Akhid Yulianto, SE, MSc (Log) 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 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

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.

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

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

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

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

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

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

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

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

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

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