1. Lecture 4

2. Network with Reservations
Connections can also make reservations apriori for its duration
Certain bandwidths will be available in certain time ranges
Time is a dimension
Networks with Advanced Reservations: The Routing Perspective
R. Guerin and A. Orda

3. Time interval is divided into slots, 0,1,2,…..
Every link l has a vector of bandwidth availabilities: [bl[0], bl[1], bl[2],……]
Given a source node s, destination node v, bandwidth requirement B, starting time t, duration u, find a path which supports this connection
If a link does not have bandwidth B in any slot in t, t+1,…t+u, remove the link
Find a path between the source and the destination in the remaining network
Complexity: O(V + E + Eu) or O(Eu) if the graph is fully connected.

4. Given a source node s, destination node v, bandwidth requirement B, starting time t, find a path which supports this connection for the maximum duration in the interval [t, t+u]
For every link l in the network, find the maximum value vl such that the link has B bandwidth available in every slot in [t, t+ vl ] and vlu
Duration of a path is the minimum value of vl in the links in the path
Find a path between the source and the destination of maximum duration.
First see whether there is a path of duration u
(Depth first Search/Breadth first search)

5. If there is, stop
Else, try with u/2.
If there is such a path, try with 3u/2
If not, try with u/4 , …..
Every time narrow the interval
``Binary search’’
O(Eu + Elog u) or O(Eu)

6. Given a source node s, destination node v, bandwidth requirement B, duration u, find a path which finishes the connection the fastest in interval [0, T].
For every path, there is an earliest time t in interval [0, T], s.t. the path has B bandwidth in all slots in [t, t+u].
Find the path with the earliest value of this start time.
1)Start with w = 0
2)Find if there exists a path which supports the connection in [w, w+u].
3)Stop if there exists one such path
4)If not, ww + 1
5)Go to (2) if wT-u

7. Complexity: O(Eu(T-u))
A faster algorithm of complexity O(E(T – u)) is in the paper.
The idea is to scan every slot in every link a constant number of times.
So the overall complexity is O(E(T – u))

8. A connection may need to transmit a fixed amount of data, but it can use different bandwidths in different slots. Find the path which completes the task in the minimum duration in interval [0, T].
A connection needs to transmit B amount of data. Say it is routed along path p. Let this path provide bi bandwidth in slot i . The task completes itself in p slots if b0 +…..+ bp  B
Computing the minimum duration is an NP-hard problem.
Heuristics

9. Overall Delay constraints
Propagation delay (l in link l)
Queueing delay (k/r in any link of bandwidth r)
A connection has a duration, a starting time, a minimum bandwidth requirement in all links in the connection, and
An additional requirement that the overall delay be less than a constant D

10. If every link of a path guarantees a bandwidth of r and there are n(p) links in the path, then depending on the traffic characteristics, the total delay in the path is at most l (+n(p))/r
The problem is to find a path which guarantees the minimum bandwidth in every slot of the duration, in every link, and has propagation delays and bandwidth r in every link in the path s.t. l (+n(p))/r

11. Uncertainty in Information
R. Guerin and A. Orda. ``QoS-based Routing in Networks with Inaccurate Information: Theory and Algorithms.''
IEEE/ACM Transaction on Networking, Vol. 7, No. 3, June 1999, pp
Propagation delay information may not be precisely known.
Available bandwidth information may not be known correctly

12. A probability distribution may be known
With certain probability, propagation delay is this
With certain probability, available bandwidth is this.
Find a path which assures delay less than D with certain probability
Most of these problems are intractable (NP-hard)