1. EE384Y: Packet Switch Architectures II
Cell Switching vs. Packet Switching
Abtin Keshavarzian
Yashar Ganjali
Department of Electrical Engineering
Stanford University
June 5, 2002
Good morning! Before officially starting the talk, there is a point that I should mention!
Please do not spend more than 10% of your time staring at the sidebar over here (point to the sidebar)! In our practice talks we noticed that it attracts more attention than what it deserves! 
Ok! In this project, which is a joint work with Abtin (point to Abtin), we try to compare the scheduling algorithms for packets vs. cells and try to investigate the existing tradeoffs.

2. Cell Switching vs. Packet Switching
Motivation
Split
Combine
All the scheduling algorithms we have seen in class are cell-based in the sense that whenever there is a packet arriving to the switch it is first …
But what if we try to switch packets instead of cells?
Of course, cell switching is much easier to be implemented in hardware, but can we gain anything by using “packet switching” instead?
This is the question that we will try to investigate in this project.
2x2 Switch
June 5, 2002
Cell Switching vs. Packet Switching

3. Cell Switching vs. Packet Switching
Outline
Background: Cells vs. Packets
Basic extensions of cell switching algorithms
Stability of packet switching algorithms
Waiting Algorithms
Non-waiting Algorithms
Stability under i.i.d. traffic
Simulation results
Ok! We will start with some basic concepts about cell switching and packet switching and then we will introduce a number of trivial ways for adapting existing cell-switching algorithms into packet-switching ones.
Then we will talk about the stability problem. We will define two classes of packet switching algorithms namely “Waiting and Non-Waiting” algorithms. We will show that under generally admissible traffic, any Non-Waiting algorithm is not stable.
We also consider the stability of packet-based scheduling algorithms under I.I.D. traffic and show that with such restrictions on the input traffic, there is a class of scheduling algorithms which are stable.
Finally we will compare the performance of the cell-based and packet-based switching algorithms.
June 5, 2002
Cell Switching vs. Packet Switching

4. Cell Switching vs. Packet Switching
Background
Cell Switching:
Fixed length cells
100% throughput using MWM for any admissible traffic pattern
Several “fast” algorithms for i.i.d. traffic
Packet Switching:
Packets of different length
Scheduling algorithms?
As every body knows in cell-switching we have
Let’s see what happen if we deal with packets instead!
June 5, 2002
Cell Switching vs. Packet Switching

5. Cell Switching vs. Packet Switching
From Cells to Packets
Algorithm 1: Consider each packet as a cell with length Lmax and use any cell-based algorithm.
Algorithm 2: Do the same as 1, except renew the input-output matching when all lines are free.
Maximum Packet Length
Current packet
The first, trivial extension that comes to mind is to use any cell-based switching algorithm and have large cells such that any packet can be fit into one cell.
We can even improve this method by renewing the input-output matchings whenever all lines are free which is algorithm 2. This can save us a little more bandwidth but
We can easily show that none of these algorithms are stable even if the input traffic is I.I.d.
So let’s take a look at a more serious idea!
Packet 1
Packet 2
Packet 3
June 5, 2002
Cell Switching vs. Packet Switching

6. Cell-Based -> Packet-Based
Packet-Based X (PBX):
Start with any cell-based algorithm X
At each time slot, keep all the lines which are in the middle of sending a packet
For all free lines, re-compute a (sub-)matching using algorithm X a b c d e f
Let’s consider any cell-based scheduling algorithms X such as MWM, Maximal matching, etc.!
We can extend X to a packet based algorithm as follows:
At the beginning we start by choosing the matching introduced by X (refer to the figure).
In the following time slot, some of the lines are still busy because they are in the middle of sending a packet, and therefore we are not going to interrupt them.
For the other “free” input-output ports (port eh and gf in the figure), we computer a new sub-matching again using X (refer to the figure).
The total matching used by the switch is the combination of these two matchings. g h
June 5, 2002
Cell Switching vs. Packet Switching

7. IS Packet-Based X Always Stable?
Under any admissible input traffic
IS Packet-Based X Always Stable?
Now the question is how does PB-X perform?
June 5, 2002
Cell Switching vs. Packet Switching

8. Cell Switching vs. Packet Switching
A Counter-example
Time
A 1,1 3 5 8
A 1,2 2 10
A 2,1 6
A 2,2 1 4 7 9
The answer is: NO!
To see this let’s look at the following example.
Let’s assume we have a 2 by 2 switch as shown in this figure, and also let’s assume we have packets of size 1, 2 and 3 arriving to the system as can be seen.
When packet one comes to the system the switch chooses this parallel matching (refer to the figure) and is forced to keep it for 3 units of time. …
We also notice that none of the input or output lines are oversubscribed in this example because …
Therefore, we can say that PB-X is not stable in general. In fact we are going to show a general result. 5 3 2 6 4 1
June 5, 2002
Cell Switching vs. Packet Switching

9. Waiting vs. Non-Waiting Algorithms
Renew the matching amongst free input-output ports at every possible time slot.
Previous example shows that no non-waiting algorithm is stable in general. 1 3
In general we can divide packet-based scheduling algorithms into two categories:
Those algorithms which start sending packets as soon as there is a packet in VOQ(i,j) and none of I, or j is busy (refer to the figure).
The class of algorithms which might keep a packet for some time, even though the corresponding input and outputs are not busy.
The first class is called “Non-Waiting” and the second “Waiting”.
We have already shown that any non-waiting algorithm is not stable in general and therefore if there is a stable scheduling algorithm it must be waiting sometime.
Waiting Algorithms:
In some time slots, do not start sending packets even if the corresponding input-output ports are free.
June 5, 2002
Cell Switching vs. Packet Switching

10. Stability of Non-Waiting Algorithms under i.i.d. Traffic
We have seen a number of cell-based switching algorithms which are stable for I.I.d traffic even though they are not stable they are not stable in general.
Now there is an interesting question here, that is what happens if the input traffic is i.i.d. for a non-waiting scheduling algorithm.
June 5, 2002
Cell Switching vs. Packet Switching

11. PB-MWM: i.i.d. traffic At time slot n, find MWM
Use the same matching for the next k time slots a d c b
Fortunately, we can show that there are non-waiting algorithms with 100% throughput under i.i.d. traffic.
To show this let’s consider the following lemma.
Let’s assume we have got a switch which uses MWM at time slot n, which is shown by the purple line in this figure, and keeps using the same matching for the next k consecutive time slots.
Let’s also assume that the yellow dashed lines show the MWM that a cell-based algorithm would use at time n+k.
At each time slot between n and n+k, we can have at most N packets (N is the size of the switch) leaving the purple matching and at most N packets arriving to the yellow matching. Therefore in k time slots, the weight of the matching which is being used is at most 2Nk away from the ideal matching.
Lemma: The weight of the matching used by 2
>=
weight{MWM at time (n+k)} - 2Nk
June 5, 2002
Cell Switching vs. Packet Switching

12. Cell Switching vs. Packet Switching
PB-MWM: i.i.d. traffic
Start with MWM at state zero
Go back to state 0 with probability at least p
1 - p
1 - p
1 - p
Now let us consider the PB-MWM algorithm.
In this algorithm we start by choosing a MWM at the beginning. We indicate this as state zero of the algorithm. We also say that the algorithm is in state I if we have chosen MWM exactly I time slots ago.
Now we have shown that for PB-MWM the probability of going back to state zero at any state I, is greater than a positive value p, and therefore, the average number of steps that it takes for the algorithm to go back to state 0 is finite. 1 2 3 p p p p
June 5, 2002
Cell Switching vs. Packet Switching

13. Stability Theorem Theorem: PB-MWM is stable for i.i.d. traffic
Using previous Lemma for PB-MWM &
Using the fact that we return to the first state in a finite number of steps on average,
we can show that
E{weight(PB_MWM)} >= weight(MWM) – const
Using the lemma which we just stated, and using the fact that the algorithm comes back to state zero very often in general, we can conclude that the expected value of the matching used by PB-MWM differs with the weight of MWM at most by a constant factor (point to the green stuff).
This leads to the main result of this project which is: PB-MWM is stable for I.I.d traffic.
Theorem: PB-MWM is stable for i.i.d. traffic
June 5, 2002
Cell Switching vs. Packet Switching

14. Cell Switching vs. Packet Switching
Simulation Results
Finally, we are going to compare the performance of Packet-based scheduling algorithms and cell-based ones.
In this diagram, the X axis shows the load arriving to the system and the Y axis is the average delay observed by packets. Here, in order to be able to compare Packet based and cell-based algorithms we consider the delay seen by the last cell of any packet.
In this example, we have an 8 by 8 switch with maximum packet length 16 and uniform traffic pattern. We also assume that the packet lengths are uniformly distributed.
As you can see, the first two algorithms which were using large cells to encapsulate packets perform very bad in general.
The red curve shows the performance of cell-based MWM and the violet curve shows that of PB-MWM. As you can see, for small traffic loads, PB-MWM performs slightly better than CB-MWM and the situation is reversed for higher loads.
June 5, 2002
Cell Switching vs. Packet Switching

15. Cell Switching vs. Packet Switching
Simulation Results
Here we have another simulation for an 8 by 8 switch with a different traffic pattern. Again we see the same behavior as before.
June 5, 2002
Cell Switching vs. Packet Switching

16. Cell Switching vs. Packet Switching
Conclusion
Non-Waiting PB-X algorithms unstable in general
PB-MWM stable for i.i.d. traffic
PB-MWM performs slightly better than CB-MWM for low traffic
June 5, 2002
Cell Switching vs. Packet Switching

17. Questions? Now that we have put everything together, Questions!
Thank you!