2. Queuing is techniques developed by the study of people standing in line to determine the optimum level of service provision. In queuing theory, mathematical formulae, or simulations, are used to calculate variables such as length of time spent standing in line and average service time, which depend on the frequency and number of arrivals and the facilities available. The results enable decisions to be made on the most cost-effective level of facilities and the most efficient organization of the process. Early developments in queuing theory were applied to the provision of telephone switching equipment but the techniques are now used in a wide variety of contexts, including machine maintenance, production lines, and air transportation. M. Sitepu 2 Antrian
6. Failure situations—machinery owners waiting for a failure to occurQueues form because resources are limited. In fact, it makes economic sense to have queues. In designing queuing systems, a balance is needed between service to customers (which means short queues and implies many servers) and cost (too many servers waste funds). Most queuing systems can be divided into individual sub-systems, consisting of entities queuing for some activity. M. Sitepu 3 Antrian
7. Queuing theory applies to any system in equilibrium, as long as nothing in the black box is creating or destroying tasks (arrivals=departures). M. Sitepu 4 Antrian
8. Queuing theory mathematics gets very complicated because it applies probability and statistics to queuing systems. What is the probability that the arriving task will find a device busy? On average, how many tasks are ahead of the task that just entered the system? The Early Derivations A single server queue is a combination of a servicing facility that accommodates one customer at a time (server) + a waiting area (queue). These components together are called a system. M. Sitepu 5 Antrian
9. The early queuing work treated the system as a single homogeneous “server,” without regard to discrete components or types of workloads. These systems were called M/M/1 queues. Later, this work was expanded to include multiple homogeneous servers inside the black box. Queuing Methods Four types of queuing techniques commonly implemented First in First Out (FIFO). Weighted Fair Queuing Custom Queuing Priority Queuing M. Sitepu 6 Antrian
10.
11. Each queue has some priority value or weight assigned to it.
13. After accounting for high priority traffic the remaining bandwidth is divided fairly among multiple queues (if any) of low priority traffic.M. Sitepu 7 Antrian
24. The queues are emptied in the order of - high, medium, normal and low. In each queue, packets are in the FIFO orderM. Sitepu 9 Antrian
25. Queuing Networks After much more work in the queuing theory field (approximately 20 years), a new technique was developed that divided computer system into networks of queues. MVA This new technique is called Mean Value Analysis. It allowed a computer system to be segregated by workload classes (transactions, arrival rates, numbers of clients) as well as components (CPU, disk, etc.) Systems were also delineated as being “open” or “closed.” M. Sitepu 10 Antrian
26. Mean value analysis is an iterative approach of solving three primary equations for class “r” workload at queue “i.” The three equations provide solutions for the residence time (response time, per class, per queue), the throughput, and the queue length (number of class “r” tasks at queue “i”). Software and hardware contention can be modeled using these techniques. M. Sitepu 11 Antrian
27. Model AntrianSederhana 1.Pendahuluan 2.Struktur Model Antrian (The Structure of Queuing Model) 3. Single-Channel Model 4. Multiple-Channel Model 5. Model BiayaMinimum (Cost Minimization Models) 6. Non-Poisson Model 7. Model Self Service Facilities 8. Model Network (Queuing Network) M. Sitepu 12 Antrian
32. 2. Struktur Model Antrian (The Structure of Queuing Model) Gambar 1. Struktur sistem antrian M. Sitepu 14 Antrian
33. Gambar 1 menunjukkan struktur umum suatu model antrian yang memiliki 2 komponen : 1) Garis tungguatauantrian (queue) 2) Fasilitaspelayanan (servicefacility) Gambar 2. Pelayanan nasabah di bank M. Sitepu 15 Antrian
37. satupompauntuk premium, satupompauntukpertamax, satupompauntuk solar denganmasing-masingmemilikigaristunggu.Langkah3 : Gunakan formula matematikataumetodesimulasi untukmenganalisa model antrian. M. Sitepu 17 Antrian
46. Notasidalamsistemantrian n = Jumlahpelanggandalamsistem λ = Jumlah rata-rata pelanggan yang datang per satuan waktu μ = Jumlah rata-rata pelanggan yang dilayani per satuanwaktu L = Jumlah rata-rata pelanggan yang diharapkan dalam sistem Lq = Jumlahpelanggan yang diharapkanmenunggudalamantrian Po = Probabilitas tidak ada pelanggan dalam sistem Pn = Probabilitas kepastian n pelanggan dalam sistem P = Tingkat intensitas fasilitas pelayanan W = Waktu yang diharapkan oleh pelanggan selama dalam sistem Wq = Waktu yang diharapkanolehpelangganselamamenunggudalamantrian 1/ μ = Waktu rata-rata pelayanan 1/ λ = Waktu rata-rata antarkedatangan S = Jumlahfasilitaspelayanan M. Sitepu 20 Antrian
47. 3. Single-channel Model Model antrian paling sederhana adalah model saluran tunggal (single-channel model) ditulis dengan notasi ’’ sistem M/M/1 ’’ M pertama : rata-rata kedatangan (distribusiprobabilitas Poisson), M kedua : tingkatpelayanan Angka 1 : jumlahfasilitaspelayanansatusaluran (one channel) Gambar 3. Sistem Single Channel M. Sitepu 21 Antrian
48. Komponen: Populasiinput takterbatasyaitujumlahkedatanganpelangganpotensialtakterbatas. Distribusi kedatangan pelanggan potensial mengikuti distribusiPoisson. Disiplin pelayanan mengikuti pedoman FCFS. Fasilitaspelayananterdiridarisalurantunggal. Distribusi pelayanan mengikuti distribusi Poisson, asumsi(λ < μ) Kapasitassistemdiasumsikantakterbatas Tidak ada penolakan maupun pengingkaran Probabilitas Poisson : M. Sitepu 22 Antrian
50. 4. Multiple-channel Model Dalam multiple-channel model, fasilitaspelayanan yang dimilikilebihdarisatu, ditulisdengannotasi ’’ sistem M/M/s ’’ Huruf (s) menyatakanjumlahfasilitaspelayanan. Contoh 1 : Bagianregistrasisuatuuniversitasmenggunakansistemkomputerdengan4 orang operator dansetiap operator melakukanpekerjaan yang sama. Rata-rata kedatanganmahasiswa yang mengikutidistribusikedatanganPoisson adalah 100 mahasiswa per jam. Setiapoperator dapatmemproses 40 registrasimahasiswa per jam denganwaktupelayanan per mahasiswa mengikuti distribusi eksponensial. M. Sitepu 24 Antrian
51. Berapapersentasewaktumahasiswatidakdalamregistrasi ? (Po) Berapa lama rata-rata mahasiswa menghabiskan waktunya di pusat registrasi? (W) Berapalama mahasiswamenungguuntukmendapatkanpelayananregistrasi atas dasar rata-rata tsb? (Lq) Berapalama rata-rata mahasiswamenunggudalamgarisantrian? (Wq) Jika ruang tunggu pusat registrasi mahasiswa hanya mampu untuk menampung5 mahasiswa, berapapersentasewaktusetiapmahasiswaberadadalamgarisantriandiluarruangan? Penyelesaian : Digunakansistem (M/M/4) Rata-rata kedatangan mahasiswa (λ) = 100 Rata-rata setiap operator dapatmelayanimahasiswa(μ)= 40 M. Sitepu 25 Antrian
52. a) Menggunakanpersamaan : Probabilitasmahasiswatidakdalammendatangipusatregistrasisebesar Po = 0.0737 atau 7.37 % b) Waktu rata-rata yang dihabiskanmahasiswadipusatregistrasi (W) PertamadihitungLqdenganpersamaan : M. Sitepu 26 Antrian
54. e) Untukmenentukanberapa lama mahasiswaberadadiluarruangantunggudilakukandenganmenghitungPnyaitumenjumlahkan: Pn= P0+P1 + P2 + P3 + P4 + P5 = 0.0737 + 0.1842 + 0.2303 + 0.1919 + 0.1200 + 0.0750 = 0.8751 samadenganproporsiwaktu yang digunakanmahasiswamenunggudidalamruanganregistrasi. Persentasewaktu yang digunakanmahasiswauntukmenunggudiluarruangan adalah 1-0.8751 = 0.1249 = 12.49 % dari waktu mahasiswa. Jika mahasiswa berada dalam sistem selama 1.8 menit, maka 87.51 % dariwaktutsbmahasiswaberadadalamruangtunggudan 12.49 % atau 0.225 menitmahasiswaberadadiluarruangtunggu. M. Sitepu 28 Antrian
55. 5. Model BiayaMinimum (Cost Minimization Models) Padabeberapaaplikasisistemantriansangatmungkinmendisainsistem yang akanmeminimumkanbiaya per satuanwaktu. Persamaan biaya total per jam : TC = SC +WC TC = Total biaya per jam SC = Biayapelayanan per jam WC = Biayamenunggu per jam per pelanggan Total biayamenunggu per jam : WC =λ (W cw) = ( λ W) cw = L cw M. Sitepu 29 Antrian
56. Total biayamenunggu per jam : WC = λ (W cw) = ( λ W) cw = L cw cw = biayamenungguper jamperpelanggan W = waktu yang dihabiskanpelanggan W cw = rata-ratabiayamenunggu per pelanggan λ = rata-ratakedatanganpelanggan per jam denganpersamaan L = λW (persamaan--) 6. Non Poisson Model Ditulis dengan notasi ’’ sistem M/G/1 ’’ M menunjukkankedatangan Poisson G merupakan rata-rata (means) waktu pelayanan Angka 1 menunjukkansistemmemilikisatu channel M. Sitepu 30 Antrian
57. Dalamsistem (M/M/1) dan (M/M/s) diasumsikanbahwatingkat Pelayanan (pelayananpelanggan per satuanwaktu) memilikiprobabilitasPoisson. Samadenganasumsibahwawaktupelayananpelangganmemilikiprobabilitaseksponensial. ProbabilitasPoisson sangat ideal untuk model kedatangan random. M. Sitepu 31 Antrian
59. Digunakanmodel antrian (M/M/s) apabilajumlahfasilitaspelayananterbatas (finite) Digunakan model antrian (M/M/) apabilajumlahfasilitaspelayanan tak terbatas (infinite) atau tidak perlu menunggu Persamaan yang digunakandalam model antrian (M/M/ ∞ ) sbb : M. Sitepu 33 Antrian
60. 8. Queuing Network Model networks dapat menggunakan sistem seri maupun sistemparalel. Sistemseriterdiridarisatusubsistemmengikutisubsistem yang lain. SetiappelangganharusmelewatisatusubsistemkemudianMelewatisubsistem yang lain sepertiregistrasimahasiswa. SistemparalelmaliputiduaataulebihsubsistemdansetiapPelanggan harus melewati satu subsistem. Sebuahsubsistemmungkinmenggunakansistemantrian (M/M/1) atausistem (M/M/s) M. Sitepu 34 Antrian
61. Gambar 4. Sistem Seri Gambar 5. SistemParalel M. Sitepu 35 Antrian