Routing Performance Analysis in Delay Tolerant Networks

Transcription

1 Routing Performance Analysis in Delay Tolerant Networks Presenter: Hao Liang Main References: [1] G. Resta and P. Santi, A framework for routing performance analysis in delay tolerant networks with application to noncooperative networks, IEEE Transactions on Parallel and Distributed Systems, vol. 23, no. 1, pp. 2-10, Jan (Conf. Ver.: IEEE SECON 09) [2] X. Zhang, G. Neglia, J. Kurose, and D. Towsley, Performance modeling of epidemic routing, Computer Networks, vol. 51, no. 10, pp , Jul BBCR VANET/DTN Subgroup Meeting

2 Outline Stochastic Coloring Process Based Method - System Model - Routing Analysis Framework - Preliminaries - Analysis of Epidemic Routing - Analysis of Two-Hops Routing Ordinary Differential Equations (ODEs) Based Method - A Brief Introduction Summary and Discussions 1

3 Stochastic Coloring Process Based Method [1] G. Resta and P. Santi, A framework for routing performance analysis in delay tolerant networks with application to noncooperative networks, IEEE Transactions on Parallel and Distributed Systems, vol. 23, no. 1, pp. 2-10, Jan

4 System Model Consider a delay-tolerant network composed of M nodes. Denote the set of nodes as Μ Among the M nodes, a node s is chosen uniformly at random as the packet source, and another node d s is chosen uniformly at random as the packet destination The goal is to analyze the performance of DTN routing protocols used to route the packet from s to d s d Μ 3

5 System Model Performance Metrics (Probability Distribution of ) Packet delivery delay: The time elapsing between the instant at which a packet is generated at s, and the instant at which a copy of the packet is first received by d Communication cost: The number of nodes holding at least one copy of the packet present in the network at the instant at which a copy of the packet is first received by d 4

6 System Model Assumptions Low load: network traffic is low, so that buffer capacity on the nodes is not an issue Transmission range: Two nodes can communicate (encounter) iff they are within distance r No contention (low node density): Any communicating pair of nodes does not interfere with any other pair communicating at the same time Fast transmissions: The duration of node encounters is always sufficient for the two nodes to exchange the content of their buffers 5

7 Routing Analysis Framework The routing analysis framework can be applied to any stateless, deterministic routing protocol for DTNs Stateless: Decisions about whether a copy of the packet should be delivered to an encountered node do not depend on a notion of state (such as history of past encounters and information related to social relationships between nodes) Deterministic: Routing decisions are not influenced by random choices ( Note: specifically, epidemic and two-hops (source spray-and-wait) routing protocols are considered in this work ) 6

8 Routing Analysis Framework The Stochastic Coloring Process Nodes in Μ \ {s,d} can be in one of the following states: Uncolored (U): nodes in this state have not yet received a copy of the packet Colored active (CA): nodes in this state have at least two copies of the packet; these nodes are allowed to deliver one or more copies of the packet to any uncolored node ( Note: copy of a packet = token of a packet (i.e., the number of allowed forwardings) ) Colored inactive (CI): nodes in this state have one copy of the packet; these nodes are allowed to delivery the only copy of the packet they have only to the final destination 7

9 Routing Analysis Framework The Stochastic Coloring Process (Cont d) Let L be the maximum number of copies of a packet (excluding possible copy delivered to the destination) allowed to circulate in the network by the routing protocol at hand The coloring process starts at time t 0 =0, when the source node is in state CA, and all other nodes are in state U The relevant events in the considered stochastic process are coloring events, representing the situation in which a node in CA state encounters a node in U state, or a node in CA/CI state encounters the destination The ith coloring event is denoted by E i, and occurs at time T i 8

10 Routing Analysis Framework The Stochastic Coloring Process (Cont d) ( Note: commonly referred to as the source spray-and-wait routing protocol ) An example: L = 4. Initially, only the source node is colored (and active) At time T 1, the first node is colored and transitions to state CI At time T 3, four copies of the packet are circulating in the network: the spray phase ends, and the wait phase starts At time T 4, the destination node is colored and the coloring process ends 9

11 Routing Analysis Framework Delay Analysis Let E d denote the event destination is colored, which occurs at time T d The goal is to characterize the probability distribution of random variable T d, i.e., for any given t > 0 10

12 Routing Analysis Framework Delay Analysis (Cont d) Denote by E d,i the event destination is the ith colored node. It is immediate to see that the E d,i s are mutually disjoint events, and Two subproblems: 1) Computing the distribution of r.v. T i, conditioned on E d,i 2) Computing the probability of event E d,i Computing probabilities P(E d,i ) is exactly what is needed to derive the communication cost distribution, which is then a byproduct of 2) 11

13 Delay Analysis (Cont d) Routing Analysis Framework Observe that T i can be expressed as a sum of random variables τ 1,..., τ i, where r.v. τ 1 =T 1 - t 0, and τ j =T j T j-1 for 1 < j i. Then The distribution of τ j random variables depends on the features of the mobility pattern!! Assumption: nodes move according to an arbitrary mobility model with exponentially distributed meeting time with rate 1/emt between arbitrary node pairs, where emt is the expected meeting time between arbitrary node pairs 12

14 Preliminaries Exponential Random Variable Use notation f(λ,x) and F(λ,x) to denote the PDF and CDF of an exponential random variable of parameter λ 13

15 Preliminaries Order Statistic of Random Variables Consider n i.i.d. continuous random variables X 1,..., X n, and let X 1,..., X n be a realization of the n random variables Now order the values of the realization in increasing order, starting from the smallest, and denote with X (1),..., X (n) the ordered values. Each of the X (i), for i = 1,..., n, can be considered as a realization of a random variable X (i), which is known as the ith order statistic of random variables X 1,..., X n Note that X (1) = min{x 1,..., X n }, X (n) = max{x 1,..., X n } 14

16 Preliminaries Order Statistic of Random Variables (Cont d) Denoting by ψ(x) and Ψ(x) the PDF and CDF, respectively, of each of the X i, the PDF of the ith order statistic of random variables X 1,..., X n is In the following, we denote by Ord(n, i, λ, x) the PDF of the ith order statistic of a set of n i.i.d. exponential random variables of parameter λ 15

17 Preliminaries 16

18 Analysis of Epidemic Routing 17

19 Analysis of Two-Hops Routing General Expression of Delay According to the analysis framework, the packet delivery delay distribution can be computed as 18

20 Analysis of Two-Hops Routing First Term The first term in the summation corresponds to the situation in which the destination is the first colored node, and the spray phase does not even start. Then, we have CDF of an exponential random variable with parameter (M-1)/emt 19

21 Analysis of Two-Hops Routing The Last Term (i.e., when i = L) When i = L, the spraying process is finished (i.e., (L 1) nodes have been colored by the source), and coloring of the destination occurs during the wait phase Note that in the wait phase the coloring process is symmetric, hence any of the L nodes currently holding a copy of the packet can color the destination Denote by S i the random variable corresponding to the first time source node s meets node i after packet generation, for i = 1,..., M 1 20

22 Analysis of Two-Hops Routing The Last Term (i.e., when i = L) (Cont d) It is easy to see that the starting time of the wait phase is random variable S (L 1), i.e., the (L 1)th order statistic of random variables S (1),..., S (M 1), whose PDF is Ord(M 1, L 1, 1/emt, x) 21

23 Analysis of Two-Hops Routing The Last Term (i.e., when i = L) (Cont d) Should be (t-x)? Conditioned on S (L 1) = x, the probability of delivering the packet to destination within time t is given by the CDF at time (TTL x) of an exponential random variable with rate L/emt, which represents the time at which the first amongst the L colored nodes meets the destination The value of P(E d,l ) is computed as 22

24 Analysis of Two-Hops Routing When 1 < i < L Event E d,i can be divided into mutually disjoint events and, corresponding to the situation in which the destination is colored by the source s or by a non-source node Conditioned on, the probability of delivering the packet to the destination within time t can be computed observing that the destination is the ith node encountered by s, and that the random time of this encounter corresponds to the ith order statistic of random variables S 1,..., S M 1 23

25 When 1 < i < L (Cont d) Analysis of Two-Hops Routing We now compute, from which can be trivially derived Use random variables to denote the time at which node h (which has been colored at time S h ) first meets the destination Conditioned on a specific value of has exponential distribution of rate 1/emt, for h = 1,..., i 1 Should be -x? 24

26 When 1 < i < L (Cont d) Analysis of Two-Hops Routing Subdivide into disjoint events, k = 1,..., i 1, where is the event the destination is colored by the k-th nonsource node The above event occurs if and only if for each h = 1,..., i 1 (with h k) and 25

27 When 1 < i < L (Cont d) Analysis of Two-Hops Routing where is the PDF of random variable, which is the sum of two independent random variables with different (and nontrivial) distributions. To circumvent this problem, we use the approximation The approximation is very accurate as long as the ratio L/M is around 0.1 or below 26

28 Analysis of Two-Hops Routing When 1 < i < L (Cont d) Then, we have Note that in the above equation we assume that pairs of events, with h h, are mutually independent, which is not true in general. By observing that the probability that the ith node met by the source is the destination is 1/(M (i 1)), we have 27

29 Analysis of Two-Hops Routing When 1 < i < L (Cont d) Should be (t-x)? In the above equation, we have used the fact that, conditioned on event, the PDF of random variable is closely approximated by the PDF of an exponential random variable of parameter (M i)/emt, corresponding to the first encounter (which, conditioned on, we know is with the kth relay node) between the destination and one of the i nodes (including the source) currently holding a copy of the packet 28

30 Analysis of Two-Hops Routing Performance Metric - Packet Delivery Delay (1 < i < L) Derived from Approximation based on an exponential random variable of parameter (M i)/emt Approximation by assuming independency Approximation based on S h S 1 29

31 Analysis of Two-Hops Routing Performance Metric - Communication Cost Only consider the event that the destination is colored by source s Correct?? 30

32 Numerical Results (Two-Hops Routing) Analysis of Two-Hops Routing M = 30 nodes are initially distributed uniformly at random in a square area of 10 km side. Nodes have a transmission range of 250 m, and move according to the random waypoint (RWP) mobility model with no pause time More accurate for smaller L/M 31

33 Analysis of Two-Hops Routing Numerical Results (Epidemic Routing) More accurate than [2] 32

34 Ordinary Differential Equations (ODEs) Based Method [2] X. Zhang, G. Neglia, J. Kurose, and D. Towsley, Performance modeling of epidemic routing, Computer Networks, vol. 51, no. 10, pp , Jul

35 ODEs Based Method Foundation: Markov Model Given n I (t), the number of infected nodes at time t, the transition rate from state n I to state n I + 1 is where N is the total number of nodes in the network (excluding the destination), and β is the pairwise meeting rate As N increases, the fraction of infected nodes (n I /N) converges asymptotically to the solution of the following equation 34

36 ODEs Based Method Delay Under Epidemic Routing Denote the CDF by P N (t) = Prob(T d < t). We have 35

37 ODEs Based Method Delay Under Epidemic Routing (Cont d) As N increases, E[n I (t)/n] converges to i(t), and P N (t) converges to the solution of 36

38 Summary and Discussions 37

39 Summary and Discussions Stochastic Coloring Process Based Method - Better estimation of packet delivery delay PDF - Complicated with lots of approximations, more accurate for smaller L/M ODEs Based Method - Simpler - Less accurate estimation of packet delivery delay PDF Limitations (Potential Improvement) - Exponentially distributed meeting time - Only for stateless routing protocols 38