A Call Admission Control for Service Differentiation and Fairness Management in WDM Grooming Networks Kayvan Mosharaf, Jerome Talim and Ioannis Lambadaris

A Call Admission Control for Service Differentiation and

Fairness Managementin WDM Grooming Networks

Kayvan Mosharaf, Jerome Talim and Ioannis LambadarisBroadNet 2004 proceeding

Presented by ZhanxiangFebruary 7, 2005

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.

Quick Review of MDP

• DTMC

• DTMDP– We focus on DTMDP because CTMDP

usually solved by discretization.

DTMC

0 1 -1 -1

-1

A DTMC { | 0,1,2...} is a discrete time discrete value random sequence such that

given , ... , the next random variable depends only on through the

transition probability

[ |

n

n n n

n n

X n

X X X X X

P X j X

2 2 0 0 -1, ,..., ] [ | ]

of the : ( ) ( )

Transition probability: ( ) ( | ) , 0

Chapman-Kolmogorov equation: ( ) ( ) ( ), , , , 0

n n n n

n j n

jk m n m

ij ik kjk

i X i X i P X j X i

pmf X p n P X j

p n P X k X j m

p m n p m p n m n i j

Originate from Professor Malathi Veeraraghavan’s slides.

DTMC0 1

00 01 02

0 1 2

Initial distribution: (0) [ (0), (0), ... ]

Transition probability matrix P:

...

... ... ... ...

...

... ... ... ...

: ( )

i i i

n

p p p

p p p

Pp p p

n step transition matrix P n P

probabilities that system

:

( ) (0) && ( ) (0)n nj i ij

i

is in state j after n transitions

p n p P p n p P Originate from Professor Malathi Veeraraghavan’s slides.

DTMC

• Two states i and j communicate if 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 periodic if 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.


DTMC - ( ) ,

. lim ( ),

: - ,

lim ( ) lim (0)

ij

j ijn

j

nj j j ij jn n

i

the n step transition probabilities p n of finite

irreducible and aperiodic MCs become independent

of i and n as n Let q p n

v long run proportion

v p n p P q

0

lim ( - )

, ( 0,1,...), & 1

n

n

j i ij ji j

P V steady state probability vector

v v p j v


Decision Theory

• 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.

Originate from David W. Kirsch’s slides

Markov Decision Process

• MDP is defined by:

State Space: SAction Space: AReward 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

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]

Next state s’

Action a

DTMC

MDP

Current state s

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.

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’)


Policy & Following a Policy

• Policy: a mapping from S to A, π : SA

• 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

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)


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.

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

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).


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

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 : SA

Network Model :: Definitions

OADM: Optical Add/Drop MultiplexerWC: wavelength converterTSI: time-slot interchangerL: # of links a WDM grooming network containsM: # of origin-destination pairs the network includesW: # of wavelengths in a fiber in each linkT: # of time slots each wavelength includesK: # of classes of traffic streamsck: traffic stream classes differ by their b/w requirementstk: # of time slots required by class ck traffic to be

establishednk: # of class ck calls currently in the system

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

Fairness definition

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

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

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)

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)


• Reward function R:

• Value function


• Optimal value function:

• Optimal Policy:


• Value iteration to compute Vn(s)


• Action decision: If Vn(AkP1s) >= Vn(AkP0s)then a=1;else a=0;

Basing on the equation below.

My understanding

• The author’s idea of using MDP is great

Example

Matlab toolbox calculation

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.

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.

Performance comparison

1

k k

i j

We define as the offered load per o-d pair;

BP as the blocking performance of class c calls;

Suppose that c and c calls experience the highest and lowest

blocking probabilities in the ne

Kk

k k

ir

j

twork, then we define fairness ratio

BPas f := ;

BP




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;

Possible Constrains

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

Backup

• Other MDP representations

Markov Assumption

• Markov Assumption:The next state’s conditional

probability depends only on a finite history of previous states (R&N)

kth order Markov Process

• Andrei Markov (1913)

The definitions are equivalent!!!

Any algorithm that makes the 1st order Markov Assumption can be applied to any Markov Process

• Markov Assumption:The next state’s conditional

probability depends only on its immediately previous state (J&B)

1st order Markov Process

Originate from David W. Kirsch’s slides

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.

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).

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.

Transition Matrix

Next state s’

Current state s

Action a

DTMC

MDP

MDP Policy

• A policy π is a mapping from S to Aπ : S A

• Assumes full observability: the new state resulting from executing an action will be known to the system

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?

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.

Bellman Equations

'

* *

'

( ) ( , ( )) ( , ( ), ') ( ')

( ) ( , ( )) ( , ( ), ') ( ')

.

.

| | tan

s S

s S

V s R s s T s s s V s

V s MAX R s s T s s s V s

is the discount factor

There is one equation for each state in S

Thus we have to solve S simul eo

.us Bellman equations

Value Iteration Algorithm

Can’t solve directly, so use an iterative approach:

1. Begin with an arbitrary utility vector V;

2. For eacheach s, calculate V*(s) from R(s,π) and V;

3. Use these new utility values V*(s) to update V;

4. Repeat steps 2-3 until V converges;

This equilibrium is a unique solution!

MDP Solution

* *

'

*

'

| |

( ) ( , ( )) ( , ( ), ') ( ')

arg ( , ( )) ( , ( ), ') ( ')

.

s S

s S

Solution to the S Bellman equations

V s MAX R s s T s s s V s

is policy

MAX R s s T s s s V s

when the V converges

Documents

A Call Admission Control for Service Differentiation and Fairness Management in WDM Grooming Networks Kayvan Mosharaf, Jerome Talim and Ioannis Lambadaris