5. Stochastic processes (1)

Transcription

1 Lec05.pp S Inroducion o Teleraffic Theory Spring 2005

2 Conens Basic conceps Poisson process 2

3 Sochasic processes () Consider some quaniy in a eleraffic (or any) sysem I ypically evolves in ime randomly Example : he number of occupied channels in a elephone link a ime or a he arrival ime of he n h cusomer Example 2: he number of packes in he buffer of a saisical muliplexer a ime or a he arrival ime of he n h cusomer This kind of evoluion is described by a sochasic process A any individual ime (or n) he sysem can be described by a random variable Thus, he sochasic process is a collecion of random variables 3

4 Sochasic processes (2) Definiion: A (real-valued) sochasic process X = (X I) is a collecion of random variables X aking values in some (real-valued) se S, X (ω) S, and indexed by a real-valued (ime) parameer I. Sochasic processes are also called random processes (or jus processes) The index se I Ris called he parameer space of he process The value se S Ris called he sae space of he process Noe: Someimes noaion X is used o refer o he whole sochasic process (insead of a single random variable relaed o he ime ) 4

5 Sochasic processes (3) Each (individual) random variable X is a mapping from he sample space Ω ino he real values R: X : Ω R, ω a X ( ω) Thus, a sochasic process X can be seen as a mapping from he sample space Ω ino he se of real-valued funcions R I (wih I as an argumen): X I : Ω R, ω a X ( ω) Each sample poin ω Ω is associaed wih a real-valued funcion X(ω). Funcion X(ω) is called a realizaion (or a pah or a rajecory) of he process. 5

6 Summary Given he sample poin ω Ω X(ω) = (X (ω) I) is a real-valued funcion (of I) Given he ime index I, X = (X (ω) ω Ω) is a random variable (as ω Ω) Given he sample poin ω Ω and he ime index I, X (ω) is a real value 6

7 Example Consider raffic process X = (X [0,T]) in a link beween wo elephone exchanges during some ime inerval [0,T] X denoes he number of occupied channels a ime Sample poin ω Ω ells us wha is he number X 0 of occupied channels a ime 0, wha are he remaining holding imes of he calls going on a ime 0, a wha imes new calls arrive, and wha are he holding imes of hese new calls. From his informaion, i is possible o consruc he realizaion X(ω) of he raffic process X Noe ha all he randomness in he process is included in he sample poin ω Given he sample poin, he realizaion of he process is jus a (deerminisic) funcion of ime 7

8 Traffic process channels channel-by-channel occupaion call holding ime nr of channels call arrival imes nr of channels occupied blocked call raffic volume ime ime 8

9 Caegories of sochasic processes Reminder: Parameer space: se I of indices I Sae space: se S of values X (ω) S Caegories: Based on he parameer space: Discree-ime processes: parameer space discree Coninuous-ime processes: parameer space coninuous Based on he sae space: Discree-sae processes: sae space discree Coninuous-sae processes: sae space coninuous In his course we will concenrae on he discree-sae processes (wih eiher a discree or a coninuous parameer space (ime)) Typical processes describe he number of cusomers in a queueing sysem (he sae space being hus S = {0,,2,...}) 9

10 Examples Discree-ime, discree-sae processes Example : he number of occupied channels in a elephone link a he arrival ime of he n h cusomer, n =,2,... Example 2: he number of packes in he buffer of a saisical muliplexer a he arrival ime of he n h cusomer, n =,2,... Coninuous-ime, discree-sae processes Example 3: he number of occupied channels in a elephone link a ime > 0 Example 4: he number of packes in he buffer of a saisical muliplexer a ime > 0 0

11 Noaion For a discree-ime process, he parameer space is ypically he se of posiive inegers, I = {,2, } Index is hen (ofen) replaced by n: X n, X n (ω) For a coninuous-ime process, he parameer space is ypically eiher a finie inerval, I = [0, T], or all nonnegaive real values, I = [0, ) In his case, index is (ofen) wrien no as a subscrip bu in parenheses: X(), X (;ω)

12 Disribuion The sochasic characerizaion of a sochasic process X is made by giving all possible finie-dimensional disribuions where,, n I, x,, x n S and n =,2,... In general, his is no an easy ask because of dependencies beween he random variables X (wih differen values of ime ) For discree-sae processes i is sufficien o consider probabiliies of he form cf. discree disribuions P{ X x, K, X x n n P{ X = x, K, X = n x n } } 2

13 Dependence The mos simple (bu no so ineresing) example of a sochasic process is such ha all he random variables X are independen of each oher. In his case P{ X x,..., X xn} = P{ X x} LP{ X n n n The mos simple non-rivial example is a discree sae Markov process. In his case P{ X = x,..., X = x } = n n P{ X = x } P{ X = x2 X = x} LP{ X = xn X = x } n 2 n n This is relaed o he so called Markov propery: Given he curren sae (of he process), he fuure (of he process) does no depend on he pas (of he process), i.e. how he process has arrived o he curren sae 3 x }

14 Saionariy Definiion: Sochasic process X is saionary if all finie-dimensional disribuions are invarian o ime shifs, ha is: P { X,, } {,, + x K X xn = P X x X x n + K n n for all, n,,, n and x,, x n Consequence: By choosing n =, we see ha all (individual) random variables X of a saionary process are idenically disribued: P{ X x} = F( x) for all I. This is called he saionary disribuion of he process. } 4

15 Sochasic processes in eleraffic heory In his course (and, more generally, in eleraffic heory) various sochasic processes are needed o describe he arrivals of cusomers o he sysem (arrival process) he sae of he sysem (sae process) Noe ha he laer is also ofen called as raffic process 5

16 Arrival process An arrival process can be described as a poin process (τ n n =,2,...) where τ n ells he arrival ime of he n h cusomer (discree-ime, coninuous-sae) non-decreasing: τ n+ τ n kaikilla n hus non-saionary! ypically i is assumed ha he inerarrival imes τ n τ n- are independen and idenically disribued (IID) renewal process hen i is sufficien o specify he inerarrival ime disribuion exponenial IID inerarrival imes Poisson process a couner process (A() 0) where A() ells he number of arrivals up o ime (coninuous-ime, discree-sae) non-decreasing: A(+ ) A() for all, 0 hus non-saionary! independen and idenically disribued (IID) incremens A(+ ) A() wih Poisson disribuion Poisson process 6

17 Sae process In simple cases he sae of he sysem is described jus by an ineger e.g. he number X() of calls or packes a ime This yields a sae process ha is coninuous-ime and discree-sae In more complicaed cases, he sae process is e.g. a vecor of inegers (cf. loss and queueing nework models) Typically we are ineresed in wheher he sae process has a saionary disribuion if so, wha i is? Alhough he sae of he sysem did no follow he saionary disribuion a ime 0, in many cases sae disribuion approaches he saionary disribuion as ends o 7

18 Conens Basic conceps Poisson process 8

19 Bernoulli process Definiion: Bernoulli process wih success probabiliy p is an infinie series (X n n =,2,...) of independen and idenical random experimens of Bernoulli ype wih success probabiliy p Bernoulli process is clearly discree-ime and discree-sae Parameer space: I = {,2, } Sae space: S = {0,} Finie dimensional disribuions (noe: X n s are IID): P{ X = x,..., X n = xn} = P{ X = x} LP{ X = n i= p xi ( p) xi = p ( Bernoulli process is saionary (saionary disribuion: Bernoulli(p)) i xi p) n i xi n = x n } 9

20 Definiion of a Poisson process Poisson process is he coninuous-ime counerpar of a Bernoulli process I is a poin process (τ n n =,2,...) where τ n ells ells he occurrence ime of he n h even, (e.g. arrival of a clien) failure in Bernoulli process is now an arrival of a clien Definiion : A poin process (τ n n =,2,...) is a Poisson process wih inensiy λ if he probabiliy ha here is an even during a shor ime inerval (, +h] is λh + o(h) independenly of he oher ime inervals o(h) refers o any funcion such ha o(h)/h 0 as h 0 new evens happen wih a consan inensiy λ: (λh + o(h))/h λ probabiliy ha here are no arrivals in (, +h] is λh + o(h) Defined as a poin process, Poisson process is discree-ime and coninuous-sae Parameer space: I = {,2, } Sae space: S = (0, ) 20

21 Poisson process, anoher definiion Consider he inerarrival ime τ n τ n- beween wo evens (τ 0 = 0) Since he inensiy ha somehing happens remains consan λ, he ending of he inerarrival ime wihin a shor period of ime (, +h], afer i has lased already he ime, does no depend on (or on oher previous arrivals) Thus, he inerarrival imes are independen and, addiionally, hey have he memoryless propery. This propery can be only he one of exponenial disribuion (of coninuous-ime disribuions) Definiion 2: A poin process (τ n n =,2,...) is a Poisson process wih inensiy λ if he inerarrival imes τ n τ n are independen and idenically disribued (IID) wih join disribuion Exp(λ) 2

22 Poisson process, ye anoher definiion () Consider finally he number of evens A() during ime inerval [0,] In a Bernoulli process, he number of successes in a fixed inerval would follow a binomial disribuion. As he ime slice ends o 0, his approaches a Poisson disribuion. Noe ha A(0)=0 Definiion 3: A couner process (A() 0) is a Poisson process wih inensiy λ if is incremens in disjoin inervals are independen and follow a Poisson disribuion as follows: A( + ) A( ) Poisson( λ ) Defined as a couner process, Poisson process is coninuous-ime and discree-sae Parameer space: I = [0, ) Sae space: S = {0,,2, } 22

23 23 5. Sochasic processes () Poisson process, ye anoher definiion (2) One dimensional disribuion: A() Poisson(λ) E[A()] = λ, D 2 [A()] = λ Finie dimensional disribuions (due o independence of disjoin inervals): Poisson process, defined as a couner process is no saionary, bu i has saionary incremens hus, i doesn have a saionary disribuion, bu independen and idenically disribued incremens } ) ( ) ( { } ) ( ) ( { } ) ( { } ) (,..., ) ( { 2 2 = = = = = = n n n n n n x x A A P x x A A P x A P x A x A P L

24 Three ways o characerize he Poisson process I is possible o show ha all hree definiions for a Poisson process are, indeed, equivalen A() τ 4 τ 3 τ τ 2 τ 3 τ 4 no even wih prob. λh+o(h) even wih prob. λh+o(h) 24

25 Properies () Propery (Sum): Le A () and A 2 () be wo independen Poisson processes wih inensiies λ and λ 2. Then he sum (superposiion) process A () + A 2 () is a Poisson process wih inensiy λ +λ 2. Proof: Consider a shor ime inerval (, +h] Probabiliy ha here are no evens in he superposiion is ( 2 λh + o( h))( λ2h + o( h)) = ( λ + λ ) h + o( h) On he oher hand, he probabiliy ha here is exacly one even is ( λh + o( h))( λ2h + o( h)) + ( λh + o( h))( λ2h + o( h)) = ( λ + λ2 ) h + o( ) h λ λ 2 λ +λ 2 25

26 Properies (2) Propery 2 (Random sampling): Le τ n be a Poisson process wih inensiy λ. Denoe by σ n he poin process resuling from a random and independen sampling (wih probabiliy p) of he poins of τ n. Then σ n is a Poisson process wih inensiy pλ. Proof: Consider a shor ime inerval (, +h] Probabiliy ha here are no evens afer he random sampling is ( λh + o( h)) + ( p)( λh + o( h)) = pλh + o( h) On he oher hand, he probabiliy ha here is exacly one even is λ pλ p ( λh + o( h)) = pλh + o( h) 26

27 Properies (3) Propery 3 (Random soring): Le τ n be a Poisson process wih inensiy λ. Denoe by σ n () he poin process resuling from a random and independen sampling (wih probabiliy p) of he poins of τ n. Denoe by σ n (2) he poin process resuling from he remaining poins. Then σ n () and σn (2) are independen Poisson processes wih inensiies λp and λ( p). Proof: Due o propery 2, i is enough o prove ha he resuling wo processes are independen. Proof will be ignored on his course. λ λp λ(-p) 27

28 Properies (4) Propery 4 (PASTA): Consider any simple (and sable) eleraffic model wih Poisson arrivals. Le X() denoe he sae of sysem a ime (coninuous-ime process) and Y n denoe he sae of he sysem seen by he nh arriving cusomer (discree-ime process). Then he saionary disribuion of X() is he same as he saionary disribuion of Y n. Thus, we can say ha arriving cusomers see he sysem in he saionary sae PASTA= Poisson Arrivals See Time Avarages PASTA propery is only valid for Poisson arrivals and i is no valid for oher arrival processes consider e.g. your own PC. Whenever you sar a new session, he sysem is idle. In he coninuous ime, however, he sysem is no only idle bu also busy (when you use i). 28

29 Example() Connecion requess arrive a a server according o a Poisson process wih inensiy requess in a minue. λ = 5 Wha is he probabiliy ha exacly 2 new requess arrive during he nex 30 seconds? Number of new arrivals during a ime inerval follows Poisson disribuion wih he parameer λ = 5 / = 2. 5 A( + 30) A( ) Poisson(2.5) P{ A( + 30) A( ) = 2} = ! e =

30 Example(2) Consider he sysem described on previous slide. A new connecion reques has jus arrived a he server. Wha is he probabiliy ha i akes more han 30 seconds before nex reques arrives? Consider he process as a poin process. The inerarrival ime follows exponenial disribuion wih parameer. λ P{ τ 5/ τ i 30} = P{ τ i+ τ i 30} = e = = i+ e Consider he process as a couner process, cf. slide 29. Now we can resae he quesion above as Wha is he probabiliy ha here are no arrivals during 30 seconds?. P{ A( + 30) A( ) = 0} = ! e = e =