phase_type_system - CSC5420 CSC5420 Phase-type Systems John...

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Unformatted text preview: CSC5420 CSC5420 Phase-type Systems John C.S. Lui http://www.cse.cuhk.edu.hk/~cslui 1 Copyright © John C.S. Lui • ¢” ˜ • ¢” – —‘ … “ ’ ‘ (&$W‰ ˆ † ‡… „ƒ † ‡| † vRqz s uh s wvt¤e µ E B ) ' EF CD23) B 42 53) ) @ ' A897 6  £ 0 1# ' % #! (&$"  § ¥ £¡ ¨¦¤¢ Recall: expo.  ¢© Decompose service time onto structured expo. distributions Let service time density function Method of Stages: Erlangian Distribution Er each a V WU x V WU S €h x a‚ y S u sP p h wvtr qig fP eca ` X V db¢YWU S Q PH TR8IG Copyright © John C.S. Lui service facility 2µ 1 2µ 2 more general than birth-death one customer in this box at a time CSC5420 2  ¢œ  ¢œ „ ƒ‚  …qi¢€ ž 1˜ › ™ ˜— (šg"– •  Œ ‹Š ¨&$&‰ ˆ } | {z ¨&$"y x “‘Œ ‹Š  ”’&$"‰ Ž ~ j  k r pn m &qo¢l i h fe (&g&d ™ CSC5420 Erlangian Distribution Er (Cont...) From transform table ) 7 1@8 '9 3 Copyright © John C.S. Lui ind. r.v. HW: can verify through transform or density func. mean same as expo.; var. half of expo. u w i w Cv u y b ‘ ˆ U† s aD w Cv u ‚ ‰U† ˆ y †‡ igp trqd c YWV T R D P IB P `X4USF "CQ!CI ‚ €x y„ sgp 4rqd c … y„ ƒ€x ig "hfed c G F DB H"ECA Therefore: 7 2 1§) ( ¥¢ 0¦ '  5634# %# ¦ &$ " ¢ ¨ !¦ ¢        ¤ ¨ ¦ ¤¢ ©§¥£¡ ‚ y ƒ€x "!4 § FE¡ C B ¥D § 07 ) ' a b¥ `X¡ Y¨ P© H Q¨ ID § Q¥ 4 ¡ G  7 c&  VWU ¥ 4 ¤T¨  © ¨¡ © © ¨ R 0 S I ¨ A @8 ( 9 © ¨ © ¨ '¨ ' § ( 0 ' §0 ¥3¤¡ ©¨ 0 1) ' §2 "!£ rµ 2 ......... rµ r © © ¨ ' § ' ¨ 6% 54$ # § ( ©¨ ' § & $£ # % § ¦¤¢ ¥ £¡  ¨    © rµ 1 Copyright © John C.S. Lui Er: r-stage Erlangian distribution Erlang distribution coeff. of var. CSC5420 4 CSC5420 M/Er /1 System # of stages to do for customer in service State description: [k,sl] transform to [s] where s is the total number of stages yet to be completed by all customers. th If the system has k customers and when the i stage of service contains the customers. Let Pj be the probability of j stages of work in the system. Since j=rk-i+1, we have: Copyright © John C.S. Lui ~~~{|{ €€}`z  ˜ v ¢ƒ u ‡y †…v „ƒ (‚r c€xv u tr  s @ w  u y  yw s D G V T q T b X V P V G f Ae VR Gb T G Y X V T R P H G D B agWWpSSgihgV dQ¤6dca`SWUSQI¢F ECA m x ys v wl 87 ‰ 5 u rpn tst6‰ qom l  kf“ i ”gf e – • ”“  ‰ pW¦Ij `hWhId ™˜—`¦†’ ‘hˆ ) (! ' 01¥ ¡ "! ! " 564# 2© %& 3 # $¥  ¡ © ¨ § § ¥£¡ ¦¤¢ ¥ ¡   @ @ 9 g R C Q q¤Q pP R D hQb ef e @ RaC'B@ ©YH ` QI R I ¨H CA U&@ 9 RW V Q G @ X#!@ ¢E G4iP R D hQb W R E g CA U&@ T T 9 gXf¤Q e$P R D dQbcPYH G FD I E CA 'B@ @ R QP I S@ ¨H G FD E j 9 555020 876431) ¥ ( Let Pj=0 for j < 0. '%&¡©§¨¦ $"#! ¡ ¡§ ©¨¦ ¥ rµ rµ rµ ¥ ¨¦   ¡ § £¡ ¤¢ rµ Copyright © John C.S. Lui Define P(z) = Σ Pj z 0 rµ 1 j=0 2 ∞ λ rµ .......... M/Er /1 System (Cont...) r r+1 ............... j-r rµ ........... λ rµ j rµ j+1 λ CSC5420 6 c ‚e h %g e ¥c t u b p y wv €x%c pi 43b u h g t h %g f ec 1¥db a p ibe s3Srq 9 C B 5 9 %8 B 6 8 `C 9 8 6 A@ E PR VT PR P IGE 8 YHXWUHSQ0HFDC B 9 8 6 75 A@9 ¨&%8 ¦ ¦ ¤¢ ¥£¡ '  $&% " ¤ ) ¡   (     ¢  " ¤  2¡  § 4310¨ ' " ¤ ) ¡   (     ¢  " # $!¤ !¡     ¨  © § M/Er /1 System (Cont...) Copyright © John C.S. Lui Define ρ = /µ, we have Since P(1)=1, therefore, using L’ Hospital rule, we have λ p0 = P0 ⇒ so still same notion of util. ⇒ try for r=1 to match M/M/1 solution CSC5420 7 CSC5420 M/Er /1 System (Cont...) For general r, look at denominator, there are (r+1) zeros. Unity is one of them; we have Therefore, we have r zeros which are z1, z2, ..., zr. We can arrange them to be 9H I©7 7 53 GF2 EE #$E 9B DC7 7 53 6A2 98 @17 7 53 642 0 1) '¦% (&¥ "" #$#"  ! ¥ e  ¥  Y £  ¨ ¦ ©§¥ £¡ ¤¢ Need to resolve this by partial fraction expansion. For Er /M/1 system, derive it at home. 8 Copyright © John C.S. Lui g hf ` f V 6U ca db` S XW VU 6¢Q T SR !§Q We have P CSC5420 Bulk Arrival Systems Let gi be the probability that the bulk size is i, for i > 0. ........ ........ λg i λg 2 ........ k-2 λg 1 k-1 k λg 2 λg 1 µ k+1 k+2 ........ µ µ µ Copyright © John C.S. Lui 3 40 ¡ 2#  ('& 9 ¨   ! ¨ ! £ " ¨ ! )   0 ¨ ¤¡ ) ¦  ¡ %$ # ¨   ! £"  ¨ ¤§¦  ¡ ¥ ¨¡ ©§¦ ¥ ¥ £¡ ¤¢  ¡ ¤¦   0  1¤¡ ¨ ! ) ¦    ‡ … „‚€ x †3ƒyw v USQ VTRP PE P I H GE 5FD C t` Y BX s ` pY d cr Y A@98q d cX` i3hgf d` !Y dc e3X b dc 43X b a `Y "X W ¥  B )  2§  A@98¡ ¥  ¥ 7  2 635¤  ¤  2  © 4310¤  "© ¥ ! © § ¥   © © ¥  ©§ ¥ ¨¦¤ £ ¡ ¢ ¥  )¥  % © (¡ £ # '&%$© Copyright © John C.S. Lui For bulk service system, try it at home. where Using P(1) = 1 and L’ Hospital rule Bulk Arrival Systems (Cont...) avg. bulk size transform for distri. of bulk sizes CSC5420 10 u p ia Q X TR X V USQ hf e d b Sg0¢cQ Q P I I Q Y X 9F Q VWTUSQ R P Ga ¦¤D ` G FD H3EC B % A % 5 4£ # @8 %¨ # ©  © # 98 ©¨ § § ¥ 3¡ 7 6 8 ¥ £¡ ¦¤¢ !1   % % 320)'('&¨ # !   © © $" ¢¨ α2 µ2 Copyright © John C.S. Lui In general, if we have R parallel stages (hyper-exponential): α1 Parallel System µ1 CSC5420 11  $    #  ¡     "¡  !  ¤   ¢ ¡  ¡      ¢ §¥ ¨¦¤ ¤ © £  ¤ © £ §¥ ¨¦¤ ¢ ¡ ¢  ¡ ¤  Copyright © John C.S. Lui Parallel System (Cont...) CSC5420 12    CSC5420 Series and Parallel System r1 α1 r1 µ1 r1 µ1 .......... .............. r1 µ1 ri ri µi αi ri µi ri µi ........ .......... rR rR µR αR rR µR rR µR .............. Copyright © John C.S. Lui @ 13 9£   ©  © H© F   GC © ¨ © ¦E© # $!   © & 1 ) 0¥ ( '©  ¡ ©  ¨ 74 8#65#3!2%  © % # ! ¥ © © ¡ ¦¤©  $"¦£ § § ¥C D$¡ B A ¥ £¡ ¦¤¢ CSC5420 Series and Parallel System (Cont...) Generalization of series-parallel server remove restrition that each stage within the same branch has same service rate th th ⇒ µij ⇒ service rate of i branch, j stage   14 Copyright © John C.S. Lui "  $¥ # "      ! ¦ ¨ § ¥£ ©¤¦¤¢ ¡ CSC5420 Series and Parallel System (Cont...) Another way to create series-parallel effect β1 α1 µ1 β2 α2 µ2 β3 ... α3 th βi αi µi βi+1... αi+1 αr βr µr 1 Before entering i stage, indepedent choice th with prob. βi proceed to i expo. service stage with prob. αi depart after r stage depart with prob. 1 th Copyright © John C.S. Lui (Cox showed that more complex transitions in above do not give more general service distributions.) (If permit complex values for riµi, then can syntherize any rational func. 15 of si, can approx. non-rat. func. can be approx. arb. close.) c 9 a`£WU XYX V 3 b6 Q 5 R S T) 8 RS A 6 PR "   ¨ #!    @ Q 6 B@ 5 @ CG G E C A PIG HFC D6 B@ 6  ©§¦ ¥£¡ ¨¢ ¤¢ 9 86 §75 420(& % 3 1 )' $ ...
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This note was uploaded on 05/18/2010 for the course COMPUTER S CSC5420 taught by Professor Lui during the Spring '10 term at CUHK.

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