ANALYSIS OF AN INFINITE QUEUE WITH MULTIPLEXED CBR INPUT: THE NÐD/D/1 115Queue size Queue size b Two streams in phase c All streams in phase Figure 8.2.. continued It is clear from this
Trang 18 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 CBR sources? Each source is at a
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)
Trang 20 1 2 3 4 5
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 Traffic Is Multiplexed
Trang 3ANALYSIS 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 the same period D 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,
Trang 4i.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
Buffer capacity
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
10 0
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
y1k:D NDD1Q k, 1000, 0.95
y2k:D NDD1Q k, 500, 0.95
y3k:D NDD1Q k, 200, 0.95
y4k: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
Trang 5HEAVY-TRAFFIC APPROXIMATION FOR THE M/D/1 QUEUE 117
DN/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
a significant impact on the results Remember that the traffic is periodic, 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 The value of N 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 traffic is close
to the service rate; this is called a ‘heavy traffic theorem’ But let’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 e2ÐxÐ
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; and although x can vary from zero up to infinity,
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
Why is this heavy-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
Trang 60 10 20 30
Buffer capacity
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
10 0
ρ = 0.95
ρ = 0.75
ρ = 0.55
k :D 0 30 ap95k:D Poisson k, 0.95
ap75k:D Poisson k, 0.75
ap55k:D Poisson k, 0.55
MD1Qheavy x, :D e2ÐxÐ
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 D 2x1
This can be rearranged to give
x D 1
2lnQx
1
Trang 7
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
Buffer capacity
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
N = 1000
N = 500
N = 200
N = 50
k :D 0 40 NDD1Qheavy x, N, :D e
2ÐxÐ
x
NC
1
x k :D k y1k:D NDD1Q k, 1000, 0.95
y2k:D NDD1Q k, 500, 0.95
y3k:D NDD1Q k, 200, 0.95
y4k:D NDD1Q k, 50, 0.95
y5k:D NDD1Qheavy k, 1000, 0.95
y6k:D NDD1Qheavy k, 500, 0.95
y7k:D NDD1Qheavy k, 200, 0.95
y8k: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
Trang 8[8.2] helps to identify explicitly the effect of the parameters:
Qx D e2x
x
NC
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
Buffer capacity
ρ = 0.95
ρ = 0.95
ρ = 0.95
100
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 y1k:D NDD1Q k, 200, 0.95
y2k:D NDD1Qheavy k, 200, 0.95
y3k:D NDD1Q k, 200, 0.75
y4k:D NDD1Qheavy k, 200, 0.75
y5k:D NDD1Q k, 200, 0.55
y6k: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) Values for
Plotting the Graph
Trang 9CELL-SCALE QUEUEING IN SWITCHES 121
approximation compares for three different loads For low utilizations, the approximate method underestimates the cell loss
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, the queue 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: as N ! 1,
the quadratic term disappears and the heavy traffic approximation to the NÐD/D/1 becomes the same as that for the M/D/1 In Chapter 14 we 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
Trang 10Source 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
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 p0D0.5, and that the cells are routed
Trang 11CELL-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 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 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