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___________________________________________________________________________ Operations Research Jan Fábry

Waiting Line ModelsWaiting Line Models

ServiceService

{Server}{Server}



SourceSource Arrival Arrival ProcessProcess

Waiting Waiting AreaArea

{Queue}{Queue}

ExitExit

Waiting Line SystemWaiting Line System

{Potential {Potential Customers}Customers}

{Customers}{Customers}



Examples of Waiting Line SystemsExamples of Waiting Line Systems

Service SystemService System CustomerCustomer ServerServer

Doctor’s consultancy roomDoctor’s consultancy room PatientPatient DoctorDoctor

BankBank ClientClient ClerkClerk

CrossingCrossing CarCar Traffic lightsTraffic lights

AirportAirport AirplaneAirplane RunwayRunway

Fire stationFire station FireFire Emergency unitEmergency unit

Telephone exchangeTelephone exchange CallCall SwitchboardSwitchboard

Service stationService station CarCar Petrol pumpPetrol pump

ServiceService

{Server}{Server}



SourceSource Arrival Arrival ProcessProcess

Waiting Waiting AreaArea

{Queue}{Queue}

ExitExit

Waiting Line SystemWaiting Line System


{Customers}{Customers}



SourceSource

Arrival ProcessArrival Process


Infinite – touristsInfinite – tourists

Finite – machines in factoryFinite – machines in factory




In batches – BUS of touristsIn batches – BUS of touristsArrivalsArrivals

Individually – patientsIndividually – patients

Scheduled – trams, trainsScheduled – trams, trainsArrivalsArrivals

Unscheduled – patientsUnscheduled – patients




TimeTime

ArrivalArrivalArrivalArrival

ArrivalArrival

Arrival rate – Arrival rate – number of arrivalsnumber of arrivals per time unitper time unit ( (POISSONPOISSON distribution) distribution)

Average arrival rate = Average arrival rate = –– average average number of arrivalsnumber of arrivals per time unitper time unit ( (meanmean of of POISSONPOISSON distribution) distribution)




TimeTime

ArrivalArrivalArrivalArrival

ArrivalArrival

Average interarrival time = 1/Average interarrival time = 1/ –– average time period beetween arrivals average time period beetween arrivals ( (meanmean of of EXPONENTIAL EXPONENTIAL distribution)distribution)

Interarrival time – Interarrival time – time period between time period between two two arrivalsarrivals ( (EXPONENTIAL EXPONENTIAL distribution)distribution)



Service ProcessService Process

ServiceService

{Server}{Server}

Service rate – Service rate – number of number of customers served customers served per time unitper time unit ( (POISSONPOISSON distribution) distribution)

Average service rate = Average service rate = –– average average number of number of customers served customers served per time unitper time unit ( (meanmean of of POISSONPOISSON distribution) distribution)




ServiceService

{Server}{Server}

Service time – Service time – time customer spends at service facilitytime customer spends at service facility ( (EXPONENTIALEXPONENTIAL distribution) distribution)

Average service time = 1/Average service time = 1/ –– average timeaverage time customers spend at service facilitycustomers spend at service facility ( (meanmean of of EXPONENTIAL EXPONENTIAL distribution)distribution)




Service configurations Service configurations (type, number and arrangement (type, number and arrangement of service facilities) of service facilities)

1. Single facility1. Single facility

Queue Server

ExitArrival




2. 2. Multiple, parallel, identical facilitieMultiple, parallel, identical facilities s (SINGLE queue)(SINGLE queue)

Queue

Servers

Arrival

Exit




2. 2. Multiple, parallel, identical facilitieMultiple, parallel, identical facilities s (MULTIPLE queue)(MULTIPLE queue)

Queues Servers

Arrival

Exit




3. 3. Multiple, parallel, but not identical facilitiesMultiple, parallel, but not identical facilities

Queues Servers

Arrival

Exit




4. Serial 4. Serial facilitiesfacilities

Queue Server

ExitArrival

Queue Queue Server Server

5. Combination of 5. Combination of facilitiesfacilities



Waiting LineWaiting Line

FCFS (First-Come, First-Served)FCFS (First-Come, First-Served)

Discipline of the queueDiscipline of the queue

ServiceService

{Server}{Server}




LCFS (Last-Come, First-Served)LCFS (Last-Come, First-Served)


ServiceService

{Server}{Server}




PRI (PRIority system)PRI (PRIority system)


ServiceService

{Server}{Server}


WaitWaitiing Line Modelsng Line Models


SIRO (Selection In Random Order)SIRO (Selection In Random Order)


ServiceService

{Server}{Server}



Analysis of Waiting Line ModelsAnalysis of Waiting Line Models

Waiting costWaiting cost

CostCost

Service cost (facility cost)Service cost (facility cost)

- cost of construction- cost of construction

- cost of operation- cost of operation

- cost of maintenance and repair- cost of maintenance and repair

- other costs (insurance, rental)- other costs (insurance, rental)




Average waiting time in the queueAverage waiting time in the queue

Time characteristicsTime characteristics

Average waiting time in the systemAverage waiting time in the system




Average number of customers in the queueAverage number of customers in the queue

Number of customersNumber of customers

Average number of customers in the systemAverage number of customers in the system




Probability of empty service facilityProbability of empty service facility

Probability characteristicsProbability characteristics

Probability of the service facility being busyProbability of the service facility being busy

PProbability of finding robability of finding NN customers in the system customers in the system

PProbability that robability that NN > > nn

PProbability of being in the system longer than time robability of being in the system longer than time tt



Classification of Waiting Line ModelsClassification of Waiting Line Models

AA // BB // CC // DD // EE // FF Kendall’s notationKendall’s notation

Probability distribution of interarrival timeProbability distribution of interarrival timeProbability distribution of service timeProbability distribution of service time

Number of parallel serversNumber of parallel servers

Queue disciplineQueue discipline

Maximum length of queueMaximum length of queue

Size of customer’s sourceSize of customer’s source


Standard Single-Server Standard Single-Server Exponential ModelExponential Model




Standard Single-Server Exponential ModelStandard Single-Server Exponential Model

Queue Server

ExitArrival

( ( M M / M / 1 / / M / 1 / FCFSFCFS // ∞∞ // ∞∞ ) )




AssumptionsAssumptions

SSingle serveringle server

IInterarrival times nterarrival times - - exponential exponential pprobability robability distribution with the mean = 1/distribution with the mean = 1/λλ

SService times ervice times - - exponential probability exponential probability distribution with the mean = 1/distribution with the mean = 1/μμ




AssumptionsAssumptions

IInfinite sourcenfinite source

UUnlimited length of queuenlimited length of queue

QQueue discipline is FCFSueue discipline is FCFS




Example – GroceryExample – Grocery

One shop assistant One shop assistant – serves 25 customers per hour (on the average) – serves 25 customers per hour (on the average)

From 8 a.m. to 6 p.m. From 8 a.m. to 6 p.m. – 18 customers per hour arrive (on the average) – 18 customers per hour arrive (on the average)





Average arrival rateAverage arrival rate λλ = 18 customers per hour = 18 customers per hour

Average service rateAverage service rate μ μ = 25 customers per hour = 25 customers per hour





Utilization of the systemUtilization of the system

72.0

– – probability that the server is busyprobability that the server is busy– – probability that there is at least 1 customer in the systemprobability that there is at least 1 customer in the system

PProbability of an empty facility (server is idlerobability of an empty facility (server is idle) )

1)0(P 28.0)0( P





AAverage waiting time in the systemverage waiting time in the system

minutes 8.6 hours 143.0 W

1

W

AAverage waiting time in the verage waiting time in the queue queue

)(

1

WWq minutes 6.2 hours 103.0 W





AAverage number of customers in the systemverage number of customers in the system

customers 57.2L

WL

AAverage number of customers in the verage number of customers in the queuequeue

customers 85.1qL)(

2

qq WL





PProbability of finding exactly robability of finding exactly NN customers in the system customers in the system

NNPNP )1()0()(

PP(0) (0) 0.2800.280

PP(1)(1) 0.2020.202

PP(2)(2) 0.1450.145

PP(3)(3) 0.1050.105





PProbability that robability that NN > > nn

PP{{N N > 0} > 0} 0.7200.720

PP{{N N > 1}> 1} 0.5180.518

PP{{N N > 2}> 2} 0.3730.373

PP{{N N > 3}> 3} 0.2690.269

1 nnNP





PProbability of being in the system longer than time robability of being in the system longer than time tt

PP{{T T > 1 min}> 1 min} 0.8900.890

PP{{T T > > 22 min} min} 0.7920.792

PP{{T T > > 33 min} min} 0.7050.705

PP{{T T > > 44 min} min} 0.6270.627

tetTP )(


Computer SimulationComputer Simulation



Analytical toolsAnalytical toolsSolutionSolution

Computer simulationComputer simulation

Computer simulationComputer simulation isis a special method a special method using computer experiments with using computer experiments with

the model of a real systemthe model of a real system


EntityEntity - - object that goes through the modelobject that goes through the model ResourceResource - - agent required by the entityagent required by the entity EventEvent -- significant change of the system significant change of the system ActivityActivity -- process between two events process between two events Generating of random valuesGenerating of random values Simulation timeSimulation time Computer simulation languageComputer simulation language AnimationAnimation


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Documents

operations research

average time customers

average number of customers

potential customers

average number of arrivals

single facility