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Waiting Line Theory 2

Waiting Line Theory 2. Akhid Yulianto, SE, MSc (Log). Poisson Probability. x = Tingkat kedatangan λ = rata rata kedatangan per periode e = 2.71828. Eksponential Probability. µ =jumlah unit yang di layani per periode e = 2.71828. M/M/1.

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Waiting Line Theory 2

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  1. Waiting Line Theory 2 Akhid Yulianto, SE, MSc (Log)

  2. Poisson Probability • x = Tingkat kedatangan • λ = rata rata kedatangan per periode • e = 2.71828

  3. Eksponential Probability • µ =jumlah unit yang di layani per periode • e = 2.71828

  4. M/M/1 • Ls = average number of units in the system (waiting and being served) • Ws = average time a unit spends in the system • Lq = average number of units waiting in the queue • Wq = Average time a unit spends waiting in the queue • Utilization factor for the system • Probability of 0 units in the system • Probability of more than k units in the system, where n is the number of units in the system

  5. Example • Tom Jones, mekanik di toko Golden Muffler, dapat memasang muffler baru dengan rata rata 3/jam (mengikuti eksponential distribution). Customer yang meminta service ini dengan rata rata kedatangan 2/ jam (poisson distribution). Pelayanan FCFS dan populasi yang tak terbatas.

  6. Analisa Waiting Line 1st • λ = 2 • µ = 3 • Ls = rata rata 2 mobil di sistem/jam • Ws = 1 jam rata rata menunggu di sistem • Lq = 1.33 mobil menunggu di garis , rata rata • Wq = 40 menit waktu menunggu per mobil. • ρ = 66.6% mekanik sibuk • P0 = 0.33 kemungkinan tidak ada mobil di sistem

  7. M/M/k Queuing System • Multiple channels (with one central waiting line) • Poisson arrival-rate distribution • Exponential service-time distribution • Unlimited maximum queue length • Infinite calling population • Examples: • Four-teller transaction counter in bank • Two-clerk returns counter in retail store

  8. M/M/S • Ls = average number of units in the system (waiting and being served) • Ws = average time a unit spends in the system • Lq = average number of units waiting in the queue • Wq = Average time a unit spends waiting in the queue • Probability of 0 units in the system

  9. Example • Toko Golden Muffler memutuskan untuk membuka garasi kedua dan menyewa mekanik kedua untuk menangani instalasi muffler. Tingkat kedatangan dan tingkat layanan sama. Analisa?

  10. Analisa waiting line 2th • Ls = 0.75 mobil di dalam sistem • Ws = 22.5 menit sebuah mobil di sistem • Lq = 0.083 mobil di antrian • Wq = 2.5 menit sebuah mobil di antrian

  11. M/D/1 • Constant service time model • Contoh: assembly line/pencucian mobil otomatis

  12. Costs • Berdasar jumlah unit customer • TC = Cw L + Cs k • TC = Total cost • Cw = cost of waiting • L = jumlah rata rata units di sistem • Cs = Service cost • k or s = channel number • L = Lq + λ µ

  13. Prinsip biaya • Bandingkan biaya yang terendah • Bisa terjadi pada perencanaan untuk penambahan channel • Atau penambahan layanan

  14. Tambahan • Buku lain punya rumus yang berbeda namun hasil perhitungan ± sama • Jadi jangan bingung

  15. Reference • Anderson, & Sweeney, 2002, Quantitative for decision making,9th edn, Sydney • Heizer, J.,& Render, B., 2006, Operation Management, 8th edn, Pearson Education, Singapore

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