- Processor:
- A processor (or server) is an active entity that can provide service to clients
- Example:
- A
postal worker
is a
processor
servicing
clients
who come to buy stamps
- A router is a processor servicing packets who come to be transmitted
- A
postal worker
is a
processor
servicing
clients
who come to buy stamps
- Ordinary processor:
- A "ordinary" processor provides service to one client at a time (i.e., one client per server).
- Property an "ordinary" server:
- Using an "ordinary" processor, exactly one job will make progress in time
- General Processor Sharing (GPS):
- The General Processor Sharing method
is the most ideal sharing
technique possible...
- The General Processor Sharing service technique is a purely theoretical technique
- The General Processor Sharing method
is the most ideal sharing
technique possible...
- How to share a processor in the
ideal manner:
- Suppose a processor (server) has the
capability of processing
M
units of work per sec.
- If there is
one job in the
processor's queue, then:
- the speed of progress of the job is M units/sec
- If there are
two jobs in the
processor's queue, then:
- the speed of progress of each job is M/2 units/sec
- If there are
three jobs in the
processor's queue, then:
- the speed of progress of each job is M/3 units/sec
- And so on...
Graphically:
- Suppose a processor (server) has the
capability of processing
M
units of work per sec.
- Note:
- Such a server is known in queueing theory as an Exponential Server
- The exponential server:
- The exponential server
can "split" itself into
an arbitrary number of "smaller"
(less capable) servers
- Each of these
"smaller" servers
can work on a different job
- All jobs in the queue of the exponential server are worked on simultaneously.
Result:
- In an exponential server,
all jobs
will
make progress in time - simultaneously
(But with a reduced service rate).
- The exponential server
can "split" itself into
an arbitrary number of "smaller"
(less capable) servers
- The
General Processor Sharing
technique
can be generalized by assigning
weights
(representing "importance" or "priority") to the
different jobs
- Weights:
- The weight assigned to
a job
represents the
amount of service
that is
proportional to the weight.
- A job with weight = w will receive w times more service than a job with weight = 1
- The weight assigned to
a job
represents the
amount of service
that is
proportional to the weight.
- Example:
- 2 jobs are being worked on
- Job 1 has
weight = 1 ,
and
- Job 2 has
weight = 2 ,
- Then job 1 will be worked on at a speed of
M/3 units/sec
while job 2 will be worked on at a speed of 2*M/3 units/sec
- 2 jobs are being worked on
- The
Fluid Flow Server
(FFS) is the
network equivalent
of the
Exponential Server
- Just like the
Exponential Server,`
the
Fluid Flow Server
is an
idealized way to share bandwidth
- NOTE:
- The result achieved by the
Fluid Flow Server
does not make sense;
- but that is NOT the point of a
Fluid Flow Server...
- The point of the
Fluid Flow Server
is to
establish the BEST level of sharing:
-
the level of sharing
achieved by the
Fluid Flow Server
will be the
standard
against which our schemes will be
measured against
- The result achieved by the
Fluid Flow Server
does not make sense;
- UnWeighted FFS:
Note:
- An Unweighted FFS transmits ONE bit from each backlogged flows in a round robin manner
- Degree of fairness achieved by FFS:
- The difference in
service
between backlogged
flows f and g
over an interval
[t1, t2],
is bounded by
one bit !!!
I.e.:
| Wf(t1,t2) - Wf(t1,t2) | ≤ 1 bit
This is clear from the operation of FFS:
- Because f and
g are
backlogged during
[t1, t2],
the FFS will
alternatedly transmit exactly 1 bit
from each flow.
- Therefore, the maximum difference in service of between flows f and g is at most 1 bit
- The difference in
service
between backlogged
flows f and g
over an interval
[t1, t2],
is bounded by
one bit !!!
- Note:
- The fairness achieved by FFS is best possible for network flows !!!
- The fairness criteria in FSS
can be
extended
to weighted fairness...
- Weighted FFS:
- Opeartion of the weighted FFS
- Transmits ri bit
from
a backlogged flow
if the flow has a reservation ("weight")
ri
- Data from backlogged flows are (again)
transmitted
in a
round robin manner
Example:
- Transmit r1 bytes from flow 1
- Transmit r2 bytes from flow 2
- ...
- Transmit rN bytes from
flow N
- Repeat...
- Transmits ri bit
from
a backlogged flow
if the flow has a reservation ("weight")
ri
- Degree of fairness achieved by weighted FFS:
- If flows f and g
are backlogged
over the interval
[t1, t2],
then:
| Wf(t1,t2) Wg(t1,t2) | | --------- - ---------- | ≤ 1 | rf rg | Or: | wf(t1,t2) - wf(t1,t2) | ≤ 1
This should be clear from the operation of FFS:
- Because f and
g are
backlogged during
[t1, t2],
the FFS will
alternatedly transmit exactly
rf bits from flow f
and
rg bits from flow g.
- Therefore, the maximum difference in normalized service between flows f and g is at most 1
- If flows f and g
are backlogged
over the interval
[t1, t2],
then:
- Normalize weights: wi
Define: ri wi = ---- R - The flow i
has a normalized
reservation weight
wi
and:
0 < wi <= 1
- In a FFS server, the
flow i
will receive
at least a
service rate gi:
wi gi = ------------------ R w1 + w1 + ... wN
Provided that: w1 + w1 + ... + wN <= 1
- Note:
- A flow i can receive more than gi fraction of the total bandwidth R if some flow j is not backlogged
- Comments:
- The
FFS is only an
idealized transmission method
(representing
the general processor sharing technique
)
- Reality:
- A router must transmit a packet atomically (unsplit) because the structure of the packet must remain intact.
- The FFS server does not send packets atomically
(it mixes bits from different packets), and hence
the FFS server
cannot be used to forward packets.
The FFS can give us an idea how good a solution is compared to the ideal solution
- The
FFS is only an
idealized transmission method
(representing
the general processor sharing technique
)
- Suppose 2 flows are sharing a common
Fluid Flow Server
The weights of the flows are equal (i.e., unweighted)
- The Packet arrival times and
Normalized Packet lengths
are as follows:
Flow 1: FLow 2: Arrival Time Packet length Arrival Time Packet length Packet 1: 1 1 0 3 Packet 2: 2 1 5 2 Packet 3: 3 2 9 2 Packet 4: 11 2 - -
- NOTE:
- A commonly used technique in these calculations is to
normalize the packet length
- Packet length are
normalized
by
dividing
the packet length
by the link speed (transmission rate)
- After normalization, 1 unit of packet length will take 1 sec of transmission time
- A commonly used technique in these calculations is to
normalize the packet length
- The following picture shows how the packets are serviced by a
Fluid Flow Server
when the weights are:
w1 = w2 = 0.5
(normalized weights)
- How to read the above diagram:
- The top part of
the figure shows:
- arrival time
- length
of each packet
- The bottom part of
the figure shows:
- (Real) time on the x-axis
- amount of data transmitted on the y-axis
of each flow
The packet being transmitted is color-coded
- The tangent of the
lines
is the service rate
- Tangent = 1 means service rate of 1
- Tangent = 0.5 means service rate of 0.5
The picture is pretty self-explanatory (I think)
- The top part of
the figure shows:
- From the figure, we can see that
the arrival times and
finish times
of the packets are as follows:
Flow 1: FLow 2: Arrival Time Finish Time Arrival Time Finish Time Packet 1: 1 3 0 5 Packet 2: 2 5 5 9 Packet 3: 3 9 9 11 Packet 4: 11 13 - -
- Suppose
the weights
of the flows are now:
- w1 = 1/3
- w2 = 2/3
- The Packet arrival times and
Normalized Packet lengths
are
unchanged
and they are
repeated here for
convenience:
Flow 1: FLow 2: Arrival Time Packet length Arrival Time Packet length Packet 1: 1 1 0 3 Packet 2: 2 1 5 2 Packet 3: 3 2 9 2 Packet 4: 11 2 - -
- The following picture shows how the packets are serviced by a
Fluid Flow Server
when the weights are:
w1 = 1/3 and
w2 = 2/3
(normalized weights)
- From the figure, we can see that
the arrival and finish times
of the packets are as follows:
Flow 1: FLow 2: Arrival Time Finish Time Arrival Time Finish Time Packet 1: 1 4 0 4 Packet 2: 2 5 5 8 Packet 3: 3 9 9 11 Packet 4: 11 13 - -
- The
Fluid Flow Server
achieves
most fair sharing
possible
- By studying how a
Fluid Flow Server
transmit packets, we may develop a
transmission scheme (scheduler)
that is
as fair as we can get to the ideal
- In the next few class notes, I will detail a few important developments in developing fair scheduling algorithms