Economic Issues in Shared Infrastructures Costas Courcoubetis Department of Computer Science, Athens University of Economics and Business Richard Weber Statistical Laboratory, University of Cambridge VISA 2009, August 17, Barcelona
Resource management in virtual facilities Resource sharing within virtual infrastructures is made complex because of the details of technology specificities. Mathematics/economics can help to highlight some key issues.
Example: scheduling a server ▶ A single server is to be shared amongst 풏 agents. Agent 풊 generates a jobs at rate 흀 풊 .
Example: scheduling a server ▶ A single server is to be shared amongst 풏 agents. Agent 풊 generates a jobs at rate 흀 풊 . ▶ Initially, agents contribute resource amounts 풚 1 , . . . , 풚 풏 , building a server of rate ∑ 풌 풚 풌 . Under FCFS scheduling all jobs have mean waiting time 1 / ( ∑ 풌 풚 풌 − ∑ 풌 흀 풌 ) .
Example: scheduling a server ▶ A single server is to be shared amongst 풏 agents. Agent 풊 generates a jobs at rate 흀 풊 . ▶ Initially, agents contribute resource amounts 풚 1 , . . . , 풚 풏 , building a server of rate ∑ 풌 풚 풌 . Under FCFS scheduling all jobs have mean waiting time 1 / ( ∑ 풌 풚 풌 − ∑ 풌 흀 풌 ) . ▶ Agent 풊 suffers a delay cost, so his net benefit is, say, 1 풏풃 풊 = 흀 풊 풓 − 휽 풊 흀 풊 − 풚 풊 ∑ 풌 풚 풌 − ∑ 풌 흀 풌 휽 풊 is private information of agent 풊 .
The key issue in this talk Agents (users) have private information (about the value of the tasks they wish to carry out). This creates a problem for efficiently sharing resources. ▶ Agents will attempt to free-ride . ▶ Obvious policies (like ‘internal market’, or ‘equal sharing’) may not be suitable.
The key issue in this talk Agents (users) have private information (about the value of the tasks they wish to carry out). This creates a problem for efficiently sharing resources. ▶ Agents will attempt to free-ride . ▶ Obvious policies (like ‘internal market’, or ‘equal sharing’) may not be suitable. How one chooses to share a facility’s resources will influence what agents reveal of their private information. We would like agents to truthfully reveal their privately held information since then we can operate the facility more efficiently.
A model of a managed shared infrastructure ▶ An infrastructure as composed of resources . (links, servers, buffers, etc.)
A model of a managed shared infrastructure ▶ An infrastructure as composed of resources . (links, servers, buffers, etc.) ▶ It can be operated in various ways , 흎 1 , 흎 2 , . . . (by scheduling, routing, bandwidth allocation, etc.)
A model of a managed shared infrastructure ▶ An infrastructure as composed of resources . (links, servers, buffers, etc.) ▶ It can be operated in various ways , 흎 1 , 흎 2 , . . . (by scheduling, routing, bandwidth allocation, etc.) ▶ The subset of agents who wish to use resources of the infrastructure, say 푺 , differs from day to day.
A model of a managed shared infrastructure ▶ An infrastructure as composed of resources . (links, servers, buffers, etc.) ▶ It can be operated in various ways , 흎 1 , 흎 2 , . . . (by scheduling, routing, bandwidth allocation, etc.) ▶ The subset of agents who wish to use resources of the infrastructure, say 푺 , differs from day to day. ▶ If operated in manner 흎 on day 풕 then agent 풊 has utility 휽 풊,풕 풖 풊 ( 흎 ) 풖 풊 ( ⋅ ) is pubic knowledge, but only agent 풊 knows 휽 풊,풕 .
A model of a managed shared infrastructure ▶ An infrastructure as composed of resources . (links, servers, buffers, etc.) ▶ It can be operated in various ways , 흎 1 , 흎 2 , . . . (by scheduling, routing, bandwidth allocation, etc.) ▶ The subset of agents who wish to use resources of the infrastructure, say 푺 , differs from day to day. ▶ If operated in manner 흎 on day 풕 then agent 풊 has utility 휽 풊,풕 풖 풊 ( 흎 ) 풖 풊 ( ⋅ ) is pubic knowledge, but only agent 풊 knows 휽 풊,풕 . ▶ 흎 is to be chosen as a function of 푺 and of the declared 휽 풕 = ( 휽 1 ,풕 , . . . , 휽 풏,풕 ) .
Agents pay for resources Agents may contribute resources to a shared infrastructure, like 풚 1 ( 휽 1 ) , . . . , 풚 풏 ( 휽 풏 ) .
Agents pay for resources Agents may contribute resources to a shared infrastructure, like 풚 1 ( 휽 1 ) , . . . , 풚 풏 ( 휽 풏 ) . Other times agents pay fees. In this case, we should like them to cover some daily operating cost, 풄 , [ ] 푬 푺,휽 풑 1 ( 푺, 휽 ) + ⋅ ⋅ ⋅ + 풑 풏 ( 푺, 휽 ) ≥ 풄
Agents pay for resources Agents may contribute resources to a shared infrastructure, like 풚 1 ( 휽 1 ) , . . . , 풚 풏 ( 휽 풏 ) . Other times agents pay fees. In this case, we should like them to cover some daily operating cost, 풄 , [ ] 푬 푺,휽 풑 1 ( 푺, 휽 ) + ⋅ ⋅ ⋅ + 풑 풏 ( 푺, 휽 ) ≥ 풄 Agent 풊 wishes to maximize his expected net benefit : [ � ] 풏풃 풊 ( 휽 풊 ) = 푬 푺,휽 휽 풊 풖 풊 ( 흎 ( 푺, 휽 )) − 풑 풊 ( 푺, 휽 ) � 휽 풊 � He may be untruthful in declaring his 휽 풊 .
The efficient frontier We wish to find Pareto optimal points of the vector [ ] ( 풏풃 1 , . . . , 풏풃 풏 ) = 푬 휽 풏풃 1 ( 휽 1 ) , . . . , 풏풃 풏 ( 휽 풏 ) 풏풃 2 0.6 0.5 0.4 풄 = 0 0.3 full information 0.2 풄 = 0 . 375 private information 0.1 풏풃 1 0.1 0.2 0.3 0.4 0.5 0.6
Maximum social welfare Suppose we wish to find the particular point that maximizes 풏풃 1 + ⋅ ⋅ ⋅ + 풏풃 풏 = 푬 푺,휽 [ 휽 1 풖 1 ( 흎 ( 푺, 휽 )) + ⋅ ⋅ ⋅ + 휽 풏 풖 풏 ( 흎 ( 푺, 휽 ))] − 풄 풏풃 2 0.3 0.2 풄 = 0 . 375 0.1 풏풃 1 0.1 0.2 0.3 We call this the ‘social welfare’.
Our infrastructure optimization problem Our infrastructure optimization problem: ▶ Say how the infrastructure will be operated for all possible subsets of users 푺 .
Our infrastructure optimization problem Our infrastructure optimization problem: ▶ Say how the infrastructure will be operated for all possible subsets of users 푺 . ▶ Say what fees will be collected from users.
Our infrastructure optimization problem Our infrastructure optimization problem: ▶ Say how the infrastructure will be operated for all possible subsets of users 푺 . ▶ Say what fees will be collected from users. Do the above, as function of declared 휽 풊 , so that: 1. Users find it in their best interest to truthfully reveal their 휽 풊 . 2. Users will see positive expected net benefit from participation. 3. Expected total fees cover the daily running cost, say 풄 . 4. Expected social welfare (total net benefit) is maximized
Example: a scalar resource ▶ 2 participants, both present on all days.
Example: a scalar resource ▶ 2 participants, both present on all days. ▶ On day 풕 , agent 풊 has utility for resource of 휽 풊,풕 풖 ( 풙 ) , assumed known to be distributed 푼 [0 , 1] .
Example: a scalar resource ▶ 2 participants, both present on all days. ▶ On day 풕 , agent 풊 has utility for resource of 휽 풊,풕 풖 ( 풙 ) , assumed known to be distributed 푼 [0 , 1] . ▶ The infrastructure provides a single resource, parameterized by a number (such as computing cycles), so operating methods correspond to allocations: { 흎 } ≡ { 풙 1 , 풙 2 : 풙 1 + 풙 2 ≤ 1 }
Example: a scalar resource ▶ 2 participants, both present on all days. ▶ On day 풕 , agent 풊 has utility for resource of 휽 풊,풕 풖 ( 풙 ) , assumed known to be distributed 푼 [0 , 1] . ▶ The infrastructure provides a single resource, parameterized by a number (such as computing cycles), so operating methods correspond to allocations: { 흎 } ≡ { 풙 1 , 풙 2 : 풙 1 + 풙 2 ≤ 1 } Suppose 풖 풊 ( 풙 ) = 풙 . Focus on one day 풕 ; with 휽 풊 = 휽 풊,풕 . ⎡ ⎤ ⎦ = 푬 [max { 휽 1 , 휽 2 } ] = 2 푬 휽 1 ,휽 2 max { 휽 1 풖 1 ( 풙 1 ) + 휽 2 풖 2 ( 풙 2 ) } ⎢ ⎥ 3 ⎣ 풙 1 ,풙 2 풙 1 + 풙 2 ≤ 1 We call this the ‘ first best ’.
The second-best solution A ‘ second-best ’ is with fee structure: { 0 , 휽 풊 < 휽 0 풑 풊 ( 휽 풊 ) = 1 2 ( 휽 2 풊 + 휽 2 0 ) , 휽 풊 ≥ 휽 0 Agent 풊 will not wish to participate if 휽 풊 < 휽 0 , since his net benefit cannot be positive. The entire resource is allocated to the agent declaring the greatest 휽 풊 , provided this is > 휽 0 . Thus, the resource is given wholly to one agent, but perhaps to neither. But both agents may pay.
The ‘solution’ This solution has the advantages that ▶ Agents are incentivized to truthful.
The ‘solution’ This solution has the advantages that ▶ Agents are incentivized to truthful. ▶ The sum of the expected payments is [ ] = 1 / 3 + 휽 2 0 − (4 / 3) 휽 3 푬 풑 1 ( 휽 1 ) + 풑 2 ( 휽 2 ) 0 .
The ‘solution’ This solution has the advantages that ▶ Agents are incentivized to truthful. ▶ The sum of the expected payments is [ ] = 1 / 3 + 휽 2 0 − (4 / 3) 휽 3 푬 풑 1 ( 휽 1 ) + 풑 2 ( 휽 2 ) 0 . ▶ The expected social welfare is decreasing in 휽 0 . But by taking 1 / 3 + 휽 2 0 − (4 / 3) 휽 3 0 = 풄 we maximize the social welfare of [ 2 ] ∑ 풏풃 1 + 풏풃 2 = 푬 휽 풊 풖 풊 ( 풙 풊 ) − 풑 풊 ( 휽 풊 ) 풊 =1 subject to covering cost 풄 .
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