Optimal Pricing in a Free Market Wireless Network

Michael J. NeelyUniversity of Southern California

http://www-rcf.usc.edu/~mjneely*Sponsored in part by DARPA IT-MANET Program and NSF Grant OCE 0520324

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INFOCOM 2007

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Time-slotted System: t {0, 1, 2, …}

Time-Varying Channels: (fading, mobility, etc.)

Sn(t) = (Sn1(t), Sn2(t), …, Snk(t))

(channel states on outgoing links of node n)

Transmission Rate Options (nodes use orthogonal channels):

n(t) = (n1(t), n2(t), …, nk(t)) n(Sn(t))

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Transmission Costs: Cntran(n(t), Sn(t))

Example:

Reception Costs: Cnbrec(nb(t))

*Example: Cnbrec(nb(t)) = { b if nb(t) > 0

{ 0 if nb(t) = 0

Cntran()

*this example is used in slides for simplicity

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For simplicity of these slides: Assume single commodity (multi-source, single sink) (multi-commodity case treated in the paper)

Un(t) = Queue Backlog in node n at time tRn(t) = *New data admitted to network at source n at time t

U3(t)

*Not all nodes are sources: Some simply act as profit-seeking relays

Node 3 (a source)

Transmit outNew source data R2(t)

Endogenous arrivals

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For simplicity of these slides: Assume single commodity (multi-source, single sink) (multi-commodity case treated in the paper)

Un(t) = Queue Backlog in node n at time tRn(t) = *New data admitted to network at source n at time t

U5(t)

Node 5 (pure relay: not a source)

Transmit outEndogenous arrivals

*Not all nodes are sources: Some simply act as profit-seeking relays

Free Market Network Pricing:

-Each node n sets its own per-unit price qn(t) for accepting endogenous data from others. (Seller Node Challenge: How to set the price?)

-Node n advertises qn(t) and the reception cost. (fixed reception cost b used in slides for simplicity)

current $: qn(t) rec. cost: n

AdvertisementData that node n alreadyneeds to deliver

Node n

expenses

revenue

Free Market Network Pricing:

“Buyer Nodes” pay handling charge + reception fee:

-Handling Charge: an(t) = an(t)qn(t)

-Reception Fee: n

Node n

$ = qn(t)rec = n

Advertisement

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Seller Node n Perspective

Node a

$ = qn(t)rec = n

Advertisement

n b

? ?

Buyer Node a Perspective

$ = qn(t)rec = n

Advertisement

Free Market Network Pricing:

Buyer Node Challenge: Where to send? How muchto send? Is advertised price acceptable?

(current transmission costs Cntran(n(t), Sn(t)) play a role,

as does the previous revenue earned for accepting data)

Node n

$ = qn(t)rec = n

Advertisement

? ?

Seller Node n Perspective

Node a

$ = qn(t)rec = n

Advertisement

n b

? ?

Buyer Node a Perspective

$ = qn(t)rec = n

Advertisement

Free Market Network Pricing:

The sources’ desire for communication is the driving economic force!

Modeling the Source Demand Functions:-Elastic Sources-Utility gn(r) = Source n “satisfaction” (in dollars) for sending at rate r bits/slot.

r

gn(r) Assumed to be:1. Convex 2. Non-Decreasing3. Max slope

Node n profit (on slot t):

n(t) = total income(t) - total cost(t) - payments(t)

Source (at node n) profit (on slot t):

n(t) = gn(Rn(t)) - qn(t)Rn(t)

Rn(t) $ = qn(t)

Source at n Node n

Node nPaymentsCosts

Income

Social Welfare Definition:

[gn( rn ) - costn ]n

Social Welfare =

where:

costn

rn = time avg admit rate from source n

= time avg external costs expended by node n (not payment oriented)

Simple Lemma: Maximizing Social Welfare…(i) …is equivalent to maximizing sum profit (sum profit = “network GDP”)(ii)…can (in principle) be achieved by a stationary randomized routing and scheduling policy

We will design 2 different pricing strategies:

1) Stochastic Greedy Pricing (SGP): - Greedy Interpretation - Guarantees Non-Negative Profit - If everyone uses SGP, Social Welfare Maxed over all alternatives (and so Sum Profit Maxed)

2) Bang-Bang Pricing (BB): - No Greedy Interpretation - Yields a “optimally balanced” profits (profit fairness…minimizes exploitation)

Prior Work:Utility Maximization for Static Networks: [Kelly: Eur. Trans. Tel. 97] [Kelly, Maulloo, Tan: J. Oper. Res. 98] [Low, Lapsley: TON 1999] [Lee, Mazumdar, Shroff: INFOCOM 2002]

Utility Maximization for Stochastic Networks: [Neely, Modiano, Li: INFOCOM 2005] [Andrews: INFOCOM 2005] [Georgiadis, Neely, Tassiulas: NOW F&T 2006] [Chen, Low, Chiang, Doyle: INFOCOM 2006]

Pricing plays only an indirect role in yieldingmax utilitysolution

For the stochastic algorithms, dynamic “prices” do not necessarily yield the non-negative profit goal!

Prior Work:Revenue Maximization for Downlinks (non-convex): [Acemoglu, Ozdaglar: CDC 2004] [Marbach, Berry: INFOCOM 2002] [Basar, Srikant: INFOCOM 2002]

Markov Decision Problems for single-network owner: [Paschalidis, Tsitsiklis TON 2000] [Lin, Shroff TON 2005]

Market Mechanisms: [Buttyan, Hubaux: MONET 2003] [Crowcroft, Gibbens, Kelly, Ostring WiOpt 2003] [Shang, Dick, Jha: Trans. Mob. Comput. 2004] [Marbach, Qui: TON 2005]

Profit is central toproblem

Need a stochastic theory for market-based network economics!

Stochastic Greedy Pricing Algorithm (SGP): (Similar to Cross-Layer-Control (CLC) Algorithm from [Neely 2003] [Neely, Modiano, Li INFOCOM 2005])

For a given Control Parameter V>0…

Pricing (SGP):

Node n

Queue Backlog Un(t)

Admission Control (SGP):

qn(t) = Un(t)/V

Max: gn(Rn(t)) - qn(t) Rn(t)

Subj. to: 0 < Rn(t) < Rmax

payment“instant utility”

Rn(t)

Stochastic Greedy Pricing Algorithm (SGP): (Similar to Cross-Layer-Control (CLC) Algorithm from [Neely 2003] [Neely, Modiano, Li INFOCOM 2005])

For a given Control Parameter V>0…

Resource Allocation & Routing (SGP): Define the modified differential price Wnb(t):

Wnb(t) = qn(t) - qb(t) - /V

where = max[max out, max in + Rmax]Maximize:

Wnb(t)nb(t) - Cnbrec(n(t)) - Cn

tran(n(t), Sn(t))

Subj. to : n(t) n(Sn(t))

b b

qn(t) qb(t)

Theorem (SGP Performance): For arbitrary S(t)processes and for any fixed parameter V>0:

(a) Un(t) < V + for all n, for all time t

(b) All nodes and sources receive non-negative profit at every instant of time t:

n() > 0=0

t1t

gn( ) - Rn()=0

t1t qn(t) Rn()

=0

t1t

> 0

Nodes:

Sources:

(a)(b) hold for any node n using SGP, even if others don’t use SGP!

Theorem (SGP Performance): For arbitrary S(t)processes and for any fixed parameter V>0:

(a) Un(t) < V + for all n, for all time t

(b) All nodes and sources receive non-negative profit at every instant of time t:

(c) If Channel States S(t) are i.i.d. over slots and if everyone uses SGP:

Social Welfare > g* - O(1/V)g* = maximum social welfare (sum profit) possible, optimized over all alternative algorithms for joint pricing, routing, resource allocation.

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Simulation of SGP:

Parameters: V= 50Dotted Links: ON/OFF Channels (Pr[ON] = 1/2) Transmission costs = 1 cent/packet, reception costs = .5 cent/packetSolid Links: Transmission costs = 1 cent/packetUtilities: g(r) = 10 log(1 + r)

Simulation of SGP:

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SGP: V=50C2 = C5 = C7 = 1g(r) = 10 log(1+r)

Simulation of SGP (increase cost of C2, C5):

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SGP: V=50C7=1C2 = C5 = 3g(r) = 10 log(1+r)

Bang-Bang Pricing (BB) Algorithm:

Objective is to Maximize:

nn ) + n( n ) ]n

Where n() and n() are concave profit metrics.

Yields a more balanced (and “fair”) profit distribution.

Quick (incomplete) description of Bang-Bang Pricing (see paper for details):Uses General Utility Optimization technique from our previous work in [Georgiadis, Neely, Tassiulas NOW F & T 2006]

BB Algorithm: Define Virtual Queues Xn(t), Yn(t) And Auxiliary Variables n(t), n(t).

Xn(t)

Yn(t)

expenses(t) + n(t) income(t)

payment(t) + n(t) gn(Rn(t))

Nodes n:

Sources n:

Pricing (BB):

qan(t) = { Qmax if Xa(t) < Xn(t) { 0 else

(Price depends on the incoming link)Distributed Auxiliary Variable Update:Each node n solves: Maximize: Vn() - Xn(t) Subject to: 0 < < Qmax

Resource Allocation Based on Mod. Diff. Backlog:

Wnb(t) = Un(t) - Ub(t) -qnb(t)[Xn(t) - Xb(t)]

Theorem (BB Performance): If all nodesUse BB with parameter V>0, then:

1) Avg. Queue Congesetion < O(V)

nn ) + n( n ) ]n

2)

> Optimal - O(1/V)

Simulation of BB:

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SGP: V=50C2 = C5 = C7 = 1g(r) = 10 log(1+r)

Simulation of BB (increase cost of C2, C5):

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SGP: V=50C7=1C2 = C5 = 3g(r) = 10 log(1+r)

Conclusions:

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SGP: V=50C7=1C2 = C5 = 3g(r) = 10 log(1+r)

1) SGP: GuaranteesNon-negative profit and Bounded queues, Regardless of actions of Other nodes. If all nodesUse SGP => Max sumProfit!2) BB: OptimallyBalanced, but has no greedy interpretation.