Stoqastikèc diadikasÐec kai...

Stoqastikèc diadikasÐec kai efarmogèc

SpuridoÔla K�nta

2 NoembrÐou 2012

Eisagwg DiadikasÐec GJ Ourèc Anamon c

PerÐgramma

• Eisagwg stic t.m./stoqastikèc diadikasÐec

• M.a.σ.q.

• DiadikasÐec Gènnhshc - Jan�tou

• Ourèc Anamon c (SumbolismoÐ - ParadeÐgmata)

• Eisagwg kèrdouc - kìstouc

• Melèth paradeÐgmatoc

Stoq. diad. kai efarmogèc

TuqaÐec Metablhtèc

An to apotèlesma tou peir�matoc eÐnai ènac arijmìc, tìte autì mporeÐ naekfrasteÐ me mia tuqaÐa metablht X ∈ R (diakrit - suneq c).K�je t.m. akoloujeÐ mia katanom h opoÐa ekfr�zetai apì thn antÐstoiqhspp. f(x) me f(x) ≥ 0 kai∑

f(x) = 1 (diakritèc)

∫f(x)dx = 1 (suneqeÐc)

Mèsh tim t.m.:

E[X] =

{ ∑xf(x), an Q diakrit ∫xf(x)dx, an Q suneq c

Poisson(λ): Gegonìta sumbaÐnoun me mèso rujmì λ gegonìta an� mon�daqrìnou. t.m. X: pl joc gegonìtwn sth mon�da tou qrìnou

f(x) = e−λλx

x!, x = 0, 1, . . .

Ekjetik (θ) katanom : X ∼ Exp(θ), f(x) = θe−θx, x ≥ 0.

E[X] = 1θ

Mèsh tim t.m.:

E[X] =

f(x) = e−λλx

x!, x = 0, 1, . . .

E[X] = 1θ

Mèsh tim t.m.:

E[X] =

f(x) = e−λλx

x!, x = 0, 1, . . .

E[X] = 1θ

Mèsh tim t.m.:

E[X] =

f(x) = e−λλx

x!, x = 0, 1, . . .

E[X] = 1θ

Stoqastik DiadikasÐa

• K�je oikogèneia tuqaÐwn metablht¸n {X(t), t ∈ T}, onom�zetaistoqastik diadikasÐa (sd).

• An T arijm simo sÔnolo (N0): sd diakritoÔ qrìnou, {Xn, n ∈ N0}.• An T uperarijm simo sÔnolo, (R+

0 ), sd suneqoÔc qrìnou,{X(t), t ≥ 0}.

• K�je dunat tim twn t.m. X(t) lègetai kat�stash thcdiadikasÐac.S = { dunat¸n katast�sewn }: q¸roc katast�sewn thcsd.

• P (X(tn) = j): pijanìthta h sd th stigm tn na brÐsketai sthnkat�stash j.

• Markobian diadikasÐa H sd {X(t), t ∈ T} eÐnai Markobian an:

P (X(tn) = j|X(tn−1) = i, . . . , X(t1)) = P (X(tn) = j|X(tn−1) = i)

• 'Otan o q.k. miac Markobian c diadikasÐac eÐnai diakritìc: Markobian alusÐda.

• K�je dunat tim twn t.m. X(t) lègetai kat�stash thcdiadikasÐac.

S = { dunat¸n katast�sewn }: q¸roc katast�sewn thcsd.

M.a.σ.q

pij(t) = P (X(t+ s) = j | X(s) = i):pijanìthta met�bashc apì thn i th stigm s, sthn j th stigm t+ s (qronik�omogen c).

P(t) =

p00(t) p01(t) p02(t) · · ·p10(t) p11(t) p12(t) · · ·p20(t) p21(t) p22(t) · · ·

......

.... . .

P(t):pÐnakac sunart sewn pijanot twn met�bashc.

H M.a.σ.q {X(t)} eÐnai pl rwc kajorismènh an dÐnetai o pÐnakac P(t), t ≥ 0

kai h arqik katanom pi(0) = P (X0 = i), i ∈ S.

M.a.σ.q

P(t) =

......

.... . .

M.a.σ.q

P(t) =

......

.... . .

M.a.σ.q

qij :rujmìc met�bashc apì thn i sth j

qij = p′ij(0+) =

{limt→0

pij(t)

t, i 6= j

limt→01−pii(t)

t, i = j

i, j ∈ S.

q00 q01 q02 · · ·q10 q11 q12 · · ·q20 q21 q22 · · ·...

......

Q:pÐnakac rujm¸n met�bashc apeirostìc genn torac thc diadikasÐac.

P(t) = eQt

M.a.σ.q

qij = p′ij(0+) =

{limt→0

pij(t)

t, i 6= j

limt→01−pii(t)

t, i = j

i, j ∈ S.

q00 q01 q02 · · ·q10 q11 q12 · · ·q20 q21 q22 · · ·...

......

P(t) = eQt

M.a.σ.q

qij = p′ij(0+) =

{limt→0

pij(t)

t, i 6= j

limt→01−pii(t)

t, i = j

i, j ∈ S.

q00 q01 q02 · · ·q10 q11 q12 · · ·q20 q21 q22 · · ·...

......

P(t) = eQt

M.a.σ.q

- Gia i 6= j, qij ≥ 0 kai eÐnai o rujmìc me ton opoÐo h diadikasÐametakineÐtai apì thn kat�stash i sthn kat�stash j.

- Gia i = j, qii ≤ 0 kai qi = −qii eÐnai o rujmìc me ton opoÐo h diadikasÐaanaqwreÐ apì thn kat�stash i.

-∑j 6=i qij = qi. To �jroisma twn stoiqeÐwn k�je gramm c tou pÐnaka Q

eÐnai mhdèn.

Ti: qrìnoc paramon c sth kat�stash i

Ti ∼ Exp(qi)

pij =qijqi, i 6= j.

M.a.σ.q

eÐnai mhdèn.

Ti ∼ Exp(qi)

pij =qijqi, i 6= j.

M.a.σ.q

eÐnai mhdèn.

Ti ∼ Exp(qi)

pij =qijqi, i 6= j.

M.a.σ.q

eÐnai mhdèn.

Ti ∼ Exp(qi)

pij =qijqi, i 6= j.

M.a.σ.q

eÐnai mhdèn.

Ti ∼ Exp(qi)

pij =qijqi, i 6= j.

M.a.σ.q

To endiafèron epikentr¸netai ston upologismì thc oriak c katanom c.

Oriak katanom : p = (p0, p1, p2, . . .), ìpou

pi = limt→∞

P (X(t) = i) = limt→∞

pi(t), i ∈ S

h opoÐa eÐnai anex�rthth apì thn arqik katanom .

ProkÔptei apì th lÔshtou sust matoc twn exis¸sewn:

piqi =∑j 6=i

pjqji, i ∈ S kai∑i∈S

pi = 1.

Oi pr¸tec exis¸seic kaloÔntai exis¸seic isorropÐac kai h deÔterh exÐswsh

kanonikopoÐhshc.

M.a.σ.q

To endiafèron epikentr¸netai ston upologismì thc oriak c katanom c.

Oriak katanom : p = (p0, p1, p2, . . .), ìpou

pi = limt→∞

P (X(t) = i) = limt→∞

pi(t), i ∈ S

h opoÐa eÐnai anex�rthth apì thn arqik katanom .ProkÔptei apì th lÔshtou sust matoc twn exis¸sewn:

piqi =∑j 6=i

pjqji, i ∈ S kai∑i∈S

pi = 1.

Oi pr¸tec exis¸seic kaloÔntai exis¸seic isorropÐac kai h deÔterh exÐswsh

kanonikopoÐhshc.

DiadikasÐa Poisson

X(t) : pl joc gegonìtwn sto (0, t] pou sumbaÐnoun me mèso rujmì λ(aparijm tria).

qij = λ, j = i+ 1 i 6= j ∈ S.

λ ))1

λ ))2

λ ))3

λ ))· · ·

−λ λ · · ·

−λ λ · · ·−λ λ · · ·

......

pn = e−λ

DiadikasÐa Poisson

qij = λ, j = i+ 1 i 6= j ∈ S.

λ ))1

λ ))2

λ ))3

λ ))· · ·

−λ λ · · ·

−λ λ · · ·−λ λ · · ·

......

pn = e−λ

DiadikasÐa Poisson

qij = λ, j = i+ 1 i 6= j ∈ S.

λ ))1

λ ))2

λ ))3

λ ))· · ·

−λ λ · · ·

−λ λ · · ·−λ λ · · ·

......

pn = e−λλn

DiadikasÐa Poisson

qij = λ, j = i+ 1 i 6= j ∈ S.

λ ))1

λ ))2

λ ))3

λ ))· · ·

−λ λ · · ·

−λ λ · · ·−λ λ · · ·

......

pn = e−λ

DiadikasÐec G-J

DiadikasÐec gènnhshc jan�tou: M.a.σ.q stic opoÐec apì k�je kat�stash hmet�bash eÐnai dunat mìno sthn prohgoÔmenh sthn epìmenh kat�stash.

λ0 ))1

λ1 ))2

λ2 ))

λ3 ))

ii · · ·

{λi, j = i+ 1µi, j = i− 1

i 6= j ∈ S. (2)

−λ0 λ0 · · ·µ1 −(λ1 + µ1) λ1 · · ·

µ2 −(λ2 + µ2) λ2 · · ·...

......

.... . .

DiadikasÐec G-J

λ0 ))1

λ1 ))2

λ2 ))

λ3 ))

ii · · ·

{λi, j = i+ 1µi, j = i− 1

i 6= j ∈ S. (2)

−λ0 λ0 · · ·µ1 −(λ1 + µ1) λ1 · · ·

µ2 −(λ2 + µ2) λ2 · · ·...

......

.... . .

DiadikasÐec G-J

λ0 ))1

λ1 ))2

λ2 ))

λ3 ))

ii · · ·

{λi, j = i+ 1µi, j = i− 1

i 6= j ∈ S. (2)

−λ0 λ0 · · ·µ1 −(λ1 + µ1) λ1 · · ·

µ2 −(λ2 + µ2) λ2 · · ·...

......

.... . .

DiadikasÐec G-J - Oriak Katanom

Exis¸seic IsorropÐac:

λ0p0 = µ1p1 (3)

(λn + µn)pn = λn−1pn−1 + µn+1pn+1, n = 1, 2, . . . (4)

ExÐswsh kanonikopoÐhshc:∑∞n=0 pn = 1.

B−1 =

∞∑n=0

λ0λ1 · · ·λn−1

µ1µ2 · · ·µn<∞ (5)

pn = Bλ0λ1 · · ·λn−1

µ1µ2 · · ·µn, n = 0, 1, . . . (6)

λ0p0 = µ1p1 (3)

B−1 =

∞∑n=0

λ0λ1 · · ·λn−1

µ1µ2 · · ·µn<∞ (5)

pn = Bλ0λ1 · · ·λn−1

µ1µ2 · · ·µn, n = 0, 1, . . . (6)

λ0p0 = µ1p1 (3)

B−1 =

∞∑n=0

λ0λ1 · · ·λn−1

µ1µ2 · · ·µn<∞ (5)

pn = Bλ0λ1 · · ·λn−1

µ1µ2 · · ·µn, n = 0, 1, . . . (6)

λ0p0 = µ1p1 (3)

B−1 =

∞∑n=0

λ0λ1 · · ·λn−1

µ1µ2 · · ·µn<∞ (5)

pn = Bλ0λ1 · · ·λn−1

µ1µ2 · · ·µn, n = 0, 1, . . . (6)

Ourèc Anamon c - ParadeÐgmata

OURA PELATES UPHRETES EXUPHRETHSH

Thlefwnikì dÐktuo Kl seic Thl. grammèc Sundi�lexh

SunergeÐo Autokin twn AutokÐnhta MhqanikoÐ Episkeu

P�rkingk AutokÐnhta Jèseic Park�risma

Aerodrìmio Aeropl�na Di�dromoi Prosg./Apog.

Tr�peza 'Atoma TamÐec Sunallag

Ourèc Anamon c - SumbolismoÐ

• / • / k / s ( )

• DiadikasÐa afÐxewn (M, D, G)• DiadikasÐa exuphret sewn (M, D, G)k: pl joc uphret¸ns: qwrhtikìthta sust matoc( ): peijarqÐa our�c (FCFS, LCFS, SIRO,...)

Q: pl joc pelat¸n sto sÔsthmaS: qrìnoc paramon c sto sÔsthmaW : qrìnoc anamon c mèqri thn exuphrèthshX: qrìnoc exuphrèthshc

Ourèc Anamon c - SumbolismoÐ

• / • / k / s ( )

• DiadikasÐa afÐxewn (M, D, G)• DiadikasÐa exuphret sewn (M, D, G)k: pl joc uphret¸ns: qwrhtikìthta sust matoc( ): peijarqÐa our�c (FCFS, LCFS, SIRO,...)

Q: pl joc pelat¸n sto sÔsthmaS: qrìnoc paramon c sto sÔsthmaW : qrìnoc anamon c mèqri thn exuphrèthshX: qrìnoc exuphrèthshc

ParadeÐgmata

p.q. M/M/1/1

- DiadikasÐa afÐxewn Poisson(λ)

- Qrìnoi exuphrèthshc Exp(µ)

- 1 uphrèthc

- qwrhtikìthta gia ènan pel�th

- peijarqÐa our�c FCFS

p.q. thlefwnik gramm qwrÐc dunatìthta anamon c kl sewn.

λ ))1

p0 =µ

λ+ µp1 =

λ+ µ

p.q. An λ = 5 kl seic thn ¸ra kai µ = 6 (dhl. di�rkeia kl shc 1/6 ¸rec=10 lept�) tìte p0 = 6

11' 0, 55 kai p1 = 5

11' 0, 45.

ParadeÐgmata

p.q. M/M/1/1

- 1 uphrèthc

λ ))1

p0 =µ

λ+ µp1 =

λ+ µ

11' 0, 55 kai p1 = 5

11' 0, 45.

ParadeÐgmata

p.q. M/M/1/1

- 1 uphrèthc

λ ))1

p0 =µ

λ+ µp1 =

λ+ µ

11' 0, 55 kai p1 = 5

11' 0, 45.

ParadeÐgmata

p.q. M/M/1/1

- 1 uphrèthc

λ ))1

p0 =µ

λ+ µp1 =

λ+ µ

11' 0, 55 kai p1 = 5

11' 0, 45.

ParadeÐgmata

- 1 uphrèthc

• �peirh qwrhtikìthta

{λ, j = i+ 1µ, j = i− 1

i 6= j ∈ S. (7)

λ ))1

iiλ ))

ii · · ·

ParadeÐgmata

- 1 uphrèthc

{λ, j = i+ 1µ, j = i− 1

i 6= j ∈ S. (7)

λ ))1

iiλ ))

ii · · ·

ParadeÐgmata

- 1 uphrèthc

{λ, j = i+ 1µ, j = i− 1

i 6= j ∈ S. (7)

λ ))1

iiλ ))

ii · · ·

B−1 =

∞∑n=0

λ0λ1 · · ·λn−1

µ1µ2 · · ·µn=

∞∑n=0

ρn =1

1− ρ (ρ < 1 Sunj kh eust�jeiac) (8)

pn = Bλ0λ1 · · ·λn−1

µ1µ2 · · ·µn= (1− ρ)ρn, n = 0, 1, 2 . . . ρ < 1 (9)

E[Q] =ρ

1− ρ =λ

µ− λ (10)

E[S] =1

µ− λ (11)

p.q. An λ = 5 kl seic thn ¸ra kai µ = 6 tìte p0 = 1− 56= 1

6' 0, 17,

E[Q] = 5, E[S] = 1.

p.q. An λ = 5 kl seic thn ¸ra kai µ = 7 (dhl. di�rkeia kl shc 1/7 ¸rec '8.5 lept�) tìte p0 = 1− 5

7' 0, 29, E[Q] = 5

2= 2.5, E[S] = 1

2= 0.5.

B−1 =

∞∑n=0

λ0λ1 · · ·λn−1

µ1µ2 · · ·µn=∞∑n=0

∞∑n=0

ρn =1

pn = Bλ0λ1 · · ·λn−1

µ1µ2 · · ·µn= (1− ρ)ρn, n = 0, 1, 2 . . . ρ < 1 (9)

E[Q] =ρ

1− ρ =λ

µ− λ (10)

E[S] =1

µ− λ (11)

6' 0, 17,

E[Q] = 5, E[S] = 1.

7' 0, 29, E[Q] = 5

2= 2.5, E[S] = 1

2= 0.5.

B−1 =

∞∑n=0

λ0λ1 · · ·λn−1

µ1µ2 · · ·µn=∞∑n=0

=∞∑n=0

pn = Bλ0λ1 · · ·λn−1

µ1µ2 · · ·µn= (1− ρ)ρn, n = 0, 1, 2 . . . ρ < 1 (9)

E[Q] =ρ

1− ρ =λ

µ− λ (10)

E[S] =1

µ− λ (11)

6' 0, 17,

E[Q] = 5, E[S] = 1.

7' 0, 29, E[Q] = 5

2= 2.5, E[S] = 1

2= 0.5.

B−1 =

∞∑n=0

λ0λ1 · · ·λn−1

µ1µ2 · · ·µn=∞∑n=0

=∞∑n=0

ρn =1

pn = Bλ0λ1 · · ·λn−1

µ1µ2 · · ·µn= (1− ρ)ρn, n = 0, 1, 2 . . . ρ < 1 (9)

E[Q] =ρ

1− ρ =λ

µ− λ (10)

E[S] =1

µ− λ (11)

6' 0, 17,

E[Q] = 5, E[S] = 1.

7' 0, 29, E[Q] = 5

2= 2.5, E[S] = 1

2= 0.5.

B−1 =

∞∑n=0

λ0λ1 · · ·λn−1

µ1µ2 · · ·µn=∞∑n=0

=∞∑n=0

ρn =1

pn = Bλ0λ1 · · ·λn−1

µ1µ2 · · ·µn

= (1− ρ)ρn, n = 0, 1, 2 . . . ρ < 1 (9)

E[Q] =ρ

1− ρ =λ

µ− λ (10)

E[S] =1

µ− λ (11)

6' 0, 17,

E[Q] = 5, E[S] = 1.

7' 0, 29, E[Q] = 5

2= 2.5, E[S] = 1

2= 0.5.

B−1 =

∞∑n=0

λ0λ1 · · ·λn−1

µ1µ2 · · ·µn=∞∑n=0

=∞∑n=0

ρn =1

pn = Bλ0λ1 · · ·λn−1

µ1µ2 · · ·µn= (1− ρ)ρn, n = 0, 1, 2 . . . ρ < 1 (9)

E[Q] =ρ

1− ρ =λ

µ− λ (10)

E[S] =1

µ− λ (11)

6' 0, 17,

E[Q] = 5, E[S] = 1.

7' 0, 29, E[Q] = 5

2= 2.5, E[S] = 1

2= 0.5.

B−1 =

∞∑n=0

λ0λ1 · · ·λn−1

µ1µ2 · · ·µn=∞∑n=0

=∞∑n=0

ρn =1

pn = Bλ0λ1 · · ·λn−1

µ1µ2 · · ·µn= (1− ρ)ρn, n = 0, 1, 2 . . . ρ < 1 (9)

E[Q] =ρ

1− ρ =λ

µ− λ (10)

E[S] =1

µ− λ (11)

6' 0, 17,

E[Q] = 5, E[S] = 1.

7' 0, 29, E[Q] = 5

2= 2.5, E[S] = 1

2= 0.5.

B−1 =

∞∑n=0

λ0λ1 · · ·λn−1

µ1µ2 · · ·µn=∞∑n=0

=∞∑n=0

ρn =1

pn = Bλ0λ1 · · ·λn−1

µ1µ2 · · ·µn= (1− ρ)ρn, n = 0, 1, 2 . . . ρ < 1 (9)

E[Q] =ρ

1− ρ =λ

µ− λ (10)

E[S] =1

µ− λ (11)

6' 0, 17,

E[Q] = 5, E[S] = 1.

7' 0, 29, E[Q] = 5

2= 2.5, E[S] = 1

2= 0.5.

B−1 =

∞∑n=0

λ0λ1 · · ·λn−1

µ1µ2 · · ·µn=∞∑n=0

=∞∑n=0

ρn =1

pn = Bλ0λ1 · · ·λn−1

µ1µ2 · · ·µn= (1− ρ)ρn, n = 0, 1, 2 . . . ρ < 1 (9)

E[Q] =ρ

1− ρ =λ

µ− λ (10)

E[S] =1

µ− λ (11)

6' 0, 17,

E[Q] = 5, E[S] = 1.

7' 0, 29, E[Q] = 5

2= 2.5, E[S] = 1

2= 0.5.

Oikonomik - Paigniojewrhtik Optik

- Eisagwg oikonomik c optik c sth JewrÐa twn Our¸n anamon c(Eisagwg dom c kèrdouc - kìstouc)

- Skopìc: BeltistopoÐhsh tou sust matoc (oi pel�tec lamb�nounapof�seic)

- 'Entaxh se paigniojewrhtikì plaÐsio (pel�tec, pel�tec - diaqeirist c)

- S: to sÔnolo twn strathgik¸n

- F : h sun�rthsh plhrwm c

paÐktec = pel�tec =⇒ ìmoioi(Ðdio S, Ðdia F ).

F (a, β) : h plhrwm enìc pel�th o opoÐoc epilègei th strathgik a ìtanìloi oi �lloi epilègoun th strathgik β.

Orismìc (Bèltisth Ap�nthsh)

Mia strathgik s lègetai ìti apoteleÐ bèltisth ap�nthsh enìc pel�th ènantimiac strathgik c s pou akoloujeÐtai apì touc upìloipouc an

F (s, s) ≥ F (a, s), ∀a ∈ S.

Paigniojewrhtikì plaÐsio

Orismìc (ShmeÐo Strathgik c IsorropÐac Nash )

Mia strathgik se kaleÐtai shmeÐo strathgik c isorropÐac (Nash),anapoteleÐ bèltisth ap�nthsh ston eautì thc, dhlad

F (se, se) ≥ F (s, se), ∀s ∈ S.

Orismìc (Kajar Strathgik KatwflÐou)

Mia kajar strathgik katwflÐou me kat¸fli n upagoreÔei mia apì ticenèrgeiec, èstw thn A1 gia ìlec tic katast�seic sto {0, 1, . . . , n− 1} kai thn�llh enèrgeia, A2, diaforetik�.

Paigniojewrhtikì plaÐsio

Orismìc (ShmeÐo Strathgik c IsorropÐac Nash )

Mia strathgik se kaleÐtai shmeÐo strathgik c isorropÐac (Nash),anapoteleÐ bèltisth ap�nthsh ston eautì thc, dhlad

F (se, se) ≥ F (s, se), ∀s ∈ S.

Orismìc (Kajar Strathgik KatwflÐou)

Mia kajar strathgik katwflÐou me kat¸fli n upagoreÔei mia apì ticenèrgeiec, èstw thn A1 gia ìlec tic katast�seic sto {0, 1, . . . , n− 1} kai thn�llh enèrgeia, A2, diaforetik�.

Montèlo

Melet�me bèltistec strathgikèc eisìdou-apoq¸rhshc ⇒ Apof�seicpelat¸n: na eisèljoun ìqi sto sÔsthma

- Kèrdoc R mon�dwn gia k�je pel�th pou exuphreteÐtai

- Kìstoc C mon�dwn an� mon�da qrìnou

EpÐpeda plhrofìrhshc

Pl rhc plhrofìrhsh - Parathr sima sust mata

KamÐa plhrofìrhsh - Mh parathr sima sust mata

Merik plhrofìrhsh - Merik¸c (mh) parathr sima sust mata

KateujÔnseic melèthc

Atomik beltistopoÐhsh (ShmeÐa isorropÐac)

Koinwnik beltistopoÐhsh

MegistopoÐhsh kèrdouc diaqeirist

Montèlo

Atomik beltistopoÐhsh

- Upojètoume ìti ìloi oi pel�tec akoloujoÔn thn Ðdia strathgik kai tosÔsthma brÐsketai se kat�stash statistik c isorropÐac.

- UpologÐzoume ta mètra apìdoshc tou sust matoc (st�simhkatanom ,E[Q],...).

- Epilègoume ènan pel�th o opoÐoc ft�nei se èna tètoio sÔsthma kaiupologÐzoume to atomikì kèrdoc (wfèleia) tou dedomènhc thcplhroforÐac pou lamb�nei, U = R− CE[S|lhfjeÐsa plhroforÐa].

- BrÐskoume ton trìpo -apìfash pou prèpei na p�rei- ¸ste naantidr�sei bèltista sthn strathgik pou akoloujoÔn oi upìloipoipel�tec (bèltisth ap�nthsh).

- Apì tic bèltistec apant seic epilègoume autèc pou eÐnai bèltistecapant seic ston eautì touc. Autèc oi apof�seic - strathgikècapoteloÔn tic strathgikèc isorropÐac pou anazhtoÔme.

Atomik beltistopoÐhsh - Par�deigma I

- Poisson(λ) afÐxeic

- Exp(µ) qrìnoi exuphrèthshc

- ∞ q¸roc anamon c, 1 uphrèthc, FCFS

- Amoib R mon�dwn gia k�je pel�th pou exuphreteÐtai

- Kìstoc anamon c C mon�dwn an� mon�da qrìnou

- PlhroforÐa: Pl joc parìntwn pelat¸n sto sÔsthma - Pl rwcparathr simo sÔsthma

Strathgik :

{0, 1, 2, . . .} 7→{

0 (anaq¸rhsh)1 (eÐsodoc)

Strathgik katwflÐou i:

(1, 1, . . . , 1︸︷︷︸, 0, 0, 0, . . .)i

- PlhroforÐa: Pl joc parìntwn pelat¸n sto sÔsthma - Pl rwcparathr simo sÔsthma

Strathgik :

{0, 1, 2, . . .} 7→{

0 (anaq¸rhsh)1 (eÐsodoc)

Strathgik katwflÐou i:

(1, 1, . . . , 1︸︷︷︸, 0, 0, 0, . . .)i

Kèrdoc pel�th pou lamb�nei thn plhroforÐa n:

R− (n+ 1)C

≥ 0.

Mpec an h jèsh thn opoÐa ja katal�beic eÐnai mikrìterh Ðsh apì:

⌊Rµ

⌋Kajar strathgik katwflÐou: ne =

⌊Rµ

� 1 2 3 ne//• • •Cµ

R− (n+ 1)C

µ≥ 0.

⌊Rµ

� 1 2 3 ne//• • •Cµ

R− (n+ 1)C

µ≥ 0.

⌊Rµ

� 1 2 3 ne//• • •Cµ

Atomik beltistopoÐhsh - Par�deigma II

- PlhroforÐa: KamÐa - Pl rwc mh parathr simo sÔsthma

Kajarèc strathgikèc: {0(anaq¸rhsh), 1(eÐsodoc)}Mikt strathgik : pijanìthta eisìdou q ∈ [0, 1]

⇒ M/M/1 Poisson(λq) diadikasÐa afÐxewn.

Kèrdoc pel�th: U(q) = R− Cµ−λq

- An R ≤ Cµ⇒ U(q) ≤ 0, ∀q ∈ [0, 1]⇒ Bèltisth ap�nthsh = 0

- An R ≥ Cµ−λ ⇒ U(q) ≥ 0, ∀q ∈ [0, 1]⇒ Bèltisth ap�nthsh = 1

- An Cµ< R < C

µ−λ ⇒ U(qe) = 0 ìpou

qe =µ

(1− C

). (12)

Bèltisth ap�nthsh = k�je q ∈ [0, 1].Bèltisth ap�nthsh ston eautì thc (dhl. isorropÐa) eÐnai mìno h qe.

� 0Rµ−CλR 1 qe//• •

Cµ−λ R

- An Cµ< R < C

qe =µ

(1− C

). (12)

� 0Rµ−CλR 1 qe//• •

Cµ−λ R

- An Cµ< R < C

qe =µ

(1− C

). (12)

� 0Rµ−CλR 1 qe//• •

Cµ−λ R

- An Cµ< R < C

qe =µ

(1− C

). (12)

� 0Rµ−CλR 1 qe//• •

Cµ−λ R

- An Cµ< R < C

qe =µ

(1− C

). (12)

� 0Rµ−CλR 1 qe//• •

Cµ−λ R

U(q) = R− Cµ−λq

- An q < qe ⇒ U(q) > U(qe) = 0 tìte bèltisth ap�nthsh 1

- An q > qe ⇒ U(q) < U(qe) = 0 tìte bèltisth ap�nthsh 0

- An q = qe ⇒ U(q) = U(qe) = 0 tìte bèltisth ap�nthsh k�jepijanìthta sto [0,1]

Bèltisth ap�nthsh

qe��

⇒ ATC

U(q) = R− Cµ−λq

- An q < qe ⇒ U(q) > U(qe) = 0 tìte bèltisth ap�nthsh 1

- An q > qe ⇒ U(q) < U(qe) = 0 tìte bèltisth ap�nthsh 0

- An q = qe ⇒ U(q) = U(qe) = 0 tìte bèltisth ap�nthsh k�jepijanìthta sto [0,1]

Bèltisth ap�nthsh

qe��

⇒ ATC

Skopìc: EÔresh strathgik c h opoÐa megistopoieÐ to koinwnikì kèrdoc an�mon�da qrìnou.

- Upojètoume ìti ìloi oi pel�tec akoloujoÔn thn Ðdia politik kaiupologÐzoume ta mètra apìdoshc tou sust matoc.

- Sun�rthsh koinwnikoÔ kèrdouc an� mon�da qrìnou:

Sk = λ∗R− CE[Q]

λ∗: pragmatikìc mèsoc rujmìc �fixhc pelat¸n.

- BrÐskoume to shmeÐo - strathgik gia thn opoÐa megistopoieÐtai hsun�rthsh.

Skopìc: EÔresh strathgik c h opoÐa megistopoieÐ to koinwnikì kèrdoc an�mon�da qrìnou.

- Upojètoume ìti ìloi oi pel�tec akoloujoÔn thn Ðdia politik kaiupologÐzoume ta mètra apìdoshc tou sust matoc.

- Sun�rthsh koinwnikoÔ kèrdouc an� mon�da qrìnou:

Sk = λ∗R− CE[Q]

λ∗: pragmatikìc mèsoc rujmìc �fixhc pelat¸n.

- BrÐskoume to shmeÐo - strathgik gia thn opoÐa megistopoieÐtai hsun�rthsh.

Koinwnik beltistopoÐhsh-Par�deigma I(sunèqeia)

Upojètoume ìti ìloi oi pel�tec akoloujoÔn thn Ðdia politik katwflÐou n(M/M/1/n).

Koinwnikì kèrdoc:

Sk(n) = λ∗R− CE[Q]

= λ1− ρn

1− ρn+1R− C

1− ρ −(n+ 1)ρn+1

1− ρn+1

). (13)

• n∗ = bν∗c, ìpou ν∗ eÐnai h monadik lÔsh thc exÐswshc

C=ν(1− ρ)− ρ(1− ρν)

(1− ρ)2 . (14)

• IsqÔei: n∗ ≤ ne.

Naor(1969)

Koinwnik beltistopoÐhsh-Par�deigma I(sunèqeia)

Upojètoume ìti ìloi oi pel�tec akoloujoÔn thn Ðdia politik katwflÐou n(M/M/1/n).

Koinwnikì kèrdoc:

Sk(n) = λ∗R− CE[Q]

= λ1− ρn

1− ρn+1R− C

1− ρ −(n+ 1)ρn+1

1− ρn+1

). (13)

• n∗ = bν∗c, ìpou ν∗ eÐnai h monadik lÔsh thc exÐswshc

C=ν(1− ρ)− ρ(1− ρν)

(1− ρ)2 . (14)

• IsqÔei: n∗ ≤ ne.

Naor(1969)

Koinwnik beltistopoÐhsh-Par�deigma II (sunèqeia)

Upojètoume ìti ìloi oi pel�tec akoloujoÔn thn Ðdia mikt politik q(M/M/1 me Poisson(λq)d.afÐxewn).

Koinwnikì kèrdoc:

Sk(q) = λ∗R− CE[Q]

= λqR− C λq

µ− λq . (15)

• An R ≤ Cµtìte q∗ = 0.

• An Cµ< R < C

(1− λ

)2tìte q∗ = µ

√CRµ

• An R ≥ Cµ

(1− λ

)2tìte q∗ = 1.

IsqÔei: q∗ ≤ qe.

Koinwnikì kèrdoc:

= λqR− C λq

µ− λq . (15)

• An Cµ< R < C

(1− λ

)2tìte q∗ = µ

√CRµ

• An R ≥ Cµ

(1− λ

)2tìte q∗ = 1.

Koinwnikì kèrdoc:

= λqR− C λq

µ− λq . (15)

• An Cµ< R < C

(1− λ

)2tìte q∗ = µ

√CRµ

• An R ≥ Cµ

(1− λ

)2tìte q∗ = 1.

Skopìc: EÔresh strathgik c h opoÐa megistopoieÐ to kèrdoc tou diaqeirist an� mon�da qrìnou.

- Upojètoume ìti o diaqeirist c epib�llei mÐa tim diodÐou p, opìte tokèrdoc k�je pel�th t¸ra eÐnai R− p.

- Lìgw thc allag c sto kèrdoc oi pel�tec tropopoioÔn thn politik touc an�loga, dhl. akoloujoÔn thn politik isorropÐac me kèrdocR− p.

- UpologÐzoume th sun�rthsh kèrdouc tou diaqeirist an� mon�daqrìnou

Sd = λ∗p.

- BrÐskoume thn politik gia thn opoÐa megistopoieÐtai h sun�rthsh.

Skopìc: EÔresh strathgik c h opoÐa megistopoieÐ to kèrdoc tou diaqeirist an� mon�da qrìnou.

- Upojètoume ìti o diaqeirist c epib�llei mÐa tim diodÐou p, opìte tokèrdoc k�je pel�th t¸ra eÐnai R− p.

- Lìgw thc allag c sto kèrdoc oi pel�tec tropopoioÔn thn politik touc an�loga, dhl. akoloujoÔn thn politik isorropÐac me kèrdocR− p.

- UpologÐzoume th sun�rthsh kèrdouc tou diaqeirist an� mon�daqrìnou

Sd = λ∗p.

- BrÐskoume thn politik gia thn opoÐa megistopoieÐtai h sun�rthsh.

MegistopoÐhsh kèrdouc diaqeirist -Par�deigmaI(sunèqeia)

UpenjÔmish: ne = bRµC c

O diaqeirist c epib�llei antÐtimo p ⇒ n = b (R−p)µCc

p(n) = R− Cn

µ. (16)

Kèrdoc diaqeirist :

Sd(n) = λ∗p(n) = λ1− ρn

1− ρn+1

(R− Cn

). (17)

• nm = bνmc ìpou νm eÐnai h monadik lÔsh thc exÐswshc

C= ν +

(1− ρν−1)(1− ρν+1)

ρν − 1(1− ρ)2 . (18)

'Ara to bèltisto antÐtimo eÐnai p(nm) = R− C nmµ

• IsqÔei:nm ≤ n∗ ≤ ne.

p(n) = R− Cn

µ. (16)

Sd(n) = λ∗p(n) = λ1− ρn

1− ρn+1

(R− Cn

). (17)

C= ν +

(1− ρν−1)(1− ρν+1)

ρν − 1(1− ρ)2 . (18)

p(n) = R− Cn

µ. (16)

Sd(n) = λ∗p(n) = λ1− ρn

1− ρn+1

(R− Cn

). (17)

C= ν +

(1− ρν−1)(1− ρν+1)

ρν − 1(1− ρ)2 . (18)

MegistopoÐhsh kèrdouc diaqeirist -Par�deigmaII(sunèqeia)

UpenjÔmish: qe =µλ

(1− C

)O diaqeirist c epib�llei antÐtimo p ⇒ q = µ

(1− C

(R−p)µ

)p(q) = R− C

µ− λq . (19)

Sd(q) = λ∗p(q) = λq

(R− C

µ− λq

ParathroÔme ìti Sd(q) = Sk(q)⇒ qm = q∗.

IsqÔei: qm ≤ q∗ ≤ qe

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