Medium Access Control for Distributed Systems Faeze Heydaryan Joint work with Yanru Tang Committee members: Prof. Rockey Luo Prof. Liuqing Yang Prof. Ali Pezeshki Prof. Haonan Wang
Coordinated vs Distributed Communication Distributed Communication Coordinated Communication • Opportunistic channel access • Joint coding optimization • Bursty short messages • long messages 1
Classical Information and Network Theory Classical Information Theory Classical Network Theory • Emphasizes on efficiency • Emphasizes on modularity Wireless network requires both • Joint coding optimization • Layering architecture efficiency and modularity. • Long messages Problems in Distributed Communication Networks • Binary transmission/ idlingde decisions at each • User coordination can be expensive. data link layer user • Power/rate adaptation not supported. 2
Enhanced Physical Link-layer Interface Current Physical-Link Layer Interface • Single transmission option Enhanced Physical-Link Layer Interface • Multiple transmission options • Possible support of power/rate adaptation 3
Distributed Channel Coding Ensemble of channel codes at the physical layer Each code corresponding to a link layer transmission option • Transmitters choose channel code individually. • Receiver knows the code ensemble, but not the coding choices. • Message decoding or collision report due to reliability requirement. Achievable Region • If the transmitters happen to choose their options inside the region , the packet can be recovered with asymptotic error probability of zero. • If the transmitters happen to choose their options outside the region, collision can be reported with asymptotic probability of one. Achievable region for multiple access system over a discrete time memoryless channel 4
Multiple Transmission Options at Link Layer Current Protocol Back off by decreasing transmission probability in response to packet collison. Efficient Approach Decrease communication rate in response to packet collision. � users have messages ( 𝐿 � is changing) Multiple access system with 𝐿 users + , 𝐿 Example � � � � If 𝑠 � � (bits/symbol) is fixed, maximum achievable sum rate is � � . � = � log � 1 + � log � 1 + �� �� � � � � If 𝑠 � � (bits/symbol) with rate adaptation, maximum achievable sum rate is � � . � = � log � 1 + � log � 1 + �� With multiple transmission options: How system should respond to success transmission and packet collision? How to support such functions at the physical layer ? We propose a distributed MAC algorithm to optimize a general utility function with/without enhanced interface. 5
Existing MAC Protocols Current Distributed MAC Algorithms ALOHA protocol Tree-splitting algorithm Back-off approach Set of colliding users Users maintain a transmission probability Non-Transmitting Transmitting users No Transmission users is successful Yes Probability decreases Probability increases Existing MAC algorithms either consider throughput optimization and/or collision channel. We consider a general channel, a general utility, with/without enhanced physical link-layer interface. 6
System Model Multiple access network with 𝐿 homogenous users. 𝐿 unknown to transmitters and receiver. Each user is backlogged with a saturated message queue. Time is slotted. Each user has 𝑁 transmission options + an idling option. � 𝒒 � 𝑢 = 𝑞 � 𝑢 𝒆 � 𝑢 , 0 ≤ 𝑒 �� 𝑢 ≤ 1, � 𝑒 �� 𝑢 = 1 ��� 𝒒 � 𝑢 : Transmission probability vector of user 𝑙 𝑞 � (𝑢) : Transmission probability of user 𝑙 𝒆 � 𝑢 : Transmission direction vector of user 𝑙 Probabilities of choosing each option if user) 𝑙 transmits) 7
Stochastic Approximation Framework Updating Rule: 𝒒 � 𝑢 + 1 = 1 − 𝛽 𝑢 𝒒 � 𝑢 + 𝛽 𝑢 𝒒 � � 𝑢 = 𝒒 � 𝑢 + 𝛽(𝑢)(𝒒 � � 𝑢 − 𝒒 � 𝑢 ) 𝑸 𝑢 = 𝒒 � 𝑢 � 𝒒 � 𝑢 � … 𝒒 � 𝑢 � � � 𝑢 − 𝑸(𝑢)) 𝑸 𝑢 + 1 = 𝑸 𝑢 + 𝛽(𝑢)(𝑸 � 𝑢 = 𝒒 � � 𝑢 � 𝒒 � � 𝑢 � … 𝒒 � � 𝑢 � � 𝑸 Target probability vector Lipschitz Continuity Condition Mean-Bias Condition � 𝑸 � − 𝑸 �(𝑸 � ) � 𝑢 �(𝑢) for all 𝑸 � and 𝑸 � Bias Term: 𝑯 𝑢 = 𝑯 𝑸 𝑢 𝑸 ≤ 𝐿 � 𝑸 � − 𝑸 � = 𝐹 � 𝑸 − 𝑸 𝑯 𝑢 ≤ 𝐿 � 𝛾(𝑢) � 𝑢 Noiseless version of 𝑸 �𝑸(�) �(𝑢) Associated Ordinary Differential Equation (ODE): = − 𝑸 𝑢 − 𝑸 �� Equilibrium of ODE 𝑸 ∗ : 𝑸 ∗ = 𝑸 �(𝑸 ∗ ) 8
Theorem 2 Theorem 1 Assumptions: Associated ODE has unique equilibrium 𝑸 ∗ • Assumptions: • ∃ 0 < 𝛽 < 𝛽 < 1, ∃ 𝑈 � ≥ 0, 𝛽 ≤ 𝛽 𝑢 ≤ 𝛽, 𝛾 𝑢 • Associated ODE has unique equilibrium 𝑸 ∗ ≤ 𝛽, ∀𝑢 > 𝑈 � 𝛽 � 𝑢 < ∞, ∑ � � � • ∑ 𝛽 𝑢 = ∞, ∑ 𝛽 𝑢 𝛾 𝑢 < ∞. ��� ��� ��� • Mean and Bias condition • Mean and Bias condition • Lipschitz Continuity condition • Lipschitz Continuity condition Conclusion: Conclusion: 𝑸 𝑢 converges weakly to 𝑸 ∗ in the sense that • 𝑸 𝑢 converges to 𝑸 ∗ with probability one. • 𝑸 𝑢 − 𝑸 ∗ ∀𝜗 > 0, ∃𝐿 � : lim sup 𝑄𝑠 ≥ 𝜗 < 𝐿 � 𝛽. �→� 9
MAC Algorithm Design Challenges Objective Design a distributed MAC algorithm to satisfy Mean-Bias and Lipschitz Contunity conditions and to place unique equilibrium of associated ODE at a point that maximizes a chosen utility function. • There is a virtual packet in each time slot. Assumptions • The receiver can estimate virtual packet success probability 𝑟 � (𝑢). • Users should obtain the same target transmission probability vector Design Choice 𝑸 ∗ = 𝟐⨂𝒒 ∗ = 𝟐⨂𝒒 �(𝒒 ∗ ) • Collision Channel: Virtual packet success probability Idling probability 𝑟 � 𝑢 = 1 − 𝑞 � 𝑢 𝑟 � 𝑢 Examples • General Channel + Random Block Coding: Reception of virtual packet Detecting whether transmission vector of real users belong to a specific region 10
Single Option: Channel Model Single transmission option 𝑁 = 1 𝒒 � = 𝑞 � , 𝑸 = 𝒒 = 𝑞 � 𝑞 � … 𝑞 � � Channel Model Real channel parameter set 𝐷 �� for 𝑘 ≥ 0 Virtual channel parameter set 𝐷 �� for 𝑘 ≥ 0 𝐷 �� ≥ 𝐷 �(���) Definition 𝐷 �� : Conditional success probability 𝐷 �� : Success probability of a virtual of a real packet should it be packet should it be transmitted in 𝐾 � � = 𝑏𝑠 min 𝐷 �� > 𝐷 �(���) + 𝜗 � � transmitted in parallel with j other parallel with j real packets. real packets. 11
Single Option: Utility Optimization Maximize a symmetric network utility �� Sum system throughput: Weighed sum system throughput ��� ��� = 𝐿 � 𝐿 − 1 = 𝐿 � 𝐿 − 1 𝑞 ��� 1 − 𝑞 ����� 𝐷 �� 𝑞 ��� 1 − 𝑞 ����� 𝐷 �� − 𝐿𝑞𝐹 𝑉 𝐿, 𝑞, 𝐷 �� 𝑉 𝐿, 𝑞, 𝐷 �� 𝑘 𝑘 ��� ��� ∗ = 𝑦 ∗ Asymptotically optimal transmission probability satisfies lim � →� 𝐿𝑞 � �→� 𝑉 𝐿, 𝑦 𝑦 ∗ = 𝑏𝑠 max lim 𝐿 , 𝐷 �� � � ∗ � ∗ Set the system equilibrium at 𝑞 ∗ = 𝑛𝑗𝑜 𝑞 ��� , � �� �� without knowing 𝐿 . ��� , 𝑞 ��� = 𝑛𝑗𝑜 1, 𝐿 𝑞 � 1 − 𝑞 ��� 𝐷 �� � Channel contention measure: 𝑟 � (𝑞, 𝐿) = ∑ ��� 𝑘 12
Single Option: Two Monotonicity Properties Contribution:Theorems � Estimated users number: 𝐿 With 𝐷 �� ≥ 𝐷 �(���) for all 𝑘 ≥ 0 , 𝑟 � 𝑞, 𝐿 is non-increasing in 𝑞. � ∗ Target transmission probability: 𝑞̂ = 𝑛𝑗𝑜 𝑞 ��� , ��� . � �� � �,� Furthermore < 0 for 𝐿 > 𝐾 � � and 𝑞𝜗(0,1) . �� If 𝑦 ∗ > 0 and 𝑐 ≥ 𝑛𝑏𝑦 1, 𝑦 ∗ − 𝛿 � � with 𝛿 � � being defined as ∗ Theoretical channel contention measure 𝑟 � � � � ���� � ∑ � �� �� �(���) ��� � ������ � : Largest integer below 𝐿 � 𝛿 � � = min 𝑂 = 𝐿 � �,��� �� ,��� ∗ �� � ���� � ∑ � �� �� �(���) ��� � ������ ∗ 𝑞̂ is non-decreasing in 𝑞̂ . then 𝑟 � ∗ 𝑞̂ = 𝑞̂ − 𝑞 ��� 𝑞 � − 𝑞̂ Furthermore if 𝑐 > 𝑛𝑏𝑦 1, 𝑦 ∗ − 𝛿 � � , 𝑟 � ∗ 𝑞̂ is strictly increasing in 𝑞̂ 𝑟 � 𝑟 � 𝑞̂ + 𝑟 ��� 𝑞̂ 𝑞 � − 𝑞 ��� 𝑞 � − 𝑞 ��� for 𝑞̂𝜗 (0, 𝑞 ��� ) . � ∗ 𝑞 � = 𝑛𝑗𝑜 𝑞 ��� , ��� , � 𝑟 � 𝑞 = � 𝑂 𝑞 � 1 − 𝑞 ��� 𝐷 �� 𝑘 ��� 13
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