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1 .5 / -- Communication Networks II (Görg) -- Transforms Using different summation and integral transformations pmf, pdf and cdf/ccdf can be transformed in such a way, that even originally discrete functions become continuously differentiable in the transformed range. The characteristic values and moments (defined in chapter.4) can be calculated through differentiation of the transforms. Most of the operations presented below, especially the convolution, can be conducted d much more easily in the transformed range.

2 .5 / 2 -- Communication Networks II (Görg) -- 5G.5. GeneratingFunction i (z-transform) )for the discrete pmf.5.. z-transform The generating function G(z) of a pmf P(k) of a non-negative RV K 0 is obtained through the following power series, which h is also called the z-transform: f ( ) ( ) ( ) ( ) ( ) Gz = P0 + P z+ P2 z 2 + K= Pkz k k=0= 0 with z = x + iy = r e Each of the powers z i has the effect of a stretching cum rotating factor for 0 < r, as the RV K 0. iϕ (.5.) Thus the power series G(z) converges at least for the z-values "on and within the unit circle". Further it holds, especially for z =, the total probability condition in the form G ( ) P( k) k = 0 = = (.5.2)

3 .5 / 3 -- Communication Networks II (Görg) -- The radius R of the convergence circle of G(z), is in general larger than. o < r < R, < R < (.5.3) However, this radius R is to be obtained from case to case. Equation (.5.) transforms the discrete function P(k) into a continuous function G(z), which turns out to be a known function in most of the cases. Through power series expansion of G(z) per equation (.5.), ) the unknown function P(k) can be obtained as the coefficient of a general member of the resulting series, compare also equation (.5.0). 0) This property justifies the name generating function. In other cases P(k) is known and the function G(z) is needed. The generating function corresponding to the binomial distribution is given by: Gz ( ) = ( q+ pz) n (.5.4)

4 .5 / 4 -- Communication Networks II (Görg) -- This function has an n-fold zero point at z = - q/p and converges even for all values of z (i.e. for 0 < r < ), due to the finite number of members. Figure.5.: Binomial i distribution: ib ti p=0.5, 05 n=8. 8 Figure.5.2: G(z) of binominal distribution for real z (i.e. z=x, y=0)

5 .5 / 5 -- Communication Networks II (Görg) Moments of a pmf from the z-transform The generating function G(z) is continuous; it can thus be differentiated w.r.t. z. This property is used to obtain the moments of P(r). The r-th derivative of G(z) has the form G r z k k k k r z k r P k k = ( r ) ( ) = k r (55) (.5.5) ( )( ) ( + ) 2 K ( ) FM r K For z =, the summation leads to the factorial moment ( ) ( ) ( ) = G r compare with equation (.4.) (.5.6) The evaluation of equations (.5.6) and (.5.5) 5) leads to the moments given below ( FM ) ( K) = kp( k) = G ( ) = k = M ( K ) k = (.5.7)

6 .5 / 6 -- Communication Networks II (Görg) -- FM2 K = k k P k G 2 = = M2 K M K k = ( ) ( ) ( ) ( ) ( ) ( ) ( ) () ( 2 ( ) + G ) () M ( K ) compare eq. (.4.) G = 2 (.5.8) ( ) ( ) ( 2 K = k k G ) ( ) compare. (.4.4) Z 2 + eq ( ) ( ) ( ) ( ) ( ) ( ) M3 K = G + 3 G 2 + G 3 ( ) (.5.9) 2 3 G ( ) ( ) ( ) ( 2 ) ( ) ( 3 Z K = k 2k 3k + 3 k G + G ) ( )

7 .5 / 7 -- Communication Networks II (Görg) -- Generating a pmf from the z-transform If z = 0 is set in eq. (.5.5.), then all elements with r k result in zero and only the element with r = k remains. From this results a method to determine P(k) by differentiating G(z) k times: ( ) ( ) ( ) ( ) Pk = k! G k 0 (.5.0)

8 .5 / 8 -- Communication Networks II (Görg) -- Simple Examples ( ) P k = k = ; 0 0; k ) For the unit impulse function (.5.0a) it is G ( z ) = { ( )} = GPn k z k

9 .5 / 9 -- Communication Networks II (Görg) -- ( ) 2) Unit impulse sequence P k = ; k IN it is 0 k ( ) z Gz = = k=0 z z < (.5.0b) 3) Geometric sequence Pk ( ) = ab k ; k IN it is 0 k k a Gz ( ) = a bz = bz k = 0 bz < (.5.0c) 4) Poisson pmf ( λt) λ t ; λ 0, t 0 it is PK ( = k) = k! k e Gz ( ) = λ e tz ( ) (.5.0d)

10 .5 / 0 -- Communication Networks II (Görg) -- Convolution Theorem (Faltungssatz) of the z-transform The convolution of the pmfs P (n) and P 2 (n) is defined as ( ) P P2 n : = Pn kp2 k=,, P2, n kp, k (.5.0e) k= 0 According to eq..5.: k= 0 n k k n 2, 2, k k n= k k = 0 ( ) GP P = P P z z (.5.0f) where z n has been split in z n-k and z k. After exchanging the sequence of taking the sums the following is obtained: ( ) k n k j GP P = P z P z = P z P z 2 2, k, n k, j 2, k k= 0 n= k j= 0 k= 0 ( ) = ( ) ( ) G P P G z G z 2 2 k (.5.0g)

11 .5 / -- Communication Networks II (Görg) -- Generating function of the discrete cumulative distribution function (cdf) Similar to using G(z) to generate the pmf P(k), also cumulative distribution functions can be generated according to eq. (.3.9/0). The generating function of the ccdf PK ( > k) = PK ( k) with k = 0,, 2,... can be given as a power series (Potenzreihe) ( ) ( ( ) ( ) ( )) ( ) ( ) ( ) ( ) 2 Qz = P0 + P0 P z+ P0 P P2 z + K k Q ( z ) = P ( K k ) z k ( ) (.5.) comparefigure.5.3:

12 .5 / 2 -- Communication Networks II (Görg) -- If the expectation k exists, then Q(z) converges on and within the unit circle, compare eq. (.5.3). The following important relation is valid: ( )( ) ( ) Qz z = Gz (.5.2) It follows for z= using the rule of l'hospital: ( ) ( ) ( ) (53) (.5.3) Q = G = k The first derivative is not defined for z=: Figure.5.3: Generating functions of the cdf Q ( ) ( ) z = ( zg ) () ( z ) Gz ( ) ( ) 2 + z (.5.4)

13 .5 / 3 -- Communication Networks II (Görg) -- Using the rule of l'hospital it follows: compare eq.(.5.8) Q () () The function R(z) for generating the cdf ( ) PK ( 2 ) () = G, resp. ( ) σ 2 = 2 Q ( ) k ( k ), 2 k is the following: Rz = PK kz k k = 0 ( ) ( ) ( ) ( ) Gz = Qz = z z (.5.5) R(z) cannot converge for z, as a sum with infinitely many components results that all tend to the value. In the calculation of random events often G(z) is obtained first. If the cdf and not the pmf P(k) is of interest, it can be directly derived from the series expansion of Q(z) resp. R(z). ) Corresponding to eq. (.5.0) 0) it follows: ( ) ( ) ( ) PK > k = k Q k 0! resp. ( ) ( ) ( ) PK k = k R k 0! (.5.6)

14 .5 / 4 -- Communication Networks II (Görg) -- (deleted) Laplace-Transform For positive RVs the Laplace transform is generally preferred. By a suitable substitution of the complex variable z in eq. (.5.) and now considering the continuous case from the generating function G(z)theLaplace-Stieltjes Transform (LST) for positive ZV is obtained, where L(s) is a function symbol [ ( ) ] = () = ( ) st Lpt Ls pte dt 0 t ( 0) 0 ; L = (.5.24) If the RV is a time RV, then the complex variable s = σ + iω has the dimension sec. (.5.25)

15 .5 / 5 -- Communication Networks II (Görg) -- The LST is linear. (.5.24) Example: For the negative exponential distribution: pdf: ( ) p t = μ e μt μ LST: Ls ( ) = s + μ ; with a pole at s = μ. (.5.26) see Figure.5.5

16 .5 / 6 -- Communication Networks II (Görg) -- Figure.5.5: μ t pdf of the negative exponential distribution: pt () = μe ( μ = s ) and corresponding LST: ( ) L s = s + μμ for s = σ real.

17 .5 / 7 -- Communication Networks II (Görg) -- From the r th derivative of L(s) with respect to s: ( ) ( ) ( ) ( ) r r r st L s = t p t e dt (.5.27) 0 result the absolute moments of p(t): M r T = r L r 0 ; with ( ) ( ) ( ) ( ) If ( k ) t M L ( = = ) ( 0 ) M k exists, then the cdf has a rational LST. PT The cdf ( ) [ ( t ) ] ( ) L P T t = s L s t has the LST: (.5.28) (.5.29) = s Ls. In general the integral results in L p( τ ) dτ ( ) Figure.5.6 also shows the LST of the ccdf PT ( > t) = PT ( t) t 0 : LPT ( > t) = [ ] L ( s ) s (.5.30)

18 .5 / 8 -- Communication Networks II (Görg) -- The inverse transform to determine p(t), resp. PT ( t) and PT ( t) inverse integral: +i σ ω ( ) ts ( )d) p t = 2π i σ i ω e L s ds > results from the the integration path has to be chosen appropriately p in the right part of the s-plane. σ=const. Is the so-called convergence generating abscissa. Practical methods for calculating the equations concern among others the series expansion of L(s) with the residue theorem and using appropriate tables [8]. (53) (.5.3) The inverse transformation relies on the poles (so-called singularities of finite order) of the complex function L(s). For bounded or sufficiently slow growing p(t) theybelongto to the left half plane, because for these p(t) L(s) converges according to eq. (.5.24) in the right half-plane of s.

19 .5 / 9 -- Communication Networks II (Görg) -- Figure.5.6: Laplace-Stieltjes transforms of cdfs

20 .5 / Communication Networks II (Görg) -- μ t 3 p()= t μ e mit μ = 2 0 With a series expansion (Reihenentwicklung) the LST can also be expressed using absolute and central moments for the e-function: sec j j s st s L ( s ) = + M ( T ) e Z ( T ) j = + j j= j! j= 2 j! (.5.32) Special case: If the pdf p(t) contains a simple pole at t=t, that can be described by the Dirac-function ( ) ( ) pt = Pδ t, then the LST of the pole can be represented by: [ ( )] Lpt Pe ts =. Obviously a discrete RV K with pmf P(K=k) can also be described by a pdf pk ( ) = Pk ( ) δ ( k) as a sequence of poles. The corresponding LST is given by: [ ( ) ] = ( = ) (.5.33) L p k P K k e sk ; (.5.34) k

21 .5 / 2 -- Communication Networks II (Görg) -- this is the same as eq. (.5.) with z The LST for the difference quotient 0 ( ) dp t dt = e s. ( ) dp t dt is: ( ) ( ) ( ) ( ) e st st st dt= pte + s pte dt= p+ 0 + sls. (.5.35) 0 0 By taking the limit the starting and end value p(0) resp. p() of the original function p(t) can be evaluated: ( ) lim sl s s ( ) dp t st = lim e dt+ p 0 = p 0 s 0 dt ( ) ( ) (.5.36) ( ) lim sl s ( t ) dp t st = lim e dt + p( 0) = p( t) + p( 0) = p( ) = 0 s dt s (.5.37) In complicated functions L(s) ) it is advisable to check if this is fulfilled in addition to the condition for total probability L(0)=. According to the definition of the LST the convolution of two RV compare eq results in

22 .5 / Communication Networks II (Görg) -- F st sτ s( t τ ) p( τ ) p2( t τ ) dτ = p( τ ) p2( t τ ) e dτdt = p( τ ) e p2( t τ ) e dt dτ If p2 () t vanishes for negative arguments, then it is true for all τ : ( ) st p t e ( τ τ ) dt L p () t = 0 2 [ ] 2, so that: [ ( ) ( ) ] = [ ( ) ] [ ( ) ] L p t p t L p t L p t 2 2 (.5.38)

23 .5 / Communication Networks II (Görg) --

24 .5 / Communication Networks II (Görg) -- Important distributions and their description with phase models.5.5. Important distributions The description used most often to describe arrival and service time cumulative distribution functions P ( T t) is the negative exponential cdf (table.5.). The corresponding distribution describes interarrival and service times. In this case a so-called Markov Process is obtained, which can be treated in a mathematically relatively simple way. Service time distributions, however, are often approximated by a constant distribution (table.5.). ) The mathematical ti treatment t t of such traffic is much more difficult, as the so-called Markov feature (independence of the history) is lost. Both distributions are limits of the Erlang k distribution (table 5 ) It is successfully Both distributions are limits of the Erlang-k-distribution (table.5.). It is successfully being used to describe arrival and service time distributions, whose distributions lie between constant and negative exponential.

25 .5 / Communication Networks II (Görg) -- A family of distribution functions without limitations is the General Erlang-distribution, consisting of a sum of weighted Erlang-k-distributions (Table.5.). Setting k n =, results in the class of hyper-exponential distribution functions (Table.5.). Erlang-k distributions and hyper-exponential distributions allow a good approximate description of many arrival and service time distributions that apply to practical environments, the mathematical treatment, however, is often quite difficult. These distributions have a smaller resp. larger variance than the neg. exponential distribution. Another method relies on approximating any type of distribution function by piecewise exponential distributions. In each single phase the Markov condition is fulfilled (table.5.). By choosing the appropriate parameters any distribution with any coefficient of variation can be approximated. Another important distribution results for n=2 and μ from the hyper-exponential exponential distribution. This distribution is called a degenerated negative exponential distribution (table.5.), because for p =0 the negative exponential distribution results.

26 .5 / Communication Networks II (Görg) --

27 .5 / Communication Networks II (Görg) -- The character of the General Erlang distribution (GE) can be interpreted in a very illustrative way, see figure.5.7 [Mor ]: Random variables of the GE type result from a transit with probability p n of one of the r parallel branches, that has k n sequential phases, that each represent one random variable with negative exponential distribution μ. It is easily seen that in the resulting distribution the model k n and expectation ( ) contains thefollowingcasesasspecialcases: special cases: n negative exponential ( r =, k n = ) constant ( r =, ) Erlang-k ( r =, < <) hyper-exponential exponential ( < r < k = ) k n k n r, k n.

28 .5 / Communication Networks II (Görg) -- requests Figure.5.7: Physical Interpretation of the General Erlang distribution r parallel phases, each with an Erlang-k distribution and mean /μ n. k n phases in one branch, each negative exponentially distributed with mean /(k n μ n ).

29 .5 / Communication Networks II (Görg) --

30 .5 / Communication Networks II (Görg) --

31 .5 / 3 -- Communication Networks II (Görg) -- type definition cdf expectation β r th absolute moment generating function resp. LST transform of the pdf constant 0 für 0 t < / μ (-point) PT ( t) = für t / μ μ = ( r) β β β μ r = Gz ()= zp= z 2-point PT ( t) = 0 für 0 t < t β = pi ti i= p für t t < t2 2 für t t 2 p i = i= 2 β ( r ) pt i ir = 2 = 2 G( z) z i p i= i i = Negative exp. μ P( T t) = e ( r ) r μ t β = r! β L ( s ) = μ + s β = β β μ

32 .5 / Communication Networks II (Görg) --

33 .5 / Communication Networks II (Görg) -- Erlang-k PT ( t) = k i k t ( k t) e μ μ β i= 0 i! = β μ r () r = ( + k )! β r ( k )! k s Ls () = ( + ) k kμ General Erlang P( T t) = r n= p n [ e kn i= 0 with knμ nt r β = n r pn i knμ nt) n= ( kn )! ]; p n? i! n= μn * r r ( knμn) pn = n= ( β ( r) = ( r + k )! hyperexponen n μ i t P( T t) = -tial pie i= of order n n with p = i= i β = n p i i= μ i β p ( r ) n p = r! i r i= μi n L( s) = pi i= μ i μ i + s Table.5.: Typical distribution functions for traffic models All cdfs are defined d for t 0 and P ( T t ) = 0 for t 0

34 .5 / Communication Networks II (Görg) --

35 .5 / Communication Networks II (Görg) --

36 .5 / Communication Networks II (Görg) -- A more general form than the Erlang-k k distribution is the so-called hypo-exponential distribution. The difference to the Erlang-k distribution is that the k exponential phases in the series do not all have the same service rate (Bedienrate) μ n, but different rates. This distribution ib ti is only mentioned here but not used in the following parts. Finally, the 2-point distribution or the more general n-point distribution should be mentioned. The latter is typically y the result of measurements in form of a cumulative frequency distribution and can be used to approximate continuous distributions.

37 .5 / Communication Networks II (Görg) -- Example: specialized Erlang distribution ib ti According to the convolution theorem the LST: Ls ( ) (n-): = μ μ + s n is a convolution of order ( ) pt = μe μt with itself, this means ( ) = ( μ μ ) pt e t n (.5.39) pdf: pt ( ) = n ( t) ( ) μ μ n! e μt ; and LST: Ls ( ) μ = s + μ n with a pole at s = μ (.5.40) p(t) is for real n the so-called Gamma pdf and for n IN the specialized Erlang pdf.

38 .5 / Communication Networks II (Görg) -- Phase models According to [Mor 58] any distribution can be represented by a suitable combination of so-called phases. If the distribution of a RV is formed by adding RVs of negative exponential distributions with equal or different parameter μ then figure.5.8 shows for those cases of the neg. exp. (M), Erlang-k (E k ) and constant (D) cdfs the corresponding phase model (compare table.5.). ) A second variant of the phase model is shown in figure.5.9, where the generated random variable is hyper-exponentially epoe y( (H r )ds distributed buedwith r phases and results s from a M-cdf with parameter μ i with the probability pi (compare table.5.). The general phase model has been given by Cox and is similar to the general Erlang distribution suitable to represent all distributions, that have a rational Laplace transform. This prerequisite is usually fulfilled in usual applications. Figure.5.0 shows equivalent variants of the model, where the following holds: i- p i = ( a i ) a k k= (.5.4)

39 .5 / Communication Networks II (Görg) -- Figure.5.8: phase model with k exponential phases, each with the rate kμ and corresponding cdf and pdf (k = : M-cdf, < k : E k -cdf, k = D-cdf)

40 .5 / Communication Networks II (Görg) -- Figure.5.9: phase model with r parallel exponential phases and corresponding cdf and pdf r = corresponds to the M distribution, r > corresponds to the H r distribution

41 .5 / 4 -- Communication Networks II (Görg) -- equivalent model: Figure.5.0: Phase model for COX distributions with n exponential phases with different rates μ i

42 .5 / Communication Networks II (Görg) -- Exercise: In addition to the just introduced phase model -- the following model holds (Figure.5.) for the General Erlang (GE) distribution Figure.5.: Model of the General Erlang (GE) distribution For which pair (k i, n) are the H n, E k, D-, and M- distributions obtained?

43 .5 / Communication Networks II (Görg) -- How to choose a distribution a) Starting Point: measured values of a real system relative frequency of the discrete RV K Value of RV K P(X x) x

44 .5 / Communication Networks II (Görg) -- b) calculation l of the distribution ib i parameters M = k P( k) = = 4.7 k M2 = k 2 P( k); σ k c) choose type of distribution: C = σ M d) approach of a suitable phase model (i.e. the corresponding distribution) 2 = M2 M 2, c < c = c > distribution type distribution type distribution type E H k M r d) require equal moments of measured and analytical distribution d2) Minimize mean squared difference of the actual measured values d2) Minimize mean squared difference of the actual measured values and those of the analytical distribution or approximate up to some remaining error

45 .5 / Communication Networks II (Görg) --

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