Utility-Optimal Scheduling in Time-Varying Wireless Networks with Delay Constraints

I-Hong Hou

P.R. Kumar

University of Illinois,

Urbana-Champaign

1/30

Wireless Networks A system with one server and N clients Links can fade Links interfere with each other Clients have strict per-packet delay bounds for

their packets Impossible to deliver all packets on-time

AP1

2

3 2 /30

Wireless Networks Each client needs a minimum throughput of on-

time packets Additional throughput for each client n increases

its utility through its utility function, Un(·)

AP1

2

3 3 /30

Conflict of Interests Server’s goal: maximize TOTAL utility while

supporting minimum throughput Server is in charge of scheduling clients Support minimum throughput of each client Offer additional throughput to maximize total utility

Each client’s goal: maximize its OWN utility Can lie about its utility function to gain more

throughput

4 /30

Overview of Results An on-line scheduling policy for the server that

achieves maximum total utility while respecting all minimum throughput requirements

A truthful auction conducted by the server that makes all clients report their true utility functions

Three applications Networks with Delay Constraints Mobile Cellular Networks Dynamic Spectrum Allocation

5 /30

Networks with Delay Constraints Each client periodically generates one packet

ever T time slotsτn = prescribed delay bound for client ntc,n = # of time slots needed for transmitting a

packet to client n under channel state c

T time slots

6 /30

Networks with Delay Constraints Each client periodically generates one packet

ever T time slots τn = prescribed delay bound for client n tn,c = # of time slots needed for transmitting a

packet to client n under channel state c

τ1 τ2 τ3

T time slots

t2,c t3,c

t1,c t3,c

t1,c

7 /30

Networks with Delay Constraints Each client periodically generates one packet

ever T time slots τn = prescribed delay bound for client n tn,c = # of time slots needed for transmitting a

packet to client n under channel state c

τ1 τ2 τ3

T time slots

t2,c t3,c

t1,c

t1,c

t2,cX 8 /30

Mobile Cellular Network α channels

Each channel between the base station and mobile fades ON or OFF

X9 /30

Mobile Cellular Network α channels

Each channel between the base station and mobile fades ON or OFF

XX

10 /30

Dynamic Spectrum Allocation One primary user and many secondary users Channel unused by the primary user can be

used by secondary users However, secondary users can interfere with

each other Schedule an interference-free allocation

1

2

3 5

4

11 /30

General Model A system with one server and N clients Time is divided into time intervals

An interval may consist of multiple time slots Server schedules a feasible set of clients in

each interval Feasibility depends on network constraints

AP1

2

3 12 /30

Network Feasibility Model c(k) = network “state” at interval k State = sets of feasible clients {c(1),c(2),c(3),…} are i.i.d. random variables

Prob{c(k)=c} = pc

AP1

2

3

{1,2}{1,3}

{1}{2,3} {1,2,3}

{1,2}{1,3}

{1,2}{2,3}

{2}{3}

13 /30

Utilities of Clients Server schedules a feasible set in each interval Suppose qn = long-term service rate provided to

client n Un(qn) = utility of client n

AP1

2

3

{1,2}{1,3}

{1}{2,3} {1,2,3}

{1,2}{1,3}

{1,2}{2,3}

{2}{3}

q1 = 3/6 q2 = 5/6

q3 = 4/6 14 /30

NUM in Wireless

Max ∑Un(qn)

s.t. Network dynamics constraints

Network feasibility constraints

qn ≥ qnEnhancing fairness or supporting minimum service requirements

15 /30

Server Scheduling Policy Server adapts λn(k) based on (qn – qn)+

In each interval, server schedules feasible set S that maximizes

Max-Weight Scheduling Policy Solves NUM without knowing pc

( ' ( ) ( ))n n nn SU q k

Favor clients that improve total utility most

Compensate under-served clients

( 1) { ( ) [ ]}n n k n nk k q q

16 /30

Concepts of Truthful Auction Clients may lie about their utility functions In each interval, each client n receives a

reward rn proportional to Un(qn) en = amount that n has to pay Each client n greedily maximizes its net reward

= rn-en Marginal utility of client n = {rn if it is served} –

{rn if it is not served} An auction is truthful if all clients report their

true marginal utility17 /30

Design of a Truthful Auction The server announces a discount dn(k) in each

interval k Each client n offers a bid bn(k) The server schedules the set S that maximizes

Each scheduled client n is charged

Theorem: For each client n, choosing bn(k) to be its marginal utility is optimal

( ( ) ( ))n nn Sb k d k

': ' ' ,max [ ( ( ) ( ))] ( ( ) ( )) ( )S n S m m m m nm S m S m n

b k d k b k d k d k

18 /30

Optimality of the Auction

Theorem: Let dn(k)≡λn(k). The auction schedules the same set as the Max-Weight Scheduling Policy

This auction design also solves the NUM problem

19 /30

Simulation Overview Compare with one state-of-the-art technique

and a random policy

Utility functions

Metrics: total utility and total penalty

( )n nnU q

( )nn nq q

1( )

nan

n n nn

qU q w

a

20 /30

Networks with Delay Constraints Each client generates one packet ever T time

slots τn = prescribed delay bound for client n tn,c = # of time slots needed for transmitting a

packet to client n under channel state c

A variation of knapsack problem Solved by dynamic programming in O(N2T)

τ1 τ2 τ3

T time slots

21 /30

Network with Delay Constraints 45 clients generate VoIP traffic at 64 kbit/sec An interval = 20 ms tn,c = 480 μs (under 11 Mb/sec)

or 610 μs (under 5.5 Mb/sec) wn = 3 + (n mod 3), an = 0.05 + 2n,

qn = 0.5+0.01(20n mod 300)

Compared against the modified-knapsack policy of [Hou and Kumar] Modified-knapsack focuses on satisfying minimum

service rate requirements only

22 /30

Simulation Results

23 /30

Mobile Cellular Network α channels Each channel between the base station and

mobile fades ON or OFF

Schedule the α ON clients with largest

X24 /30

( ) ( )n nb k d k

Mobile Cellular Networks 20 clients and one base station with three

channels wn = 1 + (n mod 3), an = 0.2 + 0.1(n mod 7),

qn = 0.05(n mod 5), Prob(n is ON) = 0.6+0.02(n mod 10)

Compared against the WNUM policy in [O’Neil, Goldsmith, and Boyd] WNUM optimizes utility on a per-interval basis

without considering long-term average

25 /30

Simulation Results

26 /30

Dynamic Spectrum Allocation One primary user and many secondary users Channel unused by the primary user can be

used by secondary users Secondary users can interfere with each other Schedules a maximum weight independent set

with weights

1

2

3 5

4

27 /30

( ) ( )n nb k d k

Dynamic Spectrum Allocation 20 clients randomly deployed in a 1X1 square wn = 1 + (n mod 3), an = 0.2 + 0.1(n mod 7),

qn = 0.05(n mod 8)

Compared against the VERITAS policy of [Zhou, Gandhi, Suri, and Zheng] VERITAS optimizes utility on a per-interval basis

without considering long-term average behavior

28 /30

Simulation Results

29 /30

Conclusions Network Utility Maximization (NUM) in wireless

Client utilities depend on long-term average throughput of on-time packets

Network constraints are dynamic with unknown distribution

Clients may lie about utility functions to gain more service

Solutions of the NUM problem: An on-line scheduling policy for the server A truthful auction design Applied the solutions to three applications

30 /30