Kayvan Mosharaf, Jerome Talim and Ioannis Lambadaris BroadNet 2004 proceeding

1. A Call Admission Control for Service Differentiation and Fairness Managementin WDM Grooming Networks Kayvan Mosharaf, Jerome Talim and Ioannis Lambadaris BroadNet 2004 proceeding Presented by Zhanxiang February 7, 2005

2. Goal & Contribution • Goal: • Fairness control and service differentiation in a WDM grooming network. Also maximizing the overall utilization. • Contributions: • An optimal CAC policy providing fairness control by using a Markov Decision Process approach; • A heuristic decomposition algorithm for multi-link and multi-wavelength network.

3. Quick Review of MDP • DTMC • DTMDP • We focus on DTMDP because CTMDP usually solved by discretization.

4. DTMC Originate from Professor Malathi Veeraraghavan’s slides.

5. DTMC Originate from Professor Malathi Veeraraghavan’s slides.

6. DTMC • Two states i and j communicateif for some n and n’, pij(n)>0 and pji(n’)>0. • A MC is Irreducible, if all of its states communicate. • A state of a MC is periodicif there exists some integer m>0 such that pii(m)>0 and some integer d>1 such that pii(n)>0 only if d|n. Originate from Professor Malathi Veeraraghavan’s slides.

7. DTMC Originate from Professor Malathi Veeraraghavan’s slides.

8. Probability Theory + Utility Theory = Decision Theory Describes what an agent should believe based on evidence. Describes what an agent wants. Describes what an agent should do. Decision Theory Originate from David W. Kirsch’s slides

9. Markov Decision Process • MDP is defined by: State Space: S Action Space: A Reward Function: R: S  {real number} Transition Function: T: SXA  S (deterministic) T: SXA  Power(S) (stochastic) The transition function describe the effect of an action in state s. In this second case the transition function has a probability distribution P(s’|s,a) on the range. Originate from David W. Kirsch’s slides and modified by Zhanxiang

10. MDP differs DTMC • MDP is like a DTMC, except the transition matrix depends on the action taken by the decision maker (a.k.a. agent) at each time step. Ps,a,s' = P [S(t+1)=s' | S(t)=s, A(t)=a] Action a MDP Current state s DTMC Next state s’

11. MDP Actions • Stochastic Actions: • T : S X A  PowerSet(S) For each state and action we specify a probability distribution over next states, P( s’ | s, a). • Deterministic Actions: • T : S X A  S For each state and action we specify a new state. Hence the transition probabilities will be 1 or 0.

12. Action Selection & Maximum Expected Utility • Assume we assign reward U(s) to each state s • Expected Utility for an action a in state s is • MEU Principle: An agent should choose an action that maximizes the agent’s EU. EU(a|s) = s’ P(s’ | s, a) U(s’) Originate from David W. Kirsch’s slides and modified by Zhanxiang

13. Policy & Following a Policy • Policy: a mapping from S to A, π : SA • Following policy procedure: 1. Determine current state s 2. Execute action π(s) 3. Repeat 1-2 Originate from David W. Kirsch’s slides modified by Zhanxiang

14. Solution to an MDP • In deterministic processes, solution is a plan. • In observable stochastic processes, solution is a policy • A policy’s quality is measured by its EU Notation: π ≡ a policy π(s) ≡ the recommended action in state s π* ≡ the optimal policy (maximum expected utility) Originate from David W. Kirsch’s slides and modified by Zhanxiang

15. Should we let U(s)=R(s)? • In the definition of MDP we introduce R(s), which obviously depends on some specific properties of a state. • Shall we let U(s)=R(s)? • Often very good at choosing single action decisions. • Not feasible for choosing action sequences, which implies R(s) is not enough to solve MDP.

16. Assigning Utility to Sequences • How to add rewards? - simple sum - mean reward rate Problem: Infinite Horizon  infinite reward - discounted rewards R(s0,s1,s2…) = R(s0) + cR(s1) + c2R(s2)… where 0<c≤1 Originate from David W. Kirsch’s slides modified by Zhanxiang

17. How to define U(s)? • Define Uπ(s) is specific to each π Uπ(s) = E(tR(st)| π, s0=s) • Define U(s)= Maxπ {Uπ(s) }= Uπ*(s) • We can calculate U(s) on the base of R(s) U(s)=R(s) +  max P(s’|s,π(s))U(s’) π s’ Bellman equation If we solve the Bellman equation for each state, we will have solved the optimal policy π* for the given MDP on the base of U(s). Originate from David W. Kirsch’s slides and modified by Zhanxiang

18. Value Iteration Algorithm • We have to solve |S| simultaneous Bellman equations • Can’t solve directly, so use an iterative approach: 1. Begin with an arbitrary utility function U0 2. For each s, calculate U(s) from R(s) and U0 3. Use these new utility values to update U0 4. Repeat steps 2-3 until U0 converges This equilibrium is a unique solution! (see R&N for proof) Originate from David W. Kirsch’s slides

19. State Space and Policy Definition in this paper • The author’s idea of using MDP is great, I’m not comfortable with state space definition and the policy definition. • If I were the author, I will define system state space and policy as follows: • S’ = S X E where S={(n1, n2, … , nk) | tknk<=T} and E={ck class call arrivals} U {ck class call departures} U {dummy events} • Policy π : SA

20. Network Model :: Definitions OADM: Optical Add/Drop Multiplexer WC: wavelength converter TSI: time-slot interchanger L: # of links a WDM grooming network contains M: # of origin-destination pairs the network includes W: # of wavelengths in a fiber in each link T: # of time slots each wavelength includes K: # of classes of traffic streams ck: traffic stream classes differ by their b/w requirements tk: # of time slots required by class ck traffic to be established nk: # of class ck calls currently in the system

21. Network model :: assumptions • For each o-d pair, class ck arrivals are distributed according to a Poisson process with rate λk. • The call holding time of class ck is exponentially distributed with mean 1/μk . Unless otherwise stated, we assume 1/μk = 1. • Any arriving call from any class is blocked when no wavelength has tk available time slots. • Blocked calls do not interfere with the system. • The switching nodes are non-blocking No preemption

22. Fairness definition • There is no significant difference between the blocking probabilities experienced by different classes of users;

23. CS & CP • Complete Sharing (CS) • No resources reserved for any class of calls; • Lower b/w requirement & higher arrival rate calls may starve calls with higher b/w requirement and lower arrival rate; • Complete Partitioning • A portion of resources is dedicated to each class of calls; • May not maximize the overall utilization of available resources. Not Fair Fair but

24. Single-link single-wavelength(0) • System stat space S: S={(n1, n2, … , nk) | tknk <= T} k • Operators: • Aks = (n1, n2, … , nk+1, … , nK) • Dks = (n1, n2, … , nk-1, … , nK) • AkPas = (n1, n2, … , nk+a, … , nK)

25. Single-link single-wavelength(1) • Sampling rate v = ([T/tk]μk+k) k • Only one single transition can occur during each time slot. • A transition can correspond to an event of • 1) Class ck call arrival • 2) Class ck call departure • 3) Fictitious or dummy event (caused by high sampling rate)

26. Single-link single-wavelength(2) Reward function R: Value function

27. Single-link single-wavelength(3) Optimal value function: Optimal Policy:

28. Single-link single-wavelength(4) Value iteration to compute Vn(s)

29. Single-link single-wavelength(5) • Action decision: If Vn(AkP1s) >= Vn(AkP0s) then a=1; else a=0; Basing on the equation below.

30. My understanding • The author’s idea of using MDP is great

31. Example

32. Matlab toolbox calculation

33. Heuristic decomposition algorithm • Step 1: For each hop i, partition the set of available wavelengths into subsets, dedicated to each of o-d pairs using hop i. • Step 2: Assume uniformly distributed among the Wm wavelengths, thus, the arrival rate of class ck for each of the Wm wavelengths is given by: λk/Wm.

34. Heuristic decomposition algorithm (2) • Step 3: Compute the CAC policy with respect to λk/Wm. • Step 4: Using the CAC policy computed in Step 3, we determine the optimal action for each of the Wm wavelengths, individually.

35. Performance comparison

36. Performance comparison

37. Performance comparison

38. Performance comparison

39. Relation to our work • We can utilize MDP to model our bandwidth allocation problem in call admission control to achieve fairness; • But in heterogeneous network the bandwidth granularity problem is still there;

40. Possible Constrains • Under some conditions the optimal policy of an MDP exists.

42. Backup • Other MDP representations

43. Markov Assumption: The next state’s conditional probability depends only on a finite history of previous states (R&N) kth order Markov Process Markov Assumption • Andrei Markov (1913) • Markov Assumption: The next state’s conditional probability depends only on its immediately previous state (J&B) 1st order Markov Process The definitions are equivalent!!! Any algorithm that makes the 1st order Markov Assumption can be applied to any Markov Process Originate from David W. Kirsch’s slides

44. MDP • A Markov Decision Process (MDP) model contains: • A set of possible world states S • A set of possible actions A • A real valued reward function R(s,a) • A description T(s,a) of each action’s effects in each state.

45. MDP differs DTMC • A Markov Decision Process (MDP) is just like a Markov Chain, except the transition matrix depends on the action taken by the decision maker (agent) at each time step. Ps,a,s' = P [S(t+1)=s' | S(t)=s, A(t)=a] • The agent receives a reward R(s,a), which depends on the action and the state. • The goal is to find a function, called a policy, which specifies which action to take in each state, so as to maximize some function of the sequence of rewards (e.g., the mean or expected discounted sum).

46. MDP Actions • Stochastic Actions: • T : S X A  PowerSet(S) For each state and action we specify a probability distribution over next states, P( s’ | s, a). • Deterministic Actions: • T : S X A  S For each state and action we specify a new state. Hence the transition probabilities will be 1 or 0.

47. Transition Matrix Action a MDP Current state s Next state s’ DTMC

48. MDP Policy • A policy π is a mapping from S to A π : S  A • Assumes full observability: the new state resulting from executing an action will be known to the system

49. Evaluating a Policy • How good is a policy π in the term of a sequence of actions? • For deterministic actions just total the rewards obtained... but result may be infinite. • For stochastic actions, instead expected total reward obtained… again typically yields infinite value. • How do we compare policies of infinite value?

50. Discounting to prefer earlier rewards • A value function, Vπ: S  Real, represents the expected objective value obtained following policy from each state in S . • Bellman equations relate the value function to itself via the problem dynamics.