Perturbation Analysis of a Variable M/M/1 Queue: A Probabilistic Approach

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Perturbation Analysis of a Variable M/M/1 Queue: A Probabilistic Approach

Nelson Antunes, Christine Fricker, Fabrice Guillemin, Philippe Robert

To cite this version:

Nelson Antunes, Christine Fricker, Fabrice Guillemin, Philippe Robert. Perturbation Analysis of a Variable M/M/1 Queue: A Probabilistic Approach. [Research Report] RR-5419, INRIA. 2005, pp.26.

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ISRN INRIA/RR--5419--FR+ENG

a p p o r t

d e r e c h e r c h e

Thème COM

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Perturbation Analysis of a Variable M / M / 1 Queue:

A Probabilistic Approach

Nelson Antunes — Christine Fricker — Fabrice Guillemin — Philippe Robert

N° 5419

Deembre 2004

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(X(t))

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M/M/1

^® p[]Hp[]btuxm]xm]7nEtu ]HVr q Ã|~[~P]Eqjr £ï°

&

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XYxmo=p[uY[o=pjk@kY[]~t~P]Ex

L(t)

j]Hq[ok]Hi¨kmY[]bqhp[\P]Exou¢npilkmo=\B]Hxi«tLk«kmr\]

t

^r qtq

M/M/1

^® ^p[]Hp[]

§&r kmYtxxmrLt[xvtLk]

λ

tuq¤im]Ex3rsn]&xvtLk]

µ

°³XY[]}Ltxrst[]

B

j]Hq[ok]HiWkmY[]|[p[xtkmrouqou¢t#[pimg_~P]Exroh

ilktuxlkr q_§&r kmYouq]npilkmo=\B]HxK+ ru]Hq

L(0) = 1

^ª

B = inf{s ≥ 0 : L(s) = 0}.

(8)

"!#$%

M/M/1

^& ^''

-Õk@rsi@t=imimp[\]HkmYtk@kmY]}izkvt[r r kzg_nEouq[rµkr o=q

λ < µ

Y[ous[iH°.XY[]&rqhLtxrtuq=k³jrsizkxmr[pjkr o=qo¢

(L(t))

^rsi

=]Eo=\B]EkmxrnHt gjrilkmxr[pjkm]7M§&r kmYM~tuxtu\B]Ekm]Hx

ρ = λ/µ

°&º[o=x

x ≥ 1

ª[kY[]Ltuxmrst[]

B x

[]Eq[oukm]Hi}kmY[]

[p[xtkmrouqbo¢[t[pimg|~P]Exrohilktuxlkr q[&§&r kmY

x

nEpilkmou\]HxiH°;G«gj]E²q[r kmrouq)ª

B 1

= B

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B

^ª

B 1

tq

B ⁰ ₁

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B

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®

p]Ep[]j]Hq[ok]Hïªh¢Ýo=x}ilYouxmkHªjhge3¦FEp[]Ep]u°

ºoux

ξ ≥ 0

^ª

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ξ

tuqG¢Ýoux

0 ≤ a < b

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[a, b]

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N _λ

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N _µ

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~xmojn]7imim]Hi

N _λ

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(X (t))

°XY]V~[xojn]Hii

(L(t))

nHtq¯P]xm]H~[xm]7il]Hq3km]H¬t=ikY[]

imoupjkr o=qou¢kmY]izkohnvYtuilkmrsnb[r0/9]Ex]Eq3kmrstï]

®

ptLkr o=q

dL(t)

= L(t) − L(t−) = N λ ([t, t + dt]) − {L(t−)>0} N µ ([t, t + dt])

= d N λ (t) − {L(t−)>0} d N µ (t),

^¶z7¸

§&Y]Ex]

L(t−)

rsi}kY[]_]¢àk#r \r kbou¢

s → L(s)

^tk

t

°#º[o=x|kY[]_x]E~xm]7il]Hq=kvtLkr o=qMo¢

®

p[]Ep]Erq[atuxm¥=oL

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®

ptLkr o=qiHª[il]H]{}ouP]Exmk 274J°

½W½ 7 ¹$j ½ Â ½ ½ -ùqkmY]#¢Ýou oL§&rq[ª[§¨]nEouqimrsj]Ex|tuq

M/M/1

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xvtLk]³Ltxg3rq[&rqkmr\]¨tuit¢Ýp[qnkmrouqou¢jimou\]³~[xojn]7imi

(X(t))

ktu¥hr q[&Ltu p[]7i)rqilo=\]Wil~tun]=ª7[]Eq[oukm]H

hg

S

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S

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S

rsibt¤²q[r km]oux#nEoup[q3ktu[]_im]k§&Y[]Eq

(X (t))

ritatx¥uoL

aoh[p[tkm]7w$o=riilo=qwWxmojn]7imiV¶·aaMwWw¨¸#o=x

S = R

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(X(t))

ri¨j]Eqokm]7hg

ν

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E x (·)

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x

^ou¢

kY[]atx¥uoLB~[xohnE]Hii

(X (t))

ªhkmY[]Hxm]E¢Ýoux]

E ν (·)

§&r9[]Eq[oukm]kmY[]b]£j~P]Hnk]HV¹tu p]§&Y[]Hq¤kY[]~xmojn]7imi

(X(t))

ri&tk}]

®

p[r r[xr p[\°

XY[]Ltxrtu[ ]

L e ^ε (t)

[]Eq[oukm]Hi@kmY[]qhp[\P]Ex¨o¢npizkou\]Exvi@tLk¨kmr\]

t

^r q¤kmY[]

M/M/1

^® p[]Ep]}§&rµkY kr \]E¦JLtxghr qil]Hxmhrsn]xtkm]u°$XY[]~[xmojnE]Hii

( L e ^ε (t), X(t))

rsitatuxm¥=oL~[xojn]7imiH°.XY]|kxtuqilr kmrouqi@o¢

kY[]#~[xojn]Hii

( L e ^ε (t))

tuxm]buru]Hqhg+ -Õ¢

L e ^ε (t) = l

^tq

X (t) = x

tLk&kr \]

t

^ª

l →

( l + 1

tLk&xvtLkm]

λ

l − 1 ⁰⁰ (µ + εp(x)) {l>0}

¢Ýo=ximou\]¢Ýp[qnkmrouq

p(x)

ouqkY[]ilktkm]_im~t=n]#o¢³kY[]]Hq3hrxmo=q[\]Eq3k

S

tuqMimou\]im\tu ~txvt\]Ekm]Ex

ε ≥ 0

^°M´áY[]Hq

p(x) > 0

ª.kmYrir \~[r ]7ikYtLk_kmY[]Hxm]Vri_tq¯tujrµkr o=qt«nEtu~tunErµkzgo¢|im]Ex3rsn]§&Y]Eq nEou\~tuxm]7kmoBkY[]e3¦FEp[]Ep[]=°W`qkY[]no=q3kmxvtxguª3§&Y[]Eq

p(x) < 0

ªhkmY[]im]Exu]Hxri§&r kmYMtiloL§«]Exxtkm]

kYtq r q¬kY[]eh¦Ep[]Hp[]u°±XY[]

®

ptuq3kmr kzg

p ⁺ (a)

¶·xm]7il~P]Hnkmru]Eg

p ⁻ (a)

¸rsiBj]²q]H¯t=i

max(p(a), 0)

¶·xm]7il~P]Hnkmru]H g

max(0, −p(a))

¸°¨Ã«k}kr \]

t ≥ 0

ªjkY[]tu[[rµkr o=qtnHt~t=nr kzg¤rsikmY[]Hxm]E¢Ýoux]

εp ⁺ (X(t))

tuq

−εp ⁻ (X(t))

rsikmY[]nHt~t=nr kzg o3izk7°

(9)

XY[]bLtxrtu[]

B e ^ε

rsi¨kmY[]bjp[xvtLkr o=q¤o¢t[pimg~]Hxmroj¤izkvtxmkmrq[§&r kmYtnpizkou\]Ex7ªukYtLkrsiEªhuru]Hq

L e ^ε (0) = 1

^ª

B e ^ε = inf{s ≥ 0 : L e ^ε (s) = 0}.

ºoux

x ≥ 1

ªhkmY[]Ltxrst[]

B e _x ^ε

j]Eqokm]7i¨kmY[]#jp[xvtLkr o=qVou¢.t_[pimg¤~]Hxmrojilktuxlkr q_§&r kmY

x

npilkmo=\B]Hxi

¶

e B ^ε ₁

= B e ^ε

^¸° -ùqkmY[]#x]Hilk&o¢kY[rsi&~t~P]Ex7ª[§«]\tu¥u]kmY[]kz§«oB¢Ýou oL§&rq[tuiimp[\~jkmrouqiK+

kY[]¢Ýp[qnkmrouq

|p(x)|

rsi&o=p[qj]73gtBnEouqilktuq3k

M > 0

^¶

H 1

¸

ε sup(|p(x)| : x ∈ S) < µ.

^¶

H 2

¸

e3kvtqtxv¬txup[\]Eq3kviEª.il]H]VaM]Eghq¯tuq¬X§«]E]7jr]L2 4Jª³imY[oL§ kmYtk

( L e ^ε (t), X(t))

ri_tq¬]Hxm=oh[rn aGtx¥uoLw@xohnE]Hii§&Y[]Eq

λ < µ

^tuq

ε

ri|ilpj©¤nEr ]Hq=k gil\t)imo_kYtLk&kY[]#x]EstLkmrouq

λ < µ

1 + ε Z

S

p ⁻ (x) ν(dx)

rsiitLkmrsil²]Hï° 5}q[]ExkmY]nouqjrµkr o=q

ρ = λ/µ < 1

ª@kmYrinEouqjr kmrouq §&r &P]imtkmrsiz²]H¯~xmoLhrsj]H

kYtLk

ε

riilpj©¤nEr ]Hq=k gMim\t5°bXY[]

®

p[]Hp[]§&r kmYMkr \]E¦JLtxghr qVil]Hxmhrsn]xvtLk]tuij]²q[]HtuoL=]§&r

P]xm]E¢Ý]Exxm]7MkotuikmY[]~P]Exmkmpxmtkm]7

®

p[]Hp[]uª[]Eq[oukm]H)ª)¢Ýoux#imY[ouxmkHª3gGw³¦Ep]Ep[]=°XY]nHtuim]

ε = 0

o=h3roupilg¤no=xmx]Him~o=q[iWkmokY[]e3¦FEp[]Ep[]=°

- +,+ %(5"!(,+ &%( 4,"# &2 /

XY]Wt=ilrsn³rsj]Ht&ou¢jkmY[]«~P]Exmkmp[xtLkr o=qtqtu gjimrinEtxxr ]7oupjk.r q#kY[rsi~tu~]Hxrsi)konEouqilkmxpnk.t|no=p[~[r q

ou¢WkY[][pimgG~P]Exr oj[iou¢¨kY[]~[xojn]Hiim]Hi

(L(t))

^tuq

( L e ^ε (t))

°VXY[rsiri[ouq[]tuib¢Ýo= oL§}iHª)~[xoL3rsj]7

kYtLk&¢Ýo=x&oukmY

®

p[]Ep]Hi«kmY]txxmrLt9~[xohnE]Hii«ri

N _λ

^°

Â.Â ·Á ³Â ½ j¹ ½ ´±]j]Hq[ok]|hg

N ⁺

kmY[]q[ouqj¦ÕY[o=\Bo=u]Hq[]Eo=pi$w$oursimimouq~[xojn]Hii@§&Yo=im]

rq3km]Hqilr kzgVrsi}=r =]Eqhg

t → εp ⁺ (X (t))

°|^¨ouqjr kmrouqt gouq

(X(t))

ªkY[]qhp[\P]Exo¢³~Pourq=kvi}ou¢

N ⁺

rqkmY[]#rq3km]HxmLt

[a, b]

^ª

0 ≤ a ≤ b

rsiw$oursimimouq§&r kmY~txvt\]Ekm]Ex

ε Z b

a

p ⁺ (X(s)) ds.

XY]V~Pourq=kviou¢

N ⁺

tx]Vj]Eqokm]7¬hg

0 < t ⁺ ₁ ≤ t ⁺ ₂ ≤ · · · ≤ t ⁺ _n · · ·

tq¯tuxm]¤nHt ]7¯tujrµkr o=qt

[]E~tuxlkp[xm]7iE°>-ùq~txmkmrsnptuxWkY[]jrsizkxmr[pjkr o=qou¢kmY]# ojnEtkmrouq

t ⁺ ₁

ou¢kmY]²xilk&~o=r q3k}o¢

N ⁺

^t¢àkm]Hx

0

rsiuru]Hqhg=ªh¢Ýoux

≥ 0

^ª

P(t ⁺ ₁ ≥ x) = P(N ⁺ ([0, x]) = 0) = E

exp

−ε Z x

0 p ⁺ (X(s)) ds

.

^¶5u¸

ej]E] xtuqj]H 2Ó4j¢Ýoux@tuqBt=nEnEoup[q3k$ouqBq[ouqj¦ÕY[o=\Bo=u]Hq[]Eo=pi)w$oursiilo=q~[xojn]Hiim]HiHªLxm]E¢Ý]Exxm]7#kmo#t=i.[oup[[g

ilkmojnvYt=izkrnw$oursimimouq~[xojn]Hiim]HiH°

(10)

"!#$%

M/M/1

^& ^''

½ Á ½ j¹ ½ ´]V[]Eq[oukm]¤hg

N ⁻

kmY[]¤~Pourq3k_~[xojn]7imio=jktur q[]7hg '# ¤kmY[]

~Pourq3k«~xmojn]7imi

N _µ

¶5eh]H]b{}ouP]Exmk 27 4Ý¸° -Õkrsi«[]²q[]7t=i@¢Ýo= oL§}iK+.Ã"~Pourq=k&tLk

s > 0

ou¢kmY]w$oursiilo=q

~xmojn]7imi

N _µ

ri«t~Pourq3k«ou¢

N ⁻

§&r kmY~[xmo=tr rµkzg

εp ⁻ (X(s))/µ

^° -ùq¤kY[ri«§t¹guª

N ⁻

ri«t_ilktkmrouqtuxmg

~Pourq3k|~[xojn]Hii&§&r kmYrq3km]Hqilr kzg

εp ⁻ (X(s))

°}Ã~o=r q3k|ou¢

N ⁻

rsi|nEtu ]HMt¤\tuxm¥=]Hj]H~txmkmp[x]u°&XY[]

~Pourq3kiou¢

N ⁻

tx]j]Eq[oukm]7hg

0 < t ⁻ ₁ ≤ t ⁻ ₂ ≤ · · · ≤ t ⁻ _n ≤ · · ·

^°@ºoux

x > 0

ªjhgVj]E²q[r kmrouq)ª

P(t ⁻ ₁ ≥ x) = E





N µ ([0,x])

Y

i=1

1 − εp ⁻ (X (s i )) µ



 ,

^¶(=¸

§&Y]Ex]

(s i )

tuxm]kmY[]#~Pourq3ki&ou¢kY[]#~o=r q3k}~[xojn]7imi

N µ

°

´ár kmYkmY]tPoLu]bq[okvtLkmrouqªjrµk}rsiq[ok}jr ©¤np[ k&kmoimY[oL§kYtLkkY[]atx¥uoL~[xojn]7imi

(e L ^ε (t))

^Ytui

kY[]it\]#jrilkmxr[pjkmrouqtui«kY[]imoupjkmrouqo¢.kmY[]ilkmojnvYt=izkrnbjr=/P]Hxm]Hq3kmrst)]

®

ptLkmrouq

de L ^ε (t) = d N λ (t) − _{e _L ε (t−)>0} d N µ + N ⁺ − N ⁻

(t),

^¶·3¸

§&YrnvYri&kmY[]tuqtou=p[]ou¢Ä

®

ptkmrouq¬¶z7¸¨¢ÝouxkmY]#w³¦Ep]Ep[]=°

)f,±. «' .) #¡.

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0

rqGkY[]¤e3¦FEp[]Ep[]tq

w@¦Ep[]Hp[]u°1-ùqkY[riil]7nkr o=q)ª§«]j]Ekm]Ex\r q]#kmY[]²xilk}k]Ex\ou¢.kY[]~PoL§¨]Hx|im]Exr ]7i&]£j~tuqilrouqrq

ε

^ou¢

kY[]]E£h~P]Hnkm]7¹tu p]#o¢

B e ^ε

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ε

^°

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t=[jr kmrouqtu@j]E~txmkmp[x]oux#]Esim]¤tMilrq[u]\tx¥u]H±j]H~txmkmp[x]u°¤XY]~[xmo=tr rµkzgkmYtkoukmY¬]E=]Eq3ki

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ε ²

ilrqnE]}kmY]rq3km]Eqilr kmr]Hi ou¢kY[]tuiilojnErtkm]HVw$oursimimouq~[xohnE]Hiil]7itx]~[xou~PouxmkmrouqtPkmo

ε

^°

ºoux

x ≥ 1

ª)kmY[]ilktr rµkzgt=imimp[\~jkr o=qib]Eqimp[x]kYtLkkmY]B]E£j~]7nkm]7GLtu p[]7ibo¢WkY[][pimgM~]Hxmr ¦ oj[i#ilktuxlkr q§&r kmY

x

nEpilkmou\]HxiHª)qtu\B]H g

E(B x )

^tuq

E( B e _x ^ε )

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t ⁺ ₁ > B e ^ε

^tq

t ⁻ ₁ > B e ^ε

^kY[]Eq

B = B e ^ε

^°

-ùqMt_²xvizk|ilkm]H~)ªj§¨]~[xoLu]kmY]¢Ýou oL§&rq[Bkm]7nvY[q[rsnEtï]E\\t[°

¨½ ¯ C '#

K

^O

ε 0 > 0

^N'#

ε < ε 0

n ≥ 1

sup

x∈S E

B e _n ^ε | X (0) = x

≤ Kn.

1KF -Õ¢$ouq]nvYo3o3il]7i

ε 0

imoBkmYtk

µ 0

= µ − ε 0 inf{p ⁻ (x) : x ∈ S} > λ,

kY[]EqnE ]7tx gVkmY[]q3p\P]Ex|ou¢@npilkmo=\B]Hxi}o¢$kmY[]w³¦Ep]Ep[]rsi|nE]Exmktur q[gim\t ]Hx&kmYtqkmY]qhp[\P]Ex

ou¢}npilkmo=\B]Hxio¢|tq

M/M/1

^® p[]Ep[]§&rµkY¯txxmrLt@xvtLk]

λ

tq¬im]ExhrnE]xtkm]

µ 0

°±^¨o=qil]

®

p[]Eq3kmguª

(11)

kY[]nouxx]Him~o=qjrq[[pimgM~]Hxmroj[ibnou\~tuxm]r qGkmY[]imtu\]§t¹guªPY[]EqnE]r kbrsib]Eq[o=p[uYkmo¤kvt¥=]

K = 1/(µ 0 − λ)

^°

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+ ½ [ÂÂ Ý Á ¨Â ½ j¹ ½

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(0, B e ^ε )

kmY]EqtLk|kr \]

B e ^ε

ªjkmY]#w³¦

®

p[]Hp[]#rsi]E\~jkzgtuq

®

p]Ep[]

L

rsi§&rµkYo=q[]npizkou\]Ex#¶·im]E]º.rup[x]H¸°

6 - 6

e B B ^ε t ⁺ ₁

0 1

w³¦FEp[]Ep[]

e3¦FEp[]Ep[]

º.rup[x] +³Ã([pimgVw$]HxmrojV§&r kmYMtqÃ&[jr kmrouqt]E~tuxlkp[x]

´±]il~P]HnErµ²PnEtg¤~[xoLu]kmY[]¢Ýo= oL§&r qB]E\\t[°

¨½ ¯ L' < = CO

C

E

(B − B e ^ε ) _{t ⁺

1 <B}

= ε E ν [p(X (0)) ⁺ ]

(µ − λ) ² + o(ε),

^¶Ju¸

'

ν

^3C<^# ^83C ^K ^O ^I#

(X (t))

1KF }´áY]EqkmY]Ex]#ri&o=q[gVouq]tu[jr kmrouqt)j]H~txmkmp[x]uª[kmY[]Ltuxmrst[]

B e ^ε

rsi&]Ekz§¨]H]Eq

t ⁺ ₁

^tq

t ⁺ ₂

^°

´±]nEtuq§&xr km]

E

(B − B e ^ε ) _{t ⁺

1 <B}

= E

(B − B e ^ε ) _{t ⁺

1 B e ^ε }

+ ∆,

^¶5=¸

§&Y]Ex]kY[]#oO/ïim]kkm]Hxm\

∆

nEtuqP]#Poup[q[]Htui«¢Ýo= oL§}i

∆ ≤ E

|B − B e ^ε |

{t ⁺ 2 B e ^ε } + _{t ⁻

1 ≤ B e ^ε ,t ⁺ ₁ ≤ B e ^ε }

.

^¶ ^¸

(12)

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M/M/1

^& ^''

]kpi¤]Hilkmr\tLk]MkmY[]G²xvizk¤k]Ex\ o¢kmY[]xmruY3km¦JYtuqilrsj]ou¢ ¶·=¸° Ä ®

ptLkr o=q¶5=¸tqkmY[]

Poupqj]H[q[]Hiio¢

p

uru]bkmYtk

P(t ⁺ ₁ ≤ B) = 1 − E exp −ε Z B

0 p ⁺ (X (s)) ds

!!

= εE Z B

0 p ⁺ (X (s)) ds

! + o(ε)

= εE(B) E ν

p ⁺ (X (0))

+ o(ε) = ε µ − λ E ν

p ⁺ (X (0)) + o(ε)

hgBr qj]E~P]Eq[]EqnE]ou¢

B

tqBkmY[]izkvtLkr o=qtxrµkzg_o¢

(X(t))

^° G«gBkmY]ilkmxouq[aGtx¥uoL~[xou~P]ExmkzgtLkWkY[]

ilkmo=~[~[rq[Bkmr\]

B e ^ε

ªnouqjrµkr o=qtg¤ouqkmY]]E=]Eq3k

{t ⁺ ₁ < B e ^ε < t ⁺ ₂ , B e ^ε < t ⁻ ₁ }

ªjkmY[]eh¦Ep[]Hp[]ilktxmki tk

B e ^ε

tqr qj]E~P]Eq[]Eq3k}[pimgV~P]Exr oj§&rµkYouq[]#npizkou\]Ex7ªhkmY[]Hxm]E¢Ýoux]

E

(B − B e ^ε ) _{t ⁺

1 B e ^ε }

= P(t ⁺ ₁ B e ^ε )

× E

(B − B e ^ε ) | t ⁺ ₁ B e ^ε

= P(t ⁺ ₁ B e ^ε )E(B 1 ).

?|oL§#ª[ilrqnE]

{t ⁺ ₁ < B e ^ε } = {t ⁺ ₁ < B}

o=qVkY[]#]E=]Eq3k

{t ⁺ ₁ < B e ^ε < t ⁺ ₂ , B e ^ε < t ⁻ ₁ }

^ªjkmY]Eq

P(t ⁺ ₁ B e ^ε ) = P(t ⁺ ₁ < B) − P(t ⁺ ₁ < B e ^ε , t ⁺ ₂ < B e ^ε )−

P(t ⁺ ₁ < B e ^ε , t ⁻ ₁ < B e ^ε ) + P(t ⁺ ₁ < B e ^ε , t ⁺ ₂ < B e ^ε , t ⁺ ₁ < B e ^ε )

= P(t ⁺ ₁ < B) + o(ε),

imrqn]&kz§«o#oux¨\oux]&]£hkmxvtyzp[\~i¨r qkY[]it\]}pilgB~P]Exr ojrsi

o(ε)

°³ejr \rtuxmguª=3gpimrq[t3trqkmY[]

ilkmxouqBatuxm¥=oL~[xmo=~]Hxlkzg=ªjouq[]=]ki«kY[]¢ÝouoL§&r q[]7izkr \tLkr o=q

E

|B − B e ^ε | _{t ⁺

2 B e ^ε }

≤ X

n≥2

E(B n )P(t ⁺ _n ≤ B e ^ε ≤ t ⁺ _n+1 , t ⁻ ₁ ≥ B e ^ε )

≤ 1 (µ − λ)

X

n≥2

nP(N ⁺ ([0, B]) = n).

-ùqj]E]7ïªH=r =]EqbkmY]We3¦Ep]Ep[]=ª

N ⁺ ([0, B])

Yt=i)tw$oursimimouqjrsizkxmr[pjkr o=q#§&rµkY#~txvt\]k]Ex

R B

0 εp ⁺ (X (s)) ds

^ª

§&YrnvYr \~[r]HikYtLk

X

n≥2

nP(N ⁺ ([0, B]) = n) =

E Z B

0 εp ⁺ (X (s)) ds

!

− E Z B

0 εp ⁺ (X(u)) du e ^−ε ^R ⁰ ^B ^p ⁺ ^(X(s)) ^ds

!

= o(ε)

(13)

tuqkY[]}²xvilkWkm]Hxm\ rq¤kY[]|xr =Y3k¨Ytuq¤imrj]o¢ -ùq[]

®

ptrµkzg¶

¸@riWkmYhpi¨q[]E= rur[]tkWkmY]}²xvizk«ouxvj]Ex

rq

ε

^°

Xo]Hilkmr\tLk]&kmY[]bil]7no=q_km]Ex\ r qkY[]}xr =Y3kWYtqilrsj]}ou¢-ùq]

®

ptr kzg¶

¸ª=§«]}q[]H]HBkmo_nouqilrsj]Ex

kY[]¤jr=/P]Hxm]Hq3k~o3imimr r rµkr ]7i¢ÝouxkmY[]ojnEtLkr o=qou¢

t ⁺ ₁

^tq

t ⁻ ₁

^° -ùq±kmY[]¤nHtuim]BkmYtk

t ⁺ ₁

^tuq

t ⁻ ₁

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[p[xmrq[

[0, B]

^tuq

B e ^ε ≥ B

ª$tLkkr \]

B

kmY[]¤w@¦Ep[]Hp[]Yt=itLk_\Bo3izk

p ≥ 0

npilkmo=\]Exvi#rµ¢kY[]Ex]

Yt¹u]&P]E]Eq

p + 1

\tx¥u]7B[]E~tuxlkp[xm]7iE°>-Õ¢

D([0, B])

rsi@kY[]|qhp[\_]HxWo¢)nEpizkou\]ExviW[p[xmrq[#kY[]|pilg

~P]Exroho¢.kmY]e3¦Ep]Ep[]=ªhkmY[]HqnE]Exmktur q[g

E

( B e ^ε − B) _{ _B _e ε ≥B,t ⁻ ₁ ≤B,t ⁺ ₁ ≤B}

≤ E E X(B) B _D([0,B])

P(t ⁻ ₁ < B, t ⁺ ₁ ≤ B ≤ t ⁺ ₂ )

≤ KE (D([0, B])) P(t ⁻ ₁ < B, t ⁺ ₁ ≤ B ≤ t ⁺ ₂ ) = o(ε),

hg ]E\\tu°¨`qkmY[]#oukmY[]Hx&Ytq)ª

E e B ^ε − B { B e ^ε <B,t ⁻ ₁ ≤B,t ⁺ ₁ ≤B}

≤ E B _{t ⁻

1 ≤B,t ⁺ ₁ ≤B}

= o(ε).

º.rqtguª

E e B ^ε − B

_{t ⁻ ₁ _≤ B e ^ε ,t ⁺ ₁ ≤ B e ^ε }

≤ E e B ^ε − B

{t ⁻ ₁ ≤B,t ⁺ ₁ ≤B}

+ E

B _{t ⁻

1 ≤B,B≤t ⁺ 1 ≤ B e ^ε }

,

§&Y]Ex]#rµknHtqP]ilYoL§&qrqtimr \rtux}§«t¹gtui&P]¢Ýo=xm]kYtLk|kmY[]stuilk&km]Hxm\-rsi

o(ε)

°|`q[]nEouqnE pj]Hi kYtLkVkmY[]Gkm]Ex\

∆

^rsi

o(ε)

^tui

ε

=o3]7ikmo

0

^° G«gpilrq[¯Ä

®

ptLkmrouq ¶53¸ª}§«]ou[ktrqkmY]j]7ilrxm]7

x]Himp[ kH°

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®

ptLkr o=qá¶·=¸b\t¹gGtu~[~P]Htx

®

prµk]¤np[\_]Hximou\]u°

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(X (t))

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[]E~P]Eqj]Hqn]7iEªh§&Y[rsnvYVYt¹u]}koP]Ytqj]HV§&rµkYnHtx]u°$XY[rsi¨rsi¨§&Y3g=ªh§¨]bYt¹u]nvY[o=im]Eqko_]E£j~[ rsnr kmg

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k]Ex\° -ùqVkY[]¢Ýou oL§&rq[ª[ilr\r stx}tuxm=p[\]Eq3ki«§&r ïq[ouk}]#]E£j~[ rsnr kmg¤¢Ýo=xm\_p[stLkm]7ï°

+ ½ ¯j ½ ÂÂ ½ j¹ ½

ejp[~[~Po=im]BqoL§ kmYtLk#kmY]Ex]Brsi#o=q[ gGouq[]\tx¥u]7Gj]H~txmkmp[x]uªr5°]u°¤tj]H~txmkmp[x]o¢¨kmY[]Ve3¦FEp[]Hp[]ri

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-ùq kmY[rsinEt=il]=ª@tLkkmY]]Hq ou¢}kmY]pilg¬~P]Exroh o¢}kY[]e3¦FEp[]Ep[]=ªWtLkBkmr\]

B

^ª@kY[]w³¦^® p[]Hp[]Ytui o=q[]nEpizkou\]ExbtqkYhpi|ilktuxlkvi|t[pilg~]Hxmrojï°}w@xoLhr[]HkmYtLk}kY[]Ex]_tuxm]#q[oV\Bo=xm]\tuxm¥=]Htq

t=[jr kmrouqtuj]H~txmkmpxm]7iWjp[xrq[

(B, B e ^ε )

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~P]ExrohiWYt=i@kY[]it\]bjrilkmxr[pjkmrouqtui@kmY[]b ]Hq[kY

B 1

ou¢tilktuq[txv[pimg~]Hxmrojï°@eh]E]º.r =p[x]JM[°

(14)

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M/M/1

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6 - 6

0 1

w³¦FEp[]Ep[]

t ⁻ ₁ B

e3¦FEp[]Ep[]

B e ^ε

º.rup[x] +@Ã G«pimgVw$]Exr oj§&rµkYMtB]E]k]H]E~tuxlkp[xm]

¨½ ¯ L' < = O CO C

E

( B e ^ε − B) _{t ⁻

1 ≤B}

= ε E ν [p ⁻ (X (0))]

(µ − λ) ² + o(ε).

^¶^M=¸

1KF JG«g¤pilrq[BkmY[]it\]#txup\B]Hq3kituiP]¢Ýo=xm]=ª[ouq[]o=jktur qi«kY[]#x]EstLkmrouq

E

( B e ^ε − B) _{t ⁻

1 ≤B}

= E

B 1 {t ⁻ ₁ ≤B, B+B 1 <min(t ⁺ ₁ ,t ⁻ ₂ )}

+ o(ε)

= E(B 1 )P(t ⁻ ₁ ≤ B) + o(ε).

Xo]Hilkmr\tLkm]

P(t ⁻ ₁ ≤ B)

ªx]HnEtu )kmYtk

(D i )

j]Eqokm]kY[]im]

®

p[]Hqn]#ou¢$j]H~txmkmpxm]7i«kmr\B]7i}rqkmY[]

S

¦Ep[]Hp[]tq

N

kmY[]_qhp[\P]Exo¢Wnpizkou\]Exvi}im]Exu]Hjpxmrq[kY[][pilg~P]Exr ojo¢@]EqkmY

B

^ªkY[]Eq

Ä ®

ptLkmrouq¶(3¸«uru]7iWkY[]#rsj]Eq3kmr kzg

P(t ⁻ ₁ ≤ B) = E



 X N i=1

εp ⁻ (X (D i )) µ

i−1 Y

j=1

1 − εp ⁻ (X (D j )) µ





= ε µ E

X N i=1

p ⁻ (X (D i ))

! + o(ε)

= ε

µ E(N)E p ⁻ (X (D 1 )) + o(ε)

hgizkvtLkr o=qtxrµkzgo¢

(X(t))

tq´tsiº[o=xm\_p[st[°ehrqn]

E(N ) = µ/(µ − λ)

¶·im]E]VÃ|~[~P]Eqjr £¸ª Ä ®

ptLkmrouq¶M3¸¨¢Ýou oL§}iH°

-ùqkmY[]B]£j~tuqimr o=qo¢@kY[]B[pimgM~]Hxmrojo¢WkmY[]Bw³¦FEp[]Ep[]=ª9kmY[]_km]Hxm\»rq

ε

riuru]HqGhgkmY[]Bkz§«o ]Hu]Hq=kvinEouqimrsizkr q[r q_ouq[gouq[]«\tx¥u]7oux.ouq[gouq[]tu[[rµkr o=qthj]E~txmkmp[x]¨jpxmrq[}kY[]«[pimg#~]Hxmroj

ou¢

L

°@XY[]q[]£hk}~[xou~Po=imrµkr o=qV¢Ýo= oL§}i¨¢Ýxmo=\¡Ä

®

ptkmrouqi¶J¸tq±¶Mu¸E°

(15)

Á WÁ · Á ÝLHLÂ ½ Á

E( B e ^ε ) = 1

µ − λ − ε E ν [p(X(0))]

(µ − λ) ² + o(ε).

^¶5=¸

Ä ®

ptkmrouq±¶·=¸Wrsi«nEouqimrsizk]Eq3k¨§&r kmYkmY[]imo¦ùnEt]H{&]7jpn]7eh]Hxmhrsn]b{}tLk]btu~[~[xo¹£hr\tLkr o=q)°Ãi«t

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M/M/1

^® p[]Ep[]§&rµkYim]ExhrnE]#xtkm]

µ + εE ν [p(X (0))]

tqGtxxmrLt)xvtLk]

λ

°-ùqMkYtLk

®

p[]Ep[]=ªPkmY[]B\]HtqM]EqkmYo¢³kY[][pilg~P]Exr ojMrsi

=r =]Eq3g

1 µ + εE ν [p(X (0))] − λ = 1

µ − λ − ε E ν [p(X (0))]

(µ − λ) ² + o(ε),

§&YrnvYVnEourqnrsj]7iW§&r kmY¤Ä

®

ptLkmrouq±¶·u¸E° -ùqkmY]|¢Ýo= oL§&r qil]7nkr o=q)ª=§«]|rqhu]7izkr 3tLkm]}kmY[]il]7no=qouxvj]Ex

k]Ex\ tqimY[oL§ákYtLk&kY[]{|ej{(tu~[~[xo¹£hr\tLkr o=qVrsiq[o\Bo=xm]Ltrsï°

) $ '&*« # ¡

ε ²

o¢kmY[]&\]7tqB[pimg~]Hxmroj

E( B e ^ε )

rsi³nEtunEp[stLkm]7ï°;-ùqBkY[]}imtu\]

§t¹gtui«¢Ýo=x¨kY[]b²xvizk&ouxvj]HxHª=kmY[rsi&noh]E©nEr ]Hq3krsix]EstLkm]7ko_kY[]]E=]Eq3k¨kYtLkkz§«o_]E£hkmxvt}yzp[\~i}ohnHnp[x

[p[xmrq[Mt[pimg~P]Exr oj±o¢«kmY]~]Hxlkp[xmtLkm]7

M/M/1

^® p]Ep[]=°ehrqn]]E£3kxt_yzp[\~inEtqP]]Er kmY[]Hx t=[jr kmrouqtuj]H~txmkmp[x]Hi}ouxnHtqnE]E]HMj]H~txmkmpxm]7i¶Ý\tx¥u]7j]E~tuxlkp[x]Hiv¸ª[kY[]Ex]tuxm]kY[xm]H]nEt=il]7ikmo

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-ùqákY[]¢Ýo= oL§&rq[ªrqouxvj]Ex¤kmo¯=]k¤kY[]

ε ²

~Po=iimr [rrµkr ]7i«¢Ýoux&kY[]# ojnHtLkmrouqo¢.kmY]#¹tuxmrst ]7i

t ⁺ ₁

^ª

t ⁺ ₂

^tuq

t ⁻ ₁

^ª

t ⁻ ₂

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t ⁺ ₃

^o=x

t ⁻ ₃

ghr]Es[i&t_km]Hxm\¡ou¢

kY[]#ouxvj]Hx

ε ³

rqVkY[]#]£j~tuqimr o=qou¢

E( B e ^ε − B)

^°

]²q]

A + = {t ⁺ ₁ ≤ B, t ⁻ ₁ ≥ t ⁺ ₁ + B _L(t ⁺

1 )−1 }.

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tj]E~txmkmp[x]¤rsi_imp[~[~[x]Hiim]H¶·q[ok]¤kmYtLk

B _L(t ⁺

1 )−1

ri_kY[]V]Eq[ukmY o¢|tpilg~P]Exr oj¬ou¢be3¦FEp[]Ep]

ilktuxlkr qBtk&kmr\B]

t 1

§&r kmY

L(t ⁺ ₁ ) − 1

nEpilkmou\]Hxiv¸°W`qkmY[]#]Hu]Eq3k

A _± = {t ⁻ ₁ ≤ B, B ≤ t ⁺ ₁ ≤ B + B 1 },

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pilgG~]HxmrojGou¢WkmY]w³¦

®

p[]Hp[]uª§&Y[]Ex]

B 1

j]Hq[ok]HibkmY][p[xtkmrouq±o¢¨kmY[]tujrµkr o=qt³pilgG~]Hxmroj

[p[]kmoBkmY]ilp[~~[xm]7imim]Hj]H~txmkmp[x]u°$º.r qt g=ª[ouqkmY[]#]Hu]Hq=k

A − = {t ⁻ ₁ ≤ B, B + B 1 ≤ t ⁺ ₁ },

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kY[]#[pimg¤~P]Exr oj

B

^°

(16)

"!#$%

M/M/1

^& ^'' ^K(

A = A + ∪ A _± ∪ A ₋

^ª=kmY[]

]E£j~[xm]7imimrouq

E(( B e ^ε − B) A ^c )

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o(ε ² )

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®

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B e ^ε = B

^¸°³XY[]

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E(( B e ^ε − B) A )

^¢Ýo=x

A ∈ {A + , A _± , A ₋ }

^ª

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B

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E(( B e ^ε − B) A + )

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[0, B]

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t ⁺ ₁

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t ⁺ ₂

ε ²

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E(B − B e ^ε )

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B − B e ^ε

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{t ⁺ ₁ ≤ B}

^ª

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t ⁺ ₂

o=x«qokH°Wej]E]º.r =p[x] ([°

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B 2

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rqj]H~]Hqj]Hq=k}Ltuxmrst[]Hijrsilkmxr [p[km]Ht=i

B 1

tuqVrqj]H~]Hqj]Hq=ko¢

B

^ª

t ⁺ ₁

^tq

t ⁺ ₂

°¨`q[]#u]Eki

E

(B − B e ^ε ) _A ₊

= E

(B − B e ^ε ) _{t ⁺

1 ≤B,t ⁻ ₁ ≥t ⁺ ₁ +B _L(t + 1 )−1 }

= E (B 2 ) P

t ⁺ ₁ < B, t ⁺ ₂ < t ⁺ ₁ + B _L(t ⁺

1 )−1 , t ⁻ ₁ ≥ t ⁺ ₁ + B _L(t ⁺

1 )−1

+ E (B 1 ) P

t ⁺ ₁ < B, t ⁺ ₂ ≥ t ⁺ ₁ + B _L(t ⁺

1 )−1 , t ⁻ ₁ ≥ t ⁺ ₁ + B _L(t ⁺

1 )−1

+ o(ε ² ).

XYri&j]7no=\B~Po=imr kmrouq]Eq3ktur si«kmYtk

E

(B − B e ^ε ) A +

= (E (B 2 ) − E (B 1 )) P

t ⁺ ₁ < B, t ⁺ ₂ < t ⁺ ₁ + B _L(t ⁺

1 )−1

+ E (B 1 )

P t ⁺ ₁ < B

− P

t ⁺ ₁ < B, t ⁻ ₁ ≤ t ⁺ ₁ + B _L(t ⁺

1 )−1

+ o(ε ² ).

^¶lH=¸

6 - 6

t ⁺ ₁ B

e3¦FEp[]Ep[]

t ⁺ ₂ B e ^ε

w³¦FEp[]Ep[]

ºrupxm]3(C+³X§¨oÃ&[[rµkr o=qt]E~txmkmp[x]Hi

ºxmo=\ Ä ®

ptkmrouqG¶lHu¸ª=o=q[]}Yt=i³ko]£j~tuqBkmYxm]H]}]£j~[x]Hiilrouqi³§&rµkYxm]7il~P]HnkWko

ε

°$XYriWri¨jouq[]

hg¤~xmoLhrq[kY[]kmYxm]H]¢Ýo= oL§&rq[ ]H\B\t=iE°

(17)

¨½ ¯ C '#

P

t ⁺ ₁ < B, t ⁺ ₂ < t ⁺ ₁ + B _L(t ⁺

1 )−1

8 < '

P

t ⁺ ₁ < B, t ⁺ ₂ < t ⁺ ₁ + B _L(t ⁺

1 )−1

= ρε ² E Z B

0 (B − v)E ν p ⁺ (X (0))p ⁺ (X(v)) !

dv + o(ε ² ),

^¶lu7¸

1KF ]k@pi@xm]7nEtjkY[]&xm]Hu]Hq[]ExvtLkr =]«j]7imnExmr~jkr o=qBou¢9tpilg_~]Hxmrojizkvtxmkmrq[#tLk³kmr\]

0

§&r kmYouq[]

nEpilkmou\]HxK+Ã«k#kmr\]

E 1

¶Ý]E£h~Pouq]Eq3kmrst gjrilkmxr[pjkm]7G§&r kmY~tuxtu\]km]Hx

λ + µ

¸ª§&r kmY~[xout[r r kzg

µ/(λ+µ)

kmY[]&[pilg~P]Exroh_rsi$²q[rsilY]Hï°@`|kmY[]Hxm§&rsim]uª§&r kmYB~[xoutu[r r kzg

λ/(λ+µ)

ª3tq]E§npilkmo=\B]Hx tuxmxru]Hi.tqBtimp[j¦Õ[pimg~P]Exr oj_o¢ïjp[xvtLkmrouq

B ¹ ₁

¶·§&rµkYBkmY[]}it\]&jrsizkxmr[pjkr o=qt=i

B 1

¸.]Hurqi³p[q3kr

kY[]qhp[\P]Exo¢$npilkmo=\]Exvi¨x]Ht=nvY[]Hi

1

^t=tur q° -ùqkmY[rsi§t¹guª3kmY[]Ltxrst[]

B

nEtuqVP]xm]H~[xm]7il]Hq3km]Htui

¢Ýo= oL§}i

B = E 0 + X H i=1

E i + B ⁱ ₁

,

^¶l7u¸

§&Y]Ex]

H

ri=]Eo=\B]EkmxrnHt g jrilkmxr[pjkm]7§&rµkY~tuxtu\B]Ekm]Hx

λ/(λ + µ)

^ª

(E i )

^tx]Mr5°^rJ°]E£h~Pouq]Eqj¦

krtu gjrsizkxmr[pjk]HG§&rµkYG~tuxtu\B]Ekm]Hx

λ + µ

^tq

(B ₁ ⁱ )

^tuxm]_r5°^rJ°ï°Ã| .kY[]Him]_xvtqjou\Ltxrtu[]Hibtx]

rqj]H~]Hqj]Hq=k7°@º[o=x

0 ≤ i ≤ H

^ª

c

s i

j]Eqokm]7i.kmY[]&]HqBo¢PkmY[]

i

kYimp[j¦Õ[pimgnEghnE ]$+

s 0 = 0

tuqïªL¢Ýo=x

i ≥ 1

^ª

s i = s i−1 +E i +B ₁ ⁱ

^ª

B = s H + E 0

c

N i

j]Hq[ok]HikY[]qhp[\P]Ex}ou¢.tuxmxr Ltuijp[xrq[BkmY[]

i

kmYMimp[j¦Õ[pimgVngjnE ]

c

s i−1 + D ⁱ ₁

^{ª)°E°H°Eª}

s i−1 + D ⁱ _N _i

tx]kY[]#rqizkvtq3kiou¢.j]H~txmkmpxm]7io¢.nEpilkmou\]Hxi&jpxmrq[BkmY[]

i

^kY

ilpj¦Jpilgngjn]u°

ºoux«kmY]yzourq=k|[rilkmxr pjkmrouqo¢kmY[]#=]Hnkoux

(N i , D ⁱ ₁ , . . . , D _N ⁱ _i )

ª[il]H]kY[]Ã}~~]Hqjr £9°¨ºrupxm]buru]Hi tuqr pilkmxvtLkr o=qou¢kY[]#tPoLu]#j]E²q[r kmrouqiH°

-Õk#rsi]7tuimgkmoil]H]kYtLk¢Ýo=xkY[]]E=]Eq3k

{t ⁺ ₁ ≤ B , t ⁺ ₂ < t ⁺ ₁ + B _L(t ⁺

1 )−1 }

kmoojnEnEp[xHª

t ⁺ ₁

^tq

t ⁺ ₂

Yt¹u]}ko_P]r qkmY]bit\]ilp[[¦J[pilg~P]Exr ojïª

[s i−1 + E i , s i ]

ªh¢Ýoux&ilo=\]

i ∈ {1, . . . , H }

°³º[oux&t²[£j]H

i

ªhkY[]#~[xoutu[r r kzgkmYtkkmY[]²xvilk&kz§¨otu[[rµkr o=qthyzp[\~i|tx]brqkmY[]

i

¦JkmYilpj¦JpilgV~]Hxmrojïª[rsi

E Z s i

s i −1 +E i

εp ⁺ (X (u))e ^−ε ^R ⁰ ^u ^p ⁺ ^(X(s)) ^ds

1 − e ^−ε ^R ^u ^si ^p ⁺ ^(X(s)) ^ds du

!

= ε ² E Z s i

s i−1 +E i

p ⁺ (X(u)) Z s i

u

p ⁺ (X(s)) ds du

!

+ o(ε ² ).

(18)

"!#$%

M/M/1

^& ^'' ⁷

-

6 - - 1

B 1 ¹

s 0 =0 E 1 s H −1 s H B

E 0

s 1

º.rup[x]'+³]Hno=\~o3ilr kmrouqo¢³t G¨pilg¤w$]Hxmroj

ehrqnE]uª

B ₁ ⁱ = s i − s i−1 − E i−1

YtuikY[]it\]jrsizkxmr[pjkr o=qtui

B

tqhgkmY[]BizkvtLkmrouqtxrµkzg¤ou¢

((X (t))

ε ²

nEtqP]#]£j~[x]Hiil]7Vtui«¢Ýo= oL§}iHª

E Z

0≤u≤v≤B

p ⁺ (X(u))p ⁺ (X(v)) du dv

= E Z

0≤u≤v≤B

E ν p ⁺ (X (0))p ⁺ (X(v − u) ) du dv

.

º.rqt g=ªimr qn]

H

ri=]Eo=\B]EkmxrnHt g[rilkmxr pjkm]7§&r kmY¬~txvt\]k]Ex

λ/(λ + µ)

^ªÄ ^® ptkmrouq¶lu7¸

¢Ýo= oL§}iH°

´±]&kmpxmqq[oL§ kmokmY]}]£j~tuqilrouqBo¢PkmY]

®

ptq3kmr kzg

P(t ⁺ ₁ ≤ B)

ª=§&Y[rsnvYri@o¢ïnoupxim]«tx]²q[]E\]Eq3k ou¢§&Ytk}YtuiP]E]HqMjouq[]rqeh]Hnkmrouq([°

Perturbation Analysis of a Variable M/M/1 Queue: A Probabilistic Approach

HAL Id: inria-00070587

https://hal.inria.fr/inria-00070587

Submitted on 19 May 2006

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