1. Survivability Quantification of Communication Services Poul E.Heegaard, Kishor S. Trivedi Advisor: Frank Y. S. Lin Presented by Y.W. Lee

2. Agenda Abstract Petri Net and Stochastic Reward Net Network of waiting lines Phase dependent performance Space decomposed model Network example Conclusion

3. Agenda Abstract Petri Net and Stochastic Reward Net Network of waiting lines Phase dependent performance Space decomposed model Network example Conclusion

4. Abstract Petri net vs Stochastic Reward Net Exchange information Independent in waiting lines Decomposition and performance

5. Abstract Objective: quantify the survivability of virtual connections in telecommunication. – service: the VC between specific peering nodes in the network. – requirement: maximum packet loss probability and end-to-end delay of non-lost packets in the VC. – undesired events: link and node failures caused by attacks.

6. Agenda Abstract Petri Net and Stochastic Reward Net Network of waiting lines Phase dependent performance Space decomposed model Network example Conclusion

7. Petri net A Petri net is one of several mathematical modeling language for the description of discrete distributed systems. A Petri net consists of places, transitions, and directed arcs.

8. Places : – Contain any non-negative number of tokens. – Input place before transition and output place after transition. Marking : – A distribution of tokens over the places of a net. Directed arcs : – Run between places and transitions. – Each arc has multiplicity. (bandwidth) Transitions : – Fire whenever there is a token at the end of all input arcs. Fire : – It consumes these tokens, and places tokens at the end of all output arcs. – Duration : 0.

9. A Petri net graph is a 4-tuple (S,T,W, ), where – S is a finite set of places – T is a finite set of transitions – S and T are disjoint – W : define an arc and each arc has an non-negative integer arc multiplicity – The preset of a transition t is the set of its input places: – Its postset is the set of its output places: – : initial marking

10. Its transition relation can be described as a pair of | S | by | T | matrices: – W −, defined by – W +, defined by Then their difference W T = W + − W − can be used to describe the reachable markings in terms of matrix multiplication.

11. For any sequence of transitions w, write o(w) for the vector that maps every transition to its number of occurrences in w. Then, we have } is a firing sequence of N

12. - ＝ =

13. SRN The SRN differ from SPN (PN with stochastic transition rate) in several key aspects : – Enable function (in transition) – Marking-dependent arc cardinalities – The ability to decide in a marking-dependent fashion whether the firing time of a transition is exponentially distributed or null

14. We define a non-parametric SRN as an 11- tuple : the inhibitor arc from p to t. when the transition t fires in marking μ,the new marking satisfies:

15.  {true, false} is the guard associated to transition t. > is a transition and irreflexive imposing a priority among transitions. is the initial marking. is the rate of the exponentially distribution for the firing time of transition t.

16. The definition of vanishing and tangible marking – Vanish : at least one transition time duration is immediate. – Tangible : all transition time duration is timed.

17. We assume a race model marking μ occupied by transition t in probability The firing probability of enabled transition t is: : In a tangible marking μ : In a vanish marking μ The notation indicates that transition t is enabled in marking μ.

18. The elements described so far define a trivariate discrete-time stochastic process: (,, ) – : the n-th transition to fire. – : the time at which n-th fire. – : the n-th marking encountered.

19. is a finite set of measure. – ρ( μ ) reward rate : the rate at which reward is accumulated when the marking is μ. – reward impulse : the instantaneous reward gained when firing transition t while in marking μ. – : compute a single value from Y(θ) as index.

20. Reward process accumulated by the SRN to time θ Number of transition fires up to time θ :E(Y(θ)) – Expect time average reward up to time θ In steady state:

21. Exchange of data Given a SRN A – Define c as a boolean marking-dependent expression (1: hold 0: not hold), : tangible marking according to whether the condition c is on or off. : steady-state probability.

22. Assume that each node is an M/M/1/-queue. is the steady state probability that node i will reject an incoming packet. （2）（2）

23. : transition rate at which condition goes from on to off. : sum of steady state probability of each marking in.

24. : the expected time to absorption when the marking in are considered absorbing, starting from steady state. : the expected time spent in each non-absorbing tangible marking before absorbing., : and of the states in.

25. = [0.1,0.2,0.3,0.4] ={1,2}, ={3,4}

26. If condition b holds in the set of marking we are interested in, we can use a modified version of the steady state vector, define as:

27. The quantities, and can be generalized to, and.

28. The following steps are needed to exploit near-independence at the SRN level – Decomposition – Import graph Fix the value of when we want to compute,if > – Iteration Acyclic Cycle

29. Agenda Abstract Petri Net and Stochastic Reward Net Network of waiting lines Phase dependent performance Space decomposed model Network example Conclusion

30. Network of waiting lines A machine shop has several departments each containing a fixed number of identical machines. Each department is a multiserver system of the usual type. Arrivals at a given department come both form other departments in the shop and from outside the shop.

31. If mean arrival rates at the various departments are properly defined. Result: steady-state distribution in which the waiting-line lengths of the departments are independent.

32. Single department : k : customer number Serve with n identical servers

33. M department ……? – Department m contain servers. – Customs arrive from outside with Poisson-type – FCFS and service time: – Once served in department m, a customer goes instantaneously to department k with P( ) ; his total service is completed with probability :

34. : the average arrival rate of customers at department m from any source. A steady-state distribution of the state of the system is given by the products

35. Agenda Abstract Petri Net and Stochastic Reward Net Network of waiting lines Phase dependent performance Space decomposed model Network example Conclusion

36. The phased recovery model describes the cycle from failure until the system is back to steady state. Each phase may have different set of available resources for the virtual connections. Represented as phase-dependent stationary routing probabilities with corresponding phase-dependent arrival rates.

37. Phased recovery model Phase IV ： After the routing information is restored the network operates in fault free mode. – Absorbing state for the purpose of survivability analysis. Phase I ： Immediately after the failure the procedure is activated but it takes some time before the rerouting is effective. – packets are routed according to the original routing scheme. –, except for the failed node i and link [i, j] where q ij (I) = 0. – The rerouting time is exponentially distributed with rate.

38. Phased recovery model Phase II ： When the rerouting is effective the link or node is still failed. – new routing scheme avoid these failed links or nodes. – the exponentially distributed repair time with rate. Phase III ： On completion of repair the system returns to failure free state but the routing is yet to change. – Phase II and III may have identical routing probabilities with. – exponentially distributed rerouting time with rate.

39. Performance metrics In this paper the performance metric M includes – The transient loss rate L(t) at time t. – The loss probability l(t) at time t. – The number of packets in the system N(t) at time t. – The mean end-to-end delay D(t) of packets that are not lost in the virtual connection.

40. Performance metrics

41. The transient probabilities,, are obtained from the composite models. : The external arrival rate.

42. Performance metrics

43. Agenda Abstract Petri Net and Stochastic Reward Net Network of waiting lines Phase dependent performance Space decomposed model Network example Conclusion

44. Exact network survivability models The modeling assumptions in the simulation and Stochastic Reward Net (SRN) models are identical with – exponentially distributed interevent times – time-independent – phase-dependent routing probabilities.

45. Stochastic Reward Net model The packets are tokens that are generated by the timed transition “arrival” into the place “InQ1”. If there are less than tokens in place “Node1” the immediate transition “Q1” is enabled. If not, the “loss1” transition is enabled and count the loss packet number. The routing is determined by probabilities on the immediate transitions out “OutQ1”.

46. The rerouting, failure and repair are modeled at the top of the SRN model where the tokens in the “phase y” places will constrain the token passing of the failed node through inhibitor arcs as illustrated in the figure.

47. We decompose the problem and approximate the global probabilities by the product.

48. Calculate the arrival rates to each node in the queuing network for VC v in node i by solving the linear system of traffic equations:

49. Assume that each node is an M/M/1/-queue. is the steady state probability that node i will reject an incoming packet. （2）（2）

50. Space decomposed model As per our space decomposition approximation we model the transient behavior in each node separately. To give an example consider the four node case in right where node j = 2 has failed and the non-failed nodes (i = 1, 3, 4).

51. all the packets that are sent to node j are lost, hence all transitions lead to state ( ) where all resources are unavailable and no packets will be served.

52. Immediately after the undesired event the network state is changed from to. no packets are lost but the arrival rates are changed to by (1)

53. Space decomposed model In the model of the failed node j the transient probability is obtained with the initial condition For the non-failed nodes (i ≠ j) the is obtained with the initial condition

54. Finally, the global state probabilities are obtained by product form approximation （3）（3）

55. Agenda Abstract Petri Net and Stochastic Reward Net Network of waiting lines Phase dependent performance Space decomposed model Network example Conclusion

56. A 4 node example The first example is a network with n = 4 nodes. The performance of the virtual connection between s = 1 and d = 4 is evaluated after the failure of node 2 at time t = 500. Each node i is an M/M/1/ni system with the parameters given in Table 1. The parameters in the phased recovery model are and.

58. (a) Loss probability, l(t)(b) Mean number, N(t) The estimated performance metrics from R = 90 simulation replica (Simulations) are compared against the analytical values of the Stochastic Reward Net model solved by SPNP (SRN model).

59. A 10 node example The directed graph G[1,10] for routing virtual connections between s = 1 and d = 10. The performance of the virtual connection is evaluated after the failure of node 4 at time t = 500. each node is an M/M/1/ni system with the parameters given in Table 2. The parameters in the phased recovery model are and.

61. (a) Loss probability, l(t) (b) Number in the system, N(t) In this example also includes a “rerouting model” which is, i.e. the probability that a packet is lost in the failed node at time t after the instant of failure.

62. A 10 node example The results in Figure 9(b) means with very low steady-state packet loss probability, the approximation is good. The transient loss probability is dominated by q(1, 4) with the decay rate equal to the reciprocal of the expected rerouting time. The same is not observed in the 4-node network because here steady-state loss probability is not negligible.

63. Agenda Abstract Petri Net and Stochastic Reward Net Network of waiting lines Phase dependent performance Space decomposed model Network example Conclusion

64. In our space decomposition model we assumed that : – Independence between the network nodes which is not a fully realistic assumption but a good approximation in networks with low loss probability and with high aggregation. – In each phase we have steady-state performance : Performance in a phase is reached quickly after the change of phase compared to the expected duration of the phase.

65. The approximation is good if there is at least two order of magnitude difference between the time granularity of the events in the performance model and in the recovery model.

66. in medium loaded (30-50%) high capacity networks (100Mbit/s -10 Gbit/s) we will observe 3-300 packets/ms, while the routing, rerouting and repair (at IP level) is in the order of 100s of ms.  This means a few hundred to several thousand packets are expected in each phase.

67. The results show that when the transient performance is dominated by impairments in a single node this decomposed, product form approximation is a viable approach.

68. The assumptions made in our models will be relaxed in the future allowing for multiple failures, general distribution and multiple virtual connections. We plan to extend the phased recovery model to include more details regarding the virtual connection management at failure and repair, and possibly to include multiple failure modes.

69. Thanks for your listening !