Queuig Systems: Lecture Amedeo R. Odoi October, 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
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
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.
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
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
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
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
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
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/
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
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
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
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
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
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.
? M/M/: Observig State Trasitio From poit : Diagram from Two oits 2 - + ( + + From poit 2: 2 ( + + + 2? 2 - + +
M/M/: Derivatio of ad 2 2,,, L, ( the, < Q ( ad Step : Step 2: Step 3: Step 4:
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