Akhid Yulianto, SE, MSc (Log)

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

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

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

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 = 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

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