SDA 3E Chapter 12

8/4/2019 SDA 3E Chapter 12

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2007 Pearson Education

Queueing Systems Customer arrivals: people, machines,

telephone calls, messages

Servers: people, machines, airportrunways, ATMs, computers

Queue (waiting line): single, parallel,

multiple with common line, series


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Service Characteristics Service process: deterministic or

probabilistic

Exponential services

mean service rate m customers/time(average service time is 1/m)

Number of servers: one or many


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Queue Characteristics

Queue discipline: order in whichcustomers are served

FCFS LCFS

Priority


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System Configuration One or more parallel servers fed by a

single queue.

Several parallel servers fed by their ownqueues.

A combination of several queues in

series.


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Performance Measures The quality of the service provided to the customer.

Waiting time in the queue Time in the system (waiting time plus service time) Completion by a deadline

The efficiency of the service operation and the cost ofproviding the service.

Average queue length Average number of customers in the system (queue

plus in service) Throughput -- the rate at which customers are served Server utilization -- percentage of time servers are

busy Percentage of customers who balk or renege



Operating Characteristics Lq=average number in the queue

L =average number in the system

Wq=average waiting time in the queue

W =average time in the system

P0=probability that the system isempty



Analytical Models Single Server

Model Assumptions

Single server

Poisson arrivals, mean rate = l Exponential services, mean rate = m

FCFS queue discipline

Other modelsArbitrary service times

Multiple servers

Finite calling populations



Single Server Model Operating

CharacteristicsAverage number in queue = Lq = l

2/[mml

Average number in system = L = lml

Average waiting time in queue = Wq = lmml

Average time in system = W = 1/(ml

Probability system is empty = 1 - lm



Example Customers arrive at an airline ticket counter

at a rate ofl = 2 customers/minute and can

be served at a rate ofm = 3 customers perminute.



Calculations Lq = =1.33 customers

L = = 2.00 customers

Wq = = 0.67 minutes

W = = 1.00 minutes

P0 = 1 2/3 = 0.33

)2(3

22

33

2

)23(3

2

23

1



Analytical Models vs.

SimulationAnalytical models provide only long-

term steady-stateresults

Simulation results show short-rermtransientbehavior


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Littles Law For any steady-state queuing system,

L = lW

Other relationships

Lq = lWq

L = Lq

+ lm

W = Wq + 1/m


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Process Simulation ConceptsCustomer arrives

Customer waits for serviceif server is busy

Customer receives service

Customer leaves

next!


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Observations If a customer arrives and the server is idle,

then service can begin immediately uponarrival.

If the server is busy when a customerarrives, then the customer cannot beginservice until the previous customer hascompleted service.

The time that a customer completes serviceequals the time service begins plus theactual service time.


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Manual Process Simulation


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Process Simulation with SimQuick SimQuickElements

Entrances where objects enter a process. Buffers places where objects can be stored

(inventory storage, queues of people or parts, andso on). Work Stations places where work is performed on

objects (machines, service personnel, and so on). Decision Points where an object goes in one of

two or more directions (outcomes of processingactivities, routings for further processing, and soon).

Exits places where objects leave a processaccording to a specified schedule.


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Process Simulation with SimQuick Statistical Distributions

Normal: Nor(mean, standard

deviation) Exponential: Exp(mean)

Uniform: Uni(lower, upper)

Constant Discrete: Dis(i), where iis the

reference to table iof the worksheet


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SimQuickControl Panel


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SimQuick Queuing Simulation

Model Customers at a car wash arrive randomly at

an average of 15 cars per hour (or one car

every 4 minutes). A car takes an average of 3minutes to wash (or 20 cars per hour)

Process flow map:


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Entrances Worksheet


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Buffers Worksheet


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Work Stations Worksheet


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Simulation Results


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Work Station Statistics Final status: status of the work station when

the simulation ends Final inventory (int. buff.), Mean inventory

(int. buff.), and Mean cycle time (int. buff.): Work cycles started: the number of times the

work station has started processing Fraction time working: utilization of the work

station Fraction time blocked: fraction of time that

the work station was waiting to pass on anobject to the next element.


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Buffer Statistics Objects leaving: number of objects that left

the buffer Final inventory: Inventory refers to the

number of objects in the buffer. Finalinventory is the number remaining at the endof the simulation

Minimum inventory; Maximum inventory;

Mean inventory: statistics on the number ofobjects during the simulation Mean cycle time: mean time that an object

spends in the buffer


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Comparison to Analytical Results


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Queues in Series with Blocking


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Buffers Worksheet with Queue

Capacities


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Grocery Store Model with

Resources


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Resources Worksheets


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Inspection Model with Decision

Points


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Decision Point Table


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Pull System Supply Chain With

Exit Schedules


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Other SimQuickFeatures Discrete distributions

Custom schedules


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Continuous Simulation ModelingA continuous simulation model defines

equations for relationships among state

variables so that the dynamic behaviorof the system over time can be studied.


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Example: Cost of Medical

Services


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Modeling Equations POPLVL(t) = POPLVL(t - 1) + GROWTH(t)

DEMAND(t) = POPLVL(t) - [MEDRATE(t - 1) - MEDRATE(t - 2)]

MEDRATE(t) = MEDRATE(t = 1) + POPLVL(t) - POPLVL(t - 1)

+ .8*[INSRATE(t - 1) - INSRATE(t - 2)]

INSRATE(t) = INSRATE(t = 1) + .10*MEDSUIT(t - 1) - [RISK(t - 1) -

RISK(t - 2)]

MEDSUIT(t) = MEDSUIT(t - 1) + [MEDRATE(t - 1) - 1]/RISK(t - 1)

RISK(t) = RISK(t - 1) + .10*[MEDSUIT(t - 1) - 1]


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Simulation Results

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SDA 3E Chapter 12