1. Bandwidth Allocation with Multiple Objectives Or
What Can We Do with Per-Flow State?
Ashish Goel Stanford University Joint work with Sung-Woo Cho
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2. Bandwidth allocation
Consider N best-effort, long-lived flows sharing the Internet
Each flow only cares about its bandwidth allocation
xi = bandwidth allocated to the i-th flow
TCP: Additive Increase Multiplicative Decrease t
Window size
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3. Primal-Dual approach to B/W allocation
Constraints: Can not exceed capacity on any link
Convert 0/1 capacity constraints into “smooth” costs
pj = price for link j
Artificial. Known variously as dual costs, shadow prices etc.
Compute the total price wi of all the links in path of i-th flow
Primal update (real):
wi high => decrease xi
wi low => increase xi
Recompute pj, wi; iterate
Primal-Dual algorithm: compute prices, decide increase/decrease rule
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4. Computing dual prices j = Fractional occupancy of link j
pj must be an increasing function of j
Example: exponential prices, pj = aj
Four flows sharing
a 1Gbps link
Suppose x1 = x2 = x3 = x4 = 100Mbps
=> pj = a0.4 = very small x1 x2
Suppose x1 = x2 = x3 = x4 = 300Mbps
=> pj = a1.2 = very large x3 x4
Notice the lack of per-flow state
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5. Computing wi Imagine a field called “dual cost” in IP header
TCP sender initializes this to 0
Every link along the way can add its own price to this header
TCP recipient sends this field back to the sender in the ACK
Sender can then use this field to modify its rate
Very powerful framework
Question: what can we do with per-flow state?
First, focus on what we can do without per-flow state
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6. Example: maximize total throughput
[Bartal, Byers, Raz; Plotkin, Shmoys, Tardos; Garg, Konemann]
Use exponential dual prices
Flow i increases xi if its total cost wi is less than 1 and decreases it otherwise
Upon convergence, the algorithm maximizes i xi
Can be made to converge very fast
polylogarithmic time
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7. Maximizing throughput: contd
Initially, x1 = x2 = x3 = 0
pe, pf ¼ 0, x1,x2, x3 increase v1 v2 v3 e f x1 x2 x3
Intermediate step: x1 = x2 = x3 = 0.45+
pe, pf > 0.5, x1,x2 increase, x3 now decreases p
p=1
p=1/2
=1
Finally, x1 = x2 = 1, x3 = 0
pe, pf = 1 , x1,x2, x3 at equilibrium 
=0.9
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8. Some perspective
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9. Example: congestion minimization
[Plotkin, Shmoys, Tardos; Garg, Konemann]
Set OSPF weights using exponential costs
Send packets on shortest paths using these weights
With fixed traffic pattern, congestion (i.e. maximum link occupancy) is minimized
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10. Example: utility maximization
[Kelly, Maullo, Tan]
Goal: maximize i Ui(xi)
Ui = utility function for flow i
Ui is assumed to be concave, non-decreasing, non-negative
Primal-dual algorithm results in utility maximization
Simple rule for updating xi
Prices pj depend only on j
Convergence understood, but not in complete detail [Johari,Tan; Vinnicombe]
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11. Example: TCP as implicit primal-dual
[Kelly, Maullo, Tan; Low, Peterson, Wang]
TCP reacts to drops
pj = drop probability, which depends only on j
TCP AIMD: reacts to wi = aggregate drop probability along the route
Example: basic TCP approximately follows the dynamics
dxi/dt = a - bxi2 wi
Can be interpreted as a primal-dual algorithm
TCP Vegas: implicitly maximizes i log xi
Another variant: maximizes i tan-1xi
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12. Bandwidth allocation: goals
Leverage existing theory
Get a “dual cost” field into the IP header
Teach primal-dual algorithms for resource allocation
Algorithmic (Goel: Topics in network algorithms, winter 2006)
Economic (Johari)
2. Does per-flow state give us additional power?
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13. Which utility function to maximize?
It is not apriori clear which utility function to maximize
i log xi
i tan-1xi
i xi
mini xi (or max-min fairness; achieved by fair queueing at each router or by the poor stealing from the rich)
Pk(x) = sum of the k smallest xi’s (Prefix functions)
All the above choices are reasonable
And there are many, many more
Need a “fair” allocation
Need to maximize efficiency
Different choices lead to dramatically different allocations
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14. n short (1-hop) connections
Example v1
vn+1 v3 v2
n-1 long connections
n short (1-hop) connections
Max-min fair solution: Everyone gets bandwidth 1/n, total ¼ 2 Max-average solution: Long connections get 0, total = n
The average of the two solutions is good for both criteria
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15. Another example How to split a pie?
But what if there are more constraints?
Alice and Eddy want only apple pie
Frank: allergic to apples
Cathy: equal portions of both pies
David: twice as much apple pie as lemon
How do we measure the “goodness” of an allocation?
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16. Goal: simultaneous optimization
Suppose the utility function is U(x1,x2,…,xN)
Canonical Utility Functions: Assume that U is
Concave (the law of diminishing returns)
Non-decreasing (more bandwidth can not harm)
Symmetric in x1,x2,…,xN
Includes all the utility functions we have seen in this talk, and all the fairness functions we found in the literature
Simultaneous optimization: (Approximately) maximize U(x) simultaneously for all canonical U
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17. Why simultaneous optimization?
Often, the utility function is poorly understood, but is likely to be canonical, e.g. customer satisfaction
Consider the following three allocations of bandwidths to two users
ha,bi
hb,ai
h(a+b)/2,(a+b)/2i
If f measures fairness, then intuitively,
f(a,b) = f(b,a) · f((a+b)/2,(a+b)/2)
Hence, f is a symmetric concave function
Simultaneous optimization promotes all reasonable measures of fairness
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18. Simultaneous optimization
[Goel, Meyerson]
A bandwidth allocation x = hx1,x2,…,xNi is an -approximation for a class of functions if for all feasible allocations y and all functions U in the class,
U(x) ¸ U(y)
Pk(x) = sum of the k smallest xi’s (prefix functions)
Theorem: x is an -approximation for the class of all canonical functions if and only if it is an -approximation for the class of all N prefix functions
Gives us some hope
Theorem: There exists an efficient centralized algorithm to ensure  = O(log N), and this is the best possible
Question: Can we obtain a TCP-like protocol that does the same thing?
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19. Why do we need per-flow state?
Initially, x1 = x2 = x3 = 0
pe, pf ¼ 0, x1,x2, x3 increase v1 v2 v3 e f x1 x2 x3
Intermediate step: x1 = x2 = x3 = 0.45+
pe, pf > 0.5, x1,x2 increase, x3 now decreases p
p=1
p=1/2
=1
Finally, x1 = x2 = 1, x3 = 0
pe, pf = 1 , x1,x2, x3 at equilibrium 
=0.9
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20. Dual prices with per-flow state
Four flows sharing
a 1Gbps link
Suppose x1 = 100, x2 = 200, x3 = 300, x4 = 400 Mbps
Assume that the link knows x1,x2,x3,x4
Dual price computation uses simple exponential costs a but with different link occupancy values  for different flows x1 x2 x3 x4
For flow 1: Pretend x1 = x2 = x3 = x4 = 100,  = 0.4, pj = a0.4
For flow 2: Pretend x1 = 100, x2 = x3 = x4 = 200,  = 0.7, pj = a0.7
For flow 3: Pretend x1 = 100, x2 = 200, x3 = x4 = 300,  = 0.9, pj = a0.9
For flow 4: x1 = 100, x2 = 200, x3 = 300, x4 = 400,  = 1.0, pj = a
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21. Primal updates with per-flow state
Really simple:
wi < 1 => increase xi
wi > 1 => decrease xi
Example rule: dxi/dt = -log wi
At equilibrium, for all canonical utility functions U, U(x) ¸ U(y) for all feasible y and with  = O(log N)
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22. n short (1-hop) connections
How good is  = O(log N)?
Pretty good, for a simultaneous optimization v1 v2 v3
vn+1
n-1 long connections
n short (1-hop) connections
aTCP-RENO ¼ n1/2
aMax-Min ¼ n
MAX-SUM = 1
aTCP-VEGAS ¼ n1/2 (*)
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23. Convergence time
Primal-dual algorithm converges exponentially fast with single bottleneck link per flow
If we start from the all zeroes point, we need only polylogarithimic number of iterations (really small)
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24. Conclusion
Principled tradeoff of different utility functions and performance measures
Extensions
Convergence time
Heterogeneity: Assume user i has individual utility function fi. Then maximize U(f1(x1), f2(x2), …, fN(xN)) for all canonical U
Clean-Slate directions
Primal-dual as a paradigm to think about networks
Original spirit of TCP: don’t over-optimize for any one feature
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