Utility-Optimal Scheduling in TimeVarying Wireless Networks with
Delay Constraints
I-Hong Hou
P.R. Kumar
University of Illinois,
Urbana-Champaign
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Wireless Networks
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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
2
1
AP
3
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Wireless Networks
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Each client needs a minimum throughput of ontime packets
Additional throughput for each client n increases
its utility through its utility function, Un(·)
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1
AP
3
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Conflict of Interests
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Server’s goal: maximize TOTAL utility while
supporting minimum throughput
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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
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Can lie about its utility function to gain more
throughput
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Overview of Results
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An on-line scheduling policy for the server that
achieves maximum total utility while respecting
all minimum throughput requirements
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A truthful auction conducted by the server that
makes all clients report their true utility
functions
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Three applications
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Networks with Delay Constraints
Mobile Cellular Networks
Dynamic Spectrum Allocation
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Networks with Delay Constraints
Each client periodically generates one packet
ever T time slots
τn = prescribed delay bound for client n
tc,n = # of time slots needed for transmitting a
packet to client n under channel state c

T time slots
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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
t1,c
t2,c
t3,c
τ1
τ2
τ3
T time slots
t1,c
t3,c
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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
t1,c
t2,c
t3,c
τ1
T time slots
t1,c
X
τ2
τ3
t2,c
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Mobile Cellular Network
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α channels
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Each channel between the base station and
mobile fades ON or OFF
X
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Mobile Cellular Network

α channels
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Each channel between the base station and
mobile fades ON or OFF
X
X
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Dynamic Spectrum Allocation
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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
2
4
1
3
5
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General Model
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A system with one server and N clients
Time is divided into time intervals
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An interval may consist of multiple time slots
Server schedules a feasible set of clients in
each interval
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Feasibility depends on network constraints
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1
AP
3
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Network Feasibility Model
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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
{1,2}
{1,3}
{1}
{2,3}
{1,2,3}
{1,2}
{1,3}
{1,2}
{2,3}
{2}
{3}
2
1
AP
3
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Utilities of Clients
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Server schedules a feasible set in each interval
Suppose qn = long-term service rate provided to
client n
Un(qn) = utility of client n
{1,2}
{1,3}
{1}
{2,3}
{1,2,3}
{1,2}
{1,3}
{2}
{3}
2
1
q1 = 3/6
{1,2}
{2,3}
AP
3
q2 = 5/6
q3 = 4/6
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NUM in Wireless
Max ∑Un(qn)
s.t.
Network dynamics constraints
Network feasibility constraints
qn ≥ qn
Enhancing fairness or supporting
minimum service requirements
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Server Scheduling Policy
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Server adapts λn(k) based on (qn – qn)+
n (k  1)  {n (k )  k [qn  q n ]}
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In each interval, server schedules feasible set S
that maximizes  (U 'n (qn )  n (k ))
nS
Favor clients that
improve total utility most
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Compensate under-served
clients
Max-Weight Scheduling Policy
Solves NUM without knowing pc
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Concepts of Truthful Auction
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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 utility
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Design of a Truthful Auction
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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

nS
(bn (k )  dn (k ))
Each scheduled client n is charged
max S ':nS ' [ mS ' (bm (k )  d m (k ))]   mS ,m  n (bm (k )  d m (k ))  d n (k )
Theorem: For each client n, choosing bn(k) to be
its marginal utility is optimal
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Optimality of the Auction
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Theorem:
Let dn(k)≡λn(k).
The auction schedules the same set as the
Max-Weight Scheduling Policy
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This auction design also solves the NUM
problem
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Simulation Overview
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Compare with one state-of-the-art technique
and a random policy
Utility functions
qn an  1
U n (qn )  wn
an
Metrics: total utility n U n (qn )

and total penalty  n (q n  qn )
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Networks with Delay Constraints

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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
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Network with Delay Constraints
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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]
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Modified-knapsack focuses on satisfying minimum
service rate requirements only
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Simulation Results
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Mobile Cellular Network
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α channels
Each channel between the base station and
mobile fades ON or OFF

Schedule the α ON clients with largest bn (k )  dn (k )

X
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Mobile Cellular Networks
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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]
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WNUM optimizes utility on a per-interval basis
without considering long-term average
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Simulation Results
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Dynamic Spectrum Allocation
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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 bn (k )  dn (k )
2
4
1
3
5
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Dynamic Spectrum Allocation
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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]
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VERITAS optimizes utility on a per-interval basis
without considering long-term average behavior
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Simulation Results
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Conclusions
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Network Utility Maximization (NUM) in wireless
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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
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Applied the solutions to three applications
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