Resource Allocation: Deterministic Analysis

1
Resource Allocation Deterministic Analysis
2
Traffic Model
Stochastic Different sample paths with
different properties Expected case analysis
Deterministic Properties applying to every sample
path worst case analysis
3
System Description
Single server service rate C
Single buffer Finite Infinite
Arrival stream
Departure Stream
Output Queued System
4
Reichs Equation
A(t) Total traffic arriving in interval (0, t)
D(t) Total traffic departing in interval (0, t)
X(t) Queue length at time t
X(t) sup0?s?t (A(t) A(s) C(t-s))
D(t) A(t) X(t) A(t) - sup0?s?t
(A(t) A(s) C(t-s)) inf0?s?t (A(s)
C(t-s))
5
Convolution
D(t) inf0?s?t (A(s) C(t-s))
inf0?s?t (A(s) B(t-s))
AB(t) inf0?s?t (A(s) B(t-s))
?(t) 0 for t ? 0 ? otherwise
A ?(t) A
?d(t)?(t-d)
A ?d(t) A(t-d)
For causal B, B ? ?, thus A B ? A ? A
6
Service curve for network elements
S(t)
D(t)
A(t)
?S(t)
S(t) and?S(t) are nonnegative, non-decreasing
causal functions such that D ? A S and D ? A
?S The first is minimum service curve and the
second the maximum service curve If D AF for
some non-negative, non-decreasing causal F, then
F is the service curve
7
Examples
packetizer
coder
Coder emits bursts of bytes at rate r
Packetizer packetizes L bits in a packet, if less
than L bits are available, then packetizes
whatever available
Maximum delay is L/r D ? A(t
L/r) Minimum service curve is S ?L/r
8
Constant rate server
S(t) C max(t, 0)
Coder Packetizer
Constant rate server S
Dp
D
A
D Dp S ? (A ?L/r ) S A
(?L/r S)
Minimum service curve is ?L/r S
9
Delay in a network element
FIFO service
Let u be the minimum time such that all the data
arriving in 0, t depart by time u Then u
infs D(s) ? A(t)
Maximum delay ?max supt?0 u t infs D(s) ?
A(t)
This is the maximum horizontal distance between A
and D
D ? (A ??max )
If A curve is shifted to the right by ?max it is
below D
10
Envelope
A function E(t) is an envelope for the arrival
function A(t) if A(t) A(u) ? E(t-u) for all u
and t, 0 ? u ? t That is A ? AE
For causal E, E ? ?, thus A E ? A ? A ?
AE Thus A AE
An envelope is sub-additive if for all u ?
t, E(t) ? E(u) E(t-u) Thus E ? EE for a
sub-additive envelope
11
Regulator
Output of a source is typically statistically
characterizable
Source output can be upper-bounded by envelopes
by passing it through a regulator
A network element is a regulator with envelope E,
if for any input arrival process A, the departure
process D satisfies D ? DE
12
Buffered Leaky Bucket Regulator
There exists a fictitious token bucket which
obtains token at the rate ?. The bucket can hold
up to ? tokens
A packet generated from a source can be released
only if the token bucket has a token, and the
token is removed after releasing the packet
13
D(t) D(u) ? ? ?(t-u)
In u, t the maximum amount of data that can be
transported is the total number of tokens
generated in u, t the total number of tokens
accumulated at u. The first quantity is
?(t-u) The second is ?
Thus D ? DE where E(t) ? ?t
It follows that a leaky bucket regulator has
envelope ? ?t
Is this envelope causal? Is it sub-additive?
14
For any arrival process A, the departure process
of the regulator satisfies D AE Thus E is the
service curve of the regulator
Proof Note down from the board
15
A process A with envelope E1 is passed through a
leaky bucket regulator with envelope E2 Show
that the resulting output has envelope E1 E2
Lets work this out!
Consider the output of a voice coder Emits bits
at the rate R A has envelope E Rt
Regulate this process by a leaky bucket regulator
with envelope E(t) ? ?t
The resulting process will have an envelope
min(Rt, ? ?t)
16
What do we do if we know the envelopes?
We can upper bound the delay of an arrival
process if its envelope and the minimum service
curve of the network element is known.
Let an arrival process A be transmitted through a
network element with minimum service curve S(t)
D is the departure process
dmax inf d E ?d ? S
17
Here, dmax is the least amount the envelope E(t)
be shifted so that it falls below he service
curve S.
The delay produced by the network element is
upper bounded in terms of dmax
D ? A ?dmax
Proof D ? A S ? A (E ?dmax
) (A E) ?dmax
? A ?dmax
18
Example computations of dmax
E(t) min(Rt, ? ?t)
S(t) ct
dmax ?(R-c)/c(R - ?)
Lets work it out!
19
Consider an arbitrary envelope E(t) and S(t) ct
Not quite rigorous!
It follows that E(t- dmax) ? ct
E(t) ? c(t dmax)
dmax ? (E(t)-ct)/c
More rigorously, dmax supt?0(E(t)-ct)/c
If delay upper bound must be less than some T,
then the minimim required service rate cmin is
such that supt?0(E(t)- cmin t)/ cmin T
It follows that cmin supt?0(E(t)/ T t)