Communication Networks II Contents

Transcription

1 3 / 1 -- Communicaion Neworks II (Görg) -- Communicaion Neworks II Conens 1 Fundamenals of probabiliy heory 2 Traffic in communicaion neworks 3 Sochasic & Markovian Processes (SP & MP) 4 Markovian Sae Processes 5 Analysis of Markovian service sysems 6 Queues for modeling of communicaion neworks 7 M/G/1 model 8 The model M/G/1/FCFS/NONPRE 9 The model M/G/1/FCFS/PRE

2 3 / 2 -- Communicaion Neworks II (Görg) Sochasic and Markovian Processes (SP & MP) X is a Random Variable ou of he characerisic domain ; is in he se of numbers I; { X, I } is called a Sochasic or Random Process. Example: = ime a) I = IN (se of naural numbers): {X} is a Sochasic Process wih discree ime. b) I = IR (se of real numbers): {X} is a Sochasic Process wih coninuous ime. A process can be described by using is sae funcion (Muserfunkion): A process can be described by using is sae funcion (Muserfunkion): I gives he variaion of values of {X} over he ime.

3 3 / 3 -- Communicaion Neworks II (Görg) -- Example: sae funcion of he finie sae process for a group of elephone lines arrivals arrivals ime X = number of occupied lines deparures Figure 3.1: sae funcion of he finie sae process

4 3 / 4 -- Communicaion Neworks II (Görg) -- Componens of he SP (Sochasic Process): Sae X = x Sae of he SP a ime Sae space {X } Se of all possible saes a ime X discree SP wih discree saes ("Chain") X coninuous SP wih coninuous saes Parameer (generally) ime i, i 12,,K discree parameer 0 coninuous parameer

5 3 / 5 -- Communicaion Neworks II (Görg) -- Sae X Parameer discree coninuous 0 coninuous X xi number of busy X x service already received lines in he group a ime, remaining service ime in sysem a ime i discree X x i i number in sysem X i x service ime a arrival a he ime insan i ime i remaining service ime in sysem a ime i

6 3 / 6 -- Communicaion Neworks II (Görg) -- Dependencies wihin he process: X saisical dependencies beween he sae variables a differen insans of ime,,,, 1 2 n n Fx PX x 1 X x 2 K X x cdf ime vecor,, K, 1 2 n sae vecor,, K, x x1 x2 x n (3.1)

7 3 / 7 -- Communicaion Neworks II (Görg) Grouping of SPs Markovian-Processes (MP) A Markovian process is a special case of SP: The nex sae solely depends on he presen sae and does no direcly depend on he previous saes. Definiion: P X x X x X x X x X x P x X x n n n n n n Xn n n n 1 1, 2 2, K, 1 1, irrelevan n 1, 2,K K ; m I m 0,, 1 K, n; mi 0 1 K n ; for any real x n. (3.2)

8 3 / 8 -- Communicaion Neworks II (Görg) -- The cdf of a random variable X 2 is deermined dfor any pair of parameers, I hrough he condiional cdf 3) (3.3) 1 F,x,,x P X x X x 2 1 An SP is an MP, if is fuure behaviour depends only on he presen sae and is no influenced by he pas. 1 2 saes pas fuure ime ime insances presen Figure 3.2: Definiion of MP

9 3 / 9 -- Communicaion Neworks II (Görg) -- I is more convenien o use he condiional probabiliy mass funcion (pmf) in place of he condiional cdf of he process f x X x X x (, ) P n n Special cases: n n1 1 (3.3a) 1) Semi-Markov-Process (SMP)is a SP, whose fuure behaviour depends also on he duraion of he presen sae. Behavior similar o ha in MP a sae ransiions: embedded Markov chain, compare M/G/1, Chap.7. 2) Embedded Markov-Chain (EMC) ime discree SP wih Markov-propery. 3) Birh and Deah Process (BDP) a special MP, where ransiions can occur only o he neighbouring saes X X 1 n1 n

10 3 / Communicaion Neworks II (Görg) Non-Markovian Processes Processes wihou Markov propery, p or in general F ( x, ) P [ X x, X x,..., X x ] 1 2 n cdf (3.4),, f x x Fx 1 2 n Saionary Process: Fx, Fx, (3.5) An SP is saionary, if is cdf is ime invarian (invarian agains a ime shif of ) ie i.e. df x, d 0.

11 3 / Communicaion Neworks II (Görg) Poin Process (PP) Figure 3.3: 3 Time inervals beween evens in a poin process Special case Renewal process (also called recurren process) T i are independen of each oher and have he same cdf (iid-variable: independen and idenically disribued) i F P T independen d of index i. T i

12 3 / Communicaion Neworks II (Görg) -- Curren erminaion rae r() : T ends in (, +) las even ime Figure 3.4: Curren erminaion rae r 0 PT T lim Gesez law of der condiional bedingen Wahrscheinlichkei. probabiliies 0 P T, T lim PT P T A lim lim 0 PT 0 P[ T ]

13 3 / Communicaion Neworks II (Görg) -- r T FT F FT FT () lim 0 1 F 1 F G ( ) FT T T T (3.6) corresponds o he curren erminaion rae of a sae wih cdf cdf ime Figure 3.4a: Illusraion of equaion (3.6)

14 3 / Communicaion Neworks II (Görg) Memorylessness of RV and Poisson Process Memorylessness of a random variable a) Definiion of memoryless propery las even ime Figure 3.5: Memorylessness of he RV T (3.7) P T x T P T x The fuure phase of he process afer a duraion of behaves independen of he pas The curren erminaion rae r() does no depend on r cons. (3.8)

15 3 / Communicaion Neworks II (Görg) --

16 3 / Communicaion Neworks II (Görg) --

17 3 / Communicaion Neworks II (Görg) -- b) Exponenial disribuion funcion Derivaion of he cdf from equaion 3.6: Which cdf has he memoryless propery? r FT 1 F T 1 F G cons. wih T T () ; F d d G T T () his leads o : d GT() GT() d homogeneous 1 s order differenial equaion Subsiuion: G e a T () ; d d G T ae a () a a leads o : ae e a G T ( ) e, F P( T ) 1e T, 0 (3.9) The neg. exp. RV is he only coninuous RV wih he memoryless propery.

18 3 / Communicaion Neworks II (Görg) -- c) Geomeric disribuion ib i (proof of memoryless propery) Figure 3.6.: Bernoulli experimens A series of Bernoulli experimens x i = 0 failure probabiliy q, x i = 1 success probabiliy p=1-q RV K = Number of failures before he firs success k PK k q pk, 01,,... geomeric disribuion (3.10)

19 3 / Communicaion Neworks II (Görg) -- Saemen: The geomeric disribuion saisfies he Markov propery. Proof: i k j q p PK kikq pq pj 0 q 1 q k q k PKk jkk, P K k j PK k PK k P K k j K k P K k k j q j q PK j k q q.e.d

20 3 / Communicaion Neworks II (Görg) -- Exponenial phases as limiing case of geomeric phases according o Bernoulli experimens: exac ime inerval, figure 3.6 k limiing case: k, 0, k p kp kp success - Erfolgsrae: rae cons. k p k k P T P T k P X k q 1 p, compare vgl. c) c) lim PT k lim 1 e, 0, k k k 0 0 d.h. F P T 1e i.e. neg. exp. VF cdf T k k

21 3 / Communicaion Neworks II (Görg) The Poisson Process a) Problem ime Figure 3.7: Poin process, Renewal process wih P T 1 e evens We look for: Disribuion of he number of evens X in any inerval (0,) P P X x, x 01,,... x

22 3 / Communicaion Neworks II (Görg) -- The inegral mehod of derivaion u I holds PTi u 1 e ; i=1 due o Markov propery, for i>1. hen P x P x 1 T T x i i, i i. 1 1 The x-h even is a ime p x P x x x ( ) DF pdf T( P Tx d i i Ti ) P [ T 1 1 x1 ] 0 0 i1 d = pdf of Erlang-x disribuion, compare , Table x 0 x 1 x1 e e d e d x 1! x 1! x 0 1: Px e ; x 0: P0 PT1 e (3.11) x! Poisson disribuion: P i i! i e ; i 01,,...; Parameer

23 3 / Communicaion Neworks II (Görg) -- c) )Inerpreaion i The number of incoming jobs wihin an inerval of he lengh a he consan arrival rae of is Poisson disribued wih he parameer. For he following is valid e 1 o, i 0 Pi e o, i 1 i e o, i 1 i! (3.12) where o() sands for he Landauer symbol wih he propery: o lim 0 ("negligible for higher orders"). 0 (3.13)

24 3 / Communicaion Neworks II (Görg) -- Figure 3.8: Poisson disribuion The Poisson disribuion ib i is defined d only for ineger i, and dfor coninuous.