Queuing Systems: Lecture 1. Amedeo R. Odoni October 10, 2001
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1 Queuig Systems: Lecture Amedeo R. Odoi October, 2
2 Topics i Queuig Theory 9. Itroductio to Queues; Little s Law; M/M/. Markovia Birth-ad-Death Queues. The M/G/ Queue ad Extesios 2. riority Queues; State Represetatios 3. ogestio ricig 4. Dyamic Behavior of Queues 5. Hypercube Queuig Model 6. The Queue Iferece Egie; sychology of Queues
3 Lecture Outlie Itroductio to queuig systems oceptual represetatio of queuig systems odes for queuig models Termiology ad otatio Little s Law ad basic relatioships Birth-ad-death processes The M/M/ queuig system State trasitio diagrams Steady-state probabilities
4 Queues Queuig Theory is the brach of operatios research cocered with waitig lies (delays/cogestio A queuig system cosists of a user source, a queue ad a service facility with oe or more idetical parallel servers A queuig etwork is a set of itercoected queuig systems Fudametal parameters of a queuig system: Demad rate apacity (service rate Demad iter-arrival times Service times Queue capacity ad disciplie (fiite vs. ifiite; FIFO/FFS, SIRO, LIFO, priorities Myriad details (feedback effects, jockeyig, etc.
5 A Geeric Queuig System Servers Arrival poit at the system Departure poit from the system Source of users/ customers Queue Arrivals process Size of user source Queue disciplie ad Queue capacity Service process Number of servers
6 Queuig etwork cosistig of five queuig systems Queueig system 2 Queueig system 3 I Queueig system oit where users make a choice oit where users merge + Queueig system 5 Out Queueig system 4
7 Applicatios of Queuig Theory Some familiar queues: _ Airport check-i _ Automated Teller Machies (ATMs _ Fast food restaurats _ O hold o a 8 phoe lie _ Urba itersectio _ Toll booths _ Aircraft i a holdig patter _ alls to the police or to utility compaies Level-of-service (LOS stadards Ecoomic aalyses ivolvig trade-offs amog operatig costs, capital ivestmets ad LOS
8 Queuig Models a Be Essetial i Aalysis of apital Ivestmets ost Total cost Optimal cost ost of buildig the capacity ost of losses due to waitig Optimal capacity Airport apacity
9 Stregths ad Weakesses of Queuig Theory Queuig models ecessarily ivolve approximatios ad simplificatio of reality Results give a sese of order of magitude, chages relative to a baselie, promisig directios i which to move losed-form results essetially limited to steady state coditios ad derived primarily (but ot solely for birth-ad-death systems ad phase systems Some useful bouds for more geeral systems at steady state Numerical solutios icreasigly viable for dyamic systems
10 A ode for Queuig Models: A/B/m Distributio of service time / / Distributio of iterarrival time Number of servers ustomers Queue Queueig System S S S S Service facility Some stadard code letters for A ad B: _ M: Negative expoetial (M stads for memoryless _ D: Determiistic _ E k :kth-order Erlag distributio _ G: Geeral distributio Model covered i this lecture: M/M/
11 Termiology ad Notatio State of system: umber of customers i queuig system Queue legth: umber of customers waitig for service N(t umber of customers i queueig system at time t (t probability that N(t is equal to l : mea arrival rate of ew customer whe N(t m : mea (combied service rate whe N(t
12 Termiology ad Notatio (2 Trasiet state: state of system at t depeds o state of system at t or o t Steady state: system is idepedet of iitial state ad t m: umber of servers (parallel service chaels If l ad the service rate per busy server are costat, the l l, m mi (m, mm Expected iterarrival time /l Expected service time /m
13 Some Expected Values of Iterest at Steady State Give: _ arrival rate _ service rate per service chael Ukows: _ L expected umber of users i queuig system _ L q expected umber of users i queue _ W expected time i queuig system per user (W E(w _ W q expected waitig time i queue per user (W q E(w q 4 ukows We eed 4 equatios
14 Little s Law Number of users A(t: cumulative arrivals to the system (t: cumulative service completios i the system A(t N(t (t L T T N ( t dt T A( T T t T N ( t dt A( T T T W T Time
15 Relatioships amog L, L q, W, W q Four ukows: L, W, L q, W q Need 4 equatios. We have the followig 3 equatios: _ L W (Little s law _ L q W q _ W W q + If we kow ay oe of the four expected values, we ca determie the three others The determiatio of L may be hard or easy depedig o the type of queuig system at had L ( : probability that customers are i the system
16 Birth-ad-Death Queuig Systems. m parallel, idetical servers. 2. Ifiite queue capacity. 3. Wheever users are i system (i queue plus i service arrivals are oisso at rate of l per uit of time. 4. Wheever users are i system, service completios are oisso at rate of m per uit of time. 5. FFS disciplie.
17 ? M/M/: Observig State Trasitio From poit : Diagram from Two oits ( + + From poit 2: 2 ( ?
18 M/M/: Derivatio of ad 2 2,,, L, ( the, < Q ( ad Step : Step 2: Step 3: Step 4:
19 M/M/: Derivatio of L, W, W q, ad L q ( ( ( ( ( ( ( ( 2 d d d d L L W ( W W q ( ( 2 q L q W
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