Antrian (Queuing)<br />M. Sitepu<br />1<br />Antrian<br />
Queuing is techniques developed by the study of people standing in line to determine the optimum level of service provisio...
Queuing theory deals with problems that involve queuing (or waiting). Some examples of queuing include:<br /><ul><li>Banks...
Computers—users waiting for a response
Public transportation—riders waiting for a bus/train
Failure situations—machinery owners waiting for a failure to occur</li></ul>Queues form because resources are limited. In ...
Queuing theory applies to any system in equilibrium, as long as nothing in the black box is creating or destroying tasks (...
Queuing theory mathematics gets very complicated because it applies probability and statistics to queuing systems.<br />Wh...
The early queuing work treated the system as a single homogeneous “server,” without regard to discrete components or types...
First In First Out (FIFO)<br />Packets are transmitted in the order in which they arrive.<br /><ul><li>Single Queue, Packe...
Each queue has some priority value or weight assigned to it.
Low volume traffic is given higher priority over high volume traffic
After accounting for high priority traffic the remaining bandwidth is divided fairly among multiple queues (if any) of low...
Custom Queuing<br /><ul><li>Separate queues maintained for separate classes of traffic.
The algorithm requires a byte count to be set per queue.
That many bytes rounded of to the nearest packet is scheduled for delivery.
This ensures that the minimum bandwidth requirement by the various classes of traffic is met.
CQ round robins through the queues, picking the required number of packets from each.
If a queue is of length 0 then the next queue is serviced.</li></ul>M. Sitepu<br />8<br />Antrian<br />
Priority Queuing<br /><ul><li>4 traffic priorities Defined.- high, medium, normal and low.
Incoming traffic is classified and enqueued in either of the 4 queues.
Classification criteria
Unclassified packets are put in the normal queue.
The queues are emptied in the order of - high, medium, normal and low. In each queue, packets are in the FIFO order</li></...
Queuing Networks<br /> After much more work in the queuing theory field (approximately 20 years), a new technique was deve...
Mean value analysis is an iterative approach of solving three primary equations for class “r” workload at queue “i.”<br />...
Model AntrianSederhana<br />1.Pendahuluan<br />2.Struktur Model Antrian (The Structure of Queuing Model)<br />3. Single-Ch...
1. Pendahuluan<br />Sistem antrian sangat diperlukan ketika para pelanggan (konsumen) menungguuntukmendapatkanjasapelayana...
Pelangganmenunggupelayanandidepankasir
Para calonpenumpang KA menunggupelayanandiloketpenjualantiket
Para pengendarakendaraanmenungguuntukmendapatkanpelayananpengisianbahanbakardi SPBU</li></ul>Teoriantriandiciptakanoleh A....
2. Struktur Model Antrian (The Structure of Queuing Model)<br />Gambar 1. Struktur sistem antrian<br />M. Sitepu<br />14<b...
Gambar 1 menunjukkan struktur umum suatu model antrian yang memiliki 2 komponen :<br />1) Garis tungguatauantrian (queue)<...
M. Sitepu<br />16<br />Antrian<br />
Prosedurteknikantrian<br />Langkah 1 : Tentukansistemantrianapa yang harusdipelajari.<br />Langkah 2 : Tentukan model antr...
tigapompauntuk premium denganmasing-masingmemilikigaristunggu
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  1. 1. Antrian (Queuing)<br />M. Sitepu<br />1<br />Antrian<br />
  2. 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.<br />M. Sitepu<br />2<br />Antrian<br />
  3. 3. Queuing theory deals with problems that involve queuing (or waiting). Some examples of queuing include:<br /><ul><li>Banks or supermarkets—customers waiting for service
  4. 4. Computers—users waiting for a response
  5. 5. Public transportation—riders waiting for a bus/train
  6. 6. Failure situations—machinery owners waiting for a failure to occur</li></ul>Queues form because resources are limited. In fact, it makes economic sense to have queues.<br />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).<br />Most queuing systems can be divided into individual sub-systems, consisting of entities queuing for some activity.<br />M. Sitepu<br />3<br />Antrian<br />
  7. 7. Queuing theory applies to any system in equilibrium, as long as nothing in the black box is creating or destroying tasks (arrivals=departures).<br />M. Sitepu<br />4<br />Antrian<br />
  8. 8. Queuing theory mathematics gets very complicated because it applies probability and statistics to queuing systems.<br />What is the probability that the arriving task will find a device busy?<br />On average, how many tasks are ahead of the task that just entered the system?<br />The Early Derivations<br />A single server queue is a combination of a servicing facility that accommodates one customer at a time (server) + a waiting area (queue).<br />These components together are called a system.<br />M. Sitepu<br />5<br />Antrian<br />
  9. 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.<br />Later, this work was expanded to include multiple homogeneous servers inside the black box.<br />Queuing Methods<br />Four types of queuing techniques commonly implemented<br />First in First Out (FIFO).<br />Weighted Fair Queuing<br />Custom Queuing<br />Priority Queuing<br />M. Sitepu<br />6<br />Antrian<br />
  10. 10. First In First Out (FIFO)<br />Packets are transmitted in the order in which they arrive.<br /><ul><li>Single Queue, Packet dropping</li></ul>Weighted Fair Queuing (WFQ)<br /><ul><li>Packets are classified into different "conversation messages“
  11. 11. Each queue has some priority value or weight assigned to it.
  12. 12. Low volume traffic is given higher priority over high volume traffic
  13. 13. After accounting for high priority traffic the remaining bandwidth is divided fairly among multiple queues (if any) of low priority traffic.</li></ul>M. Sitepu<br />7<br />Antrian<br />
  14. 14. Custom Queuing<br /><ul><li>Separate queues maintained for separate classes of traffic.
  15. 15. The algorithm requires a byte count to be set per queue.
  16. 16. That many bytes rounded of to the nearest packet is scheduled for delivery.
  17. 17. This ensures that the minimum bandwidth requirement by the various classes of traffic is met.
  18. 18. CQ round robins through the queues, picking the required number of packets from each.
  19. 19. If a queue is of length 0 then the next queue is serviced.</li></ul>M. Sitepu<br />8<br />Antrian<br />
  20. 20. Priority Queuing<br /><ul><li>4 traffic priorities Defined.- high, medium, normal and low.
  21. 21. Incoming traffic is classified and enqueued in either of the 4 queues.
  22. 22. Classification criteria
  23. 23. Unclassified packets are put in the normal queue.
  24. 24. The queues are emptied in the order of - high, medium, normal and low. In each queue, packets are in the FIFO order</li></ul>M. Sitepu<br />9<br />Antrian<br />
  25. 25. Queuing Networks<br /> 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.<br />MVA<br />This new technique is called Mean Value Analysis. It allowed a computer system to be segregated by workload classes<br />(transactions, arrival rates, numbers of clients) as well as components (CPU, disk, etc.)<br />Systems were also delineated as being “open” or “closed.”<br />M. Sitepu<br />10<br />Antrian<br />
  26. 26. Mean value analysis is an iterative approach of solving three primary equations for class “r” workload at queue “i.”<br />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”).<br />Software and hardware contention can be modeled using these techniques.<br />M. Sitepu<br />11<br />Antrian<br />
  27. 27. Model AntrianSederhana<br />1.Pendahuluan<br />2.Struktur Model Antrian (The Structure of Queuing Model)<br />3. Single-Channel Model<br />4. Multiple-Channel Model<br />5. Model BiayaMinimum (Cost Minimization Models)<br />6. Non-Poisson Model<br />7. Model Self Service Facilities<br />8. Model Network (Queuing Network)<br />M. Sitepu<br />12<br />Antrian<br />
  28. 28. 1. Pendahuluan<br />Sistem antrian sangat diperlukan ketika para pelanggan (konsumen) menungguuntukmendapatkanjasapelayanan<br />Contohpenggunaansistemantriandalammelancarkan<br />pelayanankepadapelangganataukonsumen :<br /><ul><li>Mahasiswamenungguuntukregistrasidanpembayaranuangkuliah
  29. 29. Pelangganmenunggupelayanandidepankasir
  30. 30. Para calonpenumpang KA menunggupelayanandiloketpenjualantiket
  31. 31. Para pengendarakendaraanmenungguuntukmendapatkanpelayananpengisianbahanbakardi SPBU</li></ul>Teoriantriandiciptakanoleh A.K. Erlang (ahlimatematik<br />Denmark) padatahun 1909.<br />M. Sitepu<br />13<br />Antrian<br />
  32. 32. 2. Struktur Model Antrian (The Structure of Queuing Model)<br />Gambar 1. Struktur sistem antrian<br />M. Sitepu<br />14<br />Antrian<br />
  33. 33. Gambar 1 menunjukkan struktur umum suatu model antrian yang memiliki 2 komponen :<br />1) Garis tungguatauantrian (queue)<br />2) Fasilitaspelayanan (servicefacility)<br />Gambar 2. Pelayanan nasabah di bank<br />M. Sitepu<br />15<br />Antrian<br />
  34. 34. M. Sitepu<br />16<br />Antrian<br />
  35. 35. Prosedurteknikantrian<br />Langkah 1 : Tentukansistemantrianapa yang harusdipelajari.<br />Langkah 2 : Tentukan model antrian yang sesuaidalammenggambarkansistem.<br />ContohdalamkasuspompabensindiSPBU,terdapattiga model yang dapat digunakan :<br /><ul><li>tigapompauntuk premium dengansatugaristunggu
  36. 36. tigapompauntuk premium denganmasing-masingmemilikigaristunggu
  37. 37. satupompauntuk premium, satupompauntukpertamax, satupompauntuk solar denganmasing-masingmemilikigaristunggu.</li></ul>Langkah3 : Gunakan formula matematikataumetodesimulasi<br />untukmenganalisa model antrian.<br />M. Sitepu<br />17<br />Antrian<br />
  38. 38. Komponen dalam Sistem Antrian (lihat gambar 1)<br />Populasimasukan (input population) :<br /><ul><li>Input populasiterbatas (finite input population)
  39. 39. Input populasitakterbatas (infinite input population)</li></ul>Distribusikedatangan,<br /><ul><li>Pelanggan datang setiap 5 menit (constant arrival distribution)
  40. 40. Pelanggandatangsecaraacak (arrival pattern random)</li></ul>Disiplinpelayanan,<br /><ul><li>FCFS (first come, first served) dan
  41. 41. LCFS (last come, first served)
  42. 42. Pelanggandilayani secara acak dan prioritas</li></ul>Fasilitaspelayanan, mengelompokkanfasilitaspelayananmenurutjumlah yang tersedia,<br /><ul><li>Sistemsingle-channel (gambar 3)
  43. 43. Sistemmultiple channel</li></ul>M. Sitepu<br />18<br />Antrian<br />
  44. 44. Distribusipelayanan, ditetapkanberdasarkansatucaradariduacaraberikut :<br /><ul><li>Berapabanyakpelanggan yang dapatdilayaniper satuanwaktu
  45. 45. Berapalama setiappelanggandapatdilayani</li></ul>Kapasitassistempelayanan, samadenganmemaksimumkanjumlahpelanggan yang diperkenankanmasukdalamsistem, Kapasitas terbatas atau tak terbatas.<br />Karakteristiksistemlainnya, pelanggantidakakanmasuksistem antrian jika pelanggan lain telah banyak menunggu ataumeninggalkanantrian.<br />Perilakuinidikatakansebagai reneging ataupengingkaran.<br />M. Sitepu<br />19<br />Antrian<br />
  46. 46. Notasidalamsistemantrian<br />n = Jumlahpelanggandalamsistem<br />λ = Jumlah rata-rata pelanggan yang datang per satuan waktu<br />μ = Jumlah rata-rata pelanggan yang dilayani per satuanwaktu<br />L = Jumlah rata-rata pelanggan yang diharapkan dalam sistem<br />Lq = Jumlahpelanggan yang diharapkanmenunggudalamantrian<br />Po = Probabilitas tidak ada pelanggan dalam sistem<br />Pn = Probabilitas kepastian n pelanggan dalam sistem<br />P = Tingkat intensitas fasilitas pelayanan<br />W = Waktu yang diharapkan oleh pelanggan selama dalam sistem<br />Wq = Waktu yang diharapkanolehpelangganselamamenunggudalamantrian<br />1/ μ = Waktu rata-rata pelayanan<br />1/ λ = Waktu rata-rata antarkedatangan<br />S = Jumlahfasilitaspelayanan<br />M. Sitepu<br />20<br />Antrian<br />
  47. 47. 3. Single-channel Model<br />Model antrian paling sederhana adalah model saluran tunggal<br />(single-channel model) ditulis dengan notasi ’’ sistem M/M/1 ’’<br />M pertama : rata-rata kedatangan (distribusiprobabilitas Poisson),<br />M kedua : tingkatpelayanan<br />Angka 1 : jumlahfasilitaspelayanansatusaluran (one channel)<br />Gambar 3. Sistem Single Channel<br />M. Sitepu<br />21<br />Antrian<br />
  48. 48. Komponen:<br />Populasiinput takterbatasyaitujumlahkedatanganpelangganpotensialtakterbatas.<br />Distribusi kedatangan pelanggan potensial mengikuti distribusiPoisson.<br />Disiplin pelayanan mengikuti pedoman FCFS.<br />Fasilitaspelayananterdiridarisalurantunggal.<br />Distribusi pelayanan mengikuti distribusi Poisson, asumsi(λ < μ)<br />Kapasitassistemdiasumsikantakterbatas<br />Tidak ada penolakan maupun pengingkaran<br />Probabilitas Poisson :<br />M. Sitepu<br />22<br />Antrian<br />
  49. 49. Persamaandalamsistem ( M/M/1 ) :<br />M. Sitepu<br />23<br />Antrian<br />
  50. 50. 4. Multiple-channel Model<br />Dalam multiple-channel model, fasilitaspelayanan yang dimilikilebihdarisatu, ditulisdengannotasi ’’ sistem M/M/s ’’<br />Huruf (s) menyatakanjumlahfasilitaspelayanan.<br />Contoh 1 :<br />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.<br />M. Sitepu<br />24<br />Antrian<br />
  51. 51. Berapapersentasewaktumahasiswatidakdalamregistrasi ? (Po)<br />Berapa lama rata-rata mahasiswa menghabiskan waktunya di pusat registrasi? (W)<br />Berapalama mahasiswamenungguuntukmendapatkanpelayananregistrasi atas dasar rata-rata tsb? (Lq)<br />Berapalama rata-rata mahasiswamenunggudalamgarisantrian? (Wq)<br />Jika ruang tunggu pusat registrasi mahasiswa hanya mampu untuk menampung5 mahasiswa, berapapersentasewaktusetiapmahasiswaberadadalamgarisantriandiluarruangan?<br />Penyelesaian :<br />Digunakansistem (M/M/4)<br />Rata-rata kedatangan mahasiswa (λ) = 100<br />Rata-rata setiap operator dapatmelayanimahasiswa(μ)= 40<br />M. Sitepu<br />25<br />Antrian<br />
  52. 52. a) Menggunakanpersamaan :<br />Probabilitasmahasiswatidakdalammendatangipusatregistrasisebesar Po = 0.0737 atau 7.37 %<br />b) Waktu rata-rata yang dihabiskanmahasiswadipusatregistrasi (W)<br />PertamadihitungLqdenganpersamaan :<br />M. Sitepu<br />26<br />Antrian<br />
  53. 53. c) Waktumenunggumahasiswauntukmendapatkanpelayananregistrasiadalah:<br />Lq = 0.533 jam = 31.98 menit<br />d) Waktumenunggumahasiswaselamadalamprosesantrianadalah:<br />Wq = 0.00533 jam = 0.03 menit<br />M. Sitepu<br />27<br />Antrian<br />
  54. 54. e) Untukmenentukanberapa lama mahasiswaberadadiluarruangantunggudilakukandenganmenghitungPnyaitumenjumlahkan: <br />Pn= P0+P1 + P2 + P3 + P4 + P5<br /> = 0.0737 + 0.1842 + 0.2303 + 0.1919 + 0.1200 + 0.0750<br /> = 0.8751<br />samadenganproporsiwaktu yang digunakanmahasiswamenunggudidalamruanganregistrasi.<br />Persentasewaktu yang digunakanmahasiswauntukmenunggudiluarruangan adalah 1-0.8751 = 0.1249 = 12.49 % dari waktu mahasiswa.<br />Jika mahasiswa berada dalam sistem selama 1.8 menit, maka 87.51 % dariwaktutsbmahasiswaberadadalamruangtunggudan 12.49 % atau 0.225 menitmahasiswaberadadiluarruangtunggu.<br />M. Sitepu<br />28<br />Antrian<br />
  55. 55. 5. Model BiayaMinimum (Cost Minimization Models)<br />Padabeberapaaplikasisistemantriansangatmungkinmendisainsistem yang akanmeminimumkanbiaya per satuanwaktu.<br />Persamaan biaya total per jam :<br /> TC = SC +WC<br />TC = Total biaya per jam<br />SC = Biayapelayanan per jam<br />WC = Biayamenunggu per jam per pelanggan<br />Total biayamenunggu per jam :<br />WC =λ (W cw) = ( λ W) cw = L cw<br />M. Sitepu<br />29<br />Antrian<br />
  56. 56. Total biayamenunggu per jam :<br />WC = λ (W cw) = ( λ W) cw = L cw<br />cw = biayamenungguper jamperpelanggan<br />W = waktu yang dihabiskanpelanggan<br />W cw = rata-ratabiayamenunggu per pelanggan<br />λ = rata-ratakedatanganpelanggan per jam denganpersamaan<br />L = λW (persamaan--)<br />6. Non Poisson Model<br />Ditulis dengan notasi ’’ sistem M/G/1 ’’<br />M menunjukkankedatangan Poisson<br />G merupakan rata-rata (means) waktu pelayanan<br />Angka 1 menunjukkansistemmemilikisatu channel<br />M. Sitepu<br />30<br />Antrian<br />
  57. 57. Dalamsistem (M/M/1) dan (M/M/s) diasumsikanbahwatingkat<br />Pelayanan (pelayananpelanggan per satuanwaktu) memilikiprobabilitasPoisson.<br />Samadenganasumsibahwawaktupelayananpelangganmemilikiprobabilitaseksponensial.<br />ProbabilitasPoisson sangat ideal untuk model kedatangan random.<br />M. Sitepu<br />31<br />Antrian<br />
  58. 58. 7. Self Service Facilities<br />Banyakperusahaan (pedagang) menyediakanfasilitaspelayanansendiri (self-service facilities) bagiparapelanggannyaseperti supermarket, restoran, Perpustakaan, pompabensin, teleponumum, tokobuku, dll.<br />M. Sitepu<br />32<br />Antrian<br />
  59. 59. Digunakanmodel antrian (M/M/s) apabilajumlahfasilitaspelayananterbatas (finite)<br />Digunakan model antrian (M/M/) apabilajumlahfasilitaspelayanan tak terbatas (infinite) atau tidak perlu menunggu<br />Persamaan yang digunakandalam model antrian (M/M/ ∞ ) sbb :<br />M. Sitepu<br />33<br />Antrian<br />
  60. 60. 8. Queuing Network<br />Model networks dapat menggunakan sistem seri maupun sistemparalel.<br />Sistemseriterdiridarisatusubsistemmengikutisubsistem yang lain.<br />SetiappelangganharusmelewatisatusubsistemkemudianMelewatisubsistem yang lain sepertiregistrasimahasiswa.<br />SistemparalelmaliputiduaataulebihsubsistemdansetiapPelanggan harus melewati satu subsistem.<br />Sebuahsubsistemmungkinmenggunakansistemantrian (M/M/1) atausistem (M/M/s)<br />M. Sitepu<br />34<br />Antrian<br />
  61. 61. Gambar 4. Sistem Seri<br />Gambar 5. SistemParalel<br />M. Sitepu<br />35<br />Antrian<br />
  62. 62. M. Sitepu<br />36<br />Antrian<br />

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