04_bdqueueing - CSC5420 CS599 Birth-Death Process John C.S....

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Unformatted text preview: CSC5420 CS599 Birth-Death Process John C.S. Lui http://merlot.usc.edu/cs599-s03 1 Copyright © John C.S. Lui CSC5420 Little’s Result arrivals Queueing System ¢ £¡ ¤ § ¨¦  © ¥ departures ! "¡      ¨    ¢ ¤ ⇒ let D F £E # of customers % & N(t) # % £$ & C "B $ ) ¨( 10 A @ 7 86 ' 32 54 5 29 α(t) 1 0 G PI δ(t) time t P U VT T SR R X T Y a £` b H a £` e fb g a £` x yw ⇒ total area between these two curves up to t represents the total time all customers spent in the system (in units of customer-seconds) during interval (0,t) ⇒ denote by ⇒ let λt be avg. arr. rate (cust/sec) during (0,t) p £i c d b ⇒ s tr u x v € h q WU W Q F 2 Copyright © John C.S. Lui CSC5420 Little’s Result (Cont...) © §¦¥ ⇒ ⇒ let ¡ ¢ £ = avg. # of cust. in system during (0,t)   §  ⇒ ⇒ !  " !# !$      ¨ ¤ §¦¥ ⇒ Tt = system time per cust. avg. over all customers in (0,t) ¨ ⇒ Little’s Result VU TS R 6Q W X `Y a I H FE PG HC Assume and exist ⇒ & '% 2 1 )( 30 1% A @ 87 B9 @4 5 64 exists D 6C 3 Copyright © John C.S. Lui CSC5420 General Equilibrium Solution ⇒ Assume   ¥¤   ¡ ¢ £ ¦ ¨§ (Birth-Death Process) exist (prob. of finding birth-death equil. prob. of finding k cust’s in the system system in state k) ⇒ The change in input and output flow at the lim is 0 or input flow = output flow ⇒ conservation of flow Flow rate into state k Flow rate out of state k   " $ # ! ! ! "   %  ('  & ) ) 4 53 1 20 © ¡ ⇒ 97 @86 7 86 In equil. D 2C 9 @7 B E7 B E7 F A A A G DC H ⇒ of course, I A 7B 7 ) % ! choose boundary P ⇒ Can apply conservation of flow to any subset of states (not just one state) ⇒ If counter flow into/from subset 0 through k-1: VT WUS V WT Y Ta X X TY ` I Q R 4 Copyright © John C.S. Lui ⇒ ⇒ " 4 5 F ¢E £ 6 G 8 I ¦H P ¦H IE 6 9 B7 ' 9A 1 C 9 B7 1 9A 1 C 3 ¢) D ¢@ A D 32 2  ( !  ) 1 2   ( &   9A ' ) 0 &   @ 9 ¦7 '  (     PQ FQ 8 ¦7 %& $   ¡§ ¥ # © ¨ ¡ ¢ ⇒ d ec S TR f U W ¦V d Xa X ¦V i p Y Y Y Y t q S `X WR w f V Sa Y Y b ¢a g hc ⇒ ⇒ from „ r sq … „ † u ‡ p qv g —• ˜–” ™ ‘ g — ‰ ™ li k m i jh e f• d Copyright © John C.S. Lui ¥ ¦¤ ⇒ From above can guess that general solution is General Equilibrium Solution (Cont...) ⇒ ‰ hˆ  y €x y y “ ƒ ƒ ƒ ‚  ‘ ’ (can verify with pk+1) CSC5420 5 CSC5420 Existence of Steady-State Probabilities ⇒ Existence of steady-state probabilties pk ⇒ For above expression to represent a probability distribution, usually require p0 > 0 ¨ ©¡¥ ¤ ¡ ¢ ⇒ More formally/precisely Define: £  §¡¥ ¦ $ ¦   ¡   require that system empties occationally  & 74 6 53 4 8 GD ¢C ¢C @ 9    % ⇒ All states: ergodic recurrent null transient Copyright © John C.S. Lui ! "# 1( ! 2 )0 # ' ⇒ E F B equil. prob’s A UR ¢Q ¢Q bY ¢X ¢X H S S c a ` a T T V W I P F 6 CSC5420 Existence of Steady-State Probabilities (Cont...) ⇒ Conditon for ergodicity is met whenever seqence remains below unity for some k onwards, i.e., there exists some k0 s.t. ¦§ ¨§   ©    ¤¢ 7 Copyright © John C.S. Lui ¢ £¡ ¥ CSC5420 M/M/1 System ¡ ¢ ¤ ¦¨ £ ¦§¥ ¦©    (inf. queueing space) £ "  & ' § 1      ⇒ 1 " 3 "! # $%! 2 # $%! ⇒ To be ergodic (and hence pk > 0) ⇒ P P ) 0( $ 2 4 7 86 5 C D86 9 BA @ ⇒ G HF Q U V X Q I TU I R STQ R STQ W Y y converges iff c g c vu g p s p r h qi ⇒ a H` w d € b b t h e Sfd e Sfd h b ƒ x  ‚ satisfied if ⇒ necessary and sufficient cond. for ergodic is λ < µ 8 Copyright © John C.S. Lui „ … † E @ CSC5420 M/M/1 System ¨ ©§ ⇒ ¤ (converges since λ < µ) ) ¡ ¢ £   ¤ ¥ ¦  ⇒ ⇒ let !   ¨  ¢ ! # ( ! " $ & '% 3 ⇒ util. ⇒ $ # 0 (for stability) V 5 7 86 1 2 4 examines p0 > 0 depends only on ratio ⇒ 9 @ R SQ B CA T B FE I P F H D H D T U U G R a h R U W p qi Hw: rewrite y ‚  € r s uvt Y Y `X f ge Y b w h x h i c b d p h ˆ ‡ „ …ƒ (Little’s Result) † t 0 Copyright © John C.S. Lui 1 9 ‰ CSC5420 M/M/1 System (Cont...) utilization factor ⇒ ρ = Ratio of the rate at which work enters the system and the max rate at which the system can perform this work ⇒ Single server    §©  © ¡ £¢ ¥¦¤ ¨§ ¢ ¨¢  £§ ¥¦¤  ¨¢  ¢ © ¨ ¨   £   ⇒ can do this work at rate of 1 sec./sec. amount of work each customer ings ⇒ avg: λ cust. arr. per sec. ⇒ Multiple servers now work capacity of system is m sec./sec. then ⇒ " # & %$ ⇒ As long as 76 5 9@ 8 34 (Note: above time of service rate is indep. of system state) ¥ can interpret it as Q R8 8 I S 10 ' ) 0( 7 G HF PI CDA B EC 1 Copyright © John C.S. Lui IQ 2 ¢ ! ⇒ 3 ¢2 " #! ⇒ ⇒ for ⇒ for ⇒ ƒ „‚ 4 $ ! 5 7 86 @9 % A B2 $ ' (& C 2 ' D ' 4  1 1 @ E 1 0 ) … ‰ Y £ € ¤ ¥ © ¨ ¤ ¦ § d – e ih  ¡ VU  ¥‚  ¥‚ ` ¥W ’ “‘ ” ’ ’ b ca — • — ” ` – ƒ f e ‡ ˆ† y T X ¢W ¡ ¢ x S Copyright © John C.S. Lui ˜ p g    q str ™ ˜  f d fu b £ d a ¤ X d C 2 2 F g   ¡ Y g ` ¥W  s g Q jk i X R G  IP H C C G@ G@ ⇒ Condition for ergodicity is M/M/m System where s v h l v m v b w X q a (m server case) expected fraction of busy servers 11 CSC5420 CSC5420 M/M/m System (Cont...) ⇒ Solve for p0: © ¤     ¨§     ¥ ¦ ¡ ¢ £ ¦  ¥ ¨      § ¨     © § % " & 0 1) 0 1) $# 3 3 # 2 ! 5 6 4 5 2 8 97 7 ' & ( 0 ` DY S b T A C DB F H IG F E E P Q W XA R a e U T V ` f gT V c d v © w Xi w „‚ ƒw q qr  y † qr ‚u u ts arr. cust. joins queue qr u w w xi ip st v 2 @ $ % ⇐ yr q Erlang’s C formula h € … v p s q w probability that no telephone trunk is available for an arriving call 12 Copyright © John C.S. Lui yr v‡ ip CSC5420 M/M/m/m System & " (m-server loss system) ⇒ ' ( # $" % 0 ' 1 5 5 7 86 ) 3 ¢2 2 9 ) 4 4 @ @ 7 ¡ § © ¨ G HF ¤ R SQ UC V    ⇒ ¦  7 @ ¡ ¢ ¤ ¥  A £   X P ¥B Y C DB E T C W a X b ⇒ Solve for p0: d ¥c e I ` ` ! f t ui x h g r ⇒ pm: fraction of time that all servers are busy ‡ D† ‰ ƒ  Erlang’s loss formula or Erlang’s B formula 13 ˆ ƒ „‚ … ƒ ‡† ‰ “ ‘ •“ q”’ ƒ – Copyright © John C.S. Lui ˆ “ ‘  p qd i s v w y €  CSC5420 Homework HW: Derive pk, p0 for: - M/M/1//M - M/M/∞//M ⇒ finite cust. population ⇒ and inf. server - M/M/m/K/M ⇒ finite pop., m-servers, finite storage (K) assume M > K > m 14 Copyright © John C.S. Lui ...
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