utility-optimal scheduling in time- varying wireless networks with delay constraints i-hong hou p.r....
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Utility-Optimal Scheduling in Time-Varying Wireless Networks with Delay Constraints
I-Hong Hou
P.R. Kumar
University of Illinois,
Urbana-Champaign
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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
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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
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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
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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
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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
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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
τ1 τ2 τ3
T time slots
t2,c t3,c
t1,c t3,c
t1,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
τ1 τ2 τ3
T time slots
t2,c t3,c
t1,c
t1,c
t2,cX 8 /30
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Mobile Cellular Network α channels
Each channel between the base station and mobile fades ON or OFF
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Mobile Cellular Network α channels
Each channel between the base station and mobile fades ON or OFF
XX
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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
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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
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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}
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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
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NUM in Wireless
Max ∑Un(qn)
s.t. Network dynamics constraints
Network feasibility constraints
qn ≥ qnEnhancing fairness or supporting minimum service requirements
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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
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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
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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
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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
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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
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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
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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
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Simulation Results
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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
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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
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Simulation Results
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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
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( ) ( )n nb k d k
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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
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Simulation Results
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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
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