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

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Presentasi berjudul: "Waiting Line Theory 2 Akhid Yulianto, SE, MSc (Log)"— Transcript presentasi:

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

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

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

4 M/M/1 Ls = average number of units in the system (waiting and being served) Ls = average number of units in the system (waiting and being served) Ws = average time a unit spends in the system Ws = average time a unit spends in the system Lq = average number of units waiting in the queue Lq = average number of units waiting in the queue Wq = Average time a unit spends waiting in the queue Wq = Average time a unit spends waiting in the queue Utilization factor for the system Utilization factor for the system Probability of 0 units in 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 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. 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 1 st λ = 2 λ = 2 µ = 3 µ = 3 Ls = rata rata 2 mobil di sistem/jam Ls = rata rata 2 mobil di sistem/jam Ws = 1 jam rata rata menunggu di sistem Ws = 1 jam rata rata menunggu di sistem Lq = 1.33 mobil menunggu di garis, rata rata Lq = 1.33 mobil menunggu di garis, rata rata Wq = 40 menit waktu menunggu per mobil. Wq = 40 menit waktu menunggu per mobil. ρ = 66.6% mekanik sibuk ρ = 66.6% mekanik sibuk P 0 = 0.33 kemungkinan tidak ada mobil di sistem P 0 = 0.33 kemungkinan tidak ada mobil di sistem

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

8 M/M/S Ls = average number of units in the system (waiting and being served) Ls = average number of units in the system (waiting and being served) Ws = average time a unit spends in the system Ws = average time a unit spends in the system Lq = average number of units waiting in the queue Lq = average number of units waiting in the queue Wq = Average time a unit spends waiting in the queue Wq = Average time a unit spends waiting in the queue Probability of 0 units in the system 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? 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 2 th Ls = 0.75 mobil di dalam sistem Ls = 0.75 mobil di dalam sistem Ws = 22.5 menit sebuah mobil di sistem Ws = 22.5 menit sebuah mobil di sistem Lq = mobil di antrian Lq = mobil di antrian Wq = 2.5 menit sebuah mobil di antrian Wq = 2.5 menit sebuah mobil di antrian

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

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

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

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

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


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