Appendix A Functions and Transforms
A.1 Nonlinear Transforms
Many theoretical and practical problems can be converted into easier forms if instead of the discrete or continuous distributions their different transforms are applied, which can be solved more readily. In probability theory, numerous trans- forms are applied. Denote by F the distribution function of a random variable X and by f the density function if it exists. The general form of the most frequently used transform depending on the real or complex parameter w is
∞ E wX = wx dF(x). −∞
If the density function exists, then the last Riemann-Stieltjes integral can be rewritten in the form of Riemann integral as follows: ∞ E wX = wx f(x)dx. −∞
1. In the general case, setting w = et ,t∈ R,wehavethecharacteristic function (Fourier-Stieltjes transform) ∞ ϕ(t) = E etX = e txdF(x). −∞
2. If the random variable X has a discrete distribution with values 0, 1,... and probabilities p0,p1,...corresponding to them, then setting z = w, |z| < 1, we get
© Springer Nature Switzerland AG 2019 547 L. Lakatos et al., Introduction to Queueing Systems with Telecommunication Applications, https://doi.org/10.1007/978-3-030-15142-3 548 A Functions and Transforms
∞ ∞ X x k G(z) = E z = z dF(x)= pkz , −∞ k=0
which is the generating function of X. 3. The Laplace-Stieltjes transform plays a significant role when considering random variables taking only nonnegative values (usually we consider this type of random variable in queuing theory), which we obtain with w = e−s,s≥ 0: ∞ ∼ − F (s) = E e−sX = e sxdF(x). 0
For the case of continuous distributions it can be rewritten in the form ∞ ∗ − ∼ f (s) = E e−sX = e sxf(x)dx = F (s), 0
where f ∗ denotes the Laplace transform of the density function f . 4. The generating function plays a significant role when considering discrete random variables taking only nonnegative values, which we obtain with w = z:
∞ X n f(z)= E z = fnz . n=0
The identical background of the transformations given above determines some identical properties. When considering various problems, the use of separate trans- forms may be advantageous. For example, in the case of random variables taking nonnegative integer values the z-transform, and in case of general nonnegative random variables the Laplace-Stieltjes or Laplace transform is favorable to apply. Note that we define the transforms given above for more general classes of functions than the distribution functions.
A.2 z-Transform
Let f0,f1,...be a sequence of real numbers and define the power series ∞ n 2 n f(z)= fnz = f0 + f1z + f2z + ...+ fnz + ... (A.1) n=0
It is known from the theory of power series that if the series (A.1) is not everywhere divergent except the point z = 0, then there exists a number A>0 such that the K | n| ∞ | | series (A.1) is absolute convergent ( n=0 fnz < )forall z
±A depending on the values of the parameters fi ,i= 0, 1,... The number A is called the convergence radius of the power series (A.1). By the Cauchy-Hadamard theorem, A can be given in the form
1/n A = 1/a, where a = lim sup (|fn|) . n→∞
In the last formula we set A =+∞if a = 0andA = 0ifa =+∞.Thefirst relation A =+∞means that the power series (A.1) is convergent in all points of the real line, and the second one means that Eq. (A.1) is convergent only at the point z = 0. = K n A finite power series f(z) n=0 fnz (K-order polynomial, which corre- sponds to the case fi = 0,i≥ K + 1) is convergent at all points of the real line.
Definition A.1 Let f0,f1,...be a sequence of real numbers satisfying the condi- 1/n tion a = lim sup (|fn|) < ∞. Then the power series n→∞
∞ n f(z)= fnz , |z| It is clear from this definition that if we use a discrete distribution fn,k≥ ∞ 0, fk = 1, then the z-transform of the sequence f0,f1,...is identical with the k=0 generating function G(z), which was introduced earlier. A.2.1 Main Properties of the z-Transform 1. Derivatives. If the convergence radius A does not equal 0, then the power series f(z)is an anytime differentiable function for all points |z| k ∞ d n−k f(z)= n(n − 1)...(n− k + 1)fnz ,k≥ 1. dzk n=k 2. Computing the coefficients of the z-transform. For all k = 0, 1,...the following relation is true: 1 dk f = f(z) ,k≥ 0. (A.2) k ! k k dz z=0 It is clear from relation (A.1) that if the condition A>0 holds, then the function f(z) defined by the power series (A.1) and the sequence f0,f1,... uniquely 550 A Functions and Transforms determine each other, that is, the z-transform realizes a one-to-one correspondence between the function f(z)and the sequence f0,f1,.... = ∞ n = ∞ n 3. Convolutions. Let f(z) n=0 fnz and g(z) n=0 gnz be two z- transforms determined by the sequences fn and gn, respectively. Define the sequence hn as the convolution of fn and gn,thatis, n hn = fkgn−k,n≥ 0. k=0 = ∞ n Then the z-transform h(z) n=0 hnz of the sequence h0,h1,...satisfies the equation h(z) = f(z)· g(z). A.3 Laplace-Stieltjes and Laplace Transform in General Form Let H(x), 0 ≤ x<∞ be a function of bounded variation. A function H is said to be of bounded variation on the interval [a,b], if its total variation VH ([a,b]) is bounded (finite). The total variation is defined as KP VH ([a,b]) = sup H(xP,k) − H(P,k−1) P k=1 where the supremum is taken over the set of all partitions ={ = = } P xP,0 a ∞ ∗ − ∼ h (s) = e sxh(x)dx = H (s). 0 Theorem A.1 If the integral (A.3) is convergent for s>0,thenH ∼(s), s > 0 is an analytic function, and for every positive integer k k ∞ d ∼ = −sx − k k H (s) e ( x) dH(x). ds 0 The transform H ∼ satisfies the following asymptotic relation [2]. If the inte- gral (A.3) is convergent for Res>0 and there exist constants α ≥ 0andA such that H(x) A lim = , x→∞ xα Γ(α+ 1) then the convergence ∼ lim sαH (s) = A (A.4) s→0+ holds. Theorem A.2 Assume that there exists a function h(x), x ≥ 0, and its Laplace transform h∗(s), s > 0; moreover, the function h(x) is convergent as x →∞, i.e., lim h(x) = h∞. Then x→∞ ∗ lim sh (s) = h∞. s→0+ = x ≥ = Proof Denote H(x) 0 h(s)ds, x 0. Choosing α 1wehave H(x) 1 x h∞ lim = lim h(s)ds = h∞ = , →∞ →∞ x x x x 0 Γ(1 + 1) thus by relation (A.4) the assertion of the theorem immediately follows. Theorem A.3 If there exists a Laplace transform f ∗ of the nonnegative function f(t),t ≥ 0, and there exists the finite limit lim f(x)= f0,then x→0+ ∗ lim sf (s) = f0. s→∞ 552 A Functions and Transforms Proof It is clear that ∞ ∞ − − s e sxdx = e x dx = 1 0 0 and ∞ ∞ √ −sx −y − s s √ e dx = √ e dy = e = o(1), s →∞; 1/ s s therefore, √ 1/ s ∞ ∗ −sx −sx sf (s) − f0 = s e [f(x)− f0]dx + s √ e f(x)dx + f0o(1). 0 1/ s Since there exists the finite limit lim f(x)= f0, with the notation x→∞ δ(z) = sup |f(x)− f0| → 0,z→ 0+, 0 On the other hand, for all 0 ∞ ∞ ∗ − − ∗ f (s) = e sxf(x)dx ≤ e txf(x)dx = f (t), 0 0 is true, then ∞ √ ∞ −sx −(1/2) s −(s/2)x s √ e f(x)dx ≤ se √ e f(x)dx 1/ s 1/ s √ ∞ √ − − − ∗ ≤ se (1/2) s e (s/2)xf(x)dx = se (1/2) s f (s/2) 0 √ − ∗ ≤ se (1/2) s f (1) → 0, as s →∞(s ≥ 2). Summing up the results obtained above, the assertion of the theorem follows. A.3 Laplace-Stieltjes and Laplace Transform in General Form 553 A.3.1 Examples for Laplace Transform of Some Distributions (a) Deterministic distribution (a>0, P(X = a) = 1): ∞ ∼ − − F (s) = e sxdF(x)= e sa, E(X) = a : 0 (b) B(n, p) binomial distribution: ∞ n ∼ − n − − F (s) = e sxdF(x)= pk(1 − p)n ke sk k 0 k=0 n n − − − = (pe s )k(1 − p)n k =[1 + p(e s − 1)]n, k k=0 − − − E(X) = npe s [1 + p(e s − 1)]n 1 = np. s=0 (c) Poisson distribution with parameter λ: ∞ ∞ k ∼ − − λ − F (s) = e sxdF(x)= e sk e λ k! 0 k=0 ∞ 1 − − − = (λe s )ke λ = exp{λ(e s − 1)}, k! k=0 = −s { −s − } = E(X) λe exp λ(e 1) s=0 λ. (d) Uniform distribution on the interval [a,b]: b 1 −sa −sb ∼ − 1 (e − e ), s > 0, F (s) = e sx dx = s(b−a) a b − a 1,s= 0. and by the use of the L’Hospital’s rule: 1 1 − − − − E(X) = lim − [e sa − e sb]−[sae sa − sbe sb] b − a s→0+ s2 2 1 b2se−sb − a e−sa b2 − a2 a + b = lim = = . b − a s→0+ 2s 2(b − a) 2 554 A Functions and Transforms (e) Exponential distribution with parameter λ: ∞ ∞ ∼ − − − + λ F (s) = e sxλe λxdx = λ e (s λ)xdx = , s + λ 0 0 λ 1 E(X) = = . + 2 (s λ) s=0 λ A.3.2 Sum of a Random Number of Independent Random Variables Theorem A.4 Let K be a random variable with nonnegative integer values, and K consider the sum of K random variables Y = Xn,where n=1 (1) The random variables K and {Xn,n≥ 0} are independent. (2) The distributions of the random variables Xn are identical with common distribution function F(x). ∼ Denote by FX (s) the Laplace-Stieltjes transform of Xn and by GK (z) the generating function of K. Then the Laplace-Stieltjes transform of random variable Y has the form −sY = ∼ E e GK (FX (s)). Proof Since K − = = ∼ k E exp s Xn K k FX (s) , n=1 then we obtain by the use of the formula of total expected value ∞ K K E exp −s Xn = E exp −s Xn K = k P(K = k) = = n 1 k=1 n 1 ∞ = ∼ k = = ∼ K = ∼ FX (s) P(K k) E FX (s) GK (FX (s)). k=1 A.4 Bessel and Modified Bessel Functions of the First Kind 555 A.4 Bessel and Modified Bessel Functions of the First Kind Definition A.2 The nonzero solutions of Bessel’s differential equation x2u + xu + (x2 − ν2)u = 0(A.5) are called v-order Bessel functions,wherev is a real number. Definition A.3 The solutions of Bessel’s differential equation are called Bessel functions of the first kind and denoted by Jv(x), which are nonsingular at the origin x = 0. The v-order Bessel functions of the first kind Jν(x) (v ≥ 0) can be defined by their Taylor series expansion around x = 0 as follows: ∞ (−1)k x 2k+ν Jν (x) = , (A.6) Γ(k+ ν + 1)Γ (k + 1) 2 k=0 ∞ where Γ(x)= e−t tx−1dt is the gamma function. This formula is valid, providing 0 v =−1, −2,.... The Bessel functions ∞ (−1)k x 2k−ν J−ν (x) = Γ(k+ ν + 1)Γ (k + 1) 2 k=0 is given by replacing v in Eq. (A.6) with a −v. An important special case of the Bessel function of the first kind is that of a purely imaginary argument. Definition A.4 The function ∞ 2k+ν −v 1 x Iν (x) = Jv(x) = Γ(k+ ν + 1)Γ (k + 1) 2 k=0 is called a modified v-order Bessel function of the first kind. Both the Bessel functions Jν(x) and Iν (x) can be expressed in terms of the generalized hypergeometric function 0F1(v; x) as follows [1]: 1 x ν x2 Jν (x) = F (ν + 1;− ), Γ(ν+ 1) 2 0 1 4 1 x ν x2 Iν (x) = F (ν + 1; ), Γ(ν+ 1) 2 0 1 4 556 A Functions and Transforms where ∞ Γ(ν) F (ν; x) = xk. 0 1 Γ(k+ ν)Γ (k + 1) k=0 A.5 Notations N+ Set of the nonnegative integer numbers R Set of the real numbers (R = (−∞, ∞)) Imaginary unit (2 =−1) δij Kronecker delta function, that is, δij = 1, if i = j, otherwise it equals 0 a+ Positive part of a real number a, i.e., a+ = max(a, 0) A Complementary event of A I {A} Indicator function of an event A, that is, it equals 1 if the event A occurs and otherwise it equals 0 P(A) Probability of an event A E(X) Expected value of a random variable X D(X) Variation of a random variable X S State space of a Markov chain Π (One-step) transition probability matrix of a discrete-time Markov chain Q rate matrix of a continuous-time Markov chain I Unit matrix References 1. Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I.: Integrals and Series, vol. 2. Gordon and Breach Science Publishers, New York (1986). Special functions 2. Takács, L.: Combinatorial Methods in the Theory of Stochastic Processes. Wiley, London (1967) Index A logarithmic normal distribution, 44 Adaptive traffic class, 451 normal distribution, 42 ALOHA, 472 Pareto distribution, 45 uniform distribution, 40 Weibull distribution, 45 B Continuous-time Markov chain, 124 Bandwidth sharing, 445 Convergence of random variables, 46 Bessel function, 307, 555 convergence in distribution, 46 Birth–Death process, 138, 142 convergence in mean square, 47 Birth–Death-Immigration process, 138 convergence in probability, 47 Birth rate, 141, 142 convergence with probability 1, 47 Death rate, 141, 142 weak convergence, 46 Immigration rate, 141 CSMA, 477 Burke’s theorem, 422 CSMA/CD, 477 nonpersistent, 480 C persistent, 480 Call admission control, 447 Cyclic waiting system Campbell’s theorem, 85 continuous, 506 Central limit theorem, 50 discrete, 517 Constant bit rate, 447 Continuous-time Markov chain Birth–Death process, 142 D embedded Markov chain, 131, 338 Dependent random variables, 13 infinitesimal matrix, 128 conditional distribution, 15 Markov reward model, 136 correlation, 31 reversible, 135 covariance, 30 short term behavior, 202 joint distribution function, 14 transition rate, 127 marginal distribution, 14 Continuous distributions, 40 Discrete distributions, 37 beta distribution, 42 Bernoulli distribution, 37 Erlang distribution, 42 binomial distribution, 37 exponential distribution, 41 geometric distribution, 38 gamma distribution, 42 negative binomial distribution, 39 Gaussian distribution, 42 Poisson distribution, 39 hyperexponential distribution, 41 polynomial distribution, 37 © Springer Nature Switzerland AG 2019 557 L. Lakatos et al., Introduction to Queueing Systems with Telecommunication Applications, https://doi.org/10.1007/978-3-030-15142-3 558 Index Discrete-time Markov chain, 94 P aperiodic, 104 PASTA, 293 ergodic, 122 Phase type distribution, 227 homogeneous, 96 acyclic PH distribution, 233 irreducible, 104 continuous-time, 227 positive recurrent, 112 discrete-time, 231 recurrent, 108 fitting, 235 stationary distribution, 116 hyper-Erlang distribution, 234 transient, 108 hyperexponential distribution, 234 transition probability matrix, 98 Pollaczek-Khinchin Displacement theorem, 87 mean value formula, 343 transform equation, 353 Priority service systems, 483 E Probability, 5 Elastic traffic class, 453 Q F Quasi-Birth–Death process, 248 Foster’s criterion, 342 condition of stability, 249 discrete-time QBD, 265 finite QBD, 252 G inhomogeneous QBD, 256, 264 G/M/1 Type Process, 270 level process, 258 matrix geometric distribution, 249 partially homogeneous QBD, 255 I transient measures, 258 IEEE 802.11, 482 Queueing network, 421 Indicator, 10, 220 BCMP type, 436 Inequalities, 25 closed, 431 Chebyshev inequality, 25 convolution algorithm, 433 Markov inequality, 25 Gordon-Newell type, 431 Jackson type, 425 mean value analysis, 435 K non-product form, 439 Kaufman-Roberts method, 447 open, 425 Kendall’s notation of queueing systems, 286 product form solution, 424 Kolmogorov criterion, 134 traffic based decomposition, 440 Queueing system, 286 busy period, 308 L G/M/1 queue, 372 The Laws of the large numbers, 48 G-queues, 528 Little’s law, 289 Lindley integral equation, 337 loss probability, 316 MAP/MAP/1 queue, 399 M MAP/PH/1/K queue, 399 Markov arrival process, 235 MAP/PH/1 queue, 398 batch Markov arrival process, 243 M/G/1 queue, 335 marked Markov arrival process, 247 M/M/m/m queue, 315 Markov chain, 94 Erlang B formula, 316 Markov property, 200 M/M/m queue, 311 Markov regenerative process, 217 Erlang C formula, 314 Medium access control, 471 M/M/1//N queue, 316 Memoryless property, 38, 41 M/M/∞ queue, 314 M/G/1 Type Process, 273 M/M/1 queue, 299 Index 559 M/PH/1 queue, 392 Renewal process, 179 negative arrivals, 528 delayed renewal process, 180 PASTA property, 396 renewal function, 180 PH/M/1 queue, 387 Restricted process, 159 response time, 392 retrial queuing system, 497 system time, 392 S utilization, 302 Semi-Markov process, 207 virtual waiting time, 365 Stochastic process, 63 waiting time, 392 Brownian motion Wiener process, 67 Gaussian process, 66 R higher dimensional poisson process, 81 Random access protocol, 471 Poisson process, 67 Random variable, 9 stationary process, 65 characteristic function, 27 stochastic process with independent and coefficient of variation, 25 stationary increments, 66 continuous, 12 Supplementary variable, 212, 337 probability density function, 12 Switching fabric, 457 discrete, 11 probability mass function, 11 distribution function, 10 T Defective, 11 Taboo process, 159 Fourier-Stieltjes transform, 28, 547 Takács integro-differential equation, 366 Laplace-Stieltjes transform, 29, 548 Throughtput, 465 Laplace transform, 29, 550 mean, 20 moment, 22 V probability generating function, 26, 548 Variable bit rate, 449 standard deviation, 23 variance, 23 z-transform, 27, 548 W Regenerative process, 193 Wald’s lemma, 88 regenerative point, 200 Weak law of large numbers, 26