- Email: [email protected]
Generation of crosscorrelated random processes
Signal Processing 79 (1999) 223}234
Generation of crosscorrelated random processes Aurelio La Corte* Istituto di Informatica e Telecomunicazioni, University of Catania, Viale Andrea Doria 6, 95125 Catania, Italy Received 3 November 1998; received in revised form 29 June 1999
Abstract In this paper the author proposes a method which will allow simple and fast generation of two sets of random variables with given probability mass functions (PMFs) and crosscorrelation sequences (CCS). It is based on the composition of three sets of independent and identically distributed discrete random variables, two of which are correlated with each other only when the lag is null. The method is applicable in cases where, once experimental measurements have established the statistical characteristics of two stationary random processes in terms of PMFs and CCS, it is necessary to implement a simulation model, that is, it is necessary to generate random discrete variables whose PMFs and CCS match the empirical histograms. A case study is given to apply the method proposed to modelling the relationships between the monomedia streams making up a multimedia stream. ( 1999 Published by Elsevier Science B.V. All rights reserved. Zusammenfassung In dieser Arbeit wird eine Methode vorgeschlagen, die eine einfache und schnelle Erzeugung zweier Mengen von Zufallsvariablen mit gegebenen Wahrscheinlichkeitsverteilungsfunktionen (PMFs) und Kreuzkorrelationsfolgen (CCS) erlaubt. Diese Methode beruht auf der Zusammensetzung dreier Mengen von statistisch unabhaK ngigen und identisch verteilten diskreten Zufallsvariablen, von denen zwei nur dann miteinander korreliert sind, wenn die zeitliche Verschiebung null ist. Die Methode ist anwendbar in jenen FaK llen, wo } nachdem durch experimentelle Messungen die statistischen Eigenschaften zweier stationaK rer Zufallsprozesse hinsichtlich PMFs und CCS ermittelt wurden } es notwendig ist, ein Simulationsmodell zu implementieren, d.h. zufaK llige diskrete Variablen zu erzeugen, deren PMFs und CCS mit den empirischen Histogrammen uK bereinstimmen. In einer Fallstudie wird die vorgeschlagene Methode auf die Modellierung jener Beziehungen angewandt, die zwischen den einen multimedialen Strom bildenden monomedialen StroK men bestehen. ( 1999 Published by Elsevier Science B.V. All rights reserved. Re2 sume2 L'auteur propose dans cet article une meH thode permettant la geH neH ration simple et rapide de deux ensembles de variables aleH atoires ayant des fonctions de probabiliteH massiques (PMF) et des seH quences d'inter-correH lation (CCS) donneH es. Cette meH thode est baseH e sur la composition de trois ensembles de variables aleH atoires indeH pendantes et identiquement distribueH es, dont deux sont correH leH s l'un avec l'autre uniquement lorsque l'eH cart temporel est nul. Cette meH thode est applicable dans les cas ou`, une fois que les mesures expeH rimentales ont eH tabli les caracteH ristiques statistiques de deux processus aleH atoires stationnaires en termes de PMF et de CCS, il est neH cessaire d'implanter un mode`le de simulation, c'est a` dire qu'il est neH cessaire de geH neH rer des variables aleH atoires discre`tes dont les PMF et les CCS correspondent aux histogrammes empiriques. Une eH tude speH ci"que est preH senteH e, dans laquelle la meH thode proposeH e est
* Corresponding author. Tel.: #39-0957382356; fax: #39-095338280. E-mail address: [email protected] (A. La Corte) 0165-1684/99/$ - see front matter ( 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 1 6 5 - 1 6 8 4 ( 9 9 ) 0 0 0 9 7 - 3
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A. La Corte / Signal Processing 79 (1999) 223}234
Nomenclature PMF ACS CCS Ma N, Mb N, Mh N n n n (a ) b )D n n i (a ) b ) D n r rEn i MAN"MA , A ,2, A N 1 2 M MBN"MB , B ,2, B N 1 2 L Mc N, Md N, Me N t t t DD i D@D i E[ ) ] MHN"M1,2,2, NN MH N L MH N G m "E[X ] X t MX N, MX N n t p2 X p t (x) X p(%)t (x) X P t (x) X (;) P~1 Xt U (q) XX U (q) XY U(%) (q) XY W ,W MAX MIN W S ;, ;@, ;A
probability mass function autocorrelation sequence crosscorrelation sequence component random processes product of the pair of values a and b when h "i n n n product of a and the value of the sequence Mb N which is at a distance of n n q from the corresponding a , when h "i n n set of values taken by a n set of values taken by b n compound processes set of pairs of values a and b corresponding to h "i. n n n set of pairs of values a and b corresponding to h "i. n r n expected operator set of positive integer values taken by h n subset of elements of MHN less than or equal to DqD subset of elements of MHN greater than DqD; mean of X t discrete, wide-sense stationary and ergodic random processes variance of X t PMF of MX N t experimental histogram of PMF of MX N t cumulative probability function of X t inverse function of P t (x) X ACS of X t CCS between X and > t t experimental CCS between X and > t t maximum and minimum possible values of U (q) XY E !m ) m ab a b p )p a b temporal shift of Md N t random variables uniformly distributed in [0,1]
appliqueH e a` la modeH lisation des relations entre des #ux monomeH dia concourant a` un #ux multi-meH dia. ( 1999 Published by Elsevier Science B.V. All rights reserved. Keywords: Crosscorrelation; Random number generation; Simulation; Modelling; Multimedia sources
1. Introduction The design and testing of communication systems often requires the availability of random number generators. This, for example, is the case
when a designer of resource allocation and management strategies has the problem of evaluating the e!ectiveness of a certain strategy by simulation: he needs a simulation model which can capture the basic properties of the tra$c sources through a
A. La Corte / Signal Processing 79 (1999) 223}234
limited number of descriptors, and thus a large number of random variables with statistical properties which match those of the source to be modelled. The statistical properties usually considered for monomedia tra$c sources are the probability mass function (PMF) and the autocorrelation sequence (ACS) [1,6,7,10,19]. In the case of multimedia traf"c sources, the crosscorrelation sequences (CCSs) between compound monomedia streams have to be considered. This is due to the fact that a multimedia source is usually considered as an aggregate of two or more monomedia streams correlated with each other by logical, spatial or temporal relationships [12,21,23]. The correlation among the component monomedia streams lies in the fact that the behavior of each stream (e.g. its interarrival times or emission process) depends on that of the others. As an example, in a slide show with voice commentary, the change of slides is temporally related to the speaker's comments. Taking into consideration the CCS between the monomedia streams it is possible to capture the intermedia relationships and to avoid over-optimistic or pessimistic predictions of erformance measures for queueing systems [13]. In a good method for the generation of random variables, to model a monomedia tra$c source it must be possible to "x their PMF and ACS independently, exactly matching the experimental histograms [6]. In the case of a multimedia source it must be possible to "x the PMF of each compound monomedia stream and the CCS between these monomedia streams, again exactly matching the experimental histograms. Whereas it is not generally a problem to match the experimental histograms of the PMF, it is di$cult to match those of the ACS or, worse still, the CCS, and at the same time preserve simplicity of implementation [11,14,16]. In order to generate random variables with an arbitrary PMF and ACS the transform-expand-sample (TES) method [8,18], the method based on autoregressive-to-anything (ARTA) processes [4], and the method based on normal to anything (NORTA) distribution [5] can be used. However, although these methods have a wide range of application they have not yet been extended to generate mutually correlated random processes. Indeed, there are no methods in scienti"c
225
literature whereby random processes can be generated with an arbitrary PMF and CCS. This paper proposes a novel method which allows the generation of two correlated sequences of discrete-time and discrete-state random variables. The two sets of crosscorrelated variables are generated starting from three sets of independent and identically distributed discrete random variables, two of which are correlated only when the lag is null. By "xing the PMF of the "rst two sets one obtains two sets of discrete random variables with given PMFs; by "xing the PMF of the third set one obtains a given CCS. The PMFs of the variables generated exactly matches any experimental histogram of two monomedia streams. The CCS between the two processes generated exactly matches the experimental histogram of the CCS when it has a symmetry axis and the "rst and second derivative are of a constant but opposite sign; otherwise it approximates the experimental histogram with a broken line. The ACS of each of these two generated processes has a decay similar to that of the CCS; so it can decay more slowly than an exponential one and is therefore able to take long-term intramedia dependences into account when necessary. The proposed generation algorithm is very simple to implement, and its complexity does not depend on the number of random variables which are generated. Referring to the accurate taxonomy for input models given in [15], the proposed method can be classi"ed as a discrete-state, discrete-time and stationary model. The main contributions of the paper are: f de"nition of an algorithm for the fast generation of two sets of random variables correlated with each other; f a method for simulation-based modelling of the intermedia relationships in multimedia streams. The paper is organized as follows. The next section introduces some basic de"nitions of the crosscorrelation sequences of random processes of real values. In Section 3 the method for the generation of crosscorrelated random variables is illustrated and their statistical properties are shown. In Section 4 the author indicates how the proposed method can be applied, and a numerical example is shown with reference to a multimedia source
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composed of a video stream and a data stream. In Section 5 the author's conclusions are drawn and possible further developments of the work are pointed out. Finally, Appendix A gives proof of some of the relations used in the paper. 2. Brief remarks on crosscorrelation sequences Let X and > , with t"0,$1,$2,2, be two t t time-discrete random processes of real values, that is, two ordered sets of time-discrete and real random variables. Let us assume that these processes are jointly stationary (in the wide sense) [22] and that their means and variances are "nite. The CCS between X and > is [20] t t E[X ) > ]!m ) m t t`q X Y, U (q)" (1) XY p )p X Y where E[)] indicates the expected operator, m "E[X ] and p2 are the mean and the variance X X t of X , respectively, and m "E[> ] and p2 are the Y t Y t mean and the variance of > , respectively. t The CCS de"ned in (1) is sometimes called a normalized crosscorrelation sequence (or crosscorrelation coezcient). It has the following properties: f U (q)"U (!q), XY YX f 0)DU (q)D)1 for any q value, XY f lim U (q)"0. q?= XY The CCS is a measure of the dependence between the values of the random processes X and > at t t di!erent times, separated by a distance called a lag q, i.e. a measure of the extent to which the time series X and > are similar to each other. If, for t t a given value of q, U (q)"0, X and > values XY t t with a time distance of q between them are uncorrelated, that is, from observation of X it is not easy t to predict the evolution of > after a time interval t equal to q. If, on the other hand, U (q)'0, it means XY that samples of > that have a time distance of t q from X tend to vary in the same linear direction. t Conversely, if U (q)(0, samples of > at a distance XY t of q from X tend to vary in the opposite linear t direction. The amplitude of DU (q)D is a measure of XY the extent of the linear dependence between the values of the two random processes. The two processes X and > are said to be t t uncorrelated if U (q)"0 for any q. In this case XY
E[X ) > ]"m ) m for any q. If the two prot t`q X Y cesses are statistically independent they are uncorrelated. Maximum (positive or negative) correlation between two random processes is when they are linked by a linear relation. If, in fact, there exists a q6 such that > 6 "K #K ) X , where K and t`q 1 2 t 1 K are constant values, then U (q6)"1 if K '0, 2 XY 2 and U (q6)"!1 if K (0. XY 2 3. Generation of two correlated random processes Let us consider three discrete random processes, that is, three ordered sets of quantized time-discrete random variables of real values Ma N, Mb N and Mh N. n n n Let us assume that: 1. all three processes are wide-sense stationary and ergodic; 2. the processes Ma N and Mb N are statistically inden n pendent of the process Mh N; n 3. each of the three processes Ma N, Mb N and Mh N is n n n an independent identically distributed random process, that is, it consists of a sequence of independent identically distributed random variables; 4. the processes Ma N and Mb N are correlated only n n when the lag is null, that is, E[a ) b ]" n n`m E[a ] ) E[b ]"m ) m if and only if mO0; n n a b 5. the values taken by a belong to a "nite set of n real values MAN"MA , A ,2, A N; likewise, the 1 2 M values taken by b belong to a "nite set of real n values MBN"MB , B ,2, B N and the values 1 2 L taken by h belong to a "nite set of positive n integers MHN"M1,2,2, NN. Starting from these three processes another set of three processes Mc N, Md N and Me N are built, which t t t will henceforward be referred to as `compound processesa. The "rst two are built by repeating the generic variable a and the generic variable b , n n respectively, a number of times equal to the variable h . These two compound processes are thus n formed by the ordered sequences Mc N"M2,a ,2, a , a ,2, a ,2, a , 2, a ,2N, 1 1 hgigj 2 2 n n t hgigj hgigj h1 5*.%4
h2 5*.%4
hn 5*.%4
h2 5*.%4
hn 5*.%4
Md N"M2, b ,2, b , b ,2, b ,2, b ,2, b ,2N. 1 1 hgigj 2 2 n n t hgigj hgigj h1 5*.%4
A. La Corte / Signal Processing 79 (1999) 223}234
The third compound process, Me N, is simply obt tained by shifting Md N by S samples, that is, t Me N"Md N. (2) t t`S The three compound processes are therefore discrete-state processes which change at discrete points in time. The state of these processes is based on Ma N or Mb N; the time they remain in this state, n n on the other hand, is based on Mh N. n As Ma N, Mb N and Mh N are wide-sense stationary n n n processes, Mc N, Md N and Me N are also wide-sense t t t stationary processes. The statistical properties of the compound processes are linked to the PMFs of Ma N, Mb N and n n Mh N. In fact, as demonstrated in Appendix A, the n PMF of Mc N is equal to that of Ma N, and the PMFs t n of Md N and Me N are equal to that of Mb N, that is, t t n p t (x)"p n (x), (3) c a (4) p t (x)"p t (x)"p n (x). e b d Moreover, again as demonstrated in Appendix A, the CCSs between the compound processes are linked to the PMF of Mh N by the relations n U (q)"U (q) cd dc W, q"0, " (5) W ) (1! 1h +@q@~1[1!P n (i)]), DqD*0, m i/0 h and
G
G
W, q"S, U (q)" ec W ) (1! 1h +@q~S@~1[1!P n (i)]), DqDOS, m i/0 h (6) where P n (i)"+i p n ( j) is the probability that j/0 h h h is less or equal to i, and n E[a ) b ]!m ) m n n a b. W" (7) p )p a b From relations (3) and (4) it can be seen that the PMF of Mc N only depends on the PMF of Ma N, t n while the PMFs of Md N and Me N only depend on the t t PMF of Mb N. Hence, the mean, mean quadratic and n variance values of Mc N are obviously equal to those t of Ma N, and those of Md N and Me N are equal to those n t t of Mb N. n As regards the correlation sequences, from relations (5) and (6) it can be seen that the decay of the
227
Table 1 Properties of the CCS between the compound processes Me N t and Mc N t U (q)"0 ec 0)DU (q)D)DU (S)D"DWD ec ec U (S#q)"U (S!q) ec ec 0)U (q))W if E[a ) b ]*m ) m ec n n a b W)U (q))0 if E[a ) b ])m ) m ec n n a b W U (q)!U (q#1)" h [1!P n (q!S)] m h ec ec U (q)!U (q!1)" Wh [1!P n (Dq!SD)] ec ec m h [U (q#1)!U (q)] ec ec W ![U (q)!U (q!1)]" h p n (q!S) ec ec m h (9) [U (q!1)!U (q)] ec ec ![U (q)!U (q#1)]" Wh p n (Dq!SD) m h ec ec
(1) (2) (3) (4) (5) (6) (7) (8)
for for for for for for for
DS!qD*N any q any q any q any q q*S#1 q)S!1
for q*S#1 for q)S!1
CCSs between the compound processes only depends on the PMF of Mh N. So, as Ma N and Mb N are n n n statistically independent of Mh N, the decay of the n CCSs between Mc N and Md N or between Mc N and t t t Me N can be "xed regardless of the PMFs of Ma N and t n Mb N. n The CCS between the compound processes Me N t and Mc N, U (q), has some interesting properties, t ec which are listed in Table 1 and can be derived from relation (6) and the properties of crosscorrelation sequences given in Section 2. Observing the table some remarks can be made: f U (q) is a sequence which has at most (2N!1) ec non-null samples. By increasing the size of the set MHN, namely N, two random processes can therefore be generated which have a non-null correlation up to any lag q"N#S!1; f U (q) is symmetrical to the lag S with which ec DU (q)D is maximum; ec f the sign of U (q) depends on the sign of U (S); ec ec f the sign of the "rst and second derivatives of U (q) therefore depends on the sign of W and the ec value of q with respect to S. When, in fact, q*S, if W'0, the CCS never increases as the absolute value of q increases and its second derivative is always non-negative; if, on the other hand, W(0, the CCS never decreases as the absolute value of q increases and its second derivative is always non-negative. Bearing in mind that the CCS is symmetrical to the lag S, the opposite holds for the "rst and second derivatives of the CCS when q)S.
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Similar properties to those listed in Table 1 and similar remarks hold for the CCS between the compound processes Md N and Mc N or Me N, by taking t t t into account how these processes are constructed and, again, the general properties of the CCS. Although the autocorrelation sequence lies beyond the scope of the paper, let us "nally note that the autocorrelation sequences of the compound processes have the same decay as the CCS. In [2] it is, in fact, demonstrated that E[c ) c ]!m2 c t t`q U (q)" cc p2 c 1 if q"0, " (8) 1! 1h +@q@~1[1!P n (i)] otherwise. m i/0 h By considering how the processes d and e are t t constructed, it can easily be derived that the autocorrelation sequences of Mc N, Md N and Me N are t t t equal, that is, U (q)"U (q)"U (q). dd ee cc
G
4. Application of the proposed method In order to explain how the proposed methodology for the generation of crosscorrelated random processes can be applied, let us now see how it can be applied to model a multimedia tra$c source. As mentioned in the introduction, in a communication environment a multimedia source is usually considered as an aggregate of two or more monomedia streams correlated with each other: the correlation among the component streams lies in the fact that the behavior of each stream depends on that of the others. More speci"cally, there is generally one monomedia stream (the master) which is completely independent of the others, and others (the slaves) which are dependent only on the "rst [23,13]. As an example, in a slide show with voice commentary, the change of slides is temporally related to the speaker's comments: the master is the audio stream, whereas the slave is the still picture stream. Modelling a multimedia source thus requires representation of the intermedia relationships between the master and slave streams. Let us assume that we have empirical measurements of the statistics of the multimedia source, more speci"cally the histogram of the empirical
probabilities p(%) (x) and p(%) (x) for all the values S-!7% M!45%3 that the output of the master and of the slave, respectively, can take and the empirical CCS between the slave stream and the master stream, (q), for q"[q ,2,!1,0,1,2,q ]. U(%) S-!7%,M!45%3 MIN MAX As said in Section 1, modelling a source by means of a simulation approach and a stochastic process means generating two stochastic processes Mc N and t Me N such that t p t (x)"p(%) (x), c M!45%3 (9) p t (x)"p(%) (x), e S-!7% (q). U (q)"U(%) S-!7%,M!45%3 ec To generate Mc N and Me N as de"ned in the previous t t section, it is necessary to determine the PMFs of Ma N, Mb N and Mh N and to generate them in such n n n a way that relations (9) are satis"ed. From relations (3), (4) and (7) and the "rst two equations in relation (9), the PMFs of Ma N and Mb N n n have to be equal to the corresponding experimental ones for the source, provided that Ma N and Mb N are n n correlated for a zero lag. That is, p n (x)"p(%) (x), a M!45%3 p n (x)"p(%) (x) b S-!7% with the condition E[a ) b ]!m ) m n n`m a b p )p a b (S) if m"0, W"U(e) S-!7%,M!45%3 " 0 if mO0,
G
(10) (11)
(12)
where &S' is the lag where the absolute value of (q) is maximum. U(%) S-!7%,M!45%3 Generation of Ma N and Mb N, once the corren n sponding PMFs are known, is easy to achieve when the two processes have to be uncorrelated for any lag [3,11,16]. But to apply the proposed method relation (12) has to be satis"ed, and the generation of a given b therefore has to depend on the corren sponding a . To this end the inversion method [3] n can be used, but the same random variable ; uniformly distributed in the interval [0,1] can be used to generate both a and b . Fig. 1 gives a pseudon n code algorithm based on the inversion method with which Ma N and Mb N are generated so as to satisfy n n
A. La Corte / Signal Processing 79 (1999) 223}234
229
p"W/W , and so as to be uncorrelated with MAX a probability of (1!p). The same applies when W(0. The minimum value is obtained for E[a ) b ] when, given a cern n tain ; value to generate a given a , (1!;) is used n to generate the corresponding value of b . In this n case we get
P
E[a ) b ] " n n MIN
1
; ) P~1 (;) ) P~1 (1!;) d;, (15) an bn 0 and correspondingly we have E[a ) b ] !m ) m n n MIN a b. W " (16) MIN p )p a b As regards the problem of generating Mh N, from n relations (6) and (7) in Table 1 we obtain
Fig. 1. Pseudo-code for generation of the processes Ma N and n Mb N which have given probability functions } P n (;) and P n (;) n a b }, are uncorrelated for a lag qO0 and are correlated for q"0 in such a way that relation (12) is satis"ed.
relations (10)}(12) and the hypothesis on processes Ma N and Mb N listed in Section 3. n n In order to explain the algorithm, let us assume that W'0. The maximum value for E[a ) b ] n n } corresponding to maximum correlation between Ma N and Mb N } is obtained when the same ; is used n n to generate the values of both Ma N and Mb N. In this n n case we get
P
E[a ) b ] " n n MAX
1
; ) P~1 (;) ) P~1 (;) d; an bn 0 and correspondingly we have
(13)
E[a ) b ] !m ) m n n MAX a b. (14) W " MAX p )p a b So to generate Ma N and Mb N in such a way that n n 0)W)W , they must be generated so as to MAX have maximum correlation with a probability of
U (i#S)!U (i#S#1) ec , (17) P n (i)"1! ec h U (S)!U (S#1) ec ec where &S' is the lag value with which the absolute value of U (q) is maximum. It is easy to verify that ec the probabilities obtained in relation (17) meet the probability congruence conditions [20] if the CCS satis"es relations (6)}(9) listed in Table 1, i.e. if U (q) is symmetrical to the lag S and, when q*S, it ec is a non-increasing sequence with downward convexity if W'0, or a non-decreasing sequence with upward convexity if W(0. By imposing the third equation in relation (9) and replacing in (17) we obtain P n (i) h
(i#S#1) (i#S)!U(%) U(%) S-!7%,M!45%3 . "1! S-!7%,M!45%3 (S#1) (S)!U(%) U(%) S-!7%,M!45%3 S-!7%,M!45%3 (18)
The experimental values of the crosscorrelation sequence do not usually meet conditions (6)}(9) listed in Table 1, under which the model proposed can be applied. To apply the proposed methodology, it is therefore "rst necessary to approximate the experimental sequence with a sequence which satis"es the above conditions. For the simplicity of the model not to be thwarted by the computational complexity needed to "nd the curve which best approximates the experimental crosscorrelation sequence, the approximation operation should preferably be very simple. It should also lead to a set
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Fig. 2. Experimental probability histograms of the multimedia source modelled in the numerical example and experimental crosscorrelation sequence between the video and data streams: (a) experimental PMF of the master stream (video source); (b) experimental PMF of the slave stream (data source); (c) experimental CCS which satis"es relations (6)}(9) in Table 1; (d) experimental CCS which does not satisfy relations (6)}(9) in Table 1 and approximation by means of a broken line.
MHN in which only a few elements have a non-null probability, so that the length of the simulation is not made excessive by the need to generate rare events. An extremely simple way to approximate the experimental crosscorrelation sequence is to do so with a broken line. For relations (6)}(9) listed in Table 1 to be satis"ed, the segments of this line have to be chosen in such a way that, when q*S, the slope of each segment is less than that of the previous one; when q)S, obviously, the symmetrical segments have to be chosen. Once the CCS has been approximated in this way, relation (18) is applied using the approximated values in order to obtain the PMF of h . n In order to facilitate understanding of the application of the proposed method, a numerical example is presented below. Let us assume that the
multimedia source to be modelled comprises two monomedia streams: the "rst, the master stream, is generated by a video source, which generates 30 frames per second, with a conditional replenishment interframe coding scheme. The second, the slave stream, is generated by a data source which is temporally synchronized with the master stream, and whose emission process depends on that of the master stream. Let us also assume that, from experimental measurements, calling the bit-rate durfor the master and s(%) ing the tth frame s(%) 4-!7%,t .!45%3,t and the slave stream, respectively, it was observed that: f the steady-state probability distribution of has a bell-like shape which can be aps(%) .!45%3,t proximated with a Gaussian probability density function (see Fig. 2(a)), with a mean m(%) " .!45%3
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130 000 bit/frame, and a standard deviation p(%) "57 500 bit/frame [17], which is then .!45%3 sampled and quantized at intervals equal to 500, and bounded by maximum and minimum values of s "352 500 bit/frame and s "20 000 .!9 .*/ bit/frame, respectively; f the steady-state probability distribution of is as shown in Fig. 2(b). The mean of the s(%) 4-!7%,t slave stream is m(%) "1986.7 bit/frame and the 4-!7% standard deviation is p(%) "911.9 bit/frame; 4-!7% f the CCS between the slave and the master stream has a shape which can be approximated with an exponential sequence (see Fig. 2(c)) according to the law U(e) (q)"0.7838 ) e~0.13>@q@ S-!7%,M!45%3 for q"!30,2,30. Let us note that although the graphs in Figs. 2(a) and (c) refer to the probability distribution and crosscorrelation sequence which approximate the experimental values for the video source, we will refer to them as experimental values so as not to create any confusion. The method presented to model the source entails the following steps: 1. veri"cation of the congruence of relation (17) on the experimental CCS values. In this speci"c case they are veri"ed and we also have S"0 and W"0.7838; 2. calculation of W in accordance with relation MAX (14). In this example W "0.9169; MAX 3. generation of the variables Ma N and Mb N accordn n ing to the algorithm in Fig. 1; 4. generation of the variables Mh N with a cumulatn ive probability distribution in agreement with relation (17); 5. generation of the compound processes Mc N and t Me N. t Various simulations were performed during which the variables Ma N, Mb N and Mh N were generated in n n n agreement with the probability distributions calculated above, the processes Mc N and Me N were t t constructed as outlined in Section 3, and the PMFs and CCS were calculated and compared with the expected values given in Figs. 2(a)}(c). More speci"cally, in order to determine the goodness of "t between the experimental histograms and the evid-
231
ence contained in the simulation results, we used the s2 test [9]. Considering a single run simulation of the processes Mc N and Me N corresponding to the t t generation of 5]105 samples of variables Ma N, Mb N n n and Mh N, and discarding the "rst 29 samples n in order to have a steady-state condition, the simulation results approximate the expected results with a 95% con"dence interval on the true value. Let us now assume that the experimental CCS is not the one in Fig. 2(c), but the one shown in Fig. 2(d). As can be seen, the CCS does not satisfy the congruence relations for the PMF of Mh N, so to apply n the proposed methodology it is "rst necessary to approximate the experimental CCS with a sequence that is symmetrical to the lag q"15 and, with q'15, is a non-decreasing sequence with upward convexity. In the example at hand, a simple way which also leads to a set MHN in which only a few elements have a non-null probability, is to approximate the experimental CCS sequence with a line comprising eight segments (see Fig. 2(d)), the ends of which are the coordinate points (!21,0), (!20,0), (!13,!0.0630), (!4, !0.4275), (0,!0.7407), (4,!0.4275), (13, !0.0630), (20,0), (21,0). Obviously the last segment is horizontal. Calculating the probability distribution of Mh N n according to relation (18) and using approximation by a broken line, we obtain a non-null PMF value corresponding to each point of intersection of the various segments. In the case in point, in fact, the values of MHN which have a non-null probability are H , H and H . More speci"cally, 4 13 20 we get p (4)"0.4828, p (13)"0.4023 and hn hn p n (20)"0.1149. h Having thus satis"ed all the hypotheses concerning the applicability of the method proposed, we can now generate the processes Ma N, Mb N and Mh N, n n n and therefore the compound process Mc N, Me N as in n n the previous example.
5. Conclusions In this paper a method which allows the generation of two sets of discrete random variables with given probability mass functions and crosscorrelation sequence has been presented. The method can
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be applied to modelling, by simulation, a multimedia source comprising a master and a slave stream linked by intermedia time relationships and whose experimental PMFs and CCS are known. Although the method only allows the experimental CCS to be matched exactly when the latter possesses certain properties, it has a certain general validity. This is for two reasons: "rst it makes it possible to mimic a CCS which decays more slowly than an exponential one, and so long-term dependences can be taken into account. Secondly, when the condition imposed on the experimental CCS for the method to have validity is not met, it is generally possible to approximate the sequence so as to be able to apply the method. In addition, as compared with the methods generally used to generate autocorrelated variables [4,5,8,18], the method proposed is simpler and thus faster in both the model "tting and the variable generation phases, and its complexity does not depend on the number of variables generated. It does, however, require a high computational load when the discretization time is excessively low, i.e. when continuous random processes are to be represented. The author is currently working on how to modify the method to generate the compound processes so that the ACS decay is independent of that of the CCS, and so that the ACS and CCS can also represent cyclostationary components. Appendix A A.1. Computation of the PMF of compound processes Let p (x) and p (x) be the probabilities that Mc N ct an t and Ma N are equal to x, respectively. Referring "rst n to the total probability theorem and then to Bayes' rule, exploiting the independence between Mc N and t Ma N we have n N p t (x)" + p(a "xDh "i) ) p n (i) c n n h i/1 N " + p(h "iDa "x) ) p n (x) n n a i/1 N (A.1) " + p n (x) ) p n (i)"p n (x). h a a i/1
A.2. Computation of the CCS between the compound processes Let U(%)(q) be the experimental crosscorrelation cd sequence between Mc N and Md N, obtained considert t ing k extractions from Ma N, Mb N and Mh N. As both n n n Mc N and Md N are wide-sense stationary and ergotic t t random processes, the temporal averages coincide with the ensemble averages. So, according to how these two processes are constructed and the properties the processes Ma N, Mb N and Mh N possess, n n n we have U (q)" lim U(%) cd cd k?= " lim k?= " lim k?=
A A
E(%)[c ) d ]!m(%) ) m(%) d c t t`q p(%) ) p(%) d c
B B B B
+k (h !DqD) ) a ) b i/1,hi ;@q@ i i i p(%) ) p(%) ) (+k h !DqD) d c i/1 i
A A A
DqD ) a ) b +k~l i i/1,h ;@q@ i r # lim h !DqD) p(%) ) p(%) ) (+k k?= c d i/1 i DqD ) a ) b +k~l i i/1,h x@q@ i r # lim p(%) ) p(%) ) (+k h !DqD) d k?= c i/1 i
B
m% ) m% c d , # lim p(%) ) p(%) d k?= c
(A.2)
where `la is a constant which does not depend on `ka, m% and m% is the experimental mean of Mc N and d t c Md N, respectively, p(%) and p(%) is the experimental d c t standard deviation of Mc N and Md N, respectively, t t and r'i, r(i or r"i, respectively, when q'0 q(0 or q"0. In relation (A.2), with k tending to in"nity, the upper part of the numerator summations can be approximated to k and the denominator term q can be neglected. We can therefore write U (q)" lim cd k?=
+k (h !DqD) ) a ) b i i i/1,hi ;@q@ i p(%) ) p(%) ) (+k h ) d c i/1 i
A
# lim k?=
A
+k DqD ) a ) b i r i/1,hi ;@q@ p(%) ) p(%) ) (+k h ) d c i/1 i
B
B
A. La Corte / Signal Processing 79 (1999) 223}234
A A
B
+k DqD ) a ) b i r i/1,hi x@q@ p(%) ) p(%) ) (+k h ) c d i/1 i m% ) m% c d . # lim (A.3) p(%) ) p(%) d k?= c Having set a certain value for q, we divide the set MHN into two subsets MH N and MH N, where MH N is L G L the subset of elements of MHN less than or equal to the absolute value of q, and MH N is the subset of G elements of MHN greater than the absolute value of q. Note that when 0(DqD(N then MH N" L M1,2,DqDN and MH N"MDqD#1,2, NN. If DqD*N G then MH N"M0N and MH N"MHN. G L Let us therefore consider the "rst of the four limits indicated in relation (A.3). As both a n and b are identically distributed variables inden pendent of h , and bearing in mind that n E[a ) b ]"E[a ) b ]OE[a ] ) E[b ] if m"0 n n`m n n n n and E[a ) b ]"E[a ] ) E[b ] if mO0, we get n n`m n n +k (h !DqD) ) a ) b i i i/1,hi ;@q@ i lim p(%) ) p(%)(+k h ) d c k?= i/1 i + G [(i!DqD) ) + i (a ) b )D ] i|H D@ n n i " lim p(%) ) p(%)(+k h ) d c k?= i/1 i + [(i!DqD) ) N ) E(%)[a ) b ]] i n n i|HG " lim p(%) ) p(%) ) k ) m(%) h d c k?= E[a ) b ] n n + (i!DqD) ) p (i), " (A.4) hi p )p )m c d h i|HG where (a ) b )D indicates the product of the pair of n n i values a and b when h "i, and DD indicates the n n n i set of pairs of values a and b corresponding to n n h "i. n Likewise, for the other terms in relation (21), we get # lim k?=
A
B
B
A A
+k DqD ) a ) b i r i/1,hi ;@q@ p(%) ) p(%)(+k h ) d c i/1 i DqD ) + G [+ i (a ) b ) D ] i|H D{@ n r rEn i " lim p(%) ) p(%)(k ) m(%)) c d h k?= E[a ) b ] ) DqD ) + G [p n (i)] n r i|H h " p )p )m c d h [p (i)] m ) m ) DqD ) + i|HG hn " c d p )p )m c d h
lim k?=
A
A
B
B
B
B (A.5)
233
and
A
B
h )a )b +k i/1,hi :@q@ i i r p(%) ) p(%)(+k h ) d c i/1 i + [i ) + i (a ) b ) D ] D{@ n r rEn i i|HL " lim p(%) ) p(%)(k ) m(%)) c d h k?= [i ) p n (i)] E[a ) b ] ) + h n r i|HL " p )p )m c d h [i ) p n (i)] m )m )+ h " c d i|HL , (A.6) p )p )m c d h where (a ) b ) D indicates the product of a multin r rEn i n plied by the value of the sequence Mb N which is at r a distance q from the corresponding a , when n h "i, and D@D indicates the set of pairs of values n i a and b corresponding to h when h "i. n r i i Therefore substituting (A.4)}(A.6) in (A.3) we get
lim k?=
A
B
E[a ) b ] ) + [(i!DqD) ) p n (i)] n n h i|HG U (q)" cd p )p )m c d h m )m c d ) DqD ) + [p (i)] # hn p )p )m c d h i|HG
A
B
m )m # + [i ) p n (i)] ! c d h p )p c d i|HL E[a ) b ] m ) m 1 n n ! c d " ) m p )p m h c d h
A
B
) + [(i!DqD) ) p n (i)]. (A.7) h i|HG If q"0, as H ,H, relation (A.7) becomes G E[a ) b ]!m ) m n n c d U (0)" cd p )p c d E[a ) b ]!m ) m n n a b "W. " (A.8) p )p a b The "rst relation in (5) is thus demonstrated. If, instead, qO0, indicating with P n (i) the probh ability that h is less than or equal to `ia, and as n @q@~1 + (1!P (i)) hn i/0 " H #1!p (H )!p (H ) 1 hn 1 hn 2 # 2!p (H )!p (H ) hn q~2 hn q~1
234
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" H #DqD!H ! + (p n (H ) ) [DqD!H ]) 1 1 h i i Hi |HL " DqD! + (p n (H ) ) [DqD!H ]), h i i Hi |HL we get 1 ) + [(i!DqD) ) p n (i)] h m h i|HG 1 @q@~1 "1! ) + (1!P (i)). hn m h i/0 Substituting the latter in (A.7) we get
A
B
1 @q@~1 U (q)"U (0) ) 1! ) + (1!P (i)) hn cd cd m h i/0 if DqDO0.
Thus, the second relation in (5) is also demonstrated. As Me N"Md N we get t t~S E[e ) c ]!m ) m t t`q c d U (q)" ec p )p c d E[d ) c ]!m ) m t~S t`q c d " p )p c d "U (q#S). (A.9) cd Substituting (A.9) in (5) we get (6). References [1] A. Adas, Tra$c models in broadband networks, Commun. Mag. 35 (7) (July 1997) 82}90. [2] M. Andronico, S. Casale, A. La Corte, Generation of random variables for modelling VBR multimedia sources, Telecommun. Systems } Baltzer Sci. 9 (1) (January 1998) 1}21. [3] P. Bratley, B.L. Fox, L.E. Schrage, A Guide to Simulation, 2nd Edition, Springer, New York, 1987. [4] M.C. Cario, B.L. Nelson, Numerical methods for "tting and simulating autoregressive-to-anything processes, INFORMS J. Comput. 10 (1997) 72}81. [5] M.C. Cario, B.L. Nelson, Modeling and generating random vectors with arbitrary marginal distributions and correlation matrix. http://www.iems.nwu.edu/nelsonb/ norta4.ps, April 1997.
[6] V.S. Frost, B. Melamed, Tra$c modeling for telecommunication networks, Commun. Mag. 32 (3) (March 1994) 70}81. [7] I.W. Habib, T.N. Saadawi, Multimedia tra$c characteristics in broadband networks, IEEE Commun. Mag. 30 (7) (July 1992) 48}54. [8] D.L. Jagerman, B. Melamed, The transition and autocorrelation structure of tes processes part ii: Special cases, Stochast. Models 8 (3) (1992) 499}527. [9] R. Jain, The Art of Computer Systems Performance Analysis, Wiley, New York, 1991. [10] P.R. Jelenkovic, A.A. Lazar, N. Semret, The e!ect of multiple time scales and subexponentiality in mpeg video stream on queueing behaviour, IEEE J. Selected Areas Commun. 15 (6) (August 1997) 1052}1071. [11] G.E. Johnson, Construction of particular random processes, Proc. IEEE 82 (2) (February 1994) 270}280. [12] A. La Corte, A. Lombardo, S. Palazzo, G. Schembra, Control of perceived quality of service in multimedia retrieval services: prediction-based mechanism vs. compensation bu!ers, Multimedia Systems 6 (2) (March 1998) 102}112. [13] A. La Corte, A. Lombardo, G. Schembra, An analytical paradigm to calculate multiplexer performance in an ATM multimedia environment, Comput. Networks ISDN Systems 29 (16) (December 1997) 1881}1900. [14] A.M. Law, W.D. Kelton, Simulation Modeling and Analysis, 2nd Edition, McGraw-Hill, New York, 1991. [15] L.M. Leemis, Input modeling for discrete-event simulation, In: Proceedings of Winter Simulation Conference, Orlando (USA), December 1994. [16] A. Leon-Garcia, Probability and Random Processes for Electrical Engineering, 2nd Edition, Addison-Wesley, New York, 1994. [17] B. Maglaris, D. Anastassiou, P. Sen, G. Karlsson, J.D. Robbins, Performance models of statistical multiplexing in packet video communications, IEEE Trans. Commun. 36 (7) (July 1988) 834}844. [18] B. Melamed, An overview of tes processes and modeling methodology, Lecture Notes Comput. Sci. 729 (1993) 359}393. [19] I. Nikolaidis, I.F. Akyildiz, Source characterization and statistical multiplexing in ATM networks, Technical Report GIT-CC-92/94, Georgia Institute of Technology, Atlanta, 1994. [20] M.B. Priestley, Spectral Analysis and Time Series, Vols. 1 and 2, Academic Press Inc., London, 1981. [21] R. Steinmetz, K. Nahrstedt, Multimedia } Computing, Communications and Applications, Prentice-Hall, New York, 1995. [22] C.W. Therrien, Discrete Random Signals and Statistical Signal Processing, Prentice-Hall, New York, 1992. [23] P. Venkat Rangan, H.M. Vin, S. Ramanathan, Designing an on-demand multimedia service, IEEE Commun. Mag. 3 (7) (July 1992) 56}64.
Generation of crosscorrelated random processes Aurelio La Corte* Istituto di Informatica e Telecomunicazioni, University of Catania, Viale Andrea Doria 6, 95125 Catania, Italy Received 3 November 1998; received in revised form 29 June 1999
Abstract In this paper the author proposes a method which will allow simple and fast generation of two sets of random variables with given probability mass functions (PMFs) and crosscorrelation sequences (CCS). It is based on the composition of three sets of independent and identically distributed discrete random variables, two of which are correlated with each other only when the lag is null. The method is applicable in cases where, once experimental measurements have established the statistical characteristics of two stationary random processes in terms of PMFs and CCS, it is necessary to implement a simulation model, that is, it is necessary to generate random discrete variables whose PMFs and CCS match the empirical histograms. A case study is given to apply the method proposed to modelling the relationships between the monomedia streams making up a multimedia stream. ( 1999 Published by Elsevier Science B.V. All rights reserved. Zusammenfassung In dieser Arbeit wird eine Methode vorgeschlagen, die eine einfache und schnelle Erzeugung zweier Mengen von Zufallsvariablen mit gegebenen Wahrscheinlichkeitsverteilungsfunktionen (PMFs) und Kreuzkorrelationsfolgen (CCS) erlaubt. Diese Methode beruht auf der Zusammensetzung dreier Mengen von statistisch unabhaK ngigen und identisch verteilten diskreten Zufallsvariablen, von denen zwei nur dann miteinander korreliert sind, wenn die zeitliche Verschiebung null ist. Die Methode ist anwendbar in jenen FaK llen, wo } nachdem durch experimentelle Messungen die statistischen Eigenschaften zweier stationaK rer Zufallsprozesse hinsichtlich PMFs und CCS ermittelt wurden } es notwendig ist, ein Simulationsmodell zu implementieren, d.h. zufaK llige diskrete Variablen zu erzeugen, deren PMFs und CCS mit den empirischen Histogrammen uK bereinstimmen. In einer Fallstudie wird die vorgeschlagene Methode auf die Modellierung jener Beziehungen angewandt, die zwischen den einen multimedialen Strom bildenden monomedialen StroK men bestehen. ( 1999 Published by Elsevier Science B.V. All rights reserved. Re2 sume2 L'auteur propose dans cet article une meH thode permettant la geH neH ration simple et rapide de deux ensembles de variables aleH atoires ayant des fonctions de probabiliteH massiques (PMF) et des seH quences d'inter-correH lation (CCS) donneH es. Cette meH thode est baseH e sur la composition de trois ensembles de variables aleH atoires indeH pendantes et identiquement distribueH es, dont deux sont correH leH s l'un avec l'autre uniquement lorsque l'eH cart temporel est nul. Cette meH thode est applicable dans les cas ou`, une fois que les mesures expeH rimentales ont eH tabli les caracteH ristiques statistiques de deux processus aleH atoires stationnaires en termes de PMF et de CCS, il est neH cessaire d'implanter un mode`le de simulation, c'est a` dire qu'il est neH cessaire de geH neH rer des variables aleH atoires discre`tes dont les PMF et les CCS correspondent aux histogrammes empiriques. Une eH tude speH ci"que est preH senteH e, dans laquelle la meH thode proposeH e est
* Corresponding author. Tel.: #39-0957382356; fax: #39-095338280. E-mail address: [email protected] (A. La Corte) 0165-1684/99/$ - see front matter ( 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 1 6 5 - 1 6 8 4 ( 9 9 ) 0 0 0 9 7 - 3
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A. La Corte / Signal Processing 79 (1999) 223}234
Nomenclature PMF ACS CCS Ma N, Mb N, Mh N n n n (a ) b )D n n i (a ) b ) D n r rEn i MAN"MA , A ,2, A N 1 2 M MBN"MB , B ,2, B N 1 2 L Mc N, Md N, Me N t t t DD i D@D i E[ ) ] MHN"M1,2,2, NN MH N L MH N G m "E[X ] X t MX N, MX N n t p2 X p t (x) X p(%)t (x) X P t (x) X (;) P~1 Xt U (q) XX U (q) XY U(%) (q) XY W ,W MAX MIN W S ;, ;@, ;A
probability mass function autocorrelation sequence crosscorrelation sequence component random processes product of the pair of values a and b when h "i n n n product of a and the value of the sequence Mb N which is at a distance of n n q from the corresponding a , when h "i n n set of values taken by a n set of values taken by b n compound processes set of pairs of values a and b corresponding to h "i. n n n set of pairs of values a and b corresponding to h "i. n r n expected operator set of positive integer values taken by h n subset of elements of MHN less than or equal to DqD subset of elements of MHN greater than DqD; mean of X t discrete, wide-sense stationary and ergodic random processes variance of X t PMF of MX N t experimental histogram of PMF of MX N t cumulative probability function of X t inverse function of P t (x) X ACS of X t CCS between X and > t t experimental CCS between X and > t t maximum and minimum possible values of U (q) XY E !m ) m ab a b p )p a b temporal shift of Md N t random variables uniformly distributed in [0,1]
appliqueH e a` la modeH lisation des relations entre des #ux monomeH dia concourant a` un #ux multi-meH dia. ( 1999 Published by Elsevier Science B.V. All rights reserved. Keywords: Crosscorrelation; Random number generation; Simulation; Modelling; Multimedia sources
1. Introduction The design and testing of communication systems often requires the availability of random number generators. This, for example, is the case
when a designer of resource allocation and management strategies has the problem of evaluating the e!ectiveness of a certain strategy by simulation: he needs a simulation model which can capture the basic properties of the tra$c sources through a
A. La Corte / Signal Processing 79 (1999) 223}234
limited number of descriptors, and thus a large number of random variables with statistical properties which match those of the source to be modelled. The statistical properties usually considered for monomedia tra$c sources are the probability mass function (PMF) and the autocorrelation sequence (ACS) [1,6,7,10,19]. In the case of multimedia traf"c sources, the crosscorrelation sequences (CCSs) between compound monomedia streams have to be considered. This is due to the fact that a multimedia source is usually considered as an aggregate of two or more monomedia streams correlated with each other by logical, spatial or temporal relationships [12,21,23]. The correlation among the component monomedia streams lies in the fact that the behavior of each stream (e.g. its interarrival times or emission process) depends on that of the others. As an example, in a slide show with voice commentary, the change of slides is temporally related to the speaker's comments. Taking into consideration the CCS between the monomedia streams it is possible to capture the intermedia relationships and to avoid over-optimistic or pessimistic predictions of erformance measures for queueing systems [13]. In a good method for the generation of random variables, to model a monomedia tra$c source it must be possible to "x their PMF and ACS independently, exactly matching the experimental histograms [6]. In the case of a multimedia source it must be possible to "x the PMF of each compound monomedia stream and the CCS between these monomedia streams, again exactly matching the experimental histograms. Whereas it is not generally a problem to match the experimental histograms of the PMF, it is di$cult to match those of the ACS or, worse still, the CCS, and at the same time preserve simplicity of implementation [11,14,16]. In order to generate random variables with an arbitrary PMF and ACS the transform-expand-sample (TES) method [8,18], the method based on autoregressive-to-anything (ARTA) processes [4], and the method based on normal to anything (NORTA) distribution [5] can be used. However, although these methods have a wide range of application they have not yet been extended to generate mutually correlated random processes. Indeed, there are no methods in scienti"c
225
literature whereby random processes can be generated with an arbitrary PMF and CCS. This paper proposes a novel method which allows the generation of two correlated sequences of discrete-time and discrete-state random variables. The two sets of crosscorrelated variables are generated starting from three sets of independent and identically distributed discrete random variables, two of which are correlated only when the lag is null. By "xing the PMF of the "rst two sets one obtains two sets of discrete random variables with given PMFs; by "xing the PMF of the third set one obtains a given CCS. The PMFs of the variables generated exactly matches any experimental histogram of two monomedia streams. The CCS between the two processes generated exactly matches the experimental histogram of the CCS when it has a symmetry axis and the "rst and second derivative are of a constant but opposite sign; otherwise it approximates the experimental histogram with a broken line. The ACS of each of these two generated processes has a decay similar to that of the CCS; so it can decay more slowly than an exponential one and is therefore able to take long-term intramedia dependences into account when necessary. The proposed generation algorithm is very simple to implement, and its complexity does not depend on the number of random variables which are generated. Referring to the accurate taxonomy for input models given in [15], the proposed method can be classi"ed as a discrete-state, discrete-time and stationary model. The main contributions of the paper are: f de"nition of an algorithm for the fast generation of two sets of random variables correlated with each other; f a method for simulation-based modelling of the intermedia relationships in multimedia streams. The paper is organized as follows. The next section introduces some basic de"nitions of the crosscorrelation sequences of random processes of real values. In Section 3 the method for the generation of crosscorrelated random variables is illustrated and their statistical properties are shown. In Section 4 the author indicates how the proposed method can be applied, and a numerical example is shown with reference to a multimedia source
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composed of a video stream and a data stream. In Section 5 the author's conclusions are drawn and possible further developments of the work are pointed out. Finally, Appendix A gives proof of some of the relations used in the paper. 2. Brief remarks on crosscorrelation sequences Let X and > , with t"0,$1,$2,2, be two t t time-discrete random processes of real values, that is, two ordered sets of time-discrete and real random variables. Let us assume that these processes are jointly stationary (in the wide sense) [22] and that their means and variances are "nite. The CCS between X and > is [20] t t E[X ) > ]!m ) m t t`q X Y, U (q)" (1) XY p )p X Y where E[)] indicates the expected operator, m "E[X ] and p2 are the mean and the variance X X t of X , respectively, and m "E[> ] and p2 are the Y t Y t mean and the variance of > , respectively. t The CCS de"ned in (1) is sometimes called a normalized crosscorrelation sequence (or crosscorrelation coezcient). It has the following properties: f U (q)"U (!q), XY YX f 0)DU (q)D)1 for any q value, XY f lim U (q)"0. q?= XY The CCS is a measure of the dependence between the values of the random processes X and > at t t di!erent times, separated by a distance called a lag q, i.e. a measure of the extent to which the time series X and > are similar to each other. If, for t t a given value of q, U (q)"0, X and > values XY t t with a time distance of q between them are uncorrelated, that is, from observation of X it is not easy t to predict the evolution of > after a time interval t equal to q. If, on the other hand, U (q)'0, it means XY that samples of > that have a time distance of t q from X tend to vary in the same linear direction. t Conversely, if U (q)(0, samples of > at a distance XY t of q from X tend to vary in the opposite linear t direction. The amplitude of DU (q)D is a measure of XY the extent of the linear dependence between the values of the two random processes. The two processes X and > are said to be t t uncorrelated if U (q)"0 for any q. In this case XY
E[X ) > ]"m ) m for any q. If the two prot t`q X Y cesses are statistically independent they are uncorrelated. Maximum (positive or negative) correlation between two random processes is when they are linked by a linear relation. If, in fact, there exists a q6 such that > 6 "K #K ) X , where K and t`q 1 2 t 1 K are constant values, then U (q6)"1 if K '0, 2 XY 2 and U (q6)"!1 if K (0. XY 2 3. Generation of two correlated random processes Let us consider three discrete random processes, that is, three ordered sets of quantized time-discrete random variables of real values Ma N, Mb N and Mh N. n n n Let us assume that: 1. all three processes are wide-sense stationary and ergodic; 2. the processes Ma N and Mb N are statistically inden n pendent of the process Mh N; n 3. each of the three processes Ma N, Mb N and Mh N is n n n an independent identically distributed random process, that is, it consists of a sequence of independent identically distributed random variables; 4. the processes Ma N and Mb N are correlated only n n when the lag is null, that is, E[a ) b ]" n n`m E[a ] ) E[b ]"m ) m if and only if mO0; n n a b 5. the values taken by a belong to a "nite set of n real values MAN"MA , A ,2, A N; likewise, the 1 2 M values taken by b belong to a "nite set of real n values MBN"MB , B ,2, B N and the values 1 2 L taken by h belong to a "nite set of positive n integers MHN"M1,2,2, NN. Starting from these three processes another set of three processes Mc N, Md N and Me N are built, which t t t will henceforward be referred to as `compound processesa. The "rst two are built by repeating the generic variable a and the generic variable b , n n respectively, a number of times equal to the variable h . These two compound processes are thus n formed by the ordered sequences Mc N"M2,a ,2, a , a ,2, a ,2, a , 2, a ,2N, 1 1 hgigj 2 2 n n t hgigj hgigj h1 5*.%4
h2 5*.%4
hn 5*.%4
h2 5*.%4
hn 5*.%4
Md N"M2, b ,2, b , b ,2, b ,2, b ,2, b ,2N. 1 1 hgigj 2 2 n n t hgigj hgigj h1 5*.%4
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The third compound process, Me N, is simply obt tained by shifting Md N by S samples, that is, t Me N"Md N. (2) t t`S The three compound processes are therefore discrete-state processes which change at discrete points in time. The state of these processes is based on Ma N or Mb N; the time they remain in this state, n n on the other hand, is based on Mh N. n As Ma N, Mb N and Mh N are wide-sense stationary n n n processes, Mc N, Md N and Me N are also wide-sense t t t stationary processes. The statistical properties of the compound processes are linked to the PMFs of Ma N, Mb N and n n Mh N. In fact, as demonstrated in Appendix A, the n PMF of Mc N is equal to that of Ma N, and the PMFs t n of Md N and Me N are equal to that of Mb N, that is, t t n p t (x)"p n (x), (3) c a (4) p t (x)"p t (x)"p n (x). e b d Moreover, again as demonstrated in Appendix A, the CCSs between the compound processes are linked to the PMF of Mh N by the relations n U (q)"U (q) cd dc W, q"0, " (5) W ) (1! 1h +@q@~1[1!P n (i)]), DqD*0, m i/0 h and
G
G
W, q"S, U (q)" ec W ) (1! 1h +@q~S@~1[1!P n (i)]), DqDOS, m i/0 h (6) where P n (i)"+i p n ( j) is the probability that j/0 h h h is less or equal to i, and n E[a ) b ]!m ) m n n a b. W" (7) p )p a b From relations (3) and (4) it can be seen that the PMF of Mc N only depends on the PMF of Ma N, t n while the PMFs of Md N and Me N only depend on the t t PMF of Mb N. Hence, the mean, mean quadratic and n variance values of Mc N are obviously equal to those t of Ma N, and those of Md N and Me N are equal to those n t t of Mb N. n As regards the correlation sequences, from relations (5) and (6) it can be seen that the decay of the
227
Table 1 Properties of the CCS between the compound processes Me N t and Mc N t U (q)"0 ec 0)DU (q)D)DU (S)D"DWD ec ec U (S#q)"U (S!q) ec ec 0)U (q))W if E[a ) b ]*m ) m ec n n a b W)U (q))0 if E[a ) b ])m ) m ec n n a b W U (q)!U (q#1)" h [1!P n (q!S)] m h ec ec U (q)!U (q!1)" Wh [1!P n (Dq!SD)] ec ec m h [U (q#1)!U (q)] ec ec W ![U (q)!U (q!1)]" h p n (q!S) ec ec m h (9) [U (q!1)!U (q)] ec ec ![U (q)!U (q#1)]" Wh p n (Dq!SD) m h ec ec
(1) (2) (3) (4) (5) (6) (7) (8)
for for for for for for for
DS!qD*N any q any q any q any q q*S#1 q)S!1
for q*S#1 for q)S!1
CCSs between the compound processes only depends on the PMF of Mh N. So, as Ma N and Mb N are n n n statistically independent of Mh N, the decay of the n CCSs between Mc N and Md N or between Mc N and t t t Me N can be "xed regardless of the PMFs of Ma N and t n Mb N. n The CCS between the compound processes Me N t and Mc N, U (q), has some interesting properties, t ec which are listed in Table 1 and can be derived from relation (6) and the properties of crosscorrelation sequences given in Section 2. Observing the table some remarks can be made: f U (q) is a sequence which has at most (2N!1) ec non-null samples. By increasing the size of the set MHN, namely N, two random processes can therefore be generated which have a non-null correlation up to any lag q"N#S!1; f U (q) is symmetrical to the lag S with which ec DU (q)D is maximum; ec f the sign of U (q) depends on the sign of U (S); ec ec f the sign of the "rst and second derivatives of U (q) therefore depends on the sign of W and the ec value of q with respect to S. When, in fact, q*S, if W'0, the CCS never increases as the absolute value of q increases and its second derivative is always non-negative; if, on the other hand, W(0, the CCS never decreases as the absolute value of q increases and its second derivative is always non-negative. Bearing in mind that the CCS is symmetrical to the lag S, the opposite holds for the "rst and second derivatives of the CCS when q)S.
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Similar properties to those listed in Table 1 and similar remarks hold for the CCS between the compound processes Md N and Mc N or Me N, by taking t t t into account how these processes are constructed and, again, the general properties of the CCS. Although the autocorrelation sequence lies beyond the scope of the paper, let us "nally note that the autocorrelation sequences of the compound processes have the same decay as the CCS. In [2] it is, in fact, demonstrated that E[c ) c ]!m2 c t t`q U (q)" cc p2 c 1 if q"0, " (8) 1! 1h +@q@~1[1!P n (i)] otherwise. m i/0 h By considering how the processes d and e are t t constructed, it can easily be derived that the autocorrelation sequences of Mc N, Md N and Me N are t t t equal, that is, U (q)"U (q)"U (q). dd ee cc
G
4. Application of the proposed method In order to explain how the proposed methodology for the generation of crosscorrelated random processes can be applied, let us now see how it can be applied to model a multimedia tra$c source. As mentioned in the introduction, in a communication environment a multimedia source is usually considered as an aggregate of two or more monomedia streams correlated with each other: the correlation among the component streams lies in the fact that the behavior of each stream depends on that of the others. More speci"cally, there is generally one monomedia stream (the master) which is completely independent of the others, and others (the slaves) which are dependent only on the "rst [23,13]. As an example, in a slide show with voice commentary, the change of slides is temporally related to the speaker's comments: the master is the audio stream, whereas the slave is the still picture stream. Modelling a multimedia source thus requires representation of the intermedia relationships between the master and slave streams. Let us assume that we have empirical measurements of the statistics of the multimedia source, more speci"cally the histogram of the empirical
probabilities p(%) (x) and p(%) (x) for all the values S-!7% M!45%3 that the output of the master and of the slave, respectively, can take and the empirical CCS between the slave stream and the master stream, (q), for q"[q ,2,!1,0,1,2,q ]. U(%) S-!7%,M!45%3 MIN MAX As said in Section 1, modelling a source by means of a simulation approach and a stochastic process means generating two stochastic processes Mc N and t Me N such that t p t (x)"p(%) (x), c M!45%3 (9) p t (x)"p(%) (x), e S-!7% (q). U (q)"U(%) S-!7%,M!45%3 ec To generate Mc N and Me N as de"ned in the previous t t section, it is necessary to determine the PMFs of Ma N, Mb N and Mh N and to generate them in such n n n a way that relations (9) are satis"ed. From relations (3), (4) and (7) and the "rst two equations in relation (9), the PMFs of Ma N and Mb N n n have to be equal to the corresponding experimental ones for the source, provided that Ma N and Mb N are n n correlated for a zero lag. That is, p n (x)"p(%) (x), a M!45%3 p n (x)"p(%) (x) b S-!7% with the condition E[a ) b ]!m ) m n n`m a b p )p a b (S) if m"0, W"U(e) S-!7%,M!45%3 " 0 if mO0,
G
(10) (11)
(12)
where &S' is the lag where the absolute value of (q) is maximum. U(%) S-!7%,M!45%3 Generation of Ma N and Mb N, once the corren n sponding PMFs are known, is easy to achieve when the two processes have to be uncorrelated for any lag [3,11,16]. But to apply the proposed method relation (12) has to be satis"ed, and the generation of a given b therefore has to depend on the corren sponding a . To this end the inversion method [3] n can be used, but the same random variable ; uniformly distributed in the interval [0,1] can be used to generate both a and b . Fig. 1 gives a pseudon n code algorithm based on the inversion method with which Ma N and Mb N are generated so as to satisfy n n
A. La Corte / Signal Processing 79 (1999) 223}234
229
p"W/W , and so as to be uncorrelated with MAX a probability of (1!p). The same applies when W(0. The minimum value is obtained for E[a ) b ] when, given a cern n tain ; value to generate a given a , (1!;) is used n to generate the corresponding value of b . In this n case we get
P
E[a ) b ] " n n MIN
1
; ) P~1 (;) ) P~1 (1!;) d;, (15) an bn 0 and correspondingly we have E[a ) b ] !m ) m n n MIN a b. W " (16) MIN p )p a b As regards the problem of generating Mh N, from n relations (6) and (7) in Table 1 we obtain
Fig. 1. Pseudo-code for generation of the processes Ma N and n Mb N which have given probability functions } P n (;) and P n (;) n a b }, are uncorrelated for a lag qO0 and are correlated for q"0 in such a way that relation (12) is satis"ed.
relations (10)}(12) and the hypothesis on processes Ma N and Mb N listed in Section 3. n n In order to explain the algorithm, let us assume that W'0. The maximum value for E[a ) b ] n n } corresponding to maximum correlation between Ma N and Mb N } is obtained when the same ; is used n n to generate the values of both Ma N and Mb N. In this n n case we get
P
E[a ) b ] " n n MAX
1
; ) P~1 (;) ) P~1 (;) d; an bn 0 and correspondingly we have
(13)
E[a ) b ] !m ) m n n MAX a b. (14) W " MAX p )p a b So to generate Ma N and Mb N in such a way that n n 0)W)W , they must be generated so as to MAX have maximum correlation with a probability of
U (i#S)!U (i#S#1) ec , (17) P n (i)"1! ec h U (S)!U (S#1) ec ec where &S' is the lag value with which the absolute value of U (q) is maximum. It is easy to verify that ec the probabilities obtained in relation (17) meet the probability congruence conditions [20] if the CCS satis"es relations (6)}(9) listed in Table 1, i.e. if U (q) is symmetrical to the lag S and, when q*S, it ec is a non-increasing sequence with downward convexity if W'0, or a non-decreasing sequence with upward convexity if W(0. By imposing the third equation in relation (9) and replacing in (17) we obtain P n (i) h
(i#S#1) (i#S)!U(%) U(%) S-!7%,M!45%3 . "1! S-!7%,M!45%3 (S#1) (S)!U(%) U(%) S-!7%,M!45%3 S-!7%,M!45%3 (18)
The experimental values of the crosscorrelation sequence do not usually meet conditions (6)}(9) listed in Table 1, under which the model proposed can be applied. To apply the proposed methodology, it is therefore "rst necessary to approximate the experimental sequence with a sequence which satis"es the above conditions. For the simplicity of the model not to be thwarted by the computational complexity needed to "nd the curve which best approximates the experimental crosscorrelation sequence, the approximation operation should preferably be very simple. It should also lead to a set
230
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Fig. 2. Experimental probability histograms of the multimedia source modelled in the numerical example and experimental crosscorrelation sequence between the video and data streams: (a) experimental PMF of the master stream (video source); (b) experimental PMF of the slave stream (data source); (c) experimental CCS which satis"es relations (6)}(9) in Table 1; (d) experimental CCS which does not satisfy relations (6)}(9) in Table 1 and approximation by means of a broken line.
MHN in which only a few elements have a non-null probability, so that the length of the simulation is not made excessive by the need to generate rare events. An extremely simple way to approximate the experimental crosscorrelation sequence is to do so with a broken line. For relations (6)}(9) listed in Table 1 to be satis"ed, the segments of this line have to be chosen in such a way that, when q*S, the slope of each segment is less than that of the previous one; when q)S, obviously, the symmetrical segments have to be chosen. Once the CCS has been approximated in this way, relation (18) is applied using the approximated values in order to obtain the PMF of h . n In order to facilitate understanding of the application of the proposed method, a numerical example is presented below. Let us assume that the
multimedia source to be modelled comprises two monomedia streams: the "rst, the master stream, is generated by a video source, which generates 30 frames per second, with a conditional replenishment interframe coding scheme. The second, the slave stream, is generated by a data source which is temporally synchronized with the master stream, and whose emission process depends on that of the master stream. Let us also assume that, from experimental measurements, calling the bit-rate durfor the master and s(%) ing the tth frame s(%) 4-!7%,t .!45%3,t and the slave stream, respectively, it was observed that: f the steady-state probability distribution of has a bell-like shape which can be aps(%) .!45%3,t proximated with a Gaussian probability density function (see Fig. 2(a)), with a mean m(%) " .!45%3
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130 000 bit/frame, and a standard deviation p(%) "57 500 bit/frame [17], which is then .!45%3 sampled and quantized at intervals equal to 500, and bounded by maximum and minimum values of s "352 500 bit/frame and s "20 000 .!9 .*/ bit/frame, respectively; f the steady-state probability distribution of is as shown in Fig. 2(b). The mean of the s(%) 4-!7%,t slave stream is m(%) "1986.7 bit/frame and the 4-!7% standard deviation is p(%) "911.9 bit/frame; 4-!7% f the CCS between the slave and the master stream has a shape which can be approximated with an exponential sequence (see Fig. 2(c)) according to the law U(e) (q)"0.7838 ) e~0.13>@q@ S-!7%,M!45%3 for q"!30,2,30. Let us note that although the graphs in Figs. 2(a) and (c) refer to the probability distribution and crosscorrelation sequence which approximate the experimental values for the video source, we will refer to them as experimental values so as not to create any confusion. The method presented to model the source entails the following steps: 1. veri"cation of the congruence of relation (17) on the experimental CCS values. In this speci"c case they are veri"ed and we also have S"0 and W"0.7838; 2. calculation of W in accordance with relation MAX (14). In this example W "0.9169; MAX 3. generation of the variables Ma N and Mb N accordn n ing to the algorithm in Fig. 1; 4. generation of the variables Mh N with a cumulatn ive probability distribution in agreement with relation (17); 5. generation of the compound processes Mc N and t Me N. t Various simulations were performed during which the variables Ma N, Mb N and Mh N were generated in n n n agreement with the probability distributions calculated above, the processes Mc N and Me N were t t constructed as outlined in Section 3, and the PMFs and CCS were calculated and compared with the expected values given in Figs. 2(a)}(c). More speci"cally, in order to determine the goodness of "t between the experimental histograms and the evid-
231
ence contained in the simulation results, we used the s2 test [9]. Considering a single run simulation of the processes Mc N and Me N corresponding to the t t generation of 5]105 samples of variables Ma N, Mb N n n and Mh N, and discarding the "rst 29 samples n in order to have a steady-state condition, the simulation results approximate the expected results with a 95% con"dence interval on the true value. Let us now assume that the experimental CCS is not the one in Fig. 2(c), but the one shown in Fig. 2(d). As can be seen, the CCS does not satisfy the congruence relations for the PMF of Mh N, so to apply n the proposed methodology it is "rst necessary to approximate the experimental CCS with a sequence that is symmetrical to the lag q"15 and, with q'15, is a non-decreasing sequence with upward convexity. In the example at hand, a simple way which also leads to a set MHN in which only a few elements have a non-null probability, is to approximate the experimental CCS sequence with a line comprising eight segments (see Fig. 2(d)), the ends of which are the coordinate points (!21,0), (!20,0), (!13,!0.0630), (!4, !0.4275), (0,!0.7407), (4,!0.4275), (13, !0.0630), (20,0), (21,0). Obviously the last segment is horizontal. Calculating the probability distribution of Mh N n according to relation (18) and using approximation by a broken line, we obtain a non-null PMF value corresponding to each point of intersection of the various segments. In the case in point, in fact, the values of MHN which have a non-null probability are H , H and H . More speci"cally, 4 13 20 we get p (4)"0.4828, p (13)"0.4023 and hn hn p n (20)"0.1149. h Having thus satis"ed all the hypotheses concerning the applicability of the method proposed, we can now generate the processes Ma N, Mb N and Mh N, n n n and therefore the compound process Mc N, Me N as in n n the previous example.
5. Conclusions In this paper a method which allows the generation of two sets of discrete random variables with given probability mass functions and crosscorrelation sequence has been presented. The method can
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be applied to modelling, by simulation, a multimedia source comprising a master and a slave stream linked by intermedia time relationships and whose experimental PMFs and CCS are known. Although the method only allows the experimental CCS to be matched exactly when the latter possesses certain properties, it has a certain general validity. This is for two reasons: "rst it makes it possible to mimic a CCS which decays more slowly than an exponential one, and so long-term dependences can be taken into account. Secondly, when the condition imposed on the experimental CCS for the method to have validity is not met, it is generally possible to approximate the sequence so as to be able to apply the method. In addition, as compared with the methods generally used to generate autocorrelated variables [4,5,8,18], the method proposed is simpler and thus faster in both the model "tting and the variable generation phases, and its complexity does not depend on the number of variables generated. It does, however, require a high computational load when the discretization time is excessively low, i.e. when continuous random processes are to be represented. The author is currently working on how to modify the method to generate the compound processes so that the ACS decay is independent of that of the CCS, and so that the ACS and CCS can also represent cyclostationary components. Appendix A A.1. Computation of the PMF of compound processes Let p (x) and p (x) be the probabilities that Mc N ct an t and Ma N are equal to x, respectively. Referring "rst n to the total probability theorem and then to Bayes' rule, exploiting the independence between Mc N and t Ma N we have n N p t (x)" + p(a "xDh "i) ) p n (i) c n n h i/1 N " + p(h "iDa "x) ) p n (x) n n a i/1 N (A.1) " + p n (x) ) p n (i)"p n (x). h a a i/1
A.2. Computation of the CCS between the compound processes Let U(%)(q) be the experimental crosscorrelation cd sequence between Mc N and Md N, obtained considert t ing k extractions from Ma N, Mb N and Mh N. As both n n n Mc N and Md N are wide-sense stationary and ergotic t t random processes, the temporal averages coincide with the ensemble averages. So, according to how these two processes are constructed and the properties the processes Ma N, Mb N and Mh N possess, n n n we have U (q)" lim U(%) cd cd k?= " lim k?= " lim k?=
A A
E(%)[c ) d ]!m(%) ) m(%) d c t t`q p(%) ) p(%) d c
B B B B
+k (h !DqD) ) a ) b i/1,hi ;@q@ i i i p(%) ) p(%) ) (+k h !DqD) d c i/1 i
A A A
DqD ) a ) b +k~l i i/1,h ;@q@ i r # lim h !DqD) p(%) ) p(%) ) (+k k?= c d i/1 i DqD ) a ) b +k~l i i/1,h x@q@ i r # lim p(%) ) p(%) ) (+k h !DqD) d k?= c i/1 i
B
m% ) m% c d , # lim p(%) ) p(%) d k?= c
(A.2)
where `la is a constant which does not depend on `ka, m% and m% is the experimental mean of Mc N and d t c Md N, respectively, p(%) and p(%) is the experimental d c t standard deviation of Mc N and Md N, respectively, t t and r'i, r(i or r"i, respectively, when q'0 q(0 or q"0. In relation (A.2), with k tending to in"nity, the upper part of the numerator summations can be approximated to k and the denominator term q can be neglected. We can therefore write U (q)" lim cd k?=
+k (h !DqD) ) a ) b i i i/1,hi ;@q@ i p(%) ) p(%) ) (+k h ) d c i/1 i
A
# lim k?=
A
+k DqD ) a ) b i r i/1,hi ;@q@ p(%) ) p(%) ) (+k h ) d c i/1 i
B
B
A. La Corte / Signal Processing 79 (1999) 223}234
A A
B
+k DqD ) a ) b i r i/1,hi x@q@ p(%) ) p(%) ) (+k h ) c d i/1 i m% ) m% c d . # lim (A.3) p(%) ) p(%) d k?= c Having set a certain value for q, we divide the set MHN into two subsets MH N and MH N, where MH N is L G L the subset of elements of MHN less than or equal to the absolute value of q, and MH N is the subset of G elements of MHN greater than the absolute value of q. Note that when 0(DqD(N then MH N" L M1,2,DqDN and MH N"MDqD#1,2, NN. If DqD*N G then MH N"M0N and MH N"MHN. G L Let us therefore consider the "rst of the four limits indicated in relation (A.3). As both a n and b are identically distributed variables inden pendent of h , and bearing in mind that n E[a ) b ]"E[a ) b ]OE[a ] ) E[b ] if m"0 n n`m n n n n and E[a ) b ]"E[a ] ) E[b ] if mO0, we get n n`m n n +k (h !DqD) ) a ) b i i i/1,hi ;@q@ i lim p(%) ) p(%)(+k h ) d c k?= i/1 i + G [(i!DqD) ) + i (a ) b )D ] i|H D@ n n i " lim p(%) ) p(%)(+k h ) d c k?= i/1 i + [(i!DqD) ) N ) E(%)[a ) b ]] i n n i|HG " lim p(%) ) p(%) ) k ) m(%) h d c k?= E[a ) b ] n n + (i!DqD) ) p (i), " (A.4) hi p )p )m c d h i|HG where (a ) b )D indicates the product of the pair of n n i values a and b when h "i, and DD indicates the n n n i set of pairs of values a and b corresponding to n n h "i. n Likewise, for the other terms in relation (21), we get # lim k?=
A
B
B
A A
+k DqD ) a ) b i r i/1,hi ;@q@ p(%) ) p(%)(+k h ) d c i/1 i DqD ) + G [+ i (a ) b ) D ] i|H D{@ n r rEn i " lim p(%) ) p(%)(k ) m(%)) c d h k?= E[a ) b ] ) DqD ) + G [p n (i)] n r i|H h " p )p )m c d h [p (i)] m ) m ) DqD ) + i|HG hn " c d p )p )m c d h
lim k?=
A
A
B
B
B
B (A.5)
233
and
A
B
h )a )b +k i/1,hi :@q@ i i r p(%) ) p(%)(+k h ) d c i/1 i + [i ) + i (a ) b ) D ] D{@ n r rEn i i|HL " lim p(%) ) p(%)(k ) m(%)) c d h k?= [i ) p n (i)] E[a ) b ] ) + h n r i|HL " p )p )m c d h [i ) p n (i)] m )m )+ h " c d i|HL , (A.6) p )p )m c d h where (a ) b ) D indicates the product of a multin r rEn i n plied by the value of the sequence Mb N which is at r a distance q from the corresponding a , when n h "i, and D@D indicates the set of pairs of values n i a and b corresponding to h when h "i. n r i i Therefore substituting (A.4)}(A.6) in (A.3) we get
lim k?=
A
B
E[a ) b ] ) + [(i!DqD) ) p n (i)] n n h i|HG U (q)" cd p )p )m c d h m )m c d ) DqD ) + [p (i)] # hn p )p )m c d h i|HG
A
B
m )m # + [i ) p n (i)] ! c d h p )p c d i|HL E[a ) b ] m ) m 1 n n ! c d " ) m p )p m h c d h
A
B
) + [(i!DqD) ) p n (i)]. (A.7) h i|HG If q"0, as H ,H, relation (A.7) becomes G E[a ) b ]!m ) m n n c d U (0)" cd p )p c d E[a ) b ]!m ) m n n a b "W. " (A.8) p )p a b The "rst relation in (5) is thus demonstrated. If, instead, qO0, indicating with P n (i) the probh ability that h is less than or equal to `ia, and as n @q@~1 + (1!P (i)) hn i/0 " H #1!p (H )!p (H ) 1 hn 1 hn 2 # 2!p (H )!p (H ) hn q~2 hn q~1
234
A. La Corte / Signal Processing 79 (1999) 223}234
" H #DqD!H ! + (p n (H ) ) [DqD!H ]) 1 1 h i i Hi |HL " DqD! + (p n (H ) ) [DqD!H ]), h i i Hi |HL we get 1 ) + [(i!DqD) ) p n (i)] h m h i|HG 1 @q@~1 "1! ) + (1!P (i)). hn m h i/0 Substituting the latter in (A.7) we get
A
B
1 @q@~1 U (q)"U (0) ) 1! ) + (1!P (i)) hn cd cd m h i/0 if DqDO0.
Thus, the second relation in (5) is also demonstrated. As Me N"Md N we get t t~S E[e ) c ]!m ) m t t`q c d U (q)" ec p )p c d E[d ) c ]!m ) m t~S t`q c d " p )p c d "U (q#S). (A.9) cd Substituting (A.9) in (5) we get (6). References [1] A. Adas, Tra$c models in broadband networks, Commun. Mag. 35 (7) (July 1997) 82}90. [2] M. Andronico, S. Casale, A. La Corte, Generation of random variables for modelling VBR multimedia sources, Telecommun. Systems } Baltzer Sci. 9 (1) (January 1998) 1}21. [3] P. Bratley, B.L. Fox, L.E. Schrage, A Guide to Simulation, 2nd Edition, Springer, New York, 1987. [4] M.C. Cario, B.L. Nelson, Numerical methods for "tting and simulating autoregressive-to-anything processes, INFORMS J. Comput. 10 (1997) 72}81. [5] M.C. Cario, B.L. Nelson, Modeling and generating random vectors with arbitrary marginal distributions and correlation matrix. http://www.iems.nwu.edu/nelsonb/ norta4.ps, April 1997.
[6] V.S. Frost, B. Melamed, Tra$c modeling for telecommunication networks, Commun. Mag. 32 (3) (March 1994) 70}81. [7] I.W. Habib, T.N. Saadawi, Multimedia tra$c characteristics in broadband networks, IEEE Commun. Mag. 30 (7) (July 1992) 48}54. [8] D.L. Jagerman, B. Melamed, The transition and autocorrelation structure of tes processes part ii: Special cases, Stochast. Models 8 (3) (1992) 499}527. [9] R. Jain, The Art of Computer Systems Performance Analysis, Wiley, New York, 1991. [10] P.R. Jelenkovic, A.A. Lazar, N. Semret, The e!ect of multiple time scales and subexponentiality in mpeg video stream on queueing behaviour, IEEE J. Selected Areas Commun. 15 (6) (August 1997) 1052}1071. [11] G.E. Johnson, Construction of particular random processes, Proc. IEEE 82 (2) (February 1994) 270}280. [12] A. La Corte, A. Lombardo, S. Palazzo, G. Schembra, Control of perceived quality of service in multimedia retrieval services: prediction-based mechanism vs. compensation bu!ers, Multimedia Systems 6 (2) (March 1998) 102}112. [13] A. La Corte, A. Lombardo, G. Schembra, An analytical paradigm to calculate multiplexer performance in an ATM multimedia environment, Comput. Networks ISDN Systems 29 (16) (December 1997) 1881}1900. [14] A.M. Law, W.D. Kelton, Simulation Modeling and Analysis, 2nd Edition, McGraw-Hill, New York, 1991. [15] L.M. Leemis, Input modeling for discrete-event simulation, In: Proceedings of Winter Simulation Conference, Orlando (USA), December 1994. [16] A. Leon-Garcia, Probability and Random Processes for Electrical Engineering, 2nd Edition, Addison-Wesley, New York, 1994. [17] B. Maglaris, D. Anastassiou, P. Sen, G. Karlsson, J.D. Robbins, Performance models of statistical multiplexing in packet video communications, IEEE Trans. Commun. 36 (7) (July 1988) 834}844. [18] B. Melamed, An overview of tes processes and modeling methodology, Lecture Notes Comput. Sci. 729 (1993) 359}393. [19] I. Nikolaidis, I.F. Akyildiz, Source characterization and statistical multiplexing in ATM networks, Technical Report GIT-CC-92/94, Georgia Institute of Technology, Atlanta, 1994. [20] M.B. Priestley, Spectral Analysis and Time Series, Vols. 1 and 2, Academic Press Inc., London, 1981. [21] R. Steinmetz, K. Nahrstedt, Multimedia } Computing, Communications and Applications, Prentice-Hall, New York, 1995. [22] C.W. Therrien, Discrete Random Signals and Statistical Signal Processing, Prentice-Hall, New York, 1992. [23] P. Venkat Rangan, H.M. Vin, S. Ramanathan, Designing an on-demand multimedia service, IEEE Commun. Mag. 3 (7) (July 1992) 56}64.