principles of network and system administration phần 4 ppt

The effective bandwidth of this on-off source has a simple form when the time parameter t is small compared with Ton and Toff.This is typical if the buffer is small.. Similar questions w

SOME EXAMPLES 95

approximation of the true acceptance region for a given finite N It becomes more exact

as N increases Practical experiments show excellent results for N of the order of 100 Suppose Nx is the operating point in A and g t x/  
is the constraint that is binding at this point Then t achieves the supremum in (4.9) Let s be the infimizer in the right hand

The linear constraint in (4.12) gives a good approximation to the boundary of the

acceptance region near Nx if the values of s and t which are optimizing in (4.9) do not change very much as x varies in the neighbourhood of Nx We can extend this idea further

to obtain an approximation for the entire acceptance region by approximating it locally

at a number of boundary points Optimizing the selection of such points may be a highly

nontrivial task A simple heuristic when the s and t do not vary widely over the boundary

of the acceptance region is to use a single point approximation One should choose thispoint to be in the ‘interesting’ part of the acceptance region, i.e in the part where we expectthe actual operating point to be Otherwise one may choose some centrally located pointsuch as the intersection of the acceptance region with the ray.1; 1; : : : ; 1/

In practice, points on the acceptance region and their corresponding s and t can be computed using (4.9) We start with some initial point x near 0 and keep increasing all its

components proportionally until the target CLP is reached

Let us summarize this section We have considered the problem of determining thenumber of contracts that can be handled by a single switch if a certain QoS constraint is

to be satisfied We take a model of a switch that has a buffer of size B and serves C cells per second in a first come first serve fashion There are k classes of traffic, and the switch

is multiplexing x j sources of type j , j D 1 ; : : : ; k We define the ‘effective bandwidths’ of source type j by (4.5) This is a measure of the bandwidth that the source consumes and depend upon the parameters s and t As s varies from 0 to 1, it lies between the mean rate and peak rate of the source, measured over an interval of length t Arriving cells are

lost if the buffer is full We consider a QoS constraint on the cell loss probability of the

form CLP  e
, and show that a good approximation to this constraint is given by the

inequality in (4.9) The approximation becomes exact as B, C and x j grow towards infinity

in fixed proportions For this reason, (4.9) is called the ‘many sources approximation’ At a

given ‘operating point’, Nx, the constraint has an approximation that is linear in x, given by (4.12), where s and t are the optimizing values in (4.9) when we put x D Nx on the right-

hand side The linear constraint (4.12) can be used as an approximation to the boundary of

the acceptance region at Nx We can interpret t as the most probable time over which the

buffer fills during a busy period in which overflow occurs

In some cases the acceptance region can be described by the intersection of only a finite

number of A t s We see this in the first two examples.

Example 4.2 (Gaussian input) Suppose that X j[0; t] is distributed as a Gaussian randomvariable with mean¼j t and variance¦2

j t For example, let X j[0; t] D ¼j t C¦j B.t/, where B.t/ is standard Brownian motion Then

the effective bandwidth depends upon C, and on the operating point through the mean and

variance of the superimposed sources

Things are rather special in this example After substitution for s and t and simplification,

Thus, the acceptance region is actually defined by just one linear constraint Expressions

(4.13) and (4.14) are the same because s D
=B is constant on the boundary of the acceptance region In fact, this acceptance region is exactly the region in which CLP  e
,i.e the asymptotic approximation is exact This is because the Gaussian input process is

infinitely divisible (i.e X j[0; t] has the same distribution as the sum of N i.i.d randomvariables, each with mean¼j =N and variance ¦2

j =N — for any N) Therefore the limit in

(4.8) is actually achieved

Example 4.3 (Gaussian input, long range bursts) Let us calculate the effectivebandwidth of a Gaussian source with autocorrelation This is interesting because positive

SOME EXAMPLES 97autocorrelation produces a process with long range bursts In the previous two examples

we have constructed a model in continuous time Now let us assume that time is discrete,

i.e with epochs t D 1; 2; : : : Suppose X i represents the contribution of the source in the

i th time interval and fX1; X2; : : : g is a sequence of Gaussian random variables with mean

¼, variance ¦2 and autocovariance function
k/ (which is not to be confused with the

logarithm of the CLP) Then we have Þ.s; t/ D ¼ C ¦2

t s=2, where t¦2

Notice that limt!1¦2

t D
, where
is the so-called ‘index of dispersion’; one canshow that when the sum converges,
D P11
k/ If there is positive autocorrelation

Example 4.4 (Brownian bridge model of periodic sources) In this example the acceptance

region is described by just two linear constraints

Consider a periodic source which produces a burst of size²j at times U , U C 1, U C 2, : : : , where U is uniformly distributed on the interval [0; 1] Consider the superposition

of x j such sources, with values of U chosen independently It is a random process whose value increases from 0 to x j²j over the interval [0; 1] At each time t between 0 and

1 the probability that any one source has already produced its burst is t It follows that the traffic produced by the superposition by time t  1 has a distribution of ²j times a

binomial distribution of B.x j ; t/; the distribution of this quantity is approximately normal, with mean x j²j t and variance x j²2

j t 1  t/ In fact, the superposition tends to that of ² j times a Brownian motion that starts at 0 and is conditioned to reach x j at time 1 Thissuggests that we consider a different type of source whose superposition is exactly this.Each of these sources is of the form

X j[0; t] D ²jbt c C²j Z t  btc/

where Z t/, 0  t  1, is the standard Brownian bridge having Z.t/ ¾ N.t; t.1  t// Superimposing x j such sources is an approximation for superimposing x j actual burstyperiodic sources As in Example 4.2, one can computeÞj s; t/ D ² jC²2

j s f t/[1 f t/]=2t, where f t/ D t  btc is the fractional part of t The acceptance region turns out to be

A D \ t A t D A0:5\A1, where A1 and A0:5Dare the sets of x satisfying the following

Constraints (4.15) and (4.16) correspond to .s; t/ values of 0; 1/ and 2
=B; 1=2/,

respectively For instance, if (4.15) is the active constraint, there is enough buffer to absorbthe temporary bursts (expressed in (4.16)), but these buffers fill infinitely slowly since the

average input rate ‘barely’ exceeds the service capacity If (4.16) is active, then the mostprobable time over which the buffer produces overflows is half way through each period.Brownian bridge inputs are infinitely divisible processes so, as in Example 4.2, the aboveacceptance region is exact for a simple queue fed by Brownian bridge inputs.

Example 4.5 (On-off sources) Consider a source that alternates between on and off states.

When it is on it sends at constant rate h and when it is off it sends at rate zero The successive lengths of time that it spends in the on and off states are Ton and Toff, respectively, which

can be either deterministic or random Let pon denote the probability that the source is on

If m is the mean rate of the source, then pon Dm=h The effective bandwidth of this on-off source has a simple form when the time parameter t is small compared with Ton and Toff.This is typical if the buffer is small In this case, there is only a very small probability

that the source is both on and off during a window of length t Thus, with high probability

X [0; t], which is the contribution of the source in a window of size t, takes only two values: zero if the source is off and ht if the source is on, with respective probabilities 1  m =h and m =h Then (4.5) becomes

to infinity, the effective bandwidth tends to the peak rate of the source The effective

bandwidth is an increasing function of s and thus takes values between the mean and the peak rate Smaller values of s correspond to more efficient multiplexing.

Example 4.6 (Markov modulated source) Let us take the model of an on-off source inExample 4.5 Let the successive lengths of time that the source spends in the on and off

states be i.i.d exponential random variables with means Ton and Toff This model has beenused to model voice and video traffic It can also be used to model the activity during aweb browsing session

We can generalize this model to one with even more than two states; suppose there are m states, m ½ 2 Suppose that in state i the source sends at rate¼i The state changes according

to a continuous-time Markov process with known transition matrix, i.e the holding time in

state i is exponentially distributed, say with mean 1=½ i , and given that the state is i , the next state will be j with probability P i j

The nice thing about this class of models is that it is possible to calculate the effective

bandwidth (at least numerically) Let X i[0; t] be the traffic produced over [0; t], conditional

on the source starting in state i In brief, the effective bandwidth is computed from the moment generating function of the X i[0; t], say f i t/ D E exp.s X i[0; t]/ Then it is not

hard to see that

i z/ be the Laplace transform of f i t/ The integral above is a convolution integral

and so we easily find

MULTIPLE QOS CONSTRAINTS 99One can, in principle, solve this set of linear equations and then invert the Laplacetransforms.

Consider the special case of an on-off source, with off and on phases that are exponentiallydistributed with means 1=½0 and 1=½1, respectively, and taking ¼0 D 0 and peak rate

¼1Dh We find, after some algebra,

f t/ D E

exp

where!1,!2are the two roots of!2C.½0C½1½0h/!  ½0h D 0

A discrete time model is even easier Suppose that the state changes at each epochaccording to the transition matrix .p i j/ Then the effective bandwidths satisfy the set oflinear recurrences

f i t/ D e s¼iP

j p i j f j t  1/; t D 1; 2; : : :

These recurrences have been successfully used to make numerical computations of effectivebandwidth functions

4.8 Multiple QoS constraints

As in Section 4.5, the idea of effective bandwidths extends to multiple QoS constraints.The acceptance region is then the intersection of the acceptance regions defined by eachconstraint Each constraint might correspond to a different manner of overflow Theeffective bandwidth of a stream is defined by the constraint that is active Example 4.7demonstrates how multiple constraints may result from the scheduling mechanism of priorityqueueing

Example 4.7 (Priority queueing) One way to give different qualities of service to different

classes of traffic is by priority queueing Suppose that traffic classes are partitioned into

two sets, J1 and J2 Service is FCFS, except that a class in J1is always given priority over

a class in J2 For i 2 J1 there is a QoS guarantee on delay of the form

P delay > B1=C/  e
1For all sources there is a QoS guarantee on CLP of

CLP  e
2

This gives the two constraints g1.x/  0 and g2.x/  0, where x is the vector of the

numbers of sources of the different types The acceptance region is now the intersection

of the acceptance regions corresponding to each of the constraints Assume that on the

‘interesting’ part of each constraint the values of s and t do not vary widely, being s i ; t i for

constraints i D 1; 2 Then, by approximating each constraint globally using (4.12) calculated

at a single appropriately chosen point, we obtain the effective bandwidth approximations

CLP ≤ e−γ2

vertical constraint is due to a guarantee on the delay of priority traffic The second constraint is due

to a guarantee on the CLP for both traffic types, and is approximated by a linear constraint at the

operating point Nx (shown dotted).

If K1=Þ1.s1; t1/ < K2=Þ1.s2; t2/ then the acceptance region takes the form illustrated in

Figure 4.8 Note that this approximation of the technology set is less accurate if the values

of s and t vary significantly along each constraint Then one might approximate each

constraint by tangent hyperplanes at more than one boundary point The key observation

is that our approximations will always be of the form (4.18) but with a larger number ofconstraints

in the rate are observed on timescales of order t1 and t2 (and there may be even largertimescales, but these do not show up in the small snapshot taken) Suppose we are at anoperating point, where a CLP guarantee is just satisfied and no more traffic can be packed

in the link We can ask, on what timescales are fluctuations most likely to cause bufferoverflow? In other words, which aspects of the traffic make it hard to multiplex it and hencecontribute to its effective bandwidth? Similar questions were posed in Section 4.4 when

we determined effective bandwidths for the boxes that were to be packed in an elevator.For a constraint on the technology set that is defined in terms of a constraint on CLP, theeffective bandwidth of a source depends on the timescales of the source’s burstiness thatsignificantly contribute to the event of buffer overflow Clearly, fluctuations on differenttimescales do not contribute equally, since those on short timescales may be absorbed bythe buffer This effect is captured in the definition of the effective bandwidth in (4.5) By

TRAFFIC SHAPING 101letting Þ.s; t/ depend upon the total contribution of the source in a window of length of time t, we ‘filter out’ the fluctuation that occur in timescales smaller than t For example,

in Figure 4.6 the timescale t1 is absorbed by the buffer and is not reflected in the effective

bandwidth of the source Here t2 is the dominant timescale of burstiness that constraints

the system; within t2, the timescale t1contributes its mean rate In other words, if we were

to replace our source by one obtained by averaging it over a timescale of t1, this would

have no effect on the multiplexing and it would neither increase or decrease the CLP.Traffic shaping can be used to reduce high frequency fluctuations and produce smoothertraffic A typical traffic shaper consists of a large buffer that is served at a rate smallerthan the peak rate of the stream, or of a buffer that is combined with a leaky bucket thatholds the part of traffic that is non-conforming with the traffic contract; see Section 2.2.2.One may design the shaper to add delays of the same order of magnitude as the timescales

of the fluctuations to be smoothed Another way to implement a shaper is to collect the

traffic that arrives every t time units and then transmit it during the next t time units at

a constant rate This rate will differ during each period, reflecting the variable volume ofdata to be transmitted The above discussion explains the effects of shaping mechanisms onthe effective bandwidth of the resulting traffic When buffers in the network are large (and

hence t is large), then only substantial traffic shaping can reduce the effective bandwidth

of the input traffic However, for real-time traffic using small buffers, a moderate amount

of traffic shaping can drastically reduce the effective bandwidths and thereby increase themultiplexing capability of the network

Let us look at some data for real traffic Figure 4.9 shows a 1000 epoch trace from a

MPEG-1 encoded video of Star Wars, each epoch being 40ms Note the different timescales

of burstiness Figure 4.10 plots an estimate of the effective bandwidth function for this trace

Observe that as either t becomes small or s becomes large, the effective bandwidth increases;

this corresponds to the source becoming more difficult to multiplex The explanation issimple The time averaging that takes place in the buffer in which this particular stream

is being multiplexed smoothes small traffic bursts In particular, it smoothes all fluctuation

taking place on timescales less than t The larger is t, the more the fluctuations are absorbed

and so the resulting trace can be multiplexed as easily as a smoother trace in which these

fluctuations have been averaged-out Eventually, when t is large enough, the trace is no

more difficult to multiplex than a constant bit rate source of the same mean Small values of

t occur when the buffer is small, in which case, the averaging effect is negligible, and so the

traffic is more difficult to multiplex This means a greater effective bandwidth Similarly,

when the link capacity decreases, s increases, and the value of the effective bandwidth

Star Wars Each epoch is 40 ms.

corresponding to different multiplexing contexts, the effective bandwidth of the MPEG traffic

stream can differ from about 0.5 to 2.5 Mbps

increases, again capturing the increased difficulty in multiplexing Note that for different

values of s and t, corresponding to different multiplexing contexts, the effective bandwidth

can differ substantially

4.10 Effective bandwidths for traffic contracts

We have assumed thus far that the source statistics are fully known and so exact effectivebandwidths can be computed In practice, this is not the case The network knows only thetraffic contract of the requested service This contract only partly characterizes the trafficsource, for example, through the leaky bucket constraints This poses a problem If onlythe traffic contract is known, what effective bandwidths should be used for call acceptance?There are several possibilities Each has its advantages and disadvantages:

1 If the network operator can tell which application generates the traffic, and thatapplication produces traffic with known statistics, then he can use the actual effectivebandwidth of that type of traffic Usually, however, this information is not available

2 If the traffic contract is used only by applications of a known general type (such

as video), then one can use the typical effective bandwidth for the traffic of thattype of application This concept of an ‘average’ effective bandwidth is used in flatrate charging The idea is that two applications that use the same contract should becharged the same, i.e on the basis of an average effective bandwidth for this contracttype, irrespective of whether they generate identical traffic

3 If very little information is known about the source, then it can be reasonable to usethe greatest effective bandwidth, say NÞ, that is possible under the service contract, i.e.the bandwidth of the traffic that is most difficult to multiplex, given the constraintsplaced upon that traffic by the service contract This is a conservative approach thatresults in network resources being underutilized However, it is the only approachthat enables the network to implement hard quality of service guarantees We pursuethis idea in Section 4.11

BOUNDS FOR EFFECTIVE BANDWIDTHS 103

4 One can modify the above approach to one of dynamic call acceptance by usinginformation that is obtained by monitoring the actual system One uses actualmeasurements of the performance on the links of the network to determine the actualamount of spare capacity available and then decides whether to accept a new contract

on the basis of this information and a worst-case model of the new call

Such a decision can depend on the duration of the new call and the time that it takes for theavailable capacity to change It could be that there is available capacity because a majority

of the active calls are sending traffic at less than their mean rates In this case, the existingload will tend to increase as more sources become active at greater rates, although some

of them may terminate and depart

Observe that lack of information about call statistics results in resource underutilizationand poor quality of service provisioning If the network has better information about theresource usage statistics of a new call, then it can better multiplex and load the networkmore efficiently This explains why a network operator wishes to have a good idea of thetraffic profiles of his customers But how can he obtain better information than that availablethrough the traffic contract? Since this information is to be used to accept or to reject a call,

it must be available at the time the call is set up One way to obtain more information isthrough pricing The network posts a set of possible tariffs for the same traffic contract, eachone resulting in a different charge, depending on the traffic that is actually generated duringthe contract Assuming the user has some information about the traffic he will generate,

he chooses the tariff that minimizes his expected charge His tariff choice therefore revealsimportant information to the network operator, who can use this information to obtain abetter approximation of the effective bandwidth This is an example of incentive compatiblepricing; when users optimize their tariff choices the network operator gains information thatallows him to better load and more efficiently run the network as a whole

4.11 Bounds for effective bandwidths

Suppose that a connection is policed by multiple leaky buckets, with a set of parameters

h D f.²k; þk /, k D 1; : : : ; K g Let m be the mean rate of the connection We are interested

in the greatest effective bandwidth, say NÞ.m; h/, that is possible for connection whose traffic

contract has these leaky bucket parameters and whose mean rate is m In practice, N Þ.m; h/

can be extremely difficult to calculate exactly, as we are in effect trying to determine aworst-case stochastic process But we can easily give a simple approximation for NÞ.m; h/,

which nicely shows how various timescales relate to buffer overflow Since the traffic source

is policed by leaky buckets, the maximum amount of traffic NX [0 ; t] that could be produced

in a time interval of length t is

obtaining an upper bound on NÞ.m; h/ This upper bound is

affects the source’s maximum contribution, H t/, and hence the effective bandwidth.

In QÞsb.m; h/ we can see the effects of leaky buckets on the resource usage Each

leaky bucket ²k; þk/ constrains the burstiness of the traffic on a particular timescale The

timescale of burstiness that contributes to buffer overflow is determined by the index k that

achieves the minimum in (4.19) We discuss this issue further at the end of this section.Consider the practical case of a dual leaky bucket .h; 0/ and ²; þ/ H.t/ is shown in Figure 4.11 If t is small, then H t/ D ht and the bound (4.20) reduces to

QÞon-off.m; h/ D 1

comply with the ²; þ/ leaky bucket which restricts the length of such bursts and so isworse than the worst possible source that is compliant with the above traffic contract

If one wants to obtain more accurate upper bounds on the effective bandwidth, one has touse complex computational procedures to determine the worst-case traffic In general, theworst-case traffic depends not only on the contract parameters, but also on the parameters

s and t In many cases, the worst-case traffic consists of blocks of an inverted T pattern

which repeat periodically or with random gaps, as shown in Figure 4.12 The sizes of the

blocks and gaps depend on the values of s and t This is a general form of extreme traffic

which, given the leaky bucket constraints, alternately sends at the maximum rate and at

a lesser rate (though not necessarily zero) While sending at the lesser rate it accumulatetokens so that it can again send at the maximum rate

t Dþ=.h  ²/, t D[.2t  t /² C t h] =m  2t.

DETERMINISTIC MULTIPLEXING 105

As an example, we consider the periodic pattern shown in Figure 4.12 Let X T [0; t]

denote the load produced by the inverted T pattern in t epochs This gives the inverted T approximation for the effective bandwidth bound,

interval of length tonCtoff, where tonD2t.

The above inverted T approximation is valid when the time parameter t is large compared

to the time for which the leaky bucket permits the source to send at its peak rate h; denote this time by t1Dþ=.h  ²/ If t is much smaller than t1, then worst case traffic is a simple on-off source, for which ton Dt1 and toff Dtonh =m  ton In this case, the source operatesonly at the extremes, sending at full speed or not at all and a reasonable approximation ofthe worst-case effective bandwidth is given by (4.21)

Which leaky bucket is most constraining of the effective bandwidth produced by a traffic

contract? The bound in (4.20) suggests that it depends on the timescale t This is determined

by the network Given t, the bound for the effective bandwidth depends only upon the leaky

bucket that achieves minkD1 ;:::;Kf²k t Cþkgfor the particular t that constrains the maximum contribution H t/ This suggests that a worst-case source for the above contract is hard to

multiplex because of burstiness that is controlled mainly by this leaky bucket Changing thevalue of other leaky buckets will not significantly reduce the difficulty of multiplexing thesource However, smoothening that reduces burstiness can reduce the effective bandwidth.Consider a source that is policed by two leaky buckets .h; 0/; ²; þ/ In Figure 4.11, we plot H t/ For values of the parameter t less than t1 D þ=.h  ²/ the leaky bucket which constrains the peak rate is dominant For t > t1, the leaky bucket.²; þ/ dominates.The physical explanation is that if burstiness on small timescales is causing overflows,(perhaps because there are small buffers in the network), then reducing the peak rate ofthese fluctuations will reduce the cell loss If buffers are large, then rapid fluctuations areabsorbed by the buffer and there is no advantage in reducing the peak rate However,reducing the length of the long bursts will reduce cell loss These bursts are controlledmainly by the second leaky bucket

4.12 Deterministic multiplexing

The case of deterministic multiplexing is one in which we require no cell loss, so that

D 1 Our effective bandwidth theory suggests that s D 1 (which is consistent with

D 1 in (4.12)) The effective bandwidth of a source of type j is Þ j 1; t/ D NX j[0; t]=t.Observe that the effective bandwidth of a source does not depend on the completedistribution function, but only on the NX j[0; t], the maximum value that Xj[0; t] takeswith positive probability

Let us derive the form of the acceptance region for sources policed by leaky buckets

Suppose that each type of source is policed by a single leaky bucket A source of type j

is guaranteed to satisfy the condition

X j[0; t]  ²j t Cþj ; for all t (4.23)Assume that there is positive probability of arbitrarily near equality in (4.23) for all

time windows t An example of a source that can do this is one that infinitely often

repeats the following pattern of three phases: it stays off for time þj=²j (to empty thetoken buffer), then produces an instantaneous burst of size þj, and then stays on at rate

²j for a time that is exponentially distributed with mean 1 It is not hard to see that forall ž > 0, there is a positive probability that such a source will produce a burst of size

þj within the interval [0; ž/ and then remain on at rate ²j over [ž; t], so ensuring that inthe interval [0; t] the number of cells received are at least þj C.t  ž/² j This impliesthat Þj 1; t/ D NX[0; t]=t D ² j Cþj =t So, following the notation from Section 4.6.1, as

! 1 also s ! 1, and A t D fx : gt x/  
g reduces to

k

X

j D1

x j.²j t Cþj /  tC C B

This infinite set of hyperplanes is dominated by two extreme ones That is, \t A t D A0\ A1,

where A1 and A0are regions defined, respectively, by

the first constraint is active, overflow occurs because each source of type j contributes at

its maximum allowed average rate²j IfP

j²j were to exceed C by a very small amount,

say Ž, then a buffer of any size would eventually fill, though very slowly As Ž tends to

0 we think of the buffer filling, but over infinite time when Ž D 0, hence t D 1 If the

second constraint is active, overflow occurs because all sources produce bursts at exactly

the same time In this case, a buffer of size B fills in zero time and hence t D 0 Note that

the effective bandwidth is defined as ²j or þj, depending upon which constraint is active

at the operating point (which can be compared to the wj andvj of the elevator analogy).The case of multiple leaky bucket constraints is more complex, but the results are similar

It turns out that one gets an acceptance region bounded by a finite number of linearconstraints, each of which corresponds to a particular way that the buffer can fill Wewill briefly investigate the particular case of adding a peak rate constraint, i.e each source

of type j satisfies

X j[0; t]  minfhj t; ² j t Cþjg; for all t (4.25)

In this case, following the previous reasoning, the effective bandwidth of a source of type

the source switches from.h j; 0/ to ²j; þj/

In the case j D 1 ; 2, and assuming that t1 < t2, the acceptance region is defined by

x1²1Cx2h2  C C B  x1þ1/=t for t1t  t2

x1²1Cx2²2  C C B  x1þ1x2þ2/=t for t2t

(4.27)

EXTENSION TO NETWORKS 107

These must hold for all t But as the reader can verify, it is enough that they hold for

t D t1, t D t2 and t D 1 The corresponding constraints become

x1h1Cx2h2C C B=t1 x1.²1Cþ1=t2/ C x2h2C C B=t2

x1²1Cx2²2C

Note, again, that depending upon which constraint is active at the operating point, there is

a unique description for the way the buffer fills and the time that it takes If the operatingpoint lies on the first constraint then the buffer fills by all sources producing at their peakrates over [0; t1], and the buffer becoming full at time t1(which is the time at which sources

of the first type of must reduce their rates to²1) If the second constraint is active then the

buffer first fills at time t2, by filling at rate x1h1Cx2h2C > 0 until time t1, and then at rate x1²1Cx2h2C > 0 from t1 to t2 If the last constraint is active, then the long-run average rate of the input equals C and the buffer fills infinitely slowly.

The above examples generalize to more leaky buckets Again, one has to check a finiteset of equations, similar to (4.27), at a finite number of times at which different leakybuckets become active by constraining the maximum contribution of the sources

4.13 Extension to networks

We have seen how to compute the effective bandwidths of a flow that is multiplexed at onebuffered switch But is this any use in the context of a network? Clearly, the statistics of theflow change as it passes through switches of the network, since the interdeparture times ofcells from a switch are not the same as their interarrival times Can the effective bandwidthstill characterize the flow’s contribution to rare overflow events in the network? Fortunately,the answer is yes One can show that in the limiting regime of many sources, in whichthe number of inputs increases and the service rate and buffer size increase proportionally

as in (4.8), the statistical characteristics and effective bandwidth of a traffic stream areessentially unchanged by passage through the switch To see this intuitively, observe that

as the scaling factor N increases, the aggregate of all the traffic streams looks more and more like a constant bit rate source, of a rate less than the switch bandwidth, NC This

means that with a probability approaching 1 the buffer is empty over the fixed interval

of time during which any one given traffic source is present Thus with probability alsoapproaching 1, the input and output processes are identical over the time that a traffic source

is present Note that these are limiting results: they hold in the limit as the capacity of thelinks become larger This assumption is realistic in the context of the expanding capacity

of today’s broadband networks

Using this result we can describe the technology set of the network as follows Let L be

a set of links and R be a set of routes Write j 2 r if route r uses link j Assume each

route is associated with a unique traffic contract type (We allow two routes to be identical

in their path through the network and differ only in contract type.) Let x r be the number

of contracts using route r Let B j and C j denote as before the resources of link j The

technology set is then defined by a set of constraints like

Note that, although the effective bandwidth functionÞr.Ð; Ð/ is the same along the entireroute of a traffic stream, this does not imply that the effective bandwidth is the same onall links This is because parameters of the operating point may vary along the links ofthe path, so that .s j ; t j / depend on the link j But do we expect these parameters to vary

widely inside the network? There are economic arguments that suggest not

Suppose that the marginal cost of adding some extra buffer in a link of the network is

b, and the marginal cost of adding extra capacity is a Then if the most economical switch configuration is used, we must have t j D a=b, independently of the link To prove this

consider the constrained optimization problem

it route and of the other flows A more refined analysis could take account of the precise

values of the parameters s and t along the path of the traffic stream, and use values for the

effective bandwidth that depend on the particular link

4.14 Call blocking

At the end of Section 4.3 we wondered what the effects of blocking would be if connectionsarrive with rates that have stochastic fluctuations and we adopt a call admission controlbased on (4.3) There is now a cost due to blocked calls It is reasonable to assume that when

a service request is refused there is some cost to the requester, while when it is acceptedsome positive value in generated We formulate and analyze such a model in Section 9.3.3,measuring blocking in terms of the call blocking probability This probability depends onthe technology set of the network and the rates of arrival of the various connection types (theservice requests) In this section we see how such blocking probabilities may be calculatedfrom the parameters of the system

Suppose there are J types of connection A connection of type r is associated with a route r and connections of this type arrive as a Poisson process of rate ½r and endurefor an average time of 1=¼r Let p r be the blocking probability for a connection of

type r Let x r t/ denote the number of connections of type r that are active at time

t These connections place a load on link j of Þjr x r t/ Here Þ jr is the effective

bandwidth of a connection of type r on link j It equals 0 if the connection does not use link j

By Little’s Law (stated in 4.4), the average number of connections of type r that will

be active is the product of the arrival rate for this type and its average holding time, i.e

.1  p r/½r=¼r Let²r D½r=¼r Then it is necessary to have, for all links j ,

FURTHER READING 109This places some restriction on the blocking probabilities However, it does not take account

of the statistical fluctuations

To be more accurate we could reason as follows The probability distribution of thenumber of calls in progress is

³.x/ D G.C/1Y

r

²x r r

Here, B i can be interpreted as the blocking probability for a unit of effective bandwidth on

link i This formula has a simple interpretation It is as if each such unit has a probability of

blocking that is independent of other such units on the same link and other links Clearly,

this is not so However, it motivates a determination of the p rs as the solution of

Here, E ²; C/ is Erlang’s formula for the blocking proportion of calls lost at a single link

of capacity C when they arrive at rate½ and hold for an average time 1=¼, with ² D ½=¼,

sources’ (large N ) asymptotic, which has eventually gained wide acceptance Large buffer

asymptotics did not capture the multiplexing effects due to a large number of independentsources being multiplexed, but only the effects due to the buffers Relevant referencesfor large buffer asymptotics and the corresponding effective bandwidths are de Veciana,Olivier and Walrand (1993), Elwalid and Mitra (1993), de Veciana and Walrand (1995),Courcoubetis and Weber (1995), Courcoubetis, Kesidis, Ridder, Walrand and Weber (1995)

A classic paper on the calculation of the overflow probabilities in queues handling manysources is that of Anick, Mitra and Sondhi (1982) This paper motivated much subsequentresearch in the field Weiss (1986) first derived the large system asymptotic for on-offsources The large system asymptotic in (4.8) that leads to the effective bandwidth formulaswas independently proved by Botvich and Duffield (1995), Simonian and Guilbert (1995)and Courcoubetis and Weber (1996) A refinement of this asymptotic using that Bahadur-Rao approximation is due to Likhanov and Mazumdar (1999) Some early references forthe effective bandwidth concept are Hui (1988), Courcoubetis and Walrand (1991), Kelly(1991a) and Gibbens and Hunt (1991) An excellent reference for the theory of the effectivebandwidths is Kelly (1996).

For a review of the Brownian bridge model in Section 4.4, see Hajek (1994) The proofthat the acceptance region given in (4.16) is exact for a simple queue fed by Brownian bridgeinputs can be found in Kelly (1996) The Markov modulated source model of Section 4.6has played an important role in theoretical and practical developments Anick, Mitra andSondhi (1982) show how to calculate the probabilities of buffer overflow Further details ofthe derivation of the effective bandwidth for this model can be found in Courcoubetis andWeber (1996) The material in Section 4.9 is taken from Courcoubetis, Siris and Stamoulis(1999) and Courcoubetis, Kelly and Weber (2000) Calculation of the effective bandwidthsfor real traffic traces first appeared in Gibbens (1996) The use of effective bandwidthconcepts for dimensioning network links and for solving other traffic engineering problems

is explained in Courcoubetis, Siris and Stamoulis (1999) This includes experimental resultsthat validate our effective bandwidth definition Siris (2002) maintains a nice web site onlarge deviation techniques and on-line tools for traffic engineering The extension of thesingle link models and the application of the asymptotics to networks in Section 4.13 is due

to Wischik (1999) Section 4.14 summarizes ideas from Kelly (1991b) and (1991c) Morerefined asymptotics for the blocking probabilities are described by Hunt and Kelly (1989).Extensions of the model to include priorities can be found in Berger and Whitt (1998).Issues of call-admission control are treated in Courcoubetis, Kesidis, Ridder, Walrandand Weber (1995), Gibbens, Key and Kelly (1995), Grossglauser and Tse (1999) andCourcoubetis, Dimakis and Stamoulis (2002)

Part B

Economics

Pricing Communication Networks: Economics, Technology and Modelling.

Costas Courcoubetis and Richard Weber

ISBN: 0-470-85130-9

Basic Concepts

Economics is concerned with the production, sale and purchase of commodities that are inlimited supply, and with how buyers and sellers interact in markets for them This and thefollowing chapter provide a tutorial in the economic concepts and models that are relevant topricing communications services It investigates how pricing depends on the assumptions that

we make about the market For example, we might assume that there is only one sole supplier

In formulating and analysing a number of models, we see that prices depend on the nature ofcompetition and regulation, and whether they are driven by competition, the profit-maximizingaim of a monopoly supplier, or the social welfare maximizing aim of a regulator

Section 5.1 sets out some basic definitions and describes some factors that affect pricing

It defines types of markets, and describes three different rationales that can provide ance in setting prices Section 5.2 considers the problem of a consumer who faces pricesfor a range of services The key observation is that the consumer will purchase a service up

guid-to an amount where his marginal utility equals the price Section 5.3 defines the problem

of supplier whose aim is to maximize his profit Section 5.4 concerns the problem that

is natural for a social planner: that of maximizing the total welfare of all participants inthe market We relate this to some important notions of market equilibrium and efficiency,noting that problems can arise if there is market failure due to externalities

Unfortunately, social welfare is achieved by setting prices equal to marginal cost Since themarginal costs of network services can be nearly zero, producers may not be able to cover theircosts unless they receive some additional lump-sum payment A compromise is to use Ramseyprices; these are prices which maximize total welfare subject to the constraint that producerscover their costs We consider these in Section 5.5 Section 5.6 considers maximizing socialwelfare under finite capacity constraints Section 5.7 discusses how customer demand can beinfluenced by the type of network externality that we mentioned in Chapter 1

The reader of this and the following chapter cannot expect to become an expert in alleconomic theory that is relevant to setting prices However, he will gain an appreciation offactors that affect pricing decisions and of what pricing can achieve In later chapters we usethis knowledge to show how one might derive some tariffs for communications services

5.1 Charging for services

Communication services are valuable economic commodities The prices for which theycan be sold depend on factors of demand, supply and how the market operates The key

Costas Courcoubetis and Richard Weber

ISBN: 0-470-85130-9