Queue Response to Input Correlation Functions: Continuous Spectral Analysis. University of Texas at Austin. Austin, Texas

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1 Queue Response to Input Correlation Functions: Continuous Spectral Analysis San-i Li Chia-Lin Hwang Department of Electrical and Computer Engineering University of Texas at Austin Austin, Texas August 4, 1995 Abstract This paper, together with [1] and [2], opens a new window for the study of ueueing performance in a richer, heterogeneous input environment. It oers a uniue way to understand the eect of second- and higher-order input statistics on ueues, and develops new concepts of trac measurement, network control and resource allocation for high speed networks in the freuency domain. The techniue developed in this paper applies to the analysis of ueue response to the individual eects of input power spectrum, bispectrum, trispectrum, and input rate steady state distribution. Our study provides clear evidence that of the four input statistics, the input power spectrum is most essential to ueueing analysis. Furthermore, input power in the lowfreuency band has a dominant impact on ueueing performance, whereas high-freuency power to a large extent can be neglected. The research reported here was supported by NSF under grant NCR and by Texas Advanced Research Program under grant TARP-129. This paper was presented at the 7th IEEE Computer Communicatior Workshop in Hilton Head Island, SC, Oct It was also submitted to IEEE/ACM Transactions on Networking under re-review.

2 1 Introduction Multimedia trac in high speed networks possesses two salient features: strong correlation and high burstiness. In stochastic modeling, such trac is represented by a stationary random process. In practice, one can hardly obtain an exact description of random trac; only its steadystate, second- and higher-order statistics are measurable. Steady-state statistics are dened by a distribution function. Second- and higher-order statistics are described by autocorrelation functions in the time domain, or by euivalent spectral functions in the freuency domain, such as power spectrum, bispectrum and trispectrum. Classic ueueing theory has generally ignored second- and higher-order input statistics by making a renewal assumption on message interarrival time. Limited studies are available in recent years for the ueue response to second-order input statistics, which is represented by three mutually euivalent functions: index of dispersion, autocorrelation and power spectrum [3]. In [4], the index of dispersion is used to describe the dependence of interarrival-time and service-time, and hence to derive approximations for the performance of an average ueue. The autocorrelation function is used to describe the strong time autocorrelation revealed in voice and video trac, and hence to derive the ueue steady state solutions (refer to [5, 6, 7] for more references in related work). Recently in [1], we analyzed ueue response to individual freuency components of the input power spectrum. In particular, we used the classic elements of DC, sinusoidal, rectangular, triangular and their superpositions, to build various input processes. The work in [1] explored a new concept of spectral representation of wide-band input processes in ueueing analysis. The low-freuency band of each source was identied to have a dominant impact on ueueing performance. The key problem facing us today is the lack of a general techniue to construct input processes with nontrivial second-order statistics for ueueing analysis. Moreover, to the best of our knowledge, no studies are available to identify the ueue response to higher-order input statistics, such as bispectrum and trispectrum. In this paper, we introduce a new input modeling techniue for the evaluation of ueue response to a wide range of input spectral functions. In particular, we identify the individual eect of input power spectrum, bispectrum, trispectrum and steady state distribution on ueueing performance. The study gives clear evidence that the input power spectrum is absolutely essential to ueueing analysis, as compared with the other three input statistics. Furthermore, input power in the low-freuency band has a dominant impact on ueueing performance, whereas high-freuency power to a large extent can be neglected. We use a Markov modulated Poisson process (MMPP) to model input trac for two reasons: (a) MMPP characterizes second- and higher-order input statistics, (b) the involved ueueing analysis is solvable. The underlying N-state homogeneous Markov chain is described by a transition rate ma- 1

3 trix Q. The Poisson input rate at each state is dened by an input rate vector ~ = [ 0 ; 1 ; :::; ]. We use (~; Q) to represent an MMPP input process. Assume that Q is diagonalizable. By spectral decomposition, Q = l=0 l ~g l ~ hl where l is the l-th eigenvalue of Q. We use Ref l g and Imf l g to represent the real and image part of l. ~g l and ~ h l are the right column eigenvector and left row eigenvector with respect to l. As one will see, it is the eigenstructure of Q that captures the input spectral functions. In general, nding the input spectral functions from (~; Q) is not dicult. The dicult part is its inverse: constructing (~; Q) from desired input spectral functions. Note that the MMPP input model is used here simply as a vehicle to explore the nature of ueue response to second- and higher-order input statistics. For Q to be superimposed by two-state Markov chains, a techniue most commonly adopted in recent ueueing analyses [6], [8], all the eigenvalues must be real, i.e., Imf l g = 0, 8l. The power spectrum must be monotonic decreasing, which allows no peaks in non-zero freuencies. In the time domain, each real eigenvalue represents an exponential term in the autocorrelation function. Furthermore, such an MMPP process is uniuely determined by both steady-state and second-order statistics, i.e., it does not contain any distinuishable higher-order input statistics. This extreme case is certainly insucient to cover the entire spectrum of wide-band sources in high speed networks. In [1], we took another extreme case by making Q a simple periodic chain of any size, which leads to Ref l g = 0, 8l. Hence, the power spectrum consists only of a discrete impulse series. In the time domain, each imaginary eigenvalue represents a sinusoidal term in the autocorrelation function. In principle, a continuous spectral function is asymptotically approached by a discrete one as we suciently reduce the discrete freuency interval. Such a discrete spectral representation has greatly simplied the construction of (~; Q) from input spectral functions. Discrete spectral analysis has the advantage of isolating the eect of each individual input freuency component on ueues, but it has overlooked the stochastic nature of the input process. In this paper, we use a new class of Markov chains, called a circulant, to construct Q. Using circulants gives one the freedom to choose eigenvalues from the entire complex domain. A key property of circulants is that Q is an eigenvalue-only matrix. That is, its eigenvectors are solely determined by the size of Q, independent of its structure. Both eigenvalues and eigenvectors of a circulant are expressed in closed form. More importantly, Q can be directly formulated by the discrete Fourier transform of properly selected eigenvalues. Hence, with a proper tuning of (~; Q) we are able to investigate the eect of each isolated input eigenvalue on the ueueing process. 2

4 Each input eigenvalue, in general, represents an exponential sinusoidal term in the autocorrelation functions, which corresponds to a bell-shaped component in the spectral functions. We also examine ueueing behavior by the superposition of individual eigenvalues. Since input spectral functions are essentially captured by the eigenstructure of the input process, the emphasis of this paper is placed on the impact of individual input eigenvalues on ueueing performance. We also study ueue response to the input rate steady state distribution, while keeping both power spectrum and bispectrum unchanged. Since the packet loss rate performance is of primary concern in the design of high speed networks, we consider here a nite ueueing system with the buer capacity of K in packets. For simplicity, we assume each packet service time to be exponential. The system is then modeled by an MMPP/M/1/K ueue, which has the structure of a nite QBD type. Throughout this paper, both ueue length and loss rate performance are evaluated by the QBD-Folding algorithm [9], [10]. The paper is organized as follows. Section 2 provides the general denition of the input spectral functions for MMPP. In Section 3 we show the circulant construction from desired input spectral functions. The main results of this paper are presented in Section 4 for ueue response to the input power spectrum. Section 5 provides clear evidence that ueueing performance is virtually independent of the input bispectrum. The study in Section 6 also indicates that, for a given input power spectrum, the impact of the input rate steady state distribution on ueue is unimportant. In Section 7 we furthermore nd that ueue response is much less sensitive to the input trispectrum than to the input power spectrum, although the input trispectrum has a more signicant eect on ueue than both the input bispectrum and steady state distribution. The paper is concluded in Section 8. 2 Input Spectral Functions To introduce input spectral functions in the freuency domain, we consider a generic MMPP input process a(t), described by (~; Q) in the time domain. The steady state solution vector of Q is denoted by ~ =[ 0 ; 1 ; :::], i.e., ~Q = 0 and ~~e = 1, where ~e is a unit column vector. The average input rate is given by = ~~ T, where [] T represents the transpose of []. For a stationary Markov chain, one of its eigenvalues, denoted by 0, must be 0. For the rest of the eigenvalues we have Ref l g < 0. Dene the input autocorrelation function of MMPP by R() = a(t)a(t + ). Assuming that Q is diagonalizable, one can readily express R() = () l=1 j l je Ref lgjj cos(imf l gjj + argf l g) (1) 3

5 with l = i j i i j g li h lj (2) where ~g l = [g l0 ; g l1 ; :::] T and ~ h l = [h l0 ; h l1 ; :::]. Refer to [1] for the derivation of (1) in the discretetime domain. argf l g is the principal value of the argument of l, i.e.,? < argf l g. () is the Dirac delta-function, which is eual to innity at = 0, and zero elsewhere. The existence of () in R() is due to the non-zero average input rate. At the limit as! 1, we must get R(1) eual to 2, which leads to 0 = 2 because of Ref l g < 0. Furthermore, since Q is real, all the complex eigenvalues in ~ must appear in conjugate pairs. If l is complex, we denote N?l =? l. Hence, ~ h N?l = ~ h? l and ~g N?l = ~g l?, which leads to N?l = l?. Thus, each eigenvalue contributes one term in R(n), which can be exponential, sinusoidal, or exponential sinusoidal, depending on if the eigenvalue is real, imaginary, or complex. Using R(0) = (0) + 2 in (1) on the basis of P 0 = 2, the input rate variance can be expressed by 2 = l=1 j lj cos(argf l g). On the basis of N?l = l?, 2 = Taking the Fourier transform of R(), we derive the input power spectrum with l=1 P (!) = (!) + b l (!) =?2 l 2 l +!2 and 1 2 l (3) l=1 Z +1?1 lb l (!) (4) b l (!)d! = 1: (5) The rst component in (4),, is described as background white noise, attributed to the local dynamics of Poisson arrivals in each input state. The second component, 2 2 (!), represents the DC term. Since the average arrival rate in ueueing systems must be positive, the DC term always exists in the input power spectrum. If a(t) is purely Poisson, the power spectrum will consist only of white noise and DC. In general, each eigenvalue l for l > 0 contributes a new component l b l (!) to P (!). From (5), one can interpret l as the average power contributed by l. As described in Figure 1, each such component represents a bell-shaped curve located at the central freuency! l = Imf l g and weighted by the average power l. The shape of each bell, before being weighted by l, is measured by its half power bandwidth B l =?2Ref l g. Since all the complex eigenvalues must appear in conjugate pairs, the power spectral function is always symmetric. In principle, we can construct a desired input power spectrum in a rational function form with a sucient number of eigenvalues. Figure 1 provides such an example with the superposition of two distinct eigenvalue components. 4

6 P( ω) half power bandwidth B 2 area= ψ 2 ω 2 0 ω 2 ω (rad/s) Figure 1: Input power spectrum with superposition of two eigenvalues. Obviously, the eigenstructure of Q captures both R() and P (!). Dene ~ = [ 0 ; 1 ; :::] and ~ = [ 0 ; 1 ; :::]. We use ( ~ ; ~ ) to represent the input power spectrum, where ~ is called the input power vector with respect to ~. Hence, nding ( ~ ; ~ ) in the freuency domain from (~; Q) in the time domain is not dicult. The dicult part is its inverse: constructing (~; Q) from ( ~ ; ~ ). To capture higher-order input statistics, one can dene the third-order autocorrelation function R( 1 ; 2 ) by a(t)a(t + 1 )a(t ). Note that MMPP is a completely stationary random process. Using spectral decomposition, we obtain R( 1 ; 2 ) = ( )( 1 ; 2 ) + l 1 =0 l 2 =0 l 1 ;l 2 e l 1 j 1 j+ l2 j 2 j (6) with l 1 ;l 2 = i j k i i j k g l1 ih l1 jg l2 jh l2 k (7) The derivation of (6) has been omitted here since it is very similar to that of (1). The third steady state moment of the input process is given by E[a 3 (t)] = R(0; 0), eual to 3 + ( )(0; 0). Taking the two-dimensional Fourier transform of R( 1 ; 2 ), we get the bispectrum of MMPP P (! 1 ;! 2 ) = l 1 =0 l 2 =0 l 1 ;l 2 b l1 (! 1 )b l2 (! 2 ) (8) Dene a matrix 2 by [ l1 ;l 2 ]. We use ( 2 ; ~ ) to represent the bispectrum. Two examples of MMPP bispectrum are illustrated in Figure 2 with the superposition of two eigenvalues. The same values of (! l ; B l ; l ) are used in Figures 1 and 2. In other words, for a given power spectrum, the bispectrum of the random input can be signicantly dierent. (This will be discussed in more detail in Section 3.) Note that P (! 1 ;! 2 ) must be symmetric in its arguments. Similarly, for fourth-order input statistics we dene the trispectrum of MMPP by P (! 1 ;! 2 ;! 3 ) = l 1 =0 l 2 =0 l 3 =0 l 1 ;l 2 ;l 3 b l1 (! 1 )b l2 (! 2 )b l3 (! 3 ) (9) 5

7 P(ω 1,ω 2 ) P(ω 1,ω 2 ) ω 2 ω 1 ω 2 ω 1 (a) Figure 2: Input bispectrum with two eigenvalues: (a) at 2 1 = 2 ; (b) at 2 1 = 2?. (b) with l 1 ;l 2 ;l 3 = i With 3 = [ l1 ;l 2 ;l 3 ], the input trispectrum is described by ( 3 ; ~ ). j k m i i j k m g l1 ih l1 jg l2 jh l2 kg l3 ;kh l3 ;m (10) Obviously, our trac modeling based on input spectral functions involves a so-called inverse eigenvalue problem, i.e., constructing (~; Q) in the time domain from ( ~ ; 2 ; 3 ; ~ ) in the freuency domain. This is generally very dicult, if at all possible. However, the success of using periodic chains in discrete spectral analysis [1] has induced us to nd another special class of Markov chains: the circulant chain. 3 Circulant Chain An N-state circulant chain is dened by Q = k=0 a k P k (11) where P = [p i;j ] is a permutation matrix, dened by p i;j = 1 at j = (i + 1) mod N, and 0 elsewhere. Let ~a = [a 0 ; a 1 ; :::] be the rst row in Q, subject to a 0 < 0, a j 0 for 0 < j < N and P k=0 a k = 0. Then, each next row in Q will be formed by circulating the previous row by one element to the right. Since the j-th eigenvalue of P k is represented by W jk with W 4 = e p?12=n, from (11) one can generally write for Q Dene a Fourier matrix by j = k=0 a k W jk (12) F = 1 p N [W?jk ] (13) with F?1 = F? = 1 p N [W jk ], where F? represents the conjugate of F. One can rewrite (12) in matrix form: Q = F? F; with = diag[ 0 ; 1 ; :::; ] (14) 6

8 Note that both Q and P have the same eigenvectors, described by F. Since F is independent of ~a, we call Q an eigenvalue-only matrix. As will be seen in Section 3.1, the trac descriptor ( ~ ; 2 ; 3 ; ~ ) can also be represented by ( ~ ; ~ ; ~ ), where ~ = [ 0 ; 1 ; :::] is called an input phase vector. Here we outline the main results of this section. According to our denition, the input process is characterized by (~; ~a) in the time domain and by ( ~ ; ~ ; ~ ) in the freuency domain. From Time Domain to Freuency Domain: input eigenvalue vector: ~ = p N~aF? with j =? N?j, 8j, and 0 = 0 input power vector: ~ = 1 N j~f? j 2 with j = N?j, 8j input phase vector: ~ = argf~f? g with j =? N?j, 8j From Freuency Domain to Time Domain: input modulator: ~a = 1 p N ~ F, subject to aj 0 for j > 0 and a 0 =? P j=1 a j input rate vector: ~ = p N ~ F with ~ = [ 0 ; 1 ; :::] and j = p je p?1 j, subject to j 0, 8j. The construction of ( ~ ; ~ ; ~ ) from (~;~a), or vice versa, is therefore simply carried out by taking the discrete Fourier transform. Under the subjected conditions, (~; ~a) will be uniuely determined by ( ~ ; ~ ; ~ ). Furthermore, since ~ is only related to ~a while ( ~ ; ~ ) is only dependent on ~, the design of the eigenvalues is well decoupled from the design of the power and phase vectors. One can also derive bounds on ~ for an N-state circulant. From matrix theory we know that each eigenvalue of a general matrix Q = [ ij ] must satisfy j i? ii j P j6=i ij, 8i. For circulants we always have ii = a 0 and P j6=i ij =?a 0, which leads to j i? a 0 j?a 0 ; 8i (15) with Ref i g 0 and a 0 < 0. Furthermore, it is indicated in [1], [11] that the eigenvalues of an N-state Markov chain in the discrete-time domain cannot be inside either of the two segments of the unit circle, which joints the point 1 with the points W and W?1. This can be expressed by jargf i? 1gj 2 + N for i > 0. In the continuous-time domain, this bound is transferred to jargf i gj 2 + ; for i > 0: (16) N Note that we must have jargf i gj 2 for Ref ig 0. Furthermore, for nite N we get jargf i gj > 2, which is euivalent to Ref ig 6= 0 for i > 0. 7

9 3.1 Input Spectral Functions The steady state distribution for a circulant chain must be uniform in each state, i.e., i = 1 N, 8i. From F and F? we get g li = W li and h lj = W?lj for the input eigvenvectors. Taking these results into (2) leads to l = 1 N 2 f i i W li gf j which is euivalent to ~ = 1 N j~f? j 2 in matrix form. Introducing j W?lj g (17) ~ = 1 p N ~F? (18) we get ~ = j ~ j 2. Also, we dene the phase vector ~ = argf ~ g, which yields j = je p?1 j : (19) Since ~ is real, both l and N?l must be conjugate for 0 < l < N and 0 must be real, which leads to 0 = 0, l = N?l and l =? N?l for 0 < l < N. It is ~ that uniuely determines ~ and ~. For a circulant chain, ~ must be real, i.e., argf l g = 0, 8l. In the time domain, this means that its autocorrelation function is limited to the following form: R() = () l=1 le Ref lgjj cos(imf l gjj) (20) which is only a subset of the general form in (1). Let us compare (20) with the two extreme cases in [1], [6]. For the superposition of two-state Markov chains in [6], we must have Imf l g = argf l g = 0, 8l, and its R() is limited to R() = () l=1 le ljj which is monotonic decreasing. For the time periodic chains used in [1], we have Ref l g = argf l g = 0, 8l, and so its R() is limited to R() = () l=1 l cos(imf l gjj) which is the superposition of sinusoidals. Although circulants cannot cover the entire input spectrum of wide-band sources, by introducing exponential sinusoidals to R(), they are certainly much more general than two-state and periodic Markov chains. As in (17), one can rewrite (7) by l 1 ;l 2 = 1 N 3 f i i W l 1i gf j j W?(l 1?l 2 )j gf k k W?l 2k g =? l 1 l1?l 2 l2 ; 8

10 which leads to j l1 ;l 2 j = l 1 l1?l 2 l2 ; argf l1 ;l 2 g =? l1 + l1?l 2 + l2 (21) with argf l1 ;l 2 g = 0 for l 1 = l 2 and argf l1 ;l 2 g =?argf l2 ;l 1 g for l 1 6= l 2. Since 2 is constructed by ( ~ ; ~ ), the bispectrum in (8) will be uniuely determined by ( ~ ; ~ ; ~ ). Since the power spectrum ( ~ ; ~ ) represents the second-order input statistics, one can use ( ~ ; ~ ) to distinguishably characterize the third-order input statistics. For instance, the bispectrum in Figure 2a for the two-eigenvalue superposition was designed by argf 1;2 g = argf 2;1 g = 0, which leads to 2 1 = 2 based on 1?2 =? 1. For comparison purposes, the bispectrum in Figure 2b is constructed by the same values of (! l ; B l ; l ) as in Figure 2a, except for argf 1;2 g = argf 2;1 g =. Here we obtain 2 1 = 2?. Inspection of Figures 2a,b indicates that two input processes with identical secondorder statistics can have substantially dierent third-order statistics. In Section 5 we will use ~ to examine the ueue response to the third-order input statistics. Note that in order for l1 ;l 2 6= 0 in (21), we need at least l1 6= 0 and l2 6= 0. In other words, each pair of bell components in (8), b l1 (! 1 )b l2 (! 2 ), 8l 1 ; l 2, cannot appear in the bispectrum unless both b l1 (!) and b l2 (!) exist in the power spectrum. Since each bell is contributed by an eigenvalue, it is impossible for an input eigenvalue, which makes no contribution to the power spectrum, to appear in the bispectrum. For the fourth-order input statistics, described by the trispectrum in (9), one obtains l1 ;l 2 ;l 3 =? l 1 l1?l 2 l2?l 3 l3, or j l1 ;l 2 ;l 3 j = l 1 l1?l 2 l2?l 3 l3 ; argf l1 ;l 2 ;l 3 g =? l1 + l1?l 2 + l2?l 3 + l3 (22) Hence, 3 is also constructed by ( ~ ; ~ ). Unlike l1 ;l 2 in the bispectrum, here one can have l1 ;l 2 ;l 3 6= 0 while l2 = 0. Essentially, this implies that an input eigenvalue, which does not appear in power spectrum or bispectrum, can still appear in the trispectrum. Section 7 will study ueue response to the input trispectrum. 3.2 Input Steady State Distribution Note that although the steady state probability of each state is uniform for circulant chains, the steady state distribution of the input rate is non-uniform. Denote the Poisson input rate by random variable ~, which is modulated by the circulant. We get P r(~ = l ) = 1 N, 8l in steady state. Hence, the steady state distribution of the input rate is given by P r(~ < x) = i N (23) where i represents the number of elements in ~ which are less than x. It is also possible for l = j while l 6= j. Let us permute the input rate vector ~ in natural order. The permuted one is denoted 9

11 Distribution ι ι =1 =2 1 2 N = 100 C 1 = C = input rate ι 2 ι =1 =4 Figure 3: Input rate steady state distribution in function of l 2 at N = 100 and l 1 = 1. by ~ p = [ p0 ; p1 ; :::; p ], subject to pi pk with index i < k. One can then plot the steady state distribution of the input rate process in a piece-wise, multi-step incremental form according to ~ p. We use ~ p to characterize the input rate steady state distribution. From ~ = p N ~ F, one can generally express the input rate vector ~ by k = + l=1 p l cos(2lk=n? l ); subject to k 0 for 0 k < N: (24) Hence, each eigenvalue can contribute a sinusoidal term in ~. The steady state distribution of the input rate process is then representable by the superposition of sinusoidals. Note that both l and l characterize the portion of power spectrum and bispectrum contributed by l. We have l = N?l, l =? N?l and l =? N?l for l > 0. For the DC term in ~, we always have 0 = 0, 0 = 0 and 0 = 2. Dene the suared coecient of variation for input rate by C 2 = 2 = 2. From (3) one can decompose C 2 by C 2 = l=1 C 2 l with C 2 l = l = 2 ; (25) where Cl 2 is the suared coecient of variation for input rate contributed by l. For convenience, in the following we shall use C l and l interchangeably. Consider that the circulant size, N, must be suciently large in order to allocate eigenvalues in a wide complex domain. In reality, only a few eigenvalues in ~ are important to ueueing performance. This provides us a certain degree of freedom in selection of ~ after matching the desired input power spectrum and bispectrum. With this freedom one can further tune the input rate steady state distribution. Typically, both power spectrum and bispectrum for each eigenvalue are determined by the values of ( l ; l ; l ), independent of their index l. From (24) one can change the index l for dierent sinusoidal freuency of each eigenvalue in ~. Figure 3 shows two examples of the input rate distribution composed by two eigenvalues, with respect to the index l 2 = 2; 4 at l 1 = 1 and N = 100. We have assumed (C 1 ; C 2 ) = (0:417; 0:186) 10

12 and ( 1 ; 2 ) = (0; 0). Both input power spectrum and bispectrum are similar to the one in Figures 1 and 2. It is obvious that a much wider range of input steady state distribution can be achieved when N and l 1 are tuned, or when there are more input eigenvalues. Note that the mean and variance of the input rate distribution are not changed by index tuning since they are already xed by the power spectrum. In Section 6 we will study ueue response to input rate distribution. 4 Queue Response to Input Power Spectrum Consider a system with a nite buer capacity of K in packets. The service time of each packet is assumed to be exponential at constant rate. The system is then modeled by an MMPP/M/1/K ueue with a nite QBD structure. We use the QBD-Folding algorithm developed in [9], [10] to evaluate both ueueing delay and loss rate performance. In all examples here we choose K + 1 to be a perfect power of 2 purely for the computational eciency of the algorithm. Recall that l represents the average input power contributed by l in the freuency domain. By properly assigning ~, one can arbitrarily tune the eect of each individual eigenvalue in ~. For instance, one can make l ineective in P (!) simply by assigning l = 0. In this section, we examine ueue response to the input power spectrum, rst as contributed by a single input eigenvalue, and then by the superposition of multiple eigenvalues. 4.1 Single Eigenvalue In the design of an input power vector ~, we assign l = 8 >< >: 2 if l = C 2 if l = i, N? i 0 elsewhere (26) for 0 < i < N. Recall in (4) that l is the average power contributed by l in the power spectrum. Hence, only three eigenvalues in ~ are eective in the power spectrum: 0, i and N?i. Because of 0 = 0 and N?i =? i, it is sucient to use i as a single eective eigenvalue. Without loss of generality, we assume Imf i g > 0. The subscript i is used to dene the original index of this eective eigenvalue in ~. Similarly, we have i = N?i and i =? N?i. For a single eective eigenvalue, power spectrum and bispectrum are represented by ( i ; i ) and ( i ; i ), respectively. Taking (26) in (24), k = [1 + p 2C cos(2ik=n? i )]; for 0 k < N: (27) Subject to k 0, we must have C 1 p 2. Hence, the input rate for a single eective eigenvalue is expressed by a sinusoidal plus the DC term. As described in Section 3.3, one can tune the sinusoidal 11

13 freuency in ~ by changing i=n, with no eect on the power spectrum and bispectrum. The input steady state distribution is then adjusted accordingly, while keeping its rst two moments and C 2 unchanged. For instance, taking i = N 2 at which i is real, we get k = [1+ p 2C cos(k? i )], 8k, which apparently assumes only two distinctive values. The probability of the input rate is then zero, except at these two input rates. The remaining ineective eigenvalues in ~ can be arbitrarily assigned. For a single eective eigenvalue, we can design ~a by two positive elements, described by ~a = [a 0 ; 0; :::; a k1 ; 0; :::; a k2 ; :::0] with a 0 =?a k1? a k2. The subscripts k 1 and k 2 are the index of the two elements in ~a. From (12), we obtain l = a k1 (W lk 1? 1) + a k2 (W lk 2? 1); 8l (28) For simplicity, assume k 1 = 1 and k 2 = N? 1, and so we get a k = 1?Ref l g 2 1? cos(2l=n) Imf lg ; 8l at k = 1; N? 1 (29) sin(2l=n) where the sign is taken as plus at k = 1 and minus at k = N? 1. Of course, one can determine a k1 and a k2 from a desired single eective eigenvalue Ref i g and Imf i g, subject to a k1 ; a k2 0. From tan(i=n) = [1? cos(2i=n)]= sin(2i=n), we obtain, for a 1 ; a 0,?Ref i g Imf i g jtan(i=n)j, one can graphically In our denition, Ref i g < 0 and Imf i g > 0. From argf i g = tan?1 Imfi g Ref i g show that this bound at i = 1 is euivalent to the general bound in (16). With! i = Imf i g for central freuency and B i =?2Ref i g for half-power bandwidth, the above bound becomes B i 2jtan(i=N)j! i (30) Under this bound, one can always use (29) to construct ~a from any single eective eigenvalue i, 8i. For each given N, this bound sets a lower limit on the bell bandwidth at each central freuency. In other words, the circulant size N must be suciently large for a narrowband bell to appear in the high-freuency band. To minimize N, one should always choose i = 1 for the eigenvalue index. From the viewpoint of ueueing analysis, this bound may not be stringent since the ueueing performance is dominated by low-freuency input power, whereas the high-freuency input power can generally be neglected [1]. So far in our analysis we have not explicitly dened the unit for average input rate, which can be measured by a number of bits or packets per second. The same is true for average service rate. The normalized input load is given by = =. All ueueing and loss solutions at each given are valid for dierent values of and. In the freuency domain, both! i and B i of each 12

14 _ ω /2πµ 10 1 ρ = 0.8 K = 255 C γ = B /2πµ σ ω /2πµ 10 1 ρ = 0.8 K = 255 C γ = B /2πµ L (in log 10 ) ω /2πµ ρ = 0.8 K = 255 C γ = B /2πµ Figure 4: Queue response to a single input eigenvalue at (; K; C ) = (0:8; 255; 0:707). bell are purely dened in the radian freuency. It is obvious that the eect of the input spectral properties on ueue depends on the length of average service time. For instance, a low-freuency component essentially represents a slow time-varying factor in the input rate process, which is true only by comparison with the average service time. In other words, the time variation of the input rate process must be measured in units of average service time for a given ueueing system. In the radian freuency domain, such a time normalization is euivalent to normalizing input freuency to the service freuency 2. Throughout this paper we use! i =2 and B i =2 (instead of! i and B i ) for all numerical analyses. In the numerical study we choose i = 1 and N = 200 and denote the single eigenvalue by! and B for simplicity. The system is described by (; K; C ). The analysis is limited to C p 1 2 (or, 0:7 by approximation). The performance is measured by (; ; L), where is the mean ueue length, is the ueue standard deviation and L is the average loss rate. We plot all solutions of (; ; L) as a function of!=2 and B=2 at each given (; K; C ). Note that the input phase vector and input rate distribution are not aected by the change of! and B. A typical set of system values are chosen: (; K; C ) = (0:8; 255; 0:7). Each 3-D contour in Figure 4 explores the remarkable impact of the input power spectrum on system performance. For each given B, every measure in (; ; L) increases by the reduction of!. As graphically described in Figure 1, reducing! actually shifts the bell in P (!) to the low-freuency band. As a result, more input power is moved from the highfreuency band to the low-freuency band, which inherently causes the performance to deteriorate [1]. The impact of the low central freuency! is especially strong when bandwidth B is also small. This is because the smaller the B, the more the input power is concentrated in the neighborhood of the lower!. For each given!, we also observe that all the performance measurements reach their maximum around B = 2!. This is because the input power at zero freuency, P (0), reaches its maximum when B = 2! for each given!. In other words, as B is close to 2!, more input power is located around the zero freuency, causing the ueue to increase. Note that because a single input eigenvalue is designed with N = 200 at i = 1, the results 13

15 _ ω /2πµ 10 1 ρ = 0.9 K = 255 C γ = B /2πµ σ ω /2πµ 10 1 ρ = 0.9 K = 255 C γ = B /2πµ L (in log 10 ) ω /2πµ ρ= 0.9 K = 255 C γ = B /2πµ Figure 5: Queue response to a single input eigenvalue at (; K; C ) = (0:9; 255; 0:707). in Figure 4 are limited to B 0:0314! according to (30). Such a limitation also explains the discontinuities in Figure 4, at the corner where B 2 < 10?2! for 2 > 10?2. For comparison purposes, the oor of the contours for (; ) in Figure 4 is set at (4; 4:5), which corresponds to the exact solutions of M/M/1 ueue at = 0:8. Note that our MMPP/M/1/K system becomes an M/M/1 ueue at C = 0 as K! 1. In other words, the oor represents the ueue response to the white noise input spectrum (which is Poisson). As one can see, the eect of the input power spectrum on ueue can generally be neglected when either! or B is signicantly high. In numerical plotting, we always set the oor of the contours for (; ) at the solutions for white noise. The oor for the loss rate contours is set at 10?12. Let us now observe the individual eect of (; K; C ) on (; ; L). First, let us change from 0:8 to 0:9, while keeping (K; C ) = (255; 0:7) as in Figure 4. The corresponding results are displayed in Figure 5. Obviously, all three contours of (; ; L) become wider and higher as increases. By comparison, the changes in height are less signicant than the changes in width, due to the same upper-bound eect of the nite buer size K = 255 used in Figures 4 and 5. For the same reason, all the contours tend to be more at-topped for higher. To examine the eect of buer size, let us change K from 255 to 1023, while keeping (; C ) = (0:8; 0:7) as in Figure 4. Clearly, increasing K reduces the upper-bound eect of the nite buer size. This is why the contours in Figure 6 for (; ) are substantially uplifted in comparison to those of Figure 4. Conseuently, the loss rate in Figure 6 is notably reduced, except when! and B are signicantly small. Inspection shows that when most input power is in the very low-freuency band, such as described by 1 2 (!; B)< (10?3 ; 10?2 ) in Figure 6, increasing the buer size as high as 1023 at = 0:8 is still insucient. Another interesting observation is that, as K is extended by four times from 255 to 1023, the height of both contours for (; ) also uadruples from 100 to 400. The contours for (; ) at K = 1023 graphically look narrower than those at K = 255, due purely to the vertical scaling eect of (; ). Without such a scaling, the lower part of the contours, dened by (; ) < (100; 100) at K = 1023, would virtually overlap the entire contours 14

16 _ ω /2πµ ρ = 0.8 K = 1023 C γ = B /2πµ σ ω /2πµ 10 1 ρ = 0.8 K = 1023 C γ = B /2πµ L (in log 10 ) ω /2πµ ρ = 0.8 K =1023 C γ = B /2πµ Figure 6: Queue response to a single input eigenvalue at (; K; C ) = (0:8; 1023; 0:707). _ ω /2πµ ρ = 0.8 K = 255 C γ = B /2πµ σ ω /2πµ 10 1 ρ = 0.8 K = 255 C γ = B /2πµ L (in log 10 ) ω /2πµ ρ = 0.8 K = 255 C γ = B /2πµ Figure 7: Queue response to a single input eigenvalue at (; K; C ) = (0:8; 255; 0:5). at K = 255. Next, to investigate the eect of an input rate variation coecient, we change C from 1 p 2 (or 0.7) to 1, while keeping (; K) = (0:8; 255) as in Figure 4. It is important to note that for each 2 given (!; B), reducing C from p 1 2 to 1 is euivalent to down-scaling the input power spectrum by 2 half, i.e., from the original P (!) to 1 P (!). This is why all the contours in Figure 7 at C 2 = 0:5 are narrower than those in Figure 4 at C = 0:7. On the other hand, due to the same upper bound eect of buer size K = 255 used in Figures 4 and 7, the height changes in these contours by C are insignicant. Certainly, all three contours at C = 0:5 are less at-topped, as compared to those at C = 0:7. It is obvious that Q is not uniue for each given i. For the given ( i ; i ; i ; i; N), we x the power spectrum, bispectrum and steady-state distribution of the input process. The uestion then is if the ueue response will be aected by the Q structure for the same values of ( i ; i ; i ; i; N). Let us use the same single eigenvalue i to design another circulant, given by a l = 8 >< >:?c + ( 1 N? 1) if l = 0 c + N 1 if l = 1 1 N for 1 < l < N which is dierent from (29). Based on the eective eigenvalue 1, both c and are determined by c = (31) Imf 1g sin(2=n) ; =?Ref 1g? tan(=n)imf 1 g: (32) 15

17 1 2 (!1; B1) L (10?3 ; 10?3 ) Q e-1 ~Q e (10?3 ; 10?1 ) Q e-3 ~Q e-3 (10?1 ; 10?1 ) Q e-6 ~Q e-6 (10?3 ; 10?3 ) Q e-1 ~Q e (10?3 ; 10?1 ) Q e-5 ~Q e-5 (10?1 ; 10?1 ) Q e-11 ~Q e-11 Table 1: Comparison of ueue response to two input circulants with identical ( 1 ; 1 ; 1 ; N). Because of Imf 1 g > 0 and (30), we always get c > 0 for N > 2 and 0 in (32). Compare the ueue response to both input circulants, denoted by Q and ~ Q, which are constructed by (29) and (31) from the same ( 1 ; 1 ; 1 ; N), respectively. Essentially, they represent two distinct random input processes. Listed in Table 1 are solutions for (; ; L) at = 0:8; 0:9, with the selection of 1 2 (! 1; B 1 ) eual to (10?3 ; 10?3 ), (10?3 ; 10?1 ) and (10?1 ; 10?1 ), respectively. As one can see, the dierence between the two solutions is negligible. From this example one can see the signicance of characterizing the input power spectrum in ueueing analysis. 4.2 Multi-Eigenvalue Superpositions When N is large, only a small subset of ~ are eective eigenvalues. The remainder are ineective, since they can be removed from both input power spectrum and bispectrum simply by assigning the corresponding l 's and l 's eual to zero. For convenience, we introduce an eective eigenvalue vector ~ i ~. For each conjugate pair of complex eective eigenvalues in ~, say ( i ; N?i ) with argf i g > 0 and argf N?i g < 0, only i is included in ~ i. In our denition, ~ i also excludes 0 = 0. The uestion then is how to construct ~a from ~ i, subject to the condition of a k 0 for 0 < k < N. One approach is to construct an independent circulant for each element in ~ i, as done in Section 4.1. For total M eective eigenvalues in ~ i, the aggregate input process is modulated by an M- dimensional circulant, described in tensor form by Q = M l=1 where the notation \ L " is a Kronecker sum operator of matrices [12]. Q l represents the circulant constructed by the l-th eective eigenvalue. Associated with Q l are (~a l ; ~ l ) in the time domain and 16 Q l :

18 ( ~ l ; ~ l; ~ l ) in the freuency domain. Correspondingly, we obtain ~ = M ~ l ; ~ = M l=1 l=1 The size of Q is then N N with N = Q l N l, where N l is the size of Q l. Dene ~ l ~ i = [ i1 ; i2 ; :::; im ] for 0 < i l < N l and 0 < l M: The subscript i l indicates the original index of il in ~ l. Dene i = [ ; i 2 ; :::; i M ] for the index of ~ i in ~ and also N = (N 1 ; N 2 ; :::; N M ) for the size of ~a in each dimension. Correspondingly, we introduce an eective power vector ~ i = [ i1 ; i2 ; :::; im ] and an eective phase vector ~ i = [ i1 ; i2 ; :::; im ] for ~ i 2 ~ and ~ i 2. ~ Both power spectrum and bispectrum of the aggregate input are then dened by ( ~ i ; ~ i) and ( ~ i ; ~ i ). The aggregate input rate distribution is extended from (27) to k = [1 + 2 M l=1 C il cos(2k l i l =N l? il )] subject to k 0 for 0 k l < N l (33) P with k = [k 1 ; k 2 ; :::; k M ]. As in (25), we have Ci 2 l = il = 2 M P and 2 l=1 Ci 2 l = C, 2 which leads to M 2 l=1 il =. 2 The selection of (C i1 ; C i2 ; :::; C im ) is subject to a non-negative input rate. Hence, for given values of ( ~ i ; ~ i; ~ i ), one can use both i and N to tune the input rate distribution for the same and C, without aecting power spectrum and bispectrum. Dene B il =?2Ref il g and! il = Imf il g for l = 1; :::; M. From (30) we obtain the bound condition on the selection of each N il B 2jtan(i il l=n l )j! il for l = 1; :::; M: (34) Consider a multi-rate packet video source, whose input rate process possesses strong autocorrelations at three substantially dierent time scales, i.e., in adjacent frames, lines and packets [13], [14]. Assume that the video source generates 25 frames per second, 256 lines per frame and 256 pixels per line [14]. For one byte per pixel at full rate, we get the maximum access rate eual to 13.1 Mbps. For 44 bytes per packet, it corresponds to 37,216 packets per second, which is about 6 packets per line. Therefore, the three fundamental time scales are T f = 40ms per frame, T l = T f =256 = 156s per line, and T p = T l =6 = 26s per packet. Its input power spectrum is expected to consist of three bell-shaped components, each corresponding to one of the above time scales. That is, we get Imf i1 ; i2 ; i3 g = 2(T f?1 ; T l?1 ; T p?1 ), which is euivalent to (! i1 ;! i2 ;! i3 ) = 50(1; 256; 1536) in radian freuencies. The bandwidth of each bell is designed by (B i1 ; B i2 ; B i3 ) = B(1; 75; 250), assuming B = 200 in radian freuency. Let the total input power be distributed among the three bells according to ( i1 ; i2 ; i3 ) = (1; 1 2 ; 1 P 3 ), where is a common factor related to. 2 M From 2 l=1 l =, 2 we get ( i1 ; i2 ; i3 ) = 2 (6; 3; 2), which leads 22 17

19 10log(P( ω )) λ ι 1 λ ι ω/2πγ _ Figure 8: Superposition of input power spectrum with three eigenvalues. λ ι 3 to (C i1 ; C i2 ; C i3 ) = C ( p 6=22; p 3=22; p 2=22). In order to keep the overall size N small, we choose ( ; i 2 ; i 3 ) = (1; 1; 1) and obtain (N 1 ; N 2 ; N 3 ) = (4; 6; 10) from the bound condition (34), which results in N = 240. The phase vector f i1 ; i2 ; i3 g is set at (0; 0; 0). We choose C = 0:5 to ensure a non-negative input rate in (33). Assuming an average source access rate eual to 7.04 Mbps, for 44 bytes per packet, we get = 20; 000 packets per second. Figure 8 shows the corresponding input power spectrum. Consider a ueueing system with K = 255 to support such a single video source. The original input power spectrum is characterized by the above three eigenvalues ( i1 ; i2 ; i3 ). The results are listed in Table 2 as a function of, where is changed by adjusting the service capacity while keeping the input rate process unchanged. Since the input power in the high-freuency band has an insignicant eect on ueue, one may assign C i3 = 0 to completely remove i3, which was originally contributed by packet (or pixel) autocorrelation. The remaining system contains the rst two input eigenvalues with respect to the above given ( ; i 2 ), (C i1 ; C i2 ), ( i1 ; i2 ) and (N 1 ; N 2 ). Based on 2(Ci Ci 2 2 ) = C 2, we have reduced C from 0:5 to 0:452. Importantly, N is reduced from 240 to 24, which signicantly simplies the numerical analysis. Removing i3 is similar to implementing a low pass ltering function at the input process. Also listed in Table 2 are the ueue responses to the remaining ( i1 ; i2 ). Furthermore, one may remove i2, thus eliminating the line autocorrelation of the video stream. Only a single input eigenvalue i1 will be left in the system, described by the above given ( ; C i1 ; i1 ; N 1 ). C is then reduced to 0:369 and N to 4. The corresponding results are also included in Table 2. The inspection of data in Table 2 shows almost no dierence in ueue response to the above three sets of input eigenvalues. This is because for = 20; 000 packets per second, we have the central freuency of the three bells for the video source given by 1 2 (! ;! i2 ;! i3 ) = (0:00125; 0:32; 1:92). Furthermore, with 2 (0:6; 1:0), we get! i 2 2 0:192 and! i 3 2 1:15. We know from the results in Section 4.1 that a bell component in the input power spectrum, centered in a high-freuency band such as! 2 0:1, has virtually no impact on ueues. That is why both i 2 and i3 can be 18

20 3-eig. case 2-eig. case 1-eig. case ( ; i 2 ; i 3 ) ( ; i 2 ) ( ) : L 4.48e e e : L 3.27e e e : L 1.20e e e : L 3.94e e e-2 Table 2: Queue response to video input with three eigenvalues. neglected. Our study clearly indicates the dominance of the input eigenvalues in the low-freuency band for ueueing analysis. One should realize that the strong scene correlation, observed in video statistics [14], can introduce much low-freuency power to the input process, which denitely will have a great impact on Q ueues. M Because of N = l=1 N l, such an eigenvalue composition approach is limited by the high computation cost of ueueing analysis, especially if more narrowband bells are centered in the highfreuency band. The construction of a single dimensional circulant ~a from an eective eigenvalue vector ~ i is currently under study. 5 Queue Response to Input Bispectrum The bispectrum represents third-order input statistics, which in this paper are uniuely characterized by the phase spectral vector ~ for a given input power spectrum. We have ~ = argf~f? g, with properties of 0 = 0 and l = N?l for 0 < l < N. Again, tuning ~ has no eect on the input power spectrum ( ~ ; ~ ). The bispectrum eect on ueues is therefore examined via the phase vector ~. For a single input eigenvalue, the phase information is redundant in the spectral domain. This is because one can readily show in (21) that argf l1 l 2 g = 0, 8l 1 ; l 2, hence the input bispectrum is solely determined by the power spectrum. The corresponding input rate vector ~ in (27) consists of a single sinusoidal plus DC term. Ideally, as N! 1, the single input phase angle i should have no eect on the input rate steady state distribution, since the distribution of the circulant in each state is uniform. Taking i = 1 for example, Figure 9 shows the solution of (; ; L) as a 19

21 _ 34.1 σ L (in log ) N θ N θ Figure 9: Queue response to input bispectrum with a single eigenvalue. N θ 1 function of N and 1 at (; K; C ) = (0:8; 255; 0:7) as in Figure 4, with respect to the single input eigenvalue designed by 1 2 (! 1; B 1 ) = (0:02; 0:04). Obviously, when N < 8, the eect of 1 on the ueue solutions becomes visible and appears to be periodic in nature. This is caused by the change of the input rate distribution through the variation of 1 for each given N. For two eective eigenvalues, we choose ( i1 ; i2 ) = C 2 (1; 1) for each bell with identical average power. Again, the system is designed at (; K; C ) = (0:8; 255; 0:7), as in Figure 4. Here we x ( ; i 2 ) at (1; 1) for (N 1 ; N 2 ) = (10; 10). In a numerical study, three typical sets of ( i1 ; i2 ) are selected for the construction of the two-bell power spectrum: Set 1: Two bells are selected in such a way that each bell has a virtually euivalent impact on a system's performance, designed by 1 2 (! ; B i1 ) = (10?3 ; 2 10?3 ) and 1 2 (! i 2 ; B i2 ) = (8 10?4 ; 5:2 10?4 ). Set 2: Allocate one bell in the low-freuency band and one bell in the high-freuency band, designed by 1 2 (! ; B i1 ) = (4 10?3 ; 8 10?3 ) and 1 2 (! i 2 ; B i2 ) = (1:6 10?1 ; 1:04 10?1 ). Set 3: Two bells have identical bandwidth and harmonic central freuencies, designed by 1 2 (! ; B i1 ) = (4 10?3 ; 8 10?3 ) and 1 2 (! i 2 ; B i2 ) = (8 10?3 ; 8 10?3 ). In general, the input bispectrum is found to have a negligible impact on system performance. Figure 10 displays the microscopic changes of (; ; L) as a function of i1 and i2 2 (0; 2] for Set 1. Again, the periodic wave form in Figure 10 is caused by the change of input rate distribution by varying i1 and i2. For instance, to reduce the impact of ~ i on the input rate vector ~, let us increase (N 1 ; N 2 ) from (10; 10) to (15; 15) while the rest of the parameters remain unchanged, as in Figure 10. The corresponding solutions are plotted in Figure 11, based on the same scales as in Figure 10. In comparison with Figure 10, the eect of ~ i on the system performance completely vanishes. Similar results are obtained in Figure 12 for Set 2. Since i1 the low-freuency band, the eect of i1 is associated with the bell in on system performance is more signicant than that of 20

22 _ σ L (in log ) π 2π θ i 2 θ π 2π θ i 2 θ π 2π θ i 2 θ Figure 10: Queue response to input bispectrum with two eigenvalues at (N 1 ; N 2 ) = (10; 10) in Set 1. _ σ L (in log ) π 2π θ i 2 θ π 2π θ i 2 θ π 2π θ i 2 Figure 11: Queue response to input bispectrum with two eigenvalues at (N 1 ; N 2 ) = (15; 15) in Set 1. θ i2. The same observation is made in Figure 13 for Set 3. Note that all our results are plotted within extremely small ranges in order for us to observe the microscopic eect of ~ i on ueue. One may therefore expect that the eect of the input bispectrum on ueueing performance is generally unimportant. This is also consistent with the observation made in [1] for the superposition of two input sinusoidals. 6 Queue Response to Input Rate Steady State Distribution Power spectrum and bispectrum provide only partial probabilistic information of the input rate in steady state. In many system theories, they suce for characterizing most interesting properties. As shown in this paper, the same is true for ueueing system analysis. In the above sections, we have used ( ~ i ; ~ i ) for the power spectrum and ( ~ i ; ~ i ) for the bispectrum. Both spectra are independent of index i. Hence, index i can be used to adjust the input rate steady state distribution, dened by the permuted input rate vector ~ p in Section 3.3, while keeping its rst two moments and C 2 unchanged. According to (33), changing index i is euivalent to adjusting each individual sinusoidal freuency i l N l, 8l, for the input rate vector ~. Since both il and Nl?i l are conjugate, we only need to have i l < N l 2. Consider a single input eigenvalue i, constructed by an N-state circulant. Its input rate vector 21