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Optimality of Periodwise Static Priority Policies in Real-Time Communications

Optimality of Periodwise Static Priority Policies in Real-Time Communications. I-Hong Hou , Anh Truong, Santanu Chakraborty , P.R. Kumar. Motivation. Study the scheduling policies for real-time wireless communication Each packet has a strict deadline

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Optimality of Periodwise Static Priority Policies in Real-Time Communications

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  1. Optimality of Periodwise Static Priority Policies in Real-Time Communications I-Hong Hou, Anh Truong, SantanuChakraborty, P.R. Kumar

  2. Motivation • Study the scheduling policies for real-time wireless communication • Each packet has a strict deadline • Timely-throughput: the throughput of packets that are delivered on time • Consider the unreliable nature of wireless transmissions • Previous work has proposed scheduling policies • This work: understand some properties of the policies

  3. Client-Server Model • A system with N wireless clients and one AP • Time is slotted • AP schedules all transmissions 1 2 AP 3

  4. Traffic Model • Each client generates one packet every T time slots • T time slots form an period T 1 2 AP 3

  5. Delay Bounds • Deadline for each packet = T • Packets are dropped if not delivered by the deadline • Delay of successfully delivered packet is at most T T 1 2 AP 3

  6. Channel Model • Transmissions are unreliable • A transmission to client n succeeds with probability pn T 1 p1 p2 2 AP p3 3

  7. A Scheduling Example Packet expires and is dropped Forced idleness I I S F F 1 p1 p2 S I I S 2 AP p3 I I S F S 3

  8. Timely Throughput • Timely Throughput = long-term average # of packets received in a period I I S F F 1 p1 p2 S I I S 2 AP p3 I I S F S 3

  9. Timely Throughput Requirements • Client n requires timely throughput qn • System is fulfilled if all requirements are met I I S F F 1 p1 p2 S I I S 2 AP p3 I I S F S 3

  10. Summary of the Model • Clients have strict per-packet delay bound • Clients have timely throughput requirements • Wireless transmissions are unreliable I I S F F 1 p1 p2 S I I S 2 AP p3 I I S F S 3

  11. Largest Debt First Policy • Give higher priority to client with larger “debt” 1 p1 p2 2 AP p3 3

  12. Largest Debt First Policy • Give higher priority to client with larger “debt” F S 1 p1 F p2 2 AP p3 F S 3

  13. Optimality Result • Theorem: By choosing the right definition of debt, the largest debt first policy fulfills all feasible systems • Adapt debt according to (qn - actual timely throughput) • Therefore, it is a Feasibility Optimal Policy • The AP does not need to change ordering during the period • Q: Why the AP doesn’t need to change ordering?

  14. Feasibility Constraints • How many time slots per perioddoes client n need to obtain a timely throughput of qn? Ans: • There are times that the AP is forced to be idle • Let IS = Expected number of idle time slots when the set of clients is S • Theorem: A system is feasible if and only if Time we need to work on S Time we can work on S

  15. Feasibility Constraints • How many time slots per interval does client n need to obtain a timely throughput of qn? Ans: • There are times that the AP is forced to be idle • Let IS = Expected number of idle time slots when the set of clients is S • Theorem: A system is feasible if and only if • Feasible region: The region consists of all feasible [qn]

  16. Flow of Arguments Periodwise Priority policy can be feasibility optimal Vertices of the feasible region can be achieved by some priority ordering among clients Feasible region forms a polymatroid f(S) (= T – IS ) is submodular

  17. Any feasible [qn] is a convex combination of vertices of the feasible region • Hence, it can be achieved by time-sharing among priority orderings corresponding to the vertices Periodwise Priority policy can be feasibility optimal Vertices of the feasible region can be achieved by some priority ordering among clients

  18. By [D. D. Yao, 2002] Vertices of the feasible region can be achieved by some priority ordering among clients Feasible region forms a polymatroid

  19. Feasible region forms a polymatroid • Definition of polymatroid: 1. 2. is non-decreasing 3. is submodular f(S) (= T – IS ) is submodular

  20. f(S) (= T – IS ) is submodular • Let be the expected amount of time that the AP spends on a subset A, if the AP schedules clients in A right after all packets for clients in subset B are delivered • Clearly, is non-increasing with • We can establish that is sub-modular by using this property • Therefore, there exist a periodwise priority policy that is feasibility optimal

  21. Extension for Time-Varying Channels • Wireless channels are time-varying • In the period, the channel reliability for client n is • Joint Debt-Channel Policy: Prioritize clients by (debt) • [Hou and Kumar 10] has only shown that this policy is feasibility optimal among all periodwise priority policies • Now, we can show that this policy if feasibility optimal among all policies 1 p1(t) p2(t) 2 AP p3(t) 3

  22. Conclusion • Study the scheduling policy for real-time wireless communication • Understand why that a periodwise priority policy can be feasibility optimal • It is because that the feasibility constraints form a polymatroid • Our result can be extended to time-varying wireless channels, and hence establish a previous policy is indeed feasibility optimal

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