Lecture 4

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

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.

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)

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)

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

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))

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

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

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

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. 350-364. Propagation delay information may not be precisely known. Available bandwidth information may not be known correctly

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)