Optimality of Periodwise Static Priority Policies in Real-Time Communications

I-Hong Hou, Anh Truong, Santanu Chakraborty, P.R. Kumar

1

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

2

Client-Server Model

3

A system with N wireless clients and one AP Time is slotted AP schedules all transmissions

AP

1

2

3

Traffic Model

4

Each client generates one packet every T time slots T time slots form an period

AP

1

2

3

T

Delay Bounds

5

Deadline for each packet = T Packets are dropped if not delivered by the

deadline Delay of successfully delivered packet is at

most T

AP

1

2

3

T

Channel Model

6

Transmissions are unreliable A transmission to client n succeeds with

probability pn

AP

1

2

3

T

p1

p2

p3

A Scheduling Example

7

AP

1

2

3

p1

p2

p3

F S

S

F F

Packet expires and is dropped

S

S

S I I

I I

I I

Forced idleness

Timely Throughput

Timely Throughput = long-term average # of packets received in a period

AP

1

2

3

8

p1

p2

p3

F S

S

F F

S

S

S I I

I I

I I

Timely Throughput

0.5

1.0

1.0

Timely Throughput Requirements

Client n requires timely throughput qn System is fulfilled if all requirements are

met

AP

1

2

3

9

p1

p2

p3

F S

S

F F

S

S

S I I

I I

I I

Timely Throughput

0.5

1.0

1.0

qn

0.7

0.3

0.5

Summary of the Model

Clients have strict per-packet delay bound Clients have timely throughput

requirements Wireless transmissions are unreliable

AP

1

2

3

10

p1

p2

p3

F S

S

F F

S

S

S I I

I I

I I

Largest Debt First Policy

Give higher priority to client with larger “debt”

AP

1

2

3

11

p1

p2

p3

Largest Debt First Policy

Give higher priority to client with larger “debt”

AP

1

2

3

12

p1

p2

p3

F

F

S

F S

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?

13

How many time slots per period 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

Feasibility Constraints

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∑𝑛∈𝑆

𝑞𝑛

𝑝𝑛≤𝑇 − 𝐼𝑆=: 𝑓 (𝑆)

Time we need to work on S

Time we can work on S

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]

Feasibility Constraints

15

∑𝑛∈𝑆

𝑞𝑛

𝑝𝑛≤𝑇 − 𝐼𝑆=: 𝑓 (𝑆)

Flow of Arguments

16

Periodwise Priority policy can be feasibility optimalVertices of the feasible region can be achieved by some priority ordering among clients

Feasible region forms a polymatroid

f(S) (= T – IS ) is submodular

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

17

Periodwise Priority policy can be feasibility optimalVertices of the feasible region can be achieved by some priority ordering among clients

By [D. D. Yao, 2002]

18

Vertices of the feasible region can be achieved by some priority ordering among clients

Feasible region forms a polymatroid

Definition of polymatroid: 1. 2. is non-decreasing 3. is submodular

19

Feasible region forms a polymatroid

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

20

f(S) (= T – IS ) is submodular

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

21

AP

1

2

3

p1(t)

p2(t)

p3(t)

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 optimal22