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Jointly Optimal Transmission and Probing Strategies for Multichannel SystemsPowerPoint Presentation

Jointly Optimal Transmission and Probing Strategies for Multichannel Systems

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### Jointly Optimal Transmission and Probing Strategies for Multichannel Systems

Saswati Sarkar

University of Pennsylvania

Joint work with Sudipto Guha (Upenn) and Kamesh Munagala (Duke)

Multichannel Systems Multichannel Systems

- Current wireless networks have access to multiple channels
- Frequencies in FDMA, codes in CDMA, antennas in MIMO, polarization states of antennas, access points in systems with multiple access points
- IEEE 802.11a, IEEE802.11b, IEEE802.11h

- Future wireless networks are likely to have access to larger number of channels
- Potential deregulation of spectrum
- Advances in device technology

Multichannel systems Multichannel Systems

- Transmission quality in different channels stochastically vary with time.
- A user is likely to find a channel with acceptable transmission quality when the number of channels is large,
- if he samples all the channels

- Sampling consumes energy and generates interference for other users
- Amount of information a user acquires about its channels becomes an important decision variable.

Policy Decisions Multichannel Systems

- Probing policy
- How many channels to probe?
- Which channels to probe?
- Sequence of probing

- Selection policy
- Which channel to transmit (probed channel or unprobed channel)?

System Goal Multichannel Systems

- Gain is the difference between the expected rate of successful transmission and the cost of probing
- Maximize the system gain
- Need to jointly optimize the probing and selection policies
- Policies will be adaptive
- Policies may depend on higher order statistics and cross statistics of channels.

Sense of déjà vu? Multichannel Systems

- Stopping Time problem (Bertsekas, Dynamic Programming, Vol. 1)
- Maximize reward by optimally selecting between two control actions at any given time: (a) continue or (b) stop
- IID channels (Kanodia et al, 2004, Ji et al, 2004)

- Generalized stopping time problems
- Select between multiple control actions
- Only limited results are known

Sense of déjà vu? Multichannel Systems

- Stochastic multi-armed bandit problem (Bertsekas, Dynamic Programming, Vol. 2)
- Observe the state of only one arm at each slot
- Select the arm to observe and get reward from it
- State of an arm changes only after you observe it
- State changes are stochastic
- Gittins index based policies maximize the reward (Gittins-Jones, 1974)

Sense of déjà vu? Multichannel Systems

- Adversarial multi-armed bandit problems (Auer et al, 2002)
- State of an arm may change even when you don’t observe it
- State changes are adversarial
- Some versions allow you to get reward from only one selected arm and subsequently observe the states of all arms in the slot
- Given a class of N policies and a time horizon of T slots, bound the performance loss as compared to the best of the policies in terms of O(sqrt(T * log N)).

Our Contribution Multichannel Systems

- Polynomial complexity optimal solution when channels are mutually and temporally independent and each channels has two states.
- Sigmetrics, 2006, poster paper

- Polynomial complexity approximate solution when channels have multiple states
- Approximation ratio ½ for arbitrary number of states
- Approximation ration 2/3 for 3 states

System Model Multichannel Systems

- Single user with access to n channels
- Each channel can be in one of K states, 0,…K-1.
- Channel i is in state j with probability pji
- Probability of successful transmission in state j is rj
- User knows the state of channel i in a slot only if it probes it.
- Cost of probing channel i is ci
- Channel states are temporally and mutually important

Optimal policy for K = 2 (Sigmetrics, 2006) Multichannel Systems

- Exhaustive (S, i) policy:
- probes all channels in set S,
- if no probed channel is in state 1, transmit in channel i (backup).

- Optimal policy is exhaustive (Si, i) for some i
- Si = {j: p1j(1-p1i) > cj}
- Probes all channels with high cross-uncertainty with the backup channel
- Probes channels in decreasing order of p1j/cj

- Search space for optimal policy restricted (n policies)
- Determine the optimal by explicitly evaluating the gain in the above class.

Additional challenges for K > 2 Multichannel Systems

- For K = 2, the optimal policy is static
- Actions can be determined before observation

- For K > 3, the optimal policy may be adaptive
- While probed channels are in state 0, focus on channels with high expected reward
- After observing a channel in intermediate state, focus on channels which have higher probabilities of being in the highest state
- After observing a channel in intermediate state, may select either the observed channel or the backup.

- Optimal policy is a decision tree
- Exponential number of decision trees
- Storage of decision tree consumes exponential complexity

Constant Factor Approximation Algorithm for K > 2 Multichannel Systems

- Let policy A be the best among those that always transmits in a probed channel
- Gain GA

- Let policy B be the best among those that always transmits in an unprobed channel
- Policy B is clearly the one that never probes any channel and transmits in the channel with the highest expected transmission rate.
- Gain GB

- Theorem 1: Max(GA , GB) >= ½ Maximum Gain
- The best among A and B attains half the maximum gain

Proof of Theorem 1 Multichannel Systems

- Q: Probability that the optimal policy transmit in an unprobed channel
- OPT: Expected gain of the optimal policy
- G: Expected gain of the optimal policy given that it transmits in an unprobed channel

Proof of Theorem 1 Multichannel Systems

- Modify the optimal so that it transmits in the best probed channel whenever it was using the backup
- Expected gain OPT’ which is less than or equal to GA
- Expected Gain given that the optimal uses a backup x
- OPT – OPT’ <= Q (G – x)
<= QG

<= GB

OPT <= OPT’ + GB

<= GA +GB

<= 2 max(GA , GB)

Missing link Multichannel Systems

- Need to compute the policy that maximizes gains among those that transmit only in probed channels.
- Once a channel is observed to be in state u, probe only those channels for which the expected improvement above u exceeds cost.

Special case: K = 3 Multichannel Systems

- Theorem 2: There exists an optimum policy which uses a unique backup channel.
- There exists a channel l which satisfies the following characteristic: if the optimum policy uses a backup it uses l.

Approximation algorithm for attaining 2/3 approximation ratio

- Let P(l) be the class of policies which
- never probe channel l and
- never uses any channel as a backup other than l.

- Let C be the best policy in P(l) for the best choice of l.
- Theorem 3: Max(GA , GB , GC) >= 2/3 Maximum Gain
- Best among A, B, C attains 2/3 of the optimum gain.

Open Problems ratio

- Is the optimization problem NP-hard when K exceeds 2?
- Can we get better approximation ratios?
- What happens when channels are not mutually independent?
- What happens when channels are not temporally independent?
- What happens when you can simultaneously transmit in multiple channels?
- Power control

- What happens when you need not transmit in every slot?
- Lazy scheduling

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