Pricing for Communication Networks

Network constraints and effective bandwidth

Roberto Battiti

Slides based on: Courcoubetis and Weber, Pricing Communication Networks, Wiley 2003 – chap4

k types of services, vector of services supplied

constrained to lie in a technology set X

f(x) can be monopolist’s profit, social welfare, …

Technology set• Prices can be used as control to constrain the demand within the

production capability of the network (technology set)• effective bandwidths to approximate the technology set at a given

operation point

Problem is of interest for asynchronous networks based on packetswitching

Statistical multiplexing

• Consider QoS service contract with maximum cell loss probability (CLP)• By considering peak rates hi

• With statistical multiplexing we can use a “controlled overbooking”

where mi < �i < hi

�i is the effective bandwidth

Figure 4.1 The call admission control (CAC) problem. Given the state of thesystem in terms of the active traffic contracts and a history of load measurements,should a new traffic contract of type i be admitted?

Link model

Without statistical multiplexing: x_peak = C / h1With statistical multiplexing: x_stat = C / �1

Statistical multiplexing gain (SMG) = x_stat/ x_peak = h1/�1… depends on CLP

Example

Discrete time, no buffer, C cells per epoch

x identical sources, binomial distribution (e.g. 5 Bernoulli trials, p = 0.2)Prob( k cells) = Binomial(n,k) p^k (1-p)^(n-k)Mean � = n p Variance n p (1-p)

Take a large number of sources N, mean cells N n p, variance N n p (1-p) , standard deviation � � N

N

� N � N

N

For large N….

Example (2)

Capacity

Can estimate the probability that at least a cell is lostTails of the binomial distribution (for r > �) :

Pr{X – � � r} � (� e / r ) ^r

Ex. If capacity C = 4 �, r= 3 �

Pr{at least one cell lost} � (e/3)^3 �

�=N, N=10 Pr{at least one cell lost} � (e/3)^3N � 0.05�=N, N=100 Pr{at least one cell lost} � (e/3)^3N � 1.4 10^-13�=N, N=1000 Pr{at least one cell lost} � (e/3)^3N � 3.3 10^-129

No loss

� r

Figure 4.2 The acceptance region problem. Here there are k = 2 traffic types and sources of in types i. We are interested in knowing for what ( , ) is CLP p, for say

. The triangular region close to the origin is the acceptance region defined by , which uses the peak cell rates and does not take advantage of the

statistical multiplexing.

ix

1x 2x ≤810−=p

Chxhx ≤+ 2211

Acceptance region

Call Admission control

Machanism to keep in the accceptance region, can be

-Conservative

-Dynamic based on contract parameters and on-line measurements

-Near a given operating point x can use effective bandwidth �ito characterize resource consumption / possible substitutions

Figure 4.3 The elevator can carry a total weight of at most W and volume atmost V . A box of type i has weight and volume . A box of type i has ntimes the relative effective usage of a box of type j if we are indifferent betweenpacking 1 box of type i or n boxes of type j.

iwiv

An elevator analogy

Figure 4.4 At the left the elevator is full because the volume of the boxes is V .The effective resource usage of a box of type i is vi . At the right the elevator isfull because the weight of the boxes equals W. The effective resource usage of abox of type i is wi .

An elevator analogy (2)

An elevator analogy (3)

The relative affective usage of a box depends on which constraint is active at a given operating point

How does one reach the operating point? On can steer to maximize utility

Constraint optimization…(volume c. active)

Let pi = � vi, the solution can be found in a decentralized way

each agent max.

In most cases prices are proportional to effective bandwidths

Effective bandwidth• We are interested in assessing resource substitution• Assume technology is defined by:• Assume binding constraint is:

• For a small change:• To remain in the acceptance region:

• Effective bandwidth of contract j

• Is a “substitution coefficient”, e.g. can increase type 1 by � / �1 and decrease type 2 by � / �2

Effective bandwidth as a local linear approximation to theboundary of the technology set at the operating point x

Figure 4.5 The acceptance region A is defined by two constraints. At the operating point ,which achieves the maximum of f in A, the active constraint is and so the effective bandwidths will be of the form . Note that the problem of maximizing f subject to ,where ,is also solved at

Thus we can use simpler effective bandwidth constraints, in place of the actual acceptance region constraints, in posing the optimization problem

11 )( cxg ≤xx

xjgj =∂∂= /1α*

1� =≤k

j jj Cx α j

k

j jxC α� ==

1*

x

x

• Case1: full description of each connection’s traffic• Number of contracts that can be handled by a single switch

• number of cells produced by bursty source j in t seconds, single bind. constraint

• s,t depend on operating point, link parameters (C,B), CLP,• if (C,B) is the asymptotic value of log(CLP)

Effective bandwidths for traffic streams

space parametertime parameter

Large deviations analysis of a model of single link“many sources” approx.

Figure 4.6 The operating point parameter t corresponds to the most probable time over which the buffer fills during a busy period in which overflow occurs.Here the source rate varies on two timescales and t2 is more relevant to overflow than is t1. This is because it is when the source produces at a high rate for a relatively long time, of order t2, that the buffer overflows. During such a long time, fluctuations on the t1 timescale are evened-out and do not contribute to the overflow.

Ex. Gaussian input

• Xj [0, t] Gaussian random variable with mean � tand variance Effective bandwidth

• Acceptance region:

Figure 4.8 An acceptance region defined by two constraints. There are twoclasses of traffic. The vertical constraint is due to a guarantee on the delay ofpriority traffic. The second constraint is due to a guarantee on the CLP for bothtraffic types, and is approximated by a linear constraint at the operating point (shown dotted).

Priority queing

Figure 4.9 Burstiness can be seen in this trace of 1000 epochs of a MPEG-1encoded video of Star Wars. Each epoch is 40ms.

Traffic shaping

Reduce high frequency oscillations(can reduce effective bandwidth when buffers are small)

Effective bandwidth for traffic contracts

• What if we do not know source statistics but only traffic contracts?– if known, application characteristics, typical effective

bandwidth for that application– estimate greatest effective bandwidth – dynamic call acceptance based on actual measurements

Lack of information � resource underutilization, poor QoSObtain more information through pricing(incentive compatible pricing)

Extension to networks• Statistics of the flow change as it passes through switches

(itredeparture times interarrival times)• BUT...in the limiting regime of many sources,

characteristics are essentially unchanged (as N grows looks more like a constant bit rate, with rate less than NC... With probability close to 1 teh buffer is empty)

• Technology set: verify for all links j

• �r(,) is the same along the route, but parameters of the operating point may vary... if network is reasonably optimized we may assume that they are almost constant

no.of contracts using route r

...to each traffic stream an effective bandwidth independent of route and of the other flows