Providing Delay Guarantees

Guaranteed Rate Scheduling Algorithms
- To provide some guaranteed perfromance, a server reserve a rate for a flow.
- For example, the Fluid Flow Server serving N flows simultaneously will reserve a transmission rate of
  where C is the bandwidth capacity of the transmission link.
- Based on this rate reservation , many scheduling algorithms can also provide a end-to-end propagation delay guarantee
- The end-to-end propagation delay guarantee is a deadline by which the packet of a flow will arrive at the destination

Hop-by-hop delay and End-to-end delay

We distinguish between:
- Hop-by-hop delay guarantee
- End-to-end delay guarantee
End-to-end Delay:
The End-to-end delay is the length of the time interval between
1. the time that the packet is sent at the source
2. the time that the packet is received at the destination
An End-to-end Delay Guarantee is a promise (guarantee) that a packet sent at the source at time t_send will arrive at the destination at a time less than or equal to t_send + e --- where e is a fixed constant
Hop-by-hop Delay:
The Hop-by-hop delay is the length of the time interval between
1. the time that the packet arrives at a node (hop)
2. the time that the packet leaves that node
A hop-by-hop Delay Guarantee given by a server (router) is a promise (guarantee) that a packet arriving at the server (router) at time t_arr will leave the server (router) at a time less than or equal to t_arr + d --- where d is a fixed constant

Fact:

If every router between the source and the destination is able to provide a hop-by-hop delay guarantee (in other words: the length of the time that a packet is waiting at a router is guaranteed to be less than some pre-defined guarantee value), then we can provide a end-to-end delay guarantee to packets between the source and the destination

This fact is easy to verify:

the end-to-end delay guarantee is simply the sum of the hop-to-hop delay guarantees plus the propagation delays on the links between each pair of neighbors.

x₁ is the propagation delay between the first and second router
x₂ is the propagation delay between the second and third router

Suppose a packet p is sent at time t₀ by source A

p arrives at the first router at time t₀

Suppose the delay guarantee at the first router is d₁. then p will leave the first router at time <= t₀ + d₁

p will arrive at the second router at time <= t₀ + d₁ + x₁

Suppose the delay guarantee at the second router is d₂. then p will leave the first router at time <= t₀ + d₁ + x₁ + d₂

And so on...

Definition: Guaranteed Rate Clock

Notations:

r_f = the minimum transmission data rate allocated to flow f by all routers

p_f^j = the j^th packet of flow f that arrives at a router

l_f^j = the length of the packet p_f^j
NOTE:

Since flow f is guaranteed a rate of r_f, the packet p_f^j will be transmitted in less than l_f^j / r_f seconds once the packet p_f^j begins service (packet p_f^j may need to wait for other packets of flow f that are ahead of it)

Aⁱ(p_f^j) = the arrival time of the packet p_f^j at the server (router) i

GRCⁱ(p_f^j) = the Guarantee Rate Clock value of the packet p_f^j at the server (router) i

The Guaranteed Rate Clock is a function that assigns each packet with a approximate departure time value (time stamp) at each server (router).

at server i p_f^j ------------> approximate departure time (time stamp)

The Guaranteed Rate Clock is defined as follows:

GRCⁱ(p_f^j) = max{ Aⁱ(p_f^j) , GRCⁱ(p_f^j-1) } + l_f^j / r_f

The meaning of GRCⁱ(p_f^j) :

The value GRCⁱ(p_f^j) can be interpreted as the expected departure time of packet p_f^j in server i.

When the packet p_f^j arrives (at time t = Aⁱ(p_f^j)) at the server (router) i, there are 2 possible scenarios:

There are no packet of flow f in the server
There are some packets of flow f in the server

In case (1), the expected departure time of packet p_f^j is:

Expected departure time = Aⁱ(p_f^j) + l_f^j / r_f

(The packet p_f^j arrived at time Aⁱ(p_f^j) and it takes l_f^j / r_f to transmit the packet)

In case (2), the packet p_f^j will only start service after the previous packet of flow f has departed.
The expected departure time of the previous packet of flow f is equal to GRCⁱ(p_f^j-1)
Therefore, the expected departure time of packet p_f^j is then equal to:

Expected departure time = GRCⁱ(p_f^j-1) + l_f^j / r_f

(The packet p_f^j begins service right after the previous packet of flow f finishes at time GRCⁱ(p_f^j-1) and it takes l_f^j / r_f to transmit packet p_f^j)

Combining these 2 cases will result in the expression for GRCⁱ(p_f^j) :

NOTE:

What we have formulated with the Guarantee rate Clock is an expection of the departure time of that the packet p_f^j at a server (router) i

Whether the server (router) i can meet the expection is still unknown !!!

I can tell you that "FIFO" servers will not be able to meet the expectation...
Hopefully, the WFQ, SCFQ and BSFQ servers can (and yes, they can :-))

Definition: Guaranteed Rate Server

Definition of a "Guaranteed Rate Server"::

A scheduling algorithm at server (router) i is a Guarantee Rate Server if it guarantees that the packet p_f^j (that received the Guaranteed Rate Clock value GRCⁱ(p_f^j)) will be transmitted by the time t = GRCⁱ(p_f^j) + βⁱ at the server (router) i where βⁱ is a constant

Further explanation:

As I mentioned previously, the value GRCⁱ(p_f^j) that is given to the packet p_f^j is an expectation,

It is important to realise that the value GRCⁱ(p_f^j) is NOT a promise (or guarantee)

Now, IF a scheduling algorithm can meet this promise within some fixed range/bound βⁱ, then the scheduling algorithm does provide a delay guarantee.

It is important that the range is a constant - otherwise, the promise is not really a promise
Example: a promise that you will leave before half of the people waiting does not constitute a delay guarantee, because the delay depends on how many people are waiting.

So what kind of scheduling algorithm is a "Guaranteed Rate Server" ???

The WFQ Server is a Guaranteed Rate Server

According to Goyal's paper, the departure time of the packet p_f^j in a WFQ server is bounded by:

Lⁱ_PGPS(p^j_f) = Departure (Leave) time of packet p_f^j in the "Packet-by-packet GPS" or WFQ server

lⁱ_max = the maximum length of packets served by the server i

Cⁱ = transmission capacity of the server i

Conclusion:

(In other words, WFQ is a guaranteed rate server)

For the proof, see the paper...

The SCFQ Server is a Guaranteed Rate Server

According to Goyal's paper, the departure time of the packet p_f^j in a Self-Clocked Fair Queueing server is bounded by:

Lⁱ_SCFQ(p^j_f) = Departure (Leave) time of packet p_f^j in the "Self-Clocked Fair Queueing" or SCFQ server

l^max_m = the maximum length of packets of flow m served by the server i

Cⁱ = transmission capacity of the server i

For the proof, see the paper...

The BSFQ Server is a Guaranteed Rate Server

According to the BSFQ paper, the departure time of the packet p_f^j in a Bin-Sort Fair Queueing server is bounded by:

Lⁱ_BSFQ(p^j_f) = Departure (Leave) time of packet p_f^j in the "Bin-Sort Fair Queueing" or BSFQ server

Δ = interval size of the time bins (it's a parameter of BSFQ - see: click here )

l^max_f = the maximum length of packets of flow f served by the BSFQ server

R = transmission capacity of the BSFQ server

For the proof, see the paper...