OM AHH AHH AHH OPERATIONS MANAGEMENT Presented by. Araro Jireh Ary Andana Muhamad Huda Naufal 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 SITUATIONARRIVALS IN QUEUSERVICE IN PROCESS SupermarketGrocery ShoppersCheckout clerks at cash register Highway toll boothAutomobilesCollection of tolls at booth Doctor’s officePatientsTreatment by doctor and nurses Computer systemPrograms to be runComputer processes jobs Telephone companyCallersSwitching equipment forwards calls BankCustomersTransaction handled by teller HarborShip and bargesDock 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 Performance 1. 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 service 2. 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

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

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 queue F = efficiency factor H = average number of units being served J = average number of units in working order Lq = average number of units waiting for service M = number of servers N = number of potential customers T = average service time U = average time between unit service requirements Wq = average time a unti waits in line X = 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 perbaikanBiaya/jam teknisibiaya total/jam (N - J)(N - $25/jam 10,725$87,00$25,00$112,00 20,513$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 s.d WIB

Data Jumlah Pasien Hari PraktikJumlah Pasien Yang Mendaftar Jumlah Pasien Yang Terlayani () Senin88 Selasa10 Rabu1512 Kamis99 Jum’at1613 Total5852 ()()

Perhitungan Single Server, Single Phase Model λ = = 11.6 ≈ 12 µ = = 10.4 ≈ 10

L S = = = - 6 W S = = =

L Q = = = W Q = L Q =- 0.6

ρ = = = 1.2 P 0 = = 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).