OM AHH AHH AHHOPERATIONS MANAGEMENT

Presented by.Araro JirehAry Andana

Muhamad HudaNaufal Taufik

QUEUING MODELS

Introduce of Queuing Theory

Antrian adalah kondisi apabila terdapatnya obyek yang menuju suatu

area untuk dilayani, namun kemudian menghadapi keterlambatan

disebabkan oleh mekanisme pelayanan yang mengalami kesibukan

SITUATION ARRIVALS IN QUEU SERVICE IN PROCESS

Supermarket Grocery Shoppers Checkout clerks at cash register

Highway toll booth Automobiles Collection of tolls at booth

Doctor’s office Patients Treatment by doctor and nurses

Computer system Programs to be run Computer processes jobs

Telephone company

Callers Switching equipment forwards calls

Bank Customers Transaction handled by teller

Harbor Ship and barges Dock workers load and unload

Common Queuing Situation

Characteristic of a Waiting Line System Kedatangan (Arrivals), Populasi yang akan dilayani (calling population)

yang merrupakan input pada sistem antrian

Kedatangan (arrivals) dapat dibagi kedalam tiga karakter :

1. Ukuran (Size) dari populasi kedatangan

Dibedakan atas infinite dan finite / unlimited dan limited

2. Perilaku (Behavior) dari populasi kedatangan

Perilaku customer pada antrian (patient, balk, reneging, jockeing)

3. Pola (Pattern) dari populasi kedatangan

Constant arrival pattern dan Random arrival pattern

Characteristic of a Waiting Line System• Antrian (Waiting-lines), Panjang dari sebuah antrian akan tidak terbatas

(unlimited / infinite) selama tidak adanya aturan yang mengatur atau

membatasi sebuah antrian.

• Karakteristik lain dari sebuah antrian (waiting lines) biasa disebut dengan

queue discipline yang biasa kita kenal dengan sebutan FIFO (First-In

First-Out) atau FIFS (First-In First-Serve)

Characteristic of a Waiting Line System• Servis (Services), (1) Design service system

Characteristic of a Waiting Line System

Measuring Queue’s Performance• Model antrian membantu seorang manager untuk membuat keputusan

yang dapat menyeimbangkan antara service cost dan waiting-line costs.

Berikut ini adalah besaran yang dapat digunakan untuk mengukur

performa dari sebuah antrian :

1. Rata-rata waktu customer berada dalam sebuah antrian

2. Rata-rata panjang antrian

3. Rata-rata waktu costumer berada dalam sistem (waiting time +

service time)

4. Rata-rata jumlah costumer dalam antrian

5. Utilization factor dalam sebuah sistem

6. Kemungkinan fasilitas servis dalam keadaan tidak terpakai (idle)

7. Kemungkinan jumlah spesifik customer dalam sistem

Measuring a Queue‘s Performance1. Average time that each customer or object spends in the queue

2. Average queue length

3. Average time that each customer spends in the system (waiting time plus service time)

4. Average number of customers in the system

5. Probability that the service facility will be idle

6. Utilization factor for the system

7. Probability of a specific number of customers in the system

QUEUING COSTS

Trade-off takes place between two costs :

1. The cost of providing good service2. The cost of customer or machine waiting time

The Variety of Queuing Models

• Three characteristics in common of queuing models :• 1. Poisson distribution arrivals• 2. FIFO discipline• 3. A single-service phase

Name Number Number Arrival Service Population QueueModel (Technical Name Example of Servers of Rate Time Size Discipline

in Parentheses) (Channels) Phases Pattern Pattern

A Single-server Information Single Single Poisson Exponential Unlimited FIFOsystem (M/M/1) counter at

department storeB Multiple-server Airline ticket Multi-server Single Poisson Exponential Unlimited FIFO

(M/M/S) counterC Constant service Automated Single Single Poisson Constant Unlimited FIFO

(M/D/1) car washD Limited population Shop with only a Single Single Poisson Exponential Limited FIFO

(finite population) dozen machinesthat might break

Model A (M/M/1): Single-Server Queuing Model Conditions exist in this type of system: 1. Arrivals are served on a first-in, first-out (FIFO) basis 2. Arrivals are independent of preceding arrivals, but

average number of arrivals (arrival rate) does not change

over time 3. Arrivals are described by a Poisson probability

distribution and come from an infinite (very large) population 4. Service times vary from one customer to the next

and are independent of one another, but their average rate

is known 5. Service times occur according to the negative

exponential probability distribution 6. The service rate is faster than the arrival rate

QUEUING MODELS

QUEUING MODELS

Model B (M/M/S): Multiple-Server Queuing Model

The Probability that there are zero people or units in the system

The average number of people or units in the system

M = number of servers (channels) open = average arrival rate = average service rate at each server (channel)

QUEUING MODELS

Model B (M/M/S): Multiple-Server Queuing Model

The average time a unit spends in the waiting line and being serviced

The average number of people or units in line waiting for

service

The average time a person or unit spends in the queue

waiting for service

M = number of servers (channels) open = average arrival rate = average service rate at each server (channel)

QUEUING MODELS

Model B (M/M/S): Multiple-Server Queuing Model

Sebuah bengkel muffler “Golden Muffler Shop” memutuskan untuk menambah garasi dan mempekerjakan seorang mekanik lagi di garasi tersebut.

Diketahui perkiraan kedatangan mobil adalah 2 orang per jam, dan waktu pengerjaan adalah 4 mobil per jam. Buatlah perbandingan kinerja bengkel tersebut pada waktu sebelum dan sesudah penambahan garasi.

QUEUING MODELS

Model C (M/D/1): Constant-Service-Time Model Average length of queue

Average waiting time in queue

Average number of customers in system

Average time in system

QUEUING MODELS

Model C (M/D/1): Constant-Service-Time Model Perusahaan daur ulang “Inman Recycling, Inc.” memperkirakan proses

sebuah truk menunggu giliran sebelum dapat melakukan bongkar muat adalah selam 15 menit, dengan biaya selama mengantri sebesar $60 per jam. Manajemen berencana untuk membeli kompaktor baru dengan kemampuan servis 12 truk per jam, dan diamortisasi sebesar $3 per truk. Waktu ketibaan truk selama 1 jam diperkirakan sebanyak 8 truk.

QUEUING MODELS

Model D: Limited-Population Model Service Factor

Average number waiting

Average waiting time

Average number of units running

Average number being serviced

Number in population

D = probability that a unit will have to wait in queueF = efficiency factorH = average number of units being servedJ = average number of units in working orderLq = average number of units waiting for serviceM = number of serversN = number of potential customersT = average service timeU = average time between unit service requirementsWq = average time a unti waits in lineX = service factor

QUEUING MODELS

Model D: Limited-Population Model Sebuah survey terhadap printer laser di Amerika Serikat

mengindikasikan bahwa pada setiap 5 printer memerlukan perbaikan setelah 18 jam penggunaan. Seorang teknisi dapat memperbaiki selama rata-rata 2 jam. Biaya penghentian (downtime) printer sebesar $120 per jam. Biaya teknisi adalah $25 per jam. Haruskah lembaga tersebut mempekerjakan 1 teknisi baru?

Jumlah Teknisi

Rata-rata printer yang rusak

Rata-rata biaya/jam untuk perbaikan Biaya/jam teknisi biaya total/jam

  (N - J) (N - J)($120/jam) @ $25/jam  

1 0,725 $87,00 $25,00 $112,00

2 0,513 $61,56 $50,00 $111,56

Contoh Kasus: Penerapan Single Server, Single Phase System pada praktik Dokter Umum

• Latar Belakang: Penerapan UU No 29 Tahun 2004

• Implikasi: dr. Delmi, seorang Dokter Umum, selain menjadi dokter umum di sebuah Rumah sakit swasta, juga membuka praktik di rumah setiap hari Senin s.d Jum’at jam 17.00 s.d 20.00 WIB

Data Jumlah Pasien

Hari Praktik Jumlah Pasien Yang Mendaftar

Jumlah Pasien Yang Terlayani ( )𝜇

Senin 8 8Selasa 10 10Rabu 15 12Kamis 9 9Jum’at 16 13Total 58 52

( )𝜆

Perhitungan Single Server, Single Phase Model

λ =

= 11.6≈ 12

µ =

= 10.4 ≈ 10

LS =

=

= - 6

WS =

=

=

LQ =

=

= - 7.2

WQ = LQ

= - 0.6

ρ =

=

= 1.2

P0 = = 1 – 1.2  = -0.2 

Kesimpulan

Sistem antrian (queuing) yang diterapkan oleh Praktek Umum dr. Delmi di sebuah rumah sakit swasta ini dinilai cukup efektif sejauh ini, dapat dilihat dari nilai utilization factor (rho) terhadap sistemnya bernilai di atas 1 (>1).