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  1. 1. This is the project work entitled THE TRAVELING MISER PROBLEM
  2. 2. Introduction <ul><li>While managing with user data (higher priority traffic) some amount of low priority traffic can be generated . </li></ul><ul><li>This information is not always needed in real time and often can be delayed by the network with out hurting the functionality. </li></ul><ul><li>This paper proposes a new framework to handle this low priority traffic. </li></ul>
  3. 3. <ul><li>The key idea is allowing the network nodes to delay low priority data by locally storing it. </li></ul><ul><li>This parking imposes additional load on the intermediate nodes . </li></ul><ul><li>In order to prevent an excessive load on the active nodes, we associate some time-dependent cost with parking at a given node, and seek to optimize packet parking schedules in terms of these costs. </li></ul><ul><li>We call this as traveling miser problem </li></ul>
  4. 4. Methodology <ul><li>Consider bi-directional time-dependent network G . </li></ul><ul><li>G = ( V , E , W , P , L ). </li></ul><ul><ul><li>V being a set of nodes . </li></ul></ul><ul><ul><li>E being a set of links </li></ul></ul><ul><ul><li>W being a set of link weight . </li></ul></ul><ul><ul><li>P being a set of parking weight densities . </li></ul></ul><ul><ul><li>L being a set of link delays . </li></ul></ul>
  5. 5. <ul><li>Let R be a simple path from s to d R= </li></ul><ul><li>The traveling miser problem is defined as follows: </li></ul><ul><li>A miser starts at node s at time 0. the miser is required to reach the destination d with in the integer number D of time units, 0<=D< ∞, called deadline. </li></ul>
  6. 6. <ul><li>Specifically at any given instant t, the miser is at some node v(j) Є R facing the following options for the next unit of time: </li></ul><ul><li>1.stay at that node </li></ul><ul><li> one link in the forward direction i.e towards destination. </li></ul><ul><li>3.Travel one link in reverse direction i.e towards source. </li></ul>
  7. 7. <ul><li>In order to state the problem more formally, we need the following definitions. </li></ul><ul><li>ITINERARY:- </li></ul><ul><li>I (s ,d) is a possibly non simple path from source s to destination d consisting of all the original path links plus fictitious self links. </li></ul><ul><li>ITINERARY WEIGHT:- </li></ul>
  8. 8. <ul><ul><li>TMP can be generally formulated as </li></ul></ul><ul><ul><li>-> on-line problem </li></ul></ul><ul><ul><li>-> off-line problem </li></ul></ul>
  9. 9. ON-LINE PROBLEM <ul><li>The idea of the on-line algorithm is as follows: </li></ul><ul><li>The miser is advancing towards the destination using the non-fictitious links only as long as he dose not reach a block . </li></ul><ul><li>At a block the miser has two options: </li></ul><ul><ul><ul><li>Stay at the node where the block has occurred </li></ul></ul></ul><ul><ul><ul><li>move backwards </li></ul></ul></ul>
  10. 10. <ul><li>Assuming that the miser knows an upper bound U for the block duration. </li></ul><ul><li>Strategy : pick up the cheapest node situated within roughly U/2 distance from its current location on the way back to the source, to spend the block time there. </li></ul><ul><li>This way the overall cost of the trip is minimized. </li></ul>
  11. 11. <ul><li>The concept behind the algorithm is that if the miser has to spend some time en route on the way to the destination due to block, it is better to do this using the cheapest itinerary. </li></ul><ul><li>This minimises the overall cost of trip </li></ul>
  12. 12. <ul><li>In order to explain this algorithm in detail we need the following definition: </li></ul><ul><li>Simple round trip itinerary : </li></ul><ul><ul><li>A simple round trip itinerary between the two nodes j, k Є R is an itinerary that is a concatenation of a shortest, possibly empty, itinerary that goes from node j to node k, possibly empty finite itinerary that uses only the (k, k) fictitious links, and the shortest, possibly empty, itinerary that goes from node k to node j. </li></ul></ul>
  13. 13. <ul><li>Suppose a block occurs at node vb Є R on the link leading towards destination the algorithm works in stages . </li></ul><ul><li>At stage j Є (1,logubase2)the miser chooses node v*j Є R reachable from vb with in t<=2^(j-1) such that w(i) should be minimal . </li></ul><ul><li>Then the miser follows this minimal weight simple round trip itinerary for stage. </li></ul><ul><li>In order to gain a logarithmic factor, the miser spends exponentially increasing periods of time away from the block node </li></ul>
  14. 14. <ul><li>After each such period the miser goes back to the blocked link and checks whether the block is lifted. </li></ul><ul><li>If the block is lifted then the miser goes through, other wise it proceeds to phase j+1. </li></ul>
  15. 15. Algorithm <ul><li>1. START </li></ul><ul><li>2.while next link is not congestion proceed to next hop. </li></ul><ul><li>3.If next link is congested then </li></ul><ul><li>3.1: for each phase j Є (1,logBbase2) find a node v*j that is reachable from vb with in t<=2^j-1, so that v*j yields the minimum weight round trip itinerary. </li></ul>
  16. 16. <ul><li>4. If by the end of the phase j no congestion is abate proceed to phase j+1. </li></ul><ul><li>5.Else </li></ul><ul><li>6.Proceed towards destination </li></ul><ul><li>7.STOP </li></ul>
  17. 17. OFF-LINE PROBLEM <ul><li>In off-line problem we assume that the input includes the values of link weight functions along the path for any given instance of time. </li></ul><ul><li>The offline problem is of interest for advanced planning. </li></ul><ul><li>e.g: </li></ul><ul><li>The traffic that is generated in the regular hours of the day. </li></ul>
  18. 18. <ul><li>The off-line algorithm is based on the following concatenation property of the optimal time-dependent paths. </li></ul><ul><li>Time dependent path concatenation property : </li></ul><ul><li>Every sub itinerary of an optimal itinerary is also optimal. </li></ul>
  19. 20. THANK ‘Q’