given by � � � � � ��

� � ��

� � ��

�

.


Proof

If � is a solution, then � � � � � � � � � �

and thus �

. is a solution if it hold the three conditions

(1), (2) and (3) on previous slide.


Proof of

��

and by induction on

�

,

� � �

for all

�

.

Thus .


Proof of that is -packetized

Consider some fixed

�

.

� � � � � �

is

�

-packetized forall

� �

.Let

� ��

� � ��

. Since

� � � � � � ��

,

� � � ��

is in the set

� � � � ��

� � � ��

� � � ��

� ��

.

This set is finite, thus� � �

has to be one of the

� � � �

for

� � . This shows that

� � �

is

�

-

packetized, and this is true for any time

�

.


Proof of that has � as an arrival curve

� � � � ��

� � � � � � � � ��

� � � � � � �

for all

�

,thus

� � � � � � ��

� � � � � � ��

� � ��

� � ��

� � ��

��

� � ��

� � � �

.

� � �Packetization & Guaranteed Rate Nodes – p.15/49

Conservation of concave arrival constraints

Assume an

�

-packetized flow with arrival curve �

is input to a packetized greedy shaper with cumu-

lative packet length

�

and shaping curve �. As-

sume that � and � are concave with � ��

��

and ��

�� (and ��

). Then

the output flow is still constrained by the original

arrival curve �.


Proof

is �-smooth, � � . The output flow canthen be expressed as � � �

�

� �

� ��

��

� � � � � ��

� �.

Theorem 3.1.6 gives � � � � � � � ��

��

and thus

� � satisfies

��

� � � � � �� . Thus

is � �-smooth, and thus �-smooth.


Series decomposition of shapers

Consider a tandem of packetized greedyshapers in series; assume that the shaping curve

� �

of the �th shaper is concave with

� ��

� � � �� . For

�

-packetized inputs, the

tandem is equivalent to the packetized greedy

shaper with shaping curve � � � � � � � �

.


Proof (for � )

Output of tandem of shapers is� � � ��

� � ��

� ��

� ��

��

since � � � for all �. � � ��

�

is

�

-packetized and �-smooth, thus

�

.

is -packetized and -smooth. Thus thetandem is a packetized (possible non greedy)shaper. Since is the output of the packetizedgreedy shaper, we must have .


Proof (for � )


� � ��

� ��

� ��

��

since � � � for all �. � � ��

�

is

�


�

.

�

is

�

-packetized and �-smooth. Thus thetandem is a packetized (possible non greedy)shaper. Since

�� is the output of the packetized

greedy shaper, we must have

�

.


Proof (for � )


� � ��

� ��

� ��

��

since � � � for all �. � � ��

�

is

�


�

.

�

is

�

-packetized and �-smooth. Thus thetandem is a packetized (possible non greedy)shaper. Since

�� is the output of the packetized

greedy shaper, we must have

�

.

� � � �Packetization & Guaranteed Rate Nodes – p.19/49

The Effective Bandwidth

Consider a flow with cumulative function and afixed but arbitrary delay .The effective bandwidth of a flow is given by

� ��

� � � � �

� � � � ��

� � �

with a virtual delay .

The effective bandwidth of a “good” arrival curveis given by


The Effective Bandwidth

Consider a flow with cumulative function and afixed but arbitrary delay .The effective bandwidth of a flow is given by

� ��

� � � � �

� � � � ��

� � �

with a virtual delay .The effective bandwidth of a “good” arrival curveis given by

� ��

��

� � �

��

�


Equivalent Capacity

The equivalent capacity for a flow is given by

��

� � � ��

� � � � ��

� � �

and for a “good” function � it is given by

��

��

� � �

��

�

A queue with constant rate , guarantees a max-

imum backlog of for a flow if ��

.


Effective Bandwidth and Equivalent Capacity

The definition of the effective bandwidth gives

� ��

�

� ��

�

� ��

� ��

and the definition of the equivalent capacity gives

��

�

� ��

�

��

��

�

where � � �.


Packet Scheduling

Packet scheduling is the function that decides, atevery buffer inside a network node, the serviceorder for different packets.

FIFO (first in, first out) - Packets are served inthe order of arrival.

Per flow queuingProvide isolation to flowsOffer different guarantees

Example: Generalized Processor Sharing.


Packet Scheduling






Packet Scheduling






Generalized Processor Sharing (GPS)

A GPS node serves several flows in parallel, and

has a total output rate equal to � b/s. A flow

�

is allocated a given weight, �. Call ��

,

��

� � �

the input and output functions for flow

�

. The

guarantee is that for any time

�

, the service rate

offered to flow

�

is 0 if flow

�

has no backlog,

and otherwise is equal to

��

��

�, where

� � �

is the set of backlogged flows at time

�

. Thus

��

� � � ��

�

��

��

��

�� .


GPS, PGPS and GR

A GPS node is a theoretical concept, which isnot really implementable, because it relies on afluid model, and assumes that packets areinfinitely divisible.

Practical implementations of GPS are Packet

Generalized Processor Sharing (PGPS) and

Guaranteed Rate (GR).


Packet Generalized Processor Sharing (PGPS)

PGPS emulates GPS as follows. There is one

FIFO queue per flow. The scheduler handles

packets one at a time, until it is fully transmitted, at

the system rate �. For every packet, we compute

the finished time that it would have under GPS

(the “GPS-finish-time”). Then, whenever a packet

is finished transmitting, the next packet selected

for transmission is the one with the earliest GPS-

finish-time, among all packets present.


Proposition

The finish time for PGPS is at most the finish time

of GPS plus

�� , where � is the total rate and

�

is

the maximum packet size.


Proof


Guaranteed Rate (GR) Node

Consider a node that serves a flow. Packets arenumbered in order of arrival. Let � �

,

� �

be the arrival and departure times. We say that anode is a guaranteed rate node for this flow, withrate � and delay �, if it guarantees that

� �, where

�

� � �

� � ��

� ��

�


One Way Deviation of a scheduler from GPS

We say that deviates from GPS by � if for allpacket the departure time satisfies

� � �,where � is the departure time from ahypothetical GPS node that allocates a rate

� � ��

�� to this flow (assumed to be flow 1).

Theorem: If a scheduler satisfies ,then it is GR with rate and latency .

Proof: and the rest is immediate.





� � ��


Theorem: If a scheduler satisfies

� � �,then it is GR with rate � and latency �.

Proof: and the rest is immediate.





� � ��


Theorem: If a scheduler satisfies

� � �,then it is GR with rate � and latency �.

Proof: � and the rest is immediate.


Max-Plus Representation of GR

Consider a system where packets are numbered��

�� in order of arrival. Call � ,

� the arrivaland departure times for packet , and

� the sizeof packet . Define by convention

��

. Thesystem is a GR node with rate � and latency � ifand only if for all there is some

� � � ��

�

such that

� � ��

��

� � �

�

�


Proof

� � � ��

��

�


Proof

� � � ��

��

�

� ��

��

�


Proof

� � � ��

��

�

� ��

��

�

� � � ��

��

��

��

�


Proof

� � � ��

��

�

� ��

��

�

� � � ��

��

��

��

�

� � �

� � ��

� ��

��

��

��

� � � � � �

��

��

�

� � ��


Equivalence with service curve

Consider a node with

�

-packetized input.

(1) If the node guarantees a minimum servicecurve equal to the rate-latency function

� �� , and if it is FIFO, then it is a GR nodewith rate � and latency �.

(2) Conversely, a GR node with rate andlatency is the concatenation of a servicecurve element, with service curve equal to therate-latency function , and an

-packetizer. If the GR node is FIFO, then sois the service element.


Equivalence with service curve

Consider a node with

�

-packetized input.

(1) If the node guarantees a minimum servicecurve equal to the rate-latency function

� �� , and if it is FIFO, then it is a GR nodewith rate � and latency �.

(2) Conversely, a GR node with rate � andlatency � is the concatenation of a servicecurve element, with service curve equal to therate-latency function �� , and an�

-packetizer. If the GR node is FIFO, then sois the service element.


Proof (1)Consider a service curve element

�

. Assume that the in-

put and output functions

�

and

��

are right-continuous.

Consider the virtual system

� �

made of a bit-by-bit greedy

shaper with shaping curve

�� , followed by a constant bit-by-

bit delay element. The bit-by-bit greedy shaper is a constant

bit rate server, with rate �. Thus the last bit of packet � de-

parts from it exactly at time

� ,

� � �� . The output

function of

� �

is

� � � � �� . By hypothesis,

��

, and

by the FIFO assumption, this shows that the delay in

�

is

upper bounded by the delay in

� �

. Thus

� � �� .


Proof (2)


Corollary

A GR node offers a minimum service curve

��


Delay Bound

For an �-smooth flow served in a (possible nonFIFO) GR node with rate � and latency �, thedelay for any packet is bounded by

��

� ��

�

� � � �


Proof

For any fixed , we can find a

� � such that

� ��

� � � � � ��

� . The delay for a packet is

� � � � � � .Define

� � � � �� . By hypothesis��

� � �

� ��

, where ��

is the limit tothe right of � at

�

. Thus

� � � � � � ��

� ��

� � � � �.

Now ��

� � � � � ��

� � � � .


Concatenation of GR nodes

The concatenation of GR nodes (that areFIFO per flow) with rates � � and latencies � � isGR with rate � � � � � � � � and latency

� � � ��

� �� , where

�� is the

maximum packet size for the flow.


Proof

By Theorem (2), we can decompose system

�

into a concatenation ��

�, where � offers theservice curve � �� and � is the packetizer.

Call the concatenation

��

��

��

��

��

�� .

By Theorem (2), is FIFO and provides theservice curve �� . By Theorem (1), it is GR withrate � and latency �. Now does not effect thefinish time of the last bit of every packet.


End-to-end Delay Bound

A bound on the end-to-end delay through aconcatenation of GR nodes is

�

��

� �

��

��

��

��

�

� � � � � �


A Composite Node

We consider a composite node, made of twocomponents. The former (“variable delaycomponent”) imposes to packets a delay in therange

��

��

� � ��

. The latter is FIFO andoffers to its input the packet scale rate guarantee,with rate � and latency �.

If the variable delay component is known to beFIFO, then we have a simple result.


Variable Delay as GR

Lemma:Consider a node which is known to guarantee adelay

�� . The node need not be FIFO. Call�

� � the minimum packet size.

For any � � �

, the node is GR with latency � �

��

� � � ��

� �

and rate �.


Proof

With the standard notation in this section, the

hypothesis implies that

� � �� for all

�

. We have �

��

�

� � � �� , thus

� � ��

� � � ��

��

� � � ��

� �

.


Composite GR Node with FIFO Variable Delay Component

Consider the concatenation of two nodes. The

former imposes to packets a delay�

�� . The

latter is a GR node with rate � and latency �. Both

nodes are FIFO. The concatenation of the two

nodes, in any order, is GR with rate � and latency

��

�� .


Proof

The former node is GR( � ��

��

��

� �

� �

) for

any � � � �. We know that the concatenation is

GR( ��

� ��

� � ). Let � �

go to .


GR nodes that are not FIFO per flowConsider the concatenation of two nodes. The first imposes

to packets a delay in the range

� ��

��

�

. The second is

FIFO and offers the guaranteed rate clock service to its in-

put, with rate � and latency� . The first node is not assumed

to be FIFO, so the order of packets arrivals at the second

node is not the order of packets arrivals at the first one.

Assume that the fresh input is constrained by a continuous

arrival curve � � �

. The concatenation of the two nodes, in

this order, offers to the fresh input the guaranteed rate clock

service with rate � and latency��

� .


Proof


Exercises

Problem 1.31 & 1.38

Proof of Example: GPS (after definition 2.1.1).


Documents

Packetization Guaranteed Rate Nodes · Greedy Shaper and Packetizer Consider a -packetizer and a “good” func-tion . Call the greedy shaper with shap-ing curve . Assume that there