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/ Ä 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 side of (4.11). Then @g t =@x j þ þ xDNx D stÞ j .s; t/ and so, as above, g t .x/ Ä has a linear approximation in the neighbourhood of Nx of st k X jD1 x j Þ j .s; t/ Ä st k X jD1 Nx j Þ j .s; t/ D s.Ct C B/   Dividing by st,wehave k X jD1 x j Þ j .s; t/ Ä C Ł ; where C Ł D C C 1 t  B   s Á (4.12) 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 this point to be in the ‘interesting’ part of the acceptance region, i.e. in the part where we expect the actual operating point to be. Otherwise one may choose some centrally located point such 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 the number 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.Ass 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 Ä 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,givenby (4.12), where s and t are the optimizing values in (4.9) when we put x DNx 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. 4.7 Some examples 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. 96 NETWORK CONSTRAINTS AND EFFECTIVE BANDWIDTHS Example 4.2 (Gaussian input) Suppose that X j [0; t] is distributed as a Gaussian random variable with mean ¼ j t and variance ¦ 2 j t. For example, let X j [0; t] D ¼ j t C¦ j B.t/,where B.t/ is standard Brownian motion. Then Þ.s; t/ D 1 st log E h e sX[t;0] i D 1 st log e ¼ j tsC¦ 2 j ts 2 =2 D ¼ j C ¦ 2 j s=2 Note that Þ is independent of t. Also, it tends to infinity as s increases. This is because the Gaussian source does not have a finite peak rate. When sources of k different types are multiplexed, the acceptance region, A, is defined by (4.9), which is sup t½0 inf s½0 h st P k jD1 x j .¼ j C ¦ 2 j s=2/  s.Ct C B/ i Ä The infimum with respect to s occurs at s D .C C B=t  P j x j ¼ j /= P j x j ¦ 2 j .Thisgives A to be defined by sup t½0 Ä  1 2 t  C C B=t  P k jD1 x j ¼ j Á 2 . P k jD1 x j ¦ 2 j ½ Ä in which the supremum with respect to t is achieved by t D B C  P j x j ¼ j Note that the most likely time over which buffer overflow occurs is the same time that it would take for a full buffer to empty while being fed with new input at the average rate of P j x j ¼ j . So (4.9) is just k X jD1 x j  ¼ j C ¦ 2 j s=2 Á Ä C C 1 t  B   s Á (4.13) The effective bandwidth is ¼ j C ¦ 2 j s=2, where s D 2.C  P j x j ¼ j /= P j x j ¦ 2 j . Note that 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, (4.13) becomes k X jD1 x j  ¼ j C  2B ¦ 2 j Á Ä C (4.14) 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 Ä 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. random variables, each with mean ¼ j =N and variance ¦ 2 j =N —foranyN ). Therefore the limit in (4.8) is actually achieved. Example 4.3 (Gaussian input, long range bursts) Let us calculate the effective bandwidth of a Gaussian source with autocorrelation. This is interesting because positive SOME EXAMPLES 97 autocorrelation 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 ith time interval and fX 1 ; X 2 ;:::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 t D var t X iD1 X i ! D t¦ 2 C 2[.t  1/ .1/ C .t  2/ .2/ CÐÐÐC.t  1/] Notice that lim t!1 ¦ 2 t D  ,where is the so-called ‘index of dispersion’; one can show that when the sum converges,  D P 1 1 .k/. If there is positive autocorrelation then ¦ 2 t >¦ 2 , and so the effective bandwidth is greater than it would be for an uncorrelated Gaussian process with the same variance. Similarly, if ¦ 2 t <¦ 2 the effective bandwidth is less. 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 size ² j at times U , U C 1, U C 2, ::: ,whereU 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 j ² j 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 Ä 1 has a distribution of ² j times a binomial distribution of B.x j ; t/; the distribution of this quantity is approximately normal, with mean x j ² j t and variance x j ² 2 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. This suggests 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 ² j btcC² j Z.t btc/ where Z .t/,0Ä t Ä 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 bursty periodic sources. As in Example 4.2, one can compute Þ j .s; t/ D ² j C² 2 j sf.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 A 0:5 \ A 1 ,whereA 1 and A 0:5 D are the sets of x satisfying the following two constraints: k X jD1 x j ² j Ä C (4.15) k X jD1 x j  ² j C ² 2 j  2B Á Ä B C C (4.16) 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 absorb the temporary bursts (expressed in (4.16)), but these buffers fill infinitely slowly since the 98 NETWORK CONSTRAINTS AND EFFECTIVE BANDWIDTHS average input rate ‘barely’ exceeds the service capacity. If (4.16) is active, then the most probable 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 above acceptance 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 T on and T off , respectively, which can be either deterministic or random. Let p on denote the probability that the source is on. If m is the mean rate of the source, then p on D m= h. The effective bandwidth of this on-off source has a simple form when the time parameter t is small compared with T on and T off . 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 Þ on-off .m; h/ D 1 st log h 1  m h Á C m h e sht i (4.17) This expression illustrates some of the properties of s mentioned in Section 4.6. As s approaches zero, the effective bandwidth approaches the mean rate of source. As s tends 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 in Example 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 T on and T off . This model has been used to model voice and video traffic. It can also be used to model the activity during a web 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 rate ¼ i . 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 ij . 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.sX i [0; t]/. Then it is not hard to see that f i .t/ D e ½ i t e s¼ i t C Z t 0 e s¼ i u ½ i e ½ i u P j P ij f j .t  u/ du Let f Ł i .z/ be the Laplace transform of f i .t/. The integral above is a convolution integral and so we easily find f Ł i .z/ D 1 z C ½ i  s¼ i  1 C P m jD1 P ij f Ł j .z/ Á MULTIPLE QOS CONSTRAINTS 99 One can, in principle, solve this set of linear equations and then invert the Laplace transforms. Consider the special case of an on-off source, with off and on phases that are exponentially distributed with means 1=½ 0 and 1=½ 1 , respectively, and taking ¼ 0 D 0 and peak rate ¼ 1 D h. We find, after some algebra, f .t / D E Ä exp  s Z t 0 x.s/ds ý D ½ 1 ½ 0 C ½ 1 f 0 .t/ C ½ 0 ½ 0 C ½ 1 f 1 .t/ D ½ 1 ! 2 C ½ 0 .! 2  sh/ .! 2  ! 1 /.½ 0 C ½ 1 / e ! 1 t C ½ 1 ! 1 C ½ 0 .sh  ! 1 / .! 2  ! 1 /.½ 0 C ½ 1 / e ! 2 t where ! 1 , ! 2 are the two roots of ! 2 C .½ 0 C ½ 1  ½ 0 Âh/!  ½ 0 Âh D 0. A discrete time model is even easier. Suppose that the state changes at each epoch according to the transition matrix . p ij /. Then the effective bandwidths satisfy the set of linear recurrences f i .t/ D e s¼ i P j p ij f j .t  1/; t D 1; 2;::: These recurrences have been successfully used to make numerical computations of effective bandwidth 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 each constraint. Each constraint might correspond to a different manner of overflow. The effective bandwidth of a stream is defined by the constraint that is active. Example 4.7 demonstrates how multiple constraints may result from the scheduling mechanism of priority queueing. 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, J 1 and J 2 . Service is FCFS, except that a class in J 1 is always given priority over a class in J 2 .Fori 2 J 1 there is a QoS guarantee on delay of the form P.delay > B 1 =C/ Ä e  1 For all sources there is a QoS guarantee on CLP of CLP Ä e  2 This gives the two constraints g 1 .x/ Ä 0andg 2 .x/ Ä 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 of the constraints X j2J 1 x j Þ j .s 1 ; t 1 / Ä K 1 and X j2J 1 [J 2 x j Þ j .s 2 ; t 2 / Ä K 2 (4.18) 100 NETWORK CONSTRAINTS AND EFFECTIVE BANDWIDTHS 0,0 P (delay > B 1 /C) ≤ e −g 1 A x 2 x 1 x CLP ≤ e −γ 2 Figure 4.8 An acceptance region defined by two constraints. There are two classes of traffic. The 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). where K 1 :D C C 1 t 1  B 1   1 s 1 à and K 2 :D C C 1 t 2  B   2 s 2 à For example, suppose J 1 Df1g, J 2 Df2g.Thenwehave x 1 Þ 1 .s 1 ; t 1 / Ä K 1 and x 1 Þ 1 .s 2 ; t 2 / C x 2 Þ 2 .s 2 ; t 2 / Ä K 2 If K 1 =Þ 1 .s 1 ; t 1 /<K 2 =Þ 1 .s 2 ; t 2 / 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 of constraints. 4.9 Traffic shaping It is a characteristic of broadband multimedia and data traffic that its rate can fluctuate widely. These fluctuations can occur at various superimposed frequencies, as illustrated in the right hand part of Figure 4.6. Each frequency of fluctuation defines a timescale of burstiness, i.e. an order of time over which significant changes are observed in the rate of the source, when this is averaged over time periods of the same size. In Figure 4.6, such changes in the rate are observed on timescales of order t 1 and t 2 (and there may be even larger timescales, but these do not show up in the small snapshot taken). Suppose we are at an operating 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 buffer overflow? In other words, which aspects of the traffic make it hard to multiplex it and hence contribute 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, the effective bandwidth of a source depends on the timescales of the source’s burstiness that significantly contribute to the event of buffer overflow. Clearly, fluctuations on different timescales do not contribute equally, since those on short timescales may be absorbed by the buffer. This effect is captured in the definition of the effective bandwidth in (4.5). By TRAFFIC SHAPING 101 letting 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 t 1 is absorbed by the buffer and is not reflected in the effective bandwidth of the source. Here t 2 is the dominant timescale of burstiness that constraints the system; within t 2 , the timescale t 1 contributes its mean rate. In other words, if we were to replace our source by one obtained by averaging it over a timescale of t 1 , 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 smoother traffic. A typical traffic shaper consists of a large buffer that is served at a rate smaller than the peak rate of the stream, or of a buffer that is combined with a leaky bucket that holds 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 of data to be transmitted. The above discussion explains the effects of shaping mechanisms on the 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 the multiplexing 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 is simple. 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 cells in epoch 5000 0 50 100 150 200 5200 5400 5600 5800 6000 Figure 4.9 Burstiness can be seen in this trace of 1000 epochs of a MPEG-1 encoded video of Star Wars. Each epoch is 40 ms. 102 NETWORK CONSTRAINTS AND EFFECTIVE BANDWIDTHS 0 0.5 1 1.5 2 t (secs) 0.0001 0.001 0.01 0.1 1 s(kb −1 ) 0 0.5 1 1.5 2 2.5 effective bandwidth (Mbps) Figure 4.10 Effective bandwidth of MPEG-1 traffic. Note that for different values of s and t, 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 effective bandwidths can be computed. In practice, this is not the case. The network knows only the traffic contract of the requested service. This contract only partly characterizes the traffic source, for example, through the leaky bucket constraints. This poses a problem. If only the 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 that application produces traffic with known statistics, then he can use the actual effective bandwidth 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 that type of application. This concept of an ‘average’ effective bandwidth is used in flat rate charging. The idea is that two applications that use the same contract should be charged the same, i.e. on the basis of an average effective bandwidth for this contract type, irrespective of whether they generate identical traffic. 3. If very little information is known about the source, then it can be reasonable to use the 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 constraints placed upon that traffic by the service contract. This is a conservative approach that results in network resources being underutilized. However, it is the only approach that enables the network to implement hard quality of service guarantees. We pursue this idea in Section 4.11. BOUNDS FOR EFFECTIVE BANDWIDTHS 103 4. One can modify the above approach to one of dynamic call acceptance by using information that is obtained by monitoring the actual system. One uses actual measurements of the performance on the links of the network to determine the actual amount 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 the available 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 existing load 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 underutilization and poor quality of service provisioning. If the network has better information about the resource usage statistics of a new call, then it can better multiplex and load the network more efficiently. This explains why a network operator wishes to have a good idea of the traffic profiles of his customers. But how can he obtain better information than that available through 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 is through pricing. The network posts a set of possible tariffs for the same traffic contract, each one resulting in a different charge, depending on the traffic that is actually generated during the 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 reveals important information to the network operator, who can use this information to obtain a better approximation of the effective bandwidth. This is an example of incentive compatible pricing; when users optimize their tariff choices the network operator gains information that allows 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 Df.² k ;þ k /, k D 1;:::;K g.Letm 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, NÞ.m; h/ can be extremely difficult to calculate exactly, as we are in effect trying to determine a worst-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 N X[0; t] that could be produced in a time interval of length t is N X[0; t] Ä H.t/ :D min kD1;:::;K f² k t C þ k g (4.19) One can show that E exp.sX[0; t]/ is maximized subject to EX[0; t] D mt and X [0; t] Ä H .t/ by the distribution in which X[0; t] equals 0 or H .t/ with probabilities 1  mt=H.t / and mt=H .t/, respectively. Of course there may not be actual traffic, conforming with the above leaky buckets, for which X[0; t] has this distribution, and this is why we are only obtaining an upper bound on NÞ.m; h/. This upper bound is NÞ.m; h/ Ä 1 st log Ä 1 C tm H .t/  e sH.t/  1 Ð ½ DQÞ sb .m; h/ (4.20) We call the right-hand side of (4.20) the simple bound. 104 NETWORK CONSTRAINTS AND EFFECTIVE BANDWIDTHS H ht t 1 t b + pt b Figure 4.11 A dual leaky bucket policer. Depending on the value of t, a different leaky bucket affects the source’s maximum contribution, H.t /, and hence the effective bandwidth. In QÞ sb .m; h/ we can see the effects of leaky buckets on the resource usage. Each leaky bucket .² k ;þ k / 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 QÞ on-off .m; h/ D 1 st log h 1 C m h .e sht  1/ i (4.21) We refer to this as the on-off bound, which we have already met in (4.17). This bound is valid for any value of t, since it is the effective bandwidth of an on-off source with peak rate h and mean rate m which can produce arbitrarily long bursts. Such a source does not comply with the .²; þ/ leaky bucket which restricts the length of such bursts and so is worse 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 to use complex computational procedures to determine the worst-case traffic. In general, the worst-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 accumulate tokens so that it can again send at the maximum rate. t off t on = 2t r t′ = h h−r b Figure 4.12 Periodic pattern for the inverted T approximation to a worst-case traffic. t 1 D þ=.h  ²/, t off D [.2t  t 1 /² C t 1 h]=m  2t. [...]... implication of the above is that it is reasonable to assume that each traffic stream in a network can be assigned a unique effective bandwidth, that is independent of 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... details of the derivation of the effective bandwidth for this model can be found in Courcoubetis and Weber (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 bandwidths for real traffic traces first appeared in Gibbens (1996) The use of effective bandwidth concepts for dimensioning network links and. .. number of independent sources being multiplexed, but only the effects due to the buffers Relevant references for 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) 110 NETWORK CONSTRAINTS AND EFFECTIVE... marginal cost and use it as a price, it can be difficult to predict the demand and to dimension the network accordingly There is a risk that we will build a network that is either too big or too small A pragmatic approach is to start conservatively and then expand the network only as demand justifies it Prices are used to signal the need to expand the network One starts with a small network and adjusts... Duffield (1995), Simonian and Guilbert (1995) and Courcoubetis and Weber (1996) A refinement of this asymptotic using that BahadurRao approximation is due to Likhanov and Mazumdar (1999) Some early references for the 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 effective bandwidths is Kelly... process of rate ½r and endure for an average time of 1=¼r Let pr be the blocking probability for a connection of type r Let xr 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 xr t/ Here Þjr 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),... 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, Walrand and Weber (1995), Gibbens, Key and Kelly (1995), Grossglauser and Tse (1999) and Courcoubetis, Dimakis and Stamoulis (2002) Pricing Communication Networks: Economics, Technology and Modelling Costas... details on the construction of such demand curves see Section 5.7 and Figure 5.8 5 .4. 4 Peak-load pricing The key result of Section 5 .4. 1 is that social surplus is maximized by marginal cost pricing A form of marginal cost pricing is also optimal in circumstances of so-called peak-load pricing Suppose that demand for a service is greater during peak hours, lesser during offpeak hours, and the cost depends... yi D b 5 .4. 5 Walrasian Equilibrium We now turn to two important notions of market equilibrium and efficiency The key points in this and the next section are the definitions of Walrasian equilibrium and Pareto efficiency, and the fact that they can be achieved simultaneously, as summarized at the end of Section 5 .4. 6 The reader may wish to skip the proofs and simply read the definitions, summary and remarks... terms of the call blocking probability This probability depends on the technology set of the network and the rates of arrival of the various connection types (the service requests) In this section we see how such blocking probabilities may be calculated from 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 . 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). 110 NETWORK CONSTRAINTS AND. of x satisfying the following two constraints: k X jD1 x j ² j Ä C (4. 15) k X jD1 x j  ² j C ² 2 j  2B Á Ä B C C (4. 16) Constraints (4. 15) and (4. 16) correspond to .s; t/ values of .0; 1/ and. 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 T on and T off , respectively, which can be either deterministic or random.