IS 79/89: Comutatoal Methods IS Research Smle Marova Queueg Model Nrmalya Roy Deartmet of Iformato Systems Uversty of Marylad Baltmore Couty www.umbc.edu
Queueg Theory Software QtsPlus software The software s avalable from the Wley ublc ft server ft://ft.wley.com/ublc/sc_tech_med/queueg_theory Based o the lectures we wll smulate some queueg theory models usg ths software QtsPlus cotas models the followg areas: Fte ad fte brth-death rocess models Sgle server queueg models Multle server queueg models Bul queueg models Prorty queueg models Networ queueg models
Itroducto Develo a broad class of smle queueg models theory of Brth-Death rocess A Brth-Death rocess s a secfc tye of cotuoustme Marov cha Examle of queues that ca be modeled are M/M/, M/M/c, M/M/c/K, M/M/c/c, M/M/ etc Cossts of a set of states {,,,..} deotg the oulato of some system State trasto occur as ut jums u or dow from the curret state
Brth-Death Process (BDP) Secal case of Marov cha wth restrcted trastos Rate trasto dagram for a brth-death rocess Trastos are restrcted to eghborg states oly (rocess state ca chage oly to state + or -)
Brth-Death Processes System s state The tme utl the ext arrval ( brth ) s a exoetal radom varable wth rate λ At a arrval system moves from state to state (+) The tme utl the ext dearture ( death ) s a exoetal radom varable wth rate μ At a dearture, the system moves from state to state (-) deotes the log-term fracto of tme the system s state Ths s a cotuous tme Marov cha States deote the umber of customers the system Brths corresod to customer arrvals Deaths corresod to customer deartures
Brth-Death Processes Cocet of flow balace I steady state rate of trasto out of a gve state = rate of trasto to that state LHS deotes the rate of trasto out of state RHS deotes the rate of trasto to state - + ( ) ( )
Brth-Death Processes - + Trasto out of state ca be ether uward or dowward ( ) s the log-term rate of trasto out of state Trasto to state ca occur ether from the state below (-) or the state above (+) s the log-term rate of trasto to state reresets a flow balace for the boudary state
...... Equlbrum Soluto of BDP ) ( 3 3 3 3 Prove that
Use mathematcal ducto to rove that Equlbrum Soluto of BDP...
Posso Process Secal case of brth-death rocesses Posso rocess s a coutg rocess (ure-brth rocess) havg costat deedet ad statoary cremets (costat brth rate)
Suerosto of Posso Processes Aggregatg multle Posso rocesses roduces a hgher rate Posso rocess (whose rate s sum of the rate of dvdual Posso rocesses)
Decomosto of Posso Processes Decomosg a Posso rocess stream to multle stream results multle Posso streams of lower-rate.
Stablty of a Brth-Death rocess We desre that queues do t grow ubouded Posso rocess (ustable sce the death rate s zero; sze grows ubouded) For a BDP to be stable, there must exst some state j such that μ j s greater tha λ j for all j > j
Sgle Server Queues
M/M/ queue Iterarrval tmes ad servce tmes are exoetally dstrbuted Probablty desty fuctos a( t) t e b( t) The rate of arrvals λ s fxed, regardless of the umber the system The rate of the server μ s fxed, regardless of the umber the system t e M/M/ s a brth-death rocess wth ad ( ) ( )
M/M/ queue (a smlfed BDP) M/M/ queue s a brth-death rocess wth λ = λ ad μ = μ Rate trasto dagram for the M/M/ queue The flow-balace equatos for ths system are ) ( ) ( Alteratvely, these ca be wrtte as ) (
M/M/ queue Soluto Solve for { } usg a teratve method Iteratvely use the balace equatos to obta a sequece of state robabltes,, 3,.., each terms of Verfy that cojecture s correct usg mathematcal ducto M/M/ system s a brth-death rocess wth costat brth ad death rates, we ca drectly aly wth ad The soluto of M/M/ system s
M/M/ queue Soluto Probabltes of { } must sum to for a sgle server queue where ρ s the traffc testy or utlzato Ths s cosstet wth geeral result for for all G/G/ queues G/G/ emty system robablty
M/M/ queue Steady State Soluto The full steady state soluto for M/M/ system s the geometrc robablty fucto ( ) ( ) Note the exstece of a steady state soluto deeds o the codto that Equvaletly Itutvely f the mea arrval rate > mea servce rate Server gets further ad further behd; system sze creases wthout boud over tme Why there s o steady state soluto whe
M/M/ queue Steady State Soluto Why there s o steady state soluto whe Ifte buld u As the queue grows t s more ad more dffcult for the server to decrease the queue Average servce rate s o hgher tha the average arrval rate
M/M/ Examle Cars arrve at a solated toll booth as Posso rocess wth arrval rate λ=.6 cars er mute. The servce requred by a customer s a exoetal radom varable wth exected value of /μ=.3 mutes. What are the lmtg state rob. for N, umber of cars at toll booth? Sce ths s a M/M/ queue (Posso arrvals/ deartures; sgle server) wth ρ=λ/μ=.8, the lmtg state robabltes are: = (-ρ) ρ =(.8)(.8), =,,. What s the robablty that the toll booth has zero cars steady-state? =.8
M/M/ Examle Cosder a l wth Posso arrvals (rate of 3 acets/sec) such that the tme tae to servce a acet s exoetally dstrbuted. Assume mea acet legth of 5 bytes ad l caacty of.5 Mbs. Probablty that the l s queue has, ad acets resectvely? Arrval rate λ = 3 acets/secod. L caacty =.5 Mbs Rate of servg acets μ =.5 Mbs/ (5 8) bts = 375 acets/s Traffc testy or utlzato ρ=λ\μ =.8 Probablty of oe acet queue = ρ (-ρ) =.8 (.) =.8 (ote: oe acet queue meas two acets system) Smlarly, 3 =., ad =.7
Measures of Effectveess Steady state robablty dstrbuto for the system sze Calculate the system s measure of effectveess Exected umber the system at steady state Exected umber the queue at steady state N reresets the radom varable umber of customers the system steady state L reresets ts exected value L E[ N] ( )
3...) 3 (... 3 ) ( d d d d Exected umber the system for M/M/ at steady state s L ) ( ) ( ) ( ) ( L Exected Number the System (M/M/)
Exected Number the Queue (M/M/) Number of customers queue steady state = N q L q reresets ts exected value For all sgle-chael, oe-at-a-tme servce queues Mea queue legth ) ( ) ( ] [ L N E L q q ) ( L L q ) ( L q
Exected Watg Tme Exected steady state system watg tme W ad le delay W q Use Lttle s Theorem of L ad L q ) ( L W ) ( q q L W
Summary Brth-death rocess Steady state robabltes ad erformace measures M/M/ ad M/M/C (Erlag C formula) ad M/M/ Fte buffer: M/M/c/K ad M/M//K ad M/M/c/c (Erlag s Loss formula/erlag B formula) M/M/c ad M/M/ are very useful models for multle servers queueg systems Fte buffered M/M//K ad M/M/c/c rovde useful egeerg rovsog gudeles
Questos?