Bina NusantaraModel Antrian (Waiting Line Models )

Bina NusantaraLearning OutcomesMahasiswa akan dapat menjelaskan pengertian atrian, system antrian, struktur dan analisis pola kedatangan pada sistem antrian.

Bina NusantaraOutline Materi:

Pengertian AntrianSistem AntrianStruktur dasar model antrianBentuk-bentuk model antrianAnalisis pola kedatanganContoh ..

Bina NusantaraPengertianIstilah antrian atau disiplin garis tunggu atau waiting lines menunjukan pada kondisi dimana kedatangan dipilih untuk dilayani. Prosedur yang umum diguna-kan adalah kedatangan menempati garis tunggu atas dasar yang : datang pertama dilayani pertama (FCFS), walaupun beberapa prioritas dapat mengubah pola pelayanan ini, namum analisis ini tidak mempertimbangkan kemungkinan itu.

Bina NusantaraFirst studied by A. K. Erlang in 1913.Analyzed telephone facilities.Body of knowledge called queuing theory.Queue is another name for waiting line.Decision problem:Balance cost of providing good service with cost of customers waiting.Waiting Lines

Bina NusantaraBankCustomersTellerDeposit etc.

DoctorsPatientDoctorTreatment office

Traffic CarsTrafficControlled intersection Signalpassage

Assembly linePartsWorkersAssembly

Situation Arrivals ServersService ProcessWaiting Line Examples

Bina NusantaraArrivals: Customers (people, machines, calls, etc.) that demand service.

Service System: Includes waiting line and servers.

Waiting Line (Queue): Arrivals waiting for a free server.

Servers: People or machines that provide service to the arrivals.

Waiting Line Components

Bina NusantaraSistem Antrian

Bina NusantaraCar Wash Example

Bina NusantaraQueue: Waiting line.Arrival: 1 person, machine, part, etc. that arrives and demands service.Queue discipline: Rules for determining the order that arrivals receive service.Channels: Parallel servers.Phases: Sequential stages in service.Waiting Line Terminology

Bina NusantaraInput source (population) size. Infinite: Number in service does not affect probability of a new arrival. A very large population can be treated as infinite.Finite: Number in service affects probability of a new arrival.Example: Population = 10 aircraft that may need repair.Arrival pattern.Random: Use Poisson probability distribution.Non-random: Appointments.Input Characteristics

Bina NusantaraNumber of events that occur in an interval of time.Example: Number of customers that arrive each half-hour.Discrete distribution with mean = Example: Mean arrival rate = 5/hour .Probability:

Time between arrivals has a negative exponential distribution.Poisson Distribution

Bina NusantaraLine length: Limited: Maximum number waiting is limited.Example: Limited space for waiting.Unlimited: No limit on number waiting. Queue discipline:FIFO (FCFS): First in, First out.(First come, first served). Random: Select next arrival to serve at random from those waiting. Priority: Give some arrivals priority for service.Waiting Line Characteristics

Bina NusantaraSingle channel, single phase.One server, one phase of service. Single channel, multi-phase.One server, multiple phases in service.Multi-channel, single phase.Multiple servers, one phase of service.Multi-channel, multia-phase.Multiple servers, multiple phases of service.Bentuk-bentuk Model Antrian

Bina NusantaraSingle Channel, Single Phase

Bina NusantaraCars & foodSingle Channel, Multi-PhaseArrivalsServed unitsService facilityQueueService systemPick-upWaiting carsCars in areaMcDonalds drive-throughPayService facility

Bina NusantaraMulti-Channel, Single Phase

Bina NusantaraMulti-Channel, Multi-Phase

Bina NusantaraRandom: Use Negative exponential probability distribution.Mean service rate = 6 customers/hr.Mean service time = 1/1/6 hour = 10 minutes.

Non-random: May be constant.Example: Automated car wash.Analisis Pola Pelayanan

Bina NusantaraAssumptions in the Basic ModelCustomer population is homogeneous and infinite. Queue capacity is infinite.Customers are well behaved (no balking or reneging). Arrivals are served FCFS (FIFO).Poisson arrivals.The time between arrivals follows a negative exponential distributionService times are described by the negative exponential distribution.

Bina NusantaraSteady State AssumptionsMean arrival rate , mean service rate , and the number of servers are constant.

The service rate is greater than the arrival rate.

These conditions have existed for a long time.

Bina NusantaraSimple (M/M/1).Example: Information booth at mall.Multi-channel (M/M/S).Example: Airline ticket counter.Constant Service (M/D/1).Example: Automated car wash.Limited Population.Example: Department with only 7 drills that may break down and require service.Types of Queuing Models

Bina NusantaraAverage queue time = WqAverage queue length = LqAverage time in system = WsAverage number in system = LsProbability of idle service facility = P0System utilization = Probability of more than k units in system = Pn > kPerformance Measures

Bina NusantaraGeneral Queuing Equations Given one of Ws , Wq , Ls, or Lq you can use these equations to find all the others.

Bina NusantaraType: Single server, single phase system.Input source: Infinite; no balks, no reneging.Queue: Unlimited; single line; FIFO (FCFS).Arrival distribution: Poisson.Service distribution: Negative exponential.M/M/1 Model

Bina NusantaraM/M/1 Model EquationsSystem utilization

Bina NusantaraProbability of 0 units in system, i.e., system idle:Probability of more than k units in system: Pk+101= - ==Pn>kM/M/1 Probability Equationsl( )This is also probability of k+1 or more units in system.

Bina NusantaraM/M/1 Example 1Average arrival rate is 10 per hour. Average service time is 5 minutes. = 10/hr and = 12/hr (1/ = 5 minutes = 1/12 hour)Q1: What is the average time between departures?5 minutes? 6 minutes?

Q2: What is the average wait in the system?W1s= 12/hr-10/hr=0.5 hour or 30 minutes

Bina NusantaraM/M/1 Example 1 = 10/hr and = 12/hrQ3: What is the average wait in line?W q=10 12 (12-10)=O.41667 hours = 25 minutes

Bina NusantaraM/M/1 Example 1 = 10/hr and = 12/hrQ4: What is the average number of customers in line and in the system?L q=102 12 (12-10)=4.1667 customersAlso note: = Wq= 10 0.41667 = 4.1667 L s=10 12-10=5 customers= Ws= 10 0.5 = 5

Bina NusantaraM/M/1 Example 1 = 10/hr and = 12/hrQ5: What is the fraction of time the system is empty (server is idle)?P1= - =01-1012 = = 16.67% of the timeQ6: What is the fraction of time there are more than 5 customers in the system? 6=Pn>5( )1012= 33.5% of the time

Bina NusantaraMore than 5 in the system...Note that more than 5 in the system is the same as: more than 4 in line 5 or more in line 6 or more in the system.

All are P n>5

Bina NusantaraM/M/1 Example 1 = 10/hr and = 12/hrQ7: How much time per day (8 hours) are there 6 or more customers in line?Q8: What fraction of time are there 3 or fewer customers in line? 0.335 x 480 min./day = 160.8 min. = ~2 hr 40 min. =Pn>5 0.335 so 33.5% of time there are 6 or more in line.

Bina NusantaraM/M/1 Example 2Five copy machines break down at UM St. Louis per eight hour day on average. The average service time for repair is one hour and 15 minutes. = 5/day ( = 0.625/hour) 1/ = 1.25 hours = 0.15625 days = 1 every 1.25 hours = 6.4/day Q1: What is the number of customers in the system? L5/dayS= 6.4/day-5/day= 3.57 broken copiers

Bina NusantaraM/M/1 Example 2 = 5/day(or = 0.625/hour) = 6.4/day (or = 0.8/hour)Q2: How long is the average wait in line?W5q= 6.4(6.4 - 5)= 0.558 days (or 4.46 hours)W0.625q= 0.8(0.8 - 0.625) = 4.46 hours

Bina Nusantara( )M/M/1 Example 2Q3: How much time per day are there 2 or more broken copiers waiting for the repair person? 2 or more in line = more than 2 in the system356.4 0.477x 480 min./day = 229 min. = 3 hr 49 min. Pn>2 = 0.477 (47.7% of the time) = 5/day(or = 0.625/hour) = 6.4/day (or = 0.8/hour) =

Bina NusantaraModel Antrian Ganda

Bina NusantaraLearning OutcomesMahasiswa akan dapat menghitung penyelesaian model antrian tunggal dan ganda dalam berbagai contoh aplikasi..

Bina NusantaraOutline Materi:Model Antrian Ganda M/M/CJaringan AntrianContoh Penerapan

Bina NusantaraType: Multiple servers; single-phase. Input source: Infinite; no balks, no reneging.Queue: Unlimited; multiple lines; FIFO (FCFS).Arrival distribution: Poisson.Service distribution: Negative exponential.M/M/S Model

Bina NusantaraM/M/S EquationsProbability of zero people or units in the system:Average number of people or units in the system:Average time a unit spends in the system:Note: M = number of servers in these equations

Bina NusantaraM/M/S EquationsAverage number of people or units waiting for service:Average time a person or unit spends in the queue:

Bina NusantaraM/M/2 Model EquationsAverage time in system:Average time in queue:Average # of customers in queue:Average # of customers in system:Probability the system is empty:

Bina NusantaraM/M/2 ExampleAverage arrival rate is 10 per hour. Average service time is 5 minutes for each of 2 servers. = 10/hr, = 12/hr, and S=2Q1: What is the average wait in the system?Ws=412 4(12)2 -(10)2=0.1008 hours = 6.05 minutes

Bina NusantaraM/M/2 Example = 10/hr, = 12/hr, and S=2Q2: What is the average wait in line?Also note:

so = Ws-=0.1008 - 0.0833 =0.0175 hrs Wq=(10)2 12 (212 + 10)(212 - 10)=0.0175 hrs = 1.05 minutes

Bina NusantaraM/M/2 Example = 10/hr, = 12/hr, and S=2Q3: What is the average number of customers in line and in the system?= Wq= 10/hr 0.0175 hr = 0.175 customers= Ws= 10/hr 0.1008 hr = 1.008 customers

Bina NusantaraM/M/2 Example = 10/hr and = 12/hrQ4: What is the fraction of time the system is empty (server is idle)? = 41.2% of the timeP0=212 - 10 212 + 10

Bina NusantaraM/M/1, M/M/2 and M/M/31 server2 servers3 servers Wq25 min.1.05 min.0.1333 min. (8 sec.)0.417 hr0.0175 hr0.00222 hr WS30 min.6.05 min.5.1333 min. Lq4.167 cust.0.175 cust.0.0222 cust. LS 5 cust.1.01 cust.0.855 cust. P016.7% 41.2% 43.2%

Bina NusantaraService Cost per Day = 10/hr and = 12/hr Suppose servers are paid $7/hr and work 8 hours/day and the marginal cost to serve each customer is $0.50.

M/M/1 Service cost per day = $7/hr x 8 hr/day + $0.5/cust x 10 cust/hr x 8 hr/day = $96/day

M/M/2 Service cost per day = 2 x $7/hr x 8 hr/day + $0.5/cust x 10 cust/hr x 8 hr/day = $152/day

Bina NusantaraCustomer Waiting Cost per Day = 10/hr and = 12/hr Suppose customer waiting cost is $10/hr.M/M/1 Waiting cost per day = $10/hr x 0.417 hr/cust x 10 cust/hr x 8 hr/day = $333.33/dayM/M/1 total cost = 96 + 333.33 = $429.33/day

M/M/2 Waiting cost per day = $10/hr x 0.0175 hr/cust x 10 cust/hr x 8 hr/day =$14/dayM/M/2 total cost = 152 + 14 = $166/day

Bina NusantaraUnknown Waiting CostSuppose customer waiting cost is not known = C.

M/M/1 Waiting cost per day = Cx 0.417 hr/cust x 10 cust/hr x 8 hr/day = 33.33C $/dayM/M/1 total cost = 96 + 33.33CM/M/2 Waiting cost per day = Cx 0.0175 hr/cust x 10 cust/hr x 8 hr/day =1.4C $/dayM/M/2 total cost = 152 + 1.4C M/M/2 is preferred when 152 + 1.4C < 96 + 33.33C orC > $1.754/hr

Bina NusantaraM/M/2 and M/M/3Q: How large must customer waiting cost be for M/M/3 to be preferred over M/M/2?M/M/2 total cost = 152 + 1.4C M/M/3 Waiting cost per day = Cx 0.00222 hr/cust x 10 cust/hr x 8 hr/day = 0.1776C $/dayM/M/3 total cost = 208 + 0.1776C

M/M/3 is preferred over M/M/2 when 208 + 0.1776C < 152 + 1.4C C > $45.81/hr

Bina Nusantara = Mean number of arrivals per time period.Example: 3 units/hour.

= Mean number of arrivals served per time period.Example: 4 units/hour.1/ = 15 minutes/unit.Remember: & Are RatesIf average service time is 15 minutes, then is 4 customers/hour

Bina NusantaraM/D/SConstant service time; Every service time is the same.Random (Poisson) arrivals.Limited population. Probability of arrival depends on number in service.Limited queue length.Limited space for waiting.Many others...

Other Queuing Models

Bina Nusantara

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