8 Cell-Scale Queueing dealing with the jitters CELL-SCALE QUEUEING In Chapter 4 we considered a situation in which a large collection of CBR voice sources all send their cells to a single buffer. We stated that it was reasonably accurate under certain circumstances (when the number of sources is large enough) to model the total cell-arrival process from all the voice sources as a Poisson process. Now a Poisson process is a single statistical model from which the detailed information about the behaviour of the individual sources has been lost, quite deliberately, in order to achieve simplicity. The process features a random number (a batch) of arrivals per slot (see Figure 8.1) where this batch can vary as 0, 1, 2, .,1. So we could say that in, for example, slot n C 4, the process has overloaded the queueing system because two cells have arrived – one more than the buffer can transmit. Again, in slot n C 5 the buffer has been overloaded by three cells in the slot. So the process provides short periods during which its instantaneous arrival rate is greater than the cell service rate; indeed, if this did not happen, there would be no need for a buffer. But what does this mean for our N CBRsources?Eachsourceisata constant rate of 167 cell/s, so the cell rate will never individually exceed the service rate of the buffer; and provided N ð 167 < 353 208 cell/s, the total cell rate will not do so either. The maximum number of sources is 353 208/167 D 2115 or, put another way, each source produces one cell every 2115 time slots. However, the sources are not necessarily arranged such that a cell from each one arrives in its own time slot; indeed, although the probability is not high, all the sources could be (accidentally) synchronized such that all the cells arrive in the same slot. In fact, for our example of multiplexing 2115 CBR sources, it is possible Introduction to IP and ATM Design Performance: With Applications Analysis Software, Second Edition. J M Pitts, J A Schormans Copyright © 2000 John Wiley & Sons Ltd ISBNs: 0-471-49187-X (Hardback); 0-470-84166-4 (Electronic) 114 CELL-SCALE QUEUEING 0 1 2 3 4 5 nn+1 n+2 n+3 n+4 n+5 n+6 n+7 n+8 n+9 n+10 Time slot number Number of arrivals in a slot Figure 8.1. A Random Number of Arrivals per Time Slot for any number of cells varying from 0 up to 2115 to arrive in the same slot. The queueing behaviour which arises from this is called ‘cell-scale queueing’. MULTIPLEXING CONSTANT-BIT-RATE TRAFFIC Let us now take a closer look at what happens when we have constant-bit- rate traffic multiplexed together. Figure 8.2 shows, for a simple situation, how repeating patterns develop in the arrival process – patterns which depend on the relative phases of the sources. Queue size (a) All streams out of phase Figure 8.2. Repeating Patterns in the Size of the Queue when Constant-Bit-Rate TrafficIsMultiplexed ANALYSIS OF AN INFINITE QUEUE WITH MULTIPLEXED CBR INPUT: THE NÐD/D/1 115 Queue size Queue size (b) Two streams in phase (c) All streams in phase Figure 8.2. (continued) It is clear from this picture that there are going to be circumstances where a simple ‘classical’ queueing system like the M/D/1 will not adequately model superposed CBR traffic; in particular, the arrival process is not well modelled by a Poisson process when the number of sources is small. At this point we need a fresh start with a new approach to the analysis. ANALYSIS OF AN INFINITE QUEUE WITH MULTIPLEXED CBR INPUT: THE N·D/D/1 The NÐD/D/1 queue is a basic model for CBR traffic where the input process comprises N independent periodic sources, each source with thesameperiodD. If we take our collection of 1000 CBR sources, then N D 1000, and D D 2115 time slots. The queueing analysis caters for all possible repeating patterns and their effect on the queue size. The buffer capacity is assumed to be infinite, and the cell loss probability is approximated by the probability that the queue exceeds a certain size x, 116 CELL-SCALE QUEUEING i.e. Qx. Details of the derivation can be found in [8.1]. CLP ³ Qx D N nDxC1 N! n! Ð N n! Ð n x D n Ð 1 n x D Nn Ð D N C x D n C x Let’s put some numbers in, and see how the cell loss varies with different parameters and their values. The distribution of Qx for a fixed load of 0 10203040 Buffer capacity 10 −10 10 −9 10 −8 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 Q(x) N = 1000 N = 500 N = 200 N = 50 • k:D 0 40 NDD1Q x, N,:D D N N nDxC1 combin (N, n)Ð n x D n Ð 1 n x D Nn Ð D N C x D n C x x k :D k y1 k :D NDD1Q k, 1000, 0.95 y2 k :D NDD1Q k, 500, 0.95 y3 k :D NDD1Q k, 200, 0.95 y4 k :D NDD1Q k, 50, 0.95 Figure 8.3. Results for the NÐD/D/1 Queue with a Load of 95%, and the Mathcad Code to Generate (x, y) Values for Plotting the Graph HEAVY-TRAFFIC APPROXIMATION FOR THE M/D/1 QUEUE 117 D N/D D 0.95 with numbers of sources ranging from 50 up to 1000 is given in Figure 8.3. Note how the number of inputs (sources) has such asignificant impact on the results. Remember that the trafficisperiodic, and the utilization is less than 1, so the maximum number of arrivals in any one period of the constant-bit-rate sources (as well as in any one time slot) is limited to one from each source, i.e. N.ThevalueofN limits the maximum size of the queue – if we provide N waiting spaces there would be no loss at all. The NÐD/D/1 result can be simplified when the applied trafficisclose to the service rate; this is called a ‘heavy traffictheorem’.Butlet’s first look at a useful heavy traffic result for a queueing system we already know – the M/D/1. HEAVY-TRAFFIC APPROXIMATION FOR THE M/D/1 QUEUE An approximate analysis of the M/D/1 system produces the following equation: Qx D e 2ÐxÐ 1 Details of the derivation can be found in [8.2]. The result amounts to approximating the queue length by an exponential distribution: Qx is the probability that the queue size exceeds x,and is the utilization. At first sight, this does not seem to be reasonable; the number in the queue is always an integer, whereas the exponential distribution applies to a continuous variable x;andalthoughx canvaryfromzerouptoinfinity, we are using it to represent a finite buffer size. However, it does work: Qx is a good approximation for the cell loss probability for a finite buffer of size x. In later chapters we will develop equations for Qx for discrete distributions. For this equation to be accurate, the utilization must be high. Figure 8.4 shows how it compares with our exact analysis from Chapter 7, with Poisson input traffic at different values of load. The approximate results are shown as lines through the origin. It is apparent that although the cell loss approximation safely overestimates at high utilization, it can significantly underestimate when the utilization is low. But in spite of this weakness, the major contribution that this analysis makes is to show that there is a log–linear relationship between cell loss probability and buffer capacity. Whyisthisheavy-traffic approximation so useful? We can rearrange the equation to specify any one variable in terms of the other two. Recalling the conceptual framework of the traffic–capacity–performance model from Chapter 3, we can see that the traffic is represented by (the utilization), the capacity is x (the buffer size), and the performance 118 CELL-SCALE QUEUEING 0102030 Buffer capacity 10 −10 10 −9 10 −8 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 CLP ρ = 0.95 ρ = 0.75 ρ = 0.55 k:D 0 30 ap95 k :D Poisson k, 0.95 ap75 k :D Poisson k, 0.75 ap55 k :D Poisson k, 0.55 MD1Qheavy x, :D e 2ÐxÐ 1 i:D 2 30 x k :D k Y1 i :D finiteQloss x i , ap95, 0.95 Y2 k :D MD1Qheavy x k , 0.95 Y3 i :D finiteQloss x i , ap75, 0.75 Y4 k :D MD1Qheavy x k , 0.75 Y5 i :D finiteQloss x i , ap55, 0.55 Y6 k :D MD1Qheavy x k , 0.55 Figure 8.4. Comparing the Heavy-Traffic Results for the M/D/1 with Exact Analysis of the M/D/1/K, and the Mathcad Code to Generate (x, y) Values for Plotting the Graph is Qx (the approximation to the cell loss probability). Taking natural logarithms of both sides of the equation gives lnQx D2x 1 This can be rearranged to give x D 1 2 lnQx 1 HEAVY-TRAFFIC APPROXIMATION FOR THE NÐD/D/1 QUEUE 119 and D 2x 2x lnQx We will not investigate how to use these equations just yet. The first relates to buffer dimensioning, and the second to admission control, and both these topics are dealt with in later chapters. HEAVY-TRAFFIC APPROXIMATION FOR THE N·D/D/1 QUEUE Although the exact solution for the NÐD/D/1 queue is relatively straight- forward, the following heavy-traffic approximation for the NÐD/D/1 010203040 Buffer capacity 10 −10 10 −9 10 −8 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 Q(x) N = 1000 N = 500 N = 200 N = 50 k:D 0 40 NDD1Qheavy x, N, :D e 2ÐxÐ x N C 1 x k :D k y1 k :D NDD1Q k, 1000, 0.95 y2 k :D NDD1Q k, 500, 0.95 y3 k :D NDD1Q k, 200, 0.95 y4 k :D NDD1Q k, 50, 0.95 y5 k :D NDD1Qheavy k, 1000, 0.95 y6 k :D NDD1Qheavy k, 500, 0.95 y7 k :D NDD1Qheavy k, 200, 0.95 y8 k :D NDD1Qheavy k, 50, 0.95 Figure 8.5. Comparison of Exact and Approximate Results for NÐD/D/1 at a Load of 95%, and the Mathcad Code to Generate (x, y) Values for Plotting the Graph 120 CELL-SCALE QUEUEING [8.2] helps to identify explicitly the effect of the parameters: Qx D e 2x x N C 1 Figure 8.5 shows how the approximation compares with exact results from the NÐD/D/1 analysis for a load of 95%. The approximate results are shown as lines, and the exact results as markers. In this case the approximation is in very good agreement. Figure 8.6 shows how the 0 10203040 Buffer capacity CLP ρ = 0.95 ρ = 0.95 ρ = 0.95 10 0 10 −1 10 −2 10 −3 10 −4 10 −5 10 −6 10 −7 10 −8 10 −9 10 −10 k:D 0 40 x k :D k y1 k :D NDD1Q k, 200, 0.95 y2 k :D NDD1Qheavy k, 200, 0.95 y3 k :D NDD1Q k, 200, 0.75 y4 k :D NDD1Qheavy k, 200, 0.75 y5 k :D NDD1Q k, 200, 0.55 y6 k :D NDD1Qheavy k, 200, 0.55 Figure 8.6. Comparison of Exact and Approximate Results for NÐD/D/1 for a variety of Loads, with N D 200, and the Mathcad Code to Generate (x, y)Valuesfor Plotting the Graph CELL-SCALE QUEUEING IN SWITCHES 121 approximation compares for three different loads. For low utilizations, theapproximatemethodunderestimatesthecellloss. Note that the form of the equation is similar to the approximation for the M/D/1 queue, with the addition of a quadratic term in x,thequeue size. So, for small values of x,NÐD/D/1 queues behave in a manner similar to M/D/1 queues with the same utilization. But for larger values of x the quadratic term dominates; this reduces the probability of larger queues occurring in the NÐD/D/1, compared to the same size queue in the M/D/1 system. Thus we can see how the Poisson process is a useful approximation for N CBR sources, particularly for large N:asN !1, the quadratic term disappears and the heavy traffic approximation to the NÐD/D/1becomesthesameasthatfortheM/D/1.InChapter14we revisit the M/D/1 to develop a more accurate formula for the overflow probability that both complements and extends the analysis presented in this chapter (see also [8.3]). CELL-SCALE QUEUEING IN SWITCHES It is important not to assume that cell-scale queueing arises only as a result of source multiplexing. If we now take a look at switching, we will find that the same effect arises. Consider the simple output buffered 2 ð 2 switching element shown in Figure 8.7. Here we can see a situation analogous to that of multiplexing the CBR sources. Both of the input ports into the switch carry cells coming from any number of previously multiplexed sources. Figure 8.8 shows a typical scenario; the cell streams on the input to the switching element are the output of another buffer, closer to the sources. The same queueing principle applies at the switch output buffer as at the source multiplexor: the sources may all be CBR, and the individual input ports to the switch may contain cells such that their aggregate arrival rate is less than the Figure 8.7. An Output Buffered 2 ð 2 Switching Element 122 CELL-SCALE QUEUEING Source 1 Source 2 Source N . . . Source 1 Source 2 Source N . . . 2 × 2 ATM switching element Source multiplexer Source multiplexer Figure 8.8. Cell-Scale Queueing in Switch Output Buffers Buffer capacity 1E−10 1E−09 1E−08 1E−07 1E−06 1E−05 1E−04 1E−03 1E−02 1E−01 1E+00 0 5 10 15 20 Cell loss probability Figure 8.9. Cell Loss at the Switch Output Buffer output rate of either of the switch output ports, but still there can be cell loss in the switch. Figure 8.9 shows an example of the cell loss probabilities for either of the output buffers in the switch for the scenario illustrated in Figure 8.8. This assumes that the output from each source multiplexor is a Bernoulli process, with parameter p 0 D 0.5, and that the cells are routed [...]... for buffers in the switches to cope with the cell-scale queueing behaviour This is inherent to ATM; it applies even if the network allocates the peak rate to variable-bit-rate sources Buffering is required, because multiple streams of cells are multiplexed together It is worth noting, however, that the cell-scale queueing effect (measured by the CLP against the buffer capacity) falls away very rapidly...CELL-SCALE QUEUEING IN SWITCHES 123 in equal proportions to the output buffers of the switching element Thus the cell-scale queueing in each of the output buffers can be modelled with binomial input, where M D 2 and p D 0.25 So, even if the whole of the ATM network is dedicated to carrying only CBR traffic, there is a need for buffers in the switches to cope with the cell-scale queueing... very rapidly with increasing buffer length – so we only need short buffers to cope with it, and to provide a cell loss performance in accord with traffic requirements This is not the case with the burst-scale queueing behaviour, as we will see in Chapter 9 . Copyright © 2000 John Wiley & Sons Ltd ISBNs: 0-4 7 1-4 9187-X (Hardback); 0-4 7 0-8 416 6-4 (Electronic) 114 CELL-SCALE QUEUEING 0 1 2 3 4 5 nn+1 n+2 n+3 n+4. ‘cell-scale queueing’. MULTIPLEXING CONSTANT-BIT-RATE TRAFFIC Let us now take a closer look at what happens when we have constant-bit- rate traffic multiplexed