E&CE 710 Wireless Communications Networksj25ni/ECE710/Win18-Lecture...B—buffer size E(W)—average waiting time S—service rate Q: Determine the buffer size B required, in units

E&CE 710 Wireless Communications Networks

Lecture Notes: #2

Xuemin (Sherman) ShenOffice: EIT 4155Phone: x 32691

Email: [email protected]

2018/2/28 1ECE710 Wireless Communications Networks

3. Traffic characterization

• Communication Networks are expected to carry a

variety of traffic types in integrated fashion.

• With traffic characterization, the statistical resource requirements, in terms of link and buffer capacities can be obtained.

• With traffic characterization, the queue wait times, blocking

probabilities, etc. can be determined.


Type of Traffic (M. Schwartz, 1996)• The importance of the traffic characterization

Ex. 3.1: 1 ATM cell = 53×8 = 424bits.


Cells are delivered to the network in FIFO order.B— buffer size E(W)— average waiting time S— service rate

Q: Determine the buffer size B required, in units of cells, and theaverage time E(W) a typical cell must wait.

Type of Traffic• Solution:

(1) The utilization factor ，B= 4 is suffice to handle

the cell arrivals. With nD/D/1 queue， Loss

probability = 0.

(2) with M/D/1, = 35.6 , B= 249 with loss probability= 10−9.


• Traffic characterization is not only important when designing buffers, but also in studying admission, access, and flow control.

• Three types of traffic:Voice: Continuous (Constant ) bit rate;Video: Variable bit rate;Bursty data;

Type of Traffic

5

Voice communication: interactive communication

Video: variable length of video frames

Bob speaks

Dave speaks

Frame 1 Frame 2 Frame 3 Frame 4

Video = moving picture (30 pics/s)

picture (frame) size depends on motion in pictures

Type of Traffic


Voice (audio) — real time traffic, in which bits are periodically generated.Video (Compression) — real time trafficImages, bursty data — non real-time traffic (Available bit rate)Images, bursty data (Unspecified bit rate)

Objectives: Characterize each traffic; using the models obtained to determine buffer occupancy statistics queue wait times, and blocking probabilities.

Voice service requires a peak cell rate traffic descriptor;Video service requires a peak cell rate, a average cell rate and an intrinsic burst tolerance traffic descriptor; ABR service requires a peak cell rate and a minimum cell rate traffic descriptor;UBR or “best-effort” service has no pre-specified traffic descriptor

Packet Voice Modeling


A single voice source can be well presented by a two-state process (on - off model).

— the rate of transition out of the silent state;

— the rate of transition out of talk spurt.

The speaker activity factor



Suppose there are voice sources; their initiations of calls areindependent of each other.

It is clear that the parameter must satisfy the low bound inequality

i.e., the utilization factor

The probability of active voices (out of N) follows binomial distribution



The N multiplexed independent voice sources give rise to thefollowing (N+ 1)-state birth-death model.

is called the underload state, is called the overload state. e.g.

When i > C, the queue is filling and the system is said to be in overload;When i < C, with the queue emptying, the system is said to be in underload.



By setting up the balance equations at each state

Let , then

where

Q: how to calculate the performance and design parameters such as delay and loss statistics and buffer size required?

Fluid Source Modeling of Packet Voice


Assume N sources each generating cells/sec during a talkspurt.The service capacity is cells/sec. When N and are very largerelative to the small cell size, it can be superimposed upon/on thecell arriving process during a talkspurt as one information unitflowing into the system. Thus the discreteness of the cell is removed(like a continuous flow of fluid).

Let be the average length of talk spurt and define the number of cells arriving during a talk spurt as a “unit of information”. The service capacity is normalized into by .



X (continuous RV) — buffer occupancyThe units of x are defined to be the number of unit of informationarriving during a talk spurt.Let be the state of the buffer in units of cells.

, Then,

Find the probability distribution of x (which can be converted to buffer cell distribution).



Assume an exponentially distributed on-off process for each source,and there are sources in talk spurt.

Define as the cumulative probability distribution of x, then,

P{one source moves from “off” to “on” during system isin state at }

P{one source moves from “on” to “off” during system isin state at }

P{no source changes state during system is in state at }



Add both side of the equation by and divide by

Letting , we get

Under steady state condition , , we have

Here , with

( is the net gain of the buffer in a small period to )



The solution of this set of equations is readily obtained by firstwriting them out in their complete form:

Define , then

With an (N+1)×(N+1) diagonal matrix defined as

and an (N+1)×(N+ 1) matrix defined as follows:



Multiplying by on both sides, we get

The general solution of the above equation is

with the i-th eigenvalue, the corresponding eigenvector given as the solution to the eigenvector equation

and the set of undetermined coefficients.

Since ， is a diagonal matrix, then



Let ,

with the j-th component of eigenvector



• Procedure of finding

and are the eigenvalues and eigenvectors of the matrix . They can be calculated by using Matlab.

Since (for the infinite-buffer case)

Let , we get . By setting ,

It has been proved that the number of negative eigenvalues is which is also the number values of that have to be found.



Since , for (When the system is in overload state, theprobability that the buffer is empty is zero), we have

e.g., . (5 overload states)

The probability for the buffer occupancy to be less than or equal to is

( )



The complementary probability cumulative distribution function

is also called the survivor function. It can be used to find the buffer size required for a given loss probability.

The buffer occupancy will exceed cells is given by

• Approximate Approach:D. Anick, et al. “Stochastic Theory of a Data Handling System withMultiple Sources”, Bell System tech. J. Vol. 61, No. 8, pp. 1871-

1894, 1982.Since the probability distribution is given by the sum of negativeexponentials, the exponential with the smallest negative eigenvaluewill dominate.



Let be the eigenvalue:

where

then where

Observations: since depends on or , the buffer size required to keep the buffer overflow probability to some desire probability is independent of the number of sources multiplexed, providing the capacity is scaled accordingly, i.e.,

Statistically multiplexing sources that provides an improvement in both the buffer size per source required and in the total queueing time in the buffer.

1. the number of buffer per source, , decrease as increases;2. the wait time in the buffer decreases as increases.



Example 3.2 Consider the special case of voice sources, sec and

sec .a) Find for the two cases and .

Indicate all three value on the (N+1)-state diagram.

Solution: Using , we have

N+ 1 state diagram:



b) Find and compare the eigenvalues of for the two cases (and ).

Solution:

for eigenvalues:

Note that: i) one eigenvalue is 0 in both cases; ii) the number of negative eigenvalues equals to the number of overload states.



c). Find the eigenvectors for the two cases

(i)

(ii)

Note: both parts (b) and (c) are solved using MATLAB.

d). Calculate and plot , where cells/sec.

Thus,

(ii) it can be calculated that



f). How large a buffer size is needed to have the buffer overflow probability equal to 10−2 for each of the two cases? Repeat for an overflow probability of 10−3.

• Observations:1. For the different overflow probability, the required buffer size

increases for lower overflow probability (with the same ).

2. As the traffic intensity increases, the required buffer size to meet the overflow probability constraint also increases.

Solution: From the plot of G(x), we get the buffer size in cells for overflow probability of 10−2 and 10−3.





G(x)

100

10-1

10-2

10-3

10-4

10-5

10-6

10-7

10-8

10-9

Fluid source modeling of video traffic

• Video frames to be transmitted at the common rate of 30 frames/sec are sampled and quantized, picture element by picture element, providing a constant number of bits/frame.

• North American Standard

Transmitted source rate

30×250,000×8 = 60 Mbits/sec (monochrome video)

or 3×60 Mbits/sec (color video)

2018/2/28 ECE710 Wireless Communications Networks 28

1 fame= 250,000 pixels 1 pixel= 8 bits (grey scale / monochrome video)

It has been found that there is long-range dependence over many frames of a video sequence and relatively heavy tails of the bandwidth distribution.



Compression (using a variety of coding techniques) can obtain0.5bit/pixel, i.e., source rate= 30×250,000×0.5 = 3.75Mbits/sec .

The compressed video traffic is with variable bit rate. (frame byframe) If independent video sources are multiplexed together,the average bit rate per second, and the autocovariancefunction are used to describe each source.

(continuous time)

(discrete time)

is the frame number, represents a frame n units (frames)away.



Statistical multiplexer, N video sources (M >> N)



is the multiplexed (time varying) rate. Actually changes at frame intervals (30 frames/sec) rather than continuously. The time-varying bit rate has been quantized to have the values 0, A, 2A, ···, mA bits/pixel only. The original N sources multiplexed model can be represented by an (m+ 1)-state Markov chain with state-dependent transition rates.

Markov chain representation, equivalent process



Question: How does one choose the three minisource model parameter and ?

Answer: Since

then,

The parameter represents the variance of video source , while is used to represent the variance of composite signal.



Proof:

Note that for wide-sense stationary process,

autocorrelation



Special case: independent sources with identical moments

It can be proved that the autocovariance function of the two-state minisource model is given by

Let be probability the minisource is in the ”on” state,is in the ”off” state.

Note: the two-state covariance of the two-state minisource exhibitsthe desired exponential behavior.



For stationary 2-state Markov Chain:

Define



where





Since



* letting

The multiplexed minisources must have autocovariance function



The average number of minisources ”on” is just , the average bit rate of the equivalent minisource model is .

If we can measure from video traffic (data) that

then,



We obtain

If , we have

With the composite minisource model for the video determined,one can use this video source model and the fluid-flow approachto calculate (to design the access buffer) to provide a specifiedprobability of cell loss, and to determine the resultant video accessdelay.

Define

where



Following the same analysis procedure as to voice model, we get the following equation

is the net gain of the buffer in a small timeperiod to

Re-arrange, let and use the steady state conditionwe get



Rewriting the above equation

Define

we have

where



Let

The solution of the above differential equation is

With the ith eigenvalue and the corresponding eigenvector satisfying the eigenvalue equation

Since

Letting be the (marginal) cdf that the queue occupancy is less than or equal to



The solution of the above differential equation is

The survivor function (the probability the queue occupancyis greater than x) is given by

Note that the overload states are those for which the queue istending to fill. For these states, we must have ; i.e., thebuffer cannot be empty. When or (overloadstate), the number of overload states is which is thenumber of negative eigenvalues.



Setting for the overload states, we can find

Approximate approach:

where and

(approximating the survivor function by the dominant (asymptotic)eigenvalue term only).

Example 2.3(Maglaris, B., et. al., ”Performance Models of Statistical Multiplexingin Packet Video Communications”, IEEE Trans. on Commun., vol. 36,No. 7. pp. 834-844, 1988)

Let video source be approximated by minisources.The average source bit rate is



The transmission capacity

The utilization is then 3.9/4.875 = 0.8, we get:

Approximating the survivor function by the dominant eigenvalue term only,





Bursty traffic model


Since a bursty source is one that is characterized by alternatingperiods of inactivity and activity (two-state model), this is similar tosilent/talk spurt of voice.

Difference: the period of inactivity is much longer than the activeperiod; a burst of cells is transmitted during the active period.

Two-state, on-off model for bursty traffic source

If the active period is much shorter than the inactivity period, aPoisson arrival process (bulk arrival process) can be used to modelbursty traffic.

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E&CE 710 Wireless Communications Networksj25ni/ECE710/Win18-Lecture...B—buffer size E(W)—average waiting time S—service rate Q: Determine the buffer size B required, in units