Utility Maximization for Delay Constrained QoS in Wireless

I-Hong Hou

P.R. Kumar

University of Illinois,

Urbana-Champaign1 /23

Problem Overview Every packet has a hard delay bound Timely throughput = Throughput of packets delivered

within their delay bounds qn = Timely throughput of client n Un(qn) = Utility of client n Channels are unreliable

Goal: Max ∑Un(qn) s.t. [qn] feasible under both channel unreliabilities and delay constraints

Example applications: VoIP, Network control, etc.2 /23

Client-Server Model A system with N wireless clients and one AP AP schedules all transmissions Time is slotted

AP1

2

3 3 /23

Traffic Model Group time slots into periods with τ time slots Clients generate packets at the beginning of each

period

AP1

2

3

τ

4 /23

Delay Bounds τ = Deadline Packets are dropped if not delivered by the deadline Delay of successful delivered packet is at most τ

AP1

2

3 5 /23

τ

arrival deadline

Channel Model Each transmission takes one time slot Links are unreliable Transmission for client n succeeds with probability pn

AP1

2

3

p1p2

p3

6 /23

How the System Works

AP1

2

3

SF

p1p2

p3

7 /23

SF F

SS

S

F II

I

Timely Throughput

AP1

2

3

SF

p1p2

p3

8 /23

SF F

SS

S

F II

I

Timely throughput (qn)

=

Client # Throughput

1 1

2 0.5

3 1

# of delivered packets

# of periods

Problem Formulation

Each client has an utility function, is strictly increasing, strictly concave, and

continuously differentiable

AP needs to assign [qn] to maximize total utility, subject to feasibility constraints

9 /23

( )n nU q( )n nU q

Characterization of What is Feasible The average number of time slots needed for client n to

have timely throughput qn is

Let IS = Expected number of idle time slots when the set of clients is S

Clearly, we need

Theorem: the condition is both necessary and sufficient

( ) nn n

n

qw q

p

, {1,2,..., }nS

n S n

qI S N

p

10 /23

Average # of packets delivered in a period

Average # of transmissions needed for a delivery

Optimization Problem SYSTEM:

Decompose SYSTEM into two subproblems CLIENTn: considers own utility function

ACCESS-POINT: considers feasibility constraints11 /23

1

( )

s.t. ,

over 0, 1

Max n n

n

n

N

n

Sn S

n

U q

qI S

p

q n N

Utility functions may be unknown

2N feasibility constraints

Problem DecompositionCLIENTn:

(Ψn given)

Max

over

ACCESS-POINT:

(ρn given)

Max

s.t.

over

12 /23

( )nn n

n

U

0 n n

1

logn

N

nnq

,nS

n S n

qI S

p

0nq

n

nn

nq

A Bidding GameStep 1. Each client n announces ρn

Step 2. Given [ρn], AP finds [qn] to solve ACCESS-POINT

Step 3. Client n observes qn, compute Ψn= ρn/qn.

Client n finds new ρn to solve CLIENTn

Step 4. Go to Step 2.

13 /23

Solving ACCESS-POINT ACCESS-POINT: (ρn given)

Max

s.t.

over

1

logn

N

nnq

,nS

n S n

qI S

p

0, 1n n Nq

By KKT condition:

n n

n SS n

q

p

∋

14 /23

Solving ACCESS-POINT ACCESS-POINT: (ρn given)

By KKT condition:

n n

n SS n

q

p

∋

Average # of time slots workingfor client n per period

15 /23

Solving ACCESS-POINT ACCESS-POINT: (ρn given)

By KKT condition:

n n

n SS n

q

p

∋

The more price paid, the more time slots received

16 /23

Solving ACCESS-POINT ACCESS-POINT: (ρn given)

By KKT condition:

n n

n SS n

q

p

∋

Depends on prices paid by all clients

and feasibility constraints

(Difficult to solve)

17 /23

Scheduling Policy for ACCESS-POINT

Weighted-Transmission Policy (WT): 1. Let be the total number of time slots allocated

for client n 2. Sort clients by

3. Clients with smaller get higher priorities

Theorem: WT solves the ACCESS-POINT problem Require no knowledge on channel reliabilities

( )n

n

u t

( )nu t

( )n

n

u t

18 /23

Simulation: Utility Maximization Setup:

A set of 30 clients Utility function: Parameters:

Setting 1:

Setting 2:

Evaluate the mean and variance of ( )n nn

U q

( mod3) 1, 0.3 0.1( mod5)n nn n

1( )

nn

n n nn

qqU

(50 )%np n 1, 0.3 0.1( mod5)n n n (20 )%np n

19 /23

Evaluated Policies WT policies and bidding game (WT-Bid)

WT policies without bidding game (WT-NoBid)

Randomly assign priorities (Rand)

Clients with larger get higher priorities, break ties randomly (P-Rand)

n

20 /23

Simulation Results: Mean

WT-Bid has highest total utility21 /23

Simulation Results: Variance

WT-Bid has small variance22 /23

Conclusion Formulate and solve the problem of utility

maximization for delay-constrained wireless networks

Propose a scheduling policy to solve ACCESS-POINT

23 /23

AP1 2p1

p2

τ

arrival deadline

CLIENTn

SYSTEM

ACCESS-POINT

Ψn ρn

24

Another work on scheduling delay-constrained packets with time-varying channels, different delay bounds, and rate adaptation will be presented in

TS60: WIRELESS NETWORK SCHEDULING 3