MODEL ANTRIAN Problem : adanya perbedaan antara jumlah (#) permintaan layanan dengan kapasitas server Jumlah permintaan lebih besar server antrian lama dan panjang pelanggan pindah Penambahan server penambahan biaya •
Model antrian tidak menjawab secara langsung hanya memberi informasi untuk pertimbangan dalam pengambilan keputusan. Informasi yang dapat diperoleh : • Persentase waktu idle Jumlah rata-rata pelanggan yang antri pada setiap saat • Rata-rata waktu tunggu dalam antrian • Sebaran distribusi # pelanggan dalam antrian • • Sebaran distribusi waktu yang dihabiskan pelanggan dalam antrian Komponen-komponen Komponen-komponen suatu sistem antrian • Customer • Server Proses kedatangan •
Proses pelayanan • Sumber input Aturan dalam antrian (queue • discipline) •
Sistem Antrian
Populasi awal
Customer
Antrian
Mekanisme Pelayanan
keluar
Notasi antrian : 1/2/3/4/5/6 1 : Proses kedatangan (M, D, E k , G) 4 : queue discipline : FCFS, LCFS, RSS, GD 2 : Proses Pelayanan (M, D, E k , G) 5 : maksimum banyaknya customer di sistem 3 : banyaknya server parallel 6 : ukuran populasi dari mana customer berasal M markovian (dist eksponensial) eksponensial) D deterministik Ek dist Erlang G General (selain eksponensial) FCFS first come first service = FIFO (first in first out) antrian loket pembelian karcis LCFS last come first service = LIFO (last in first out) RSS random service selection Example : 1) M/M/5/FCFS/50/ 2) M/M/1 3) M/M M/M/s untuk untuk s>1 s>1
4)
5) 6)
M/M/1/N M/M/1/4 M/M/s/N untuk s>1 G/G/1
Rumus-rumus Model antrian M/M/1 : rate kedatangan dan : rate pelayanan Peluang banyaknya customer di sistem Po = 1=1- ( / ) dan Pn = n (1- ) 1
Peluang banyaknya customer di antrian qo = 1- 2 dan qn = n+1 (1- ) Rata-rata banyaknya customer L = /(1- ) (di sistem) dan Lq = 2/(1- ) (di antrian) Rata-rata waktu tunggu W = 1/( - ) (di sistem) dan W q = /( ( - )) (di antrian) Example 1: Secara rata-rata 10 mobil datang setiap jam untuk server tunggal berupa drive-in teller. Rata-rata waktu pelayanan adalah 4 menit. Waktu kedatangan dan waktu pelayanan keduanya memiliki distribusi eksponesial. 1. Berapa peluang teller idle ? 2. Berapa rata-rata # customer pada antrian? 3. Berapa rata-rata waktu di dalam antrian? 4. Secara rata-rata, berapa # customer di system? Jawab : M/M/1, =10 mobil/jam dan = 60/4 = 15 mobil/jam 1. Po = 1/3 2. Lq = 4/3 3. Wq = 8 menit 4. L=2 Example 2: Rata-rata seorang pasien tiba dalam selang waktu ½ jam dan rata-rata waktu yang diperlukan dokter untuk memeriksa seorang pasien = 20 menit. Jika unit waktu adalah jam maka hitunglah : 1. Rata-rata waktu antar kedatangan = 1/ 2. Rata-rata waktu pelayanan = 1/ 3. Rata-rata pasien yang datang = 4. Rata-rata pasien yang selesai diperiksa = Jawab : M/M/1, = 2 orang/jam dan = 3 orang/jam
JENIS-JENIS ANTRIAN 1. Antrian tunggal dengan satu server
loket
2. Antrian tunggal dengan beberapa server (paralel) Server identik server berupa mesin Server non-identik bank • •
3.
Antrian tunggal dengan server seri
surat-surat 1
4.
Antrian ganda
2
3
supermarket
2
SIMULASI dengan AWESIM 1. Node CREATE
2.
Node ACTIVITY
sebagai penghubung atau server
3. Node QUEUE
3
4.
Node TERMINATE
4
SIMULASI ANTRIAN dengan AWESIM 1. Antrian tunggal dengan 1 server
Reading control ANTRI11 ... 1 GEN,"eto","antri1_1", 3-12-2003,50,YES,YES; 2 LIMITS,,,,,,,400; 3 NETWORK,READ; 4 FIN;
Reading network file ANTRI1_1.MNT ... 1 masukan: CREATE,EXPON(3); 2 ACTIVITY; 3 antri: QUEUE,1, , ,NONE; 4 ACTIVITY,1,EXPON(4), , ,1,"server1"; 5 selesai: TERMINATE,100; 6 END;
** FILE STATISTICS ** File Number Where Created 1 QUEUE ANTRI File
Number 1
Average Length 16.5879162
Maximum Average Length 35.1670416
** SERVICE ACTIVITY STATISTICS ** Activity Label Server Average Number Capacity Utilization 1 server1 1 0.9721436
Standard Deviation 8.0198691 Average Wait Time 48.0491364 Standard Deviation 0.0374964
Average Blockage 0.0000000
** AweSim! MULTIPLE RUN SUMMARY REPORT ** ** FILE STATISTICS ** File Number 1 QUEUE
Activity Number 1
Where Created ANTRI
Average Length 1.2191018
standard Deviation 0.8363926
File Number
1 ** SERVICE ACTIVITY STATISTICS ** Label Server Average Capacity Utilization 1 0.6580955 Server 1
Maximum Average Length 4.9688312 Activity Number 1
Average Wait Time 3.5853552 Standard Deviation 0.0868101
Average Blockage 0.0000000
2. Antrian tunggal dengan 3 server identik
Reading network file ANTRI1_3.MNT ... 1 masukan: CREATE,EXPON(3); 2 ACTIVITY; 3 antri: QUEUE,1, , ,NONE; 4 ACTIVITY,1,EXPON(4), , ,3,"server1"; 5 selesai: TERMINATE,100; 6 END; 5
** FILE STATISTICS ** File Number 1 QUEUE
Activity Number 1
Where Created ANTRI
Average Length 0.1175473
standard Deviation 0.1040447
File Number
1 ** SERVICE ACTIVITY STATISTICS ** Label Server Average Capacity Utilization 3 1.2889142 Server 1
Maximum Average Length 0.4593855 Activity Number 1
Average Wait Time 0.3506529 Standard Deviation 0.1805306
Average Blockage 0.0000000
3. Antrian tunggal dengan 3 server non identik
Reading network file ANT1_3NI.MNT ... 1 #line 0 ANT1_3NI.NET 1 #line 0 ANT1_3NI.NET 1 masukan: CREATE,EXPON(3); 2 ACTIVITY; 3 antri: QUEUE,1, , ,NONE,{pilih}; 4 pilih: SELECT,RANDOM,RANDOM,NONE,{antri}; 5 ACTIVITY,1,EXPON(4),prob(1/3), ,1,"server1"; 6 ACTIVITY,2,EXPON(5),prob(1/3),"selesai2",1,"server2"; 7 ACTIVITY,3,EXPON(6),prob(1/3),"selesai3",1,"server3"; 8 selesai: TERMINATE,100; 9 selesai2: TERMINATE,100; 10 selesai3: TERMINATE,100; 11 END; ** FILE STATISTICS ** File Where Average Standard File Maximum Average Number Created Length Deviation Number Average Wait Time 1 QUEUE ANTRI 0.3587746 0.2131394 Length 1 1.1387428 1.0644167 ** SERVICE ACTIVITY STATISTICS ** Activity Label Server Number Capacity 1 server1 1 2 server2 1 3 server3 1
Average Utilization 0.5017950 0.5334371 0.5723760
6
ANTRIAN GANDA 1. ANTRIAN TUNGGAL DENGAN 1 SERVER
Reading network file ANTG_1.MNT ... 1 masukan: CREATE,EXPON(2); 2 ACTIVITY; 3 antri: QUEUE,1, , ,NONE;
4 ACTIVITY,1,EXPON(4), , ,1,"server1"; 5 selesai: TERMINATE,100; 6 END;
** AweSim! MULTIPLE RUN SUMMARY REPORT ** ** FILE STATISTICS ** File Number 1 QUEUE
Where Average Created Length ANTRI 53.9313330
Standard Deviation 13.9328900
File Number 1
Maximum Average Length 90.7033304
Average Wait Time 104.0707066
** SERVICE ACTIVITY STATISTICS ** Activity Number 1
Label
Server Capacity server1 1
Average Utilization 0.9940099
Activity Number 1
Standard Deviation 0.0097114
Average Blockage 0.0000000
2. ANTRIAN GANDA DENGAN 2 SERVER
Reading control ANTG_2 ... 1 GEN,"eto","antrian ganda dg 2 server",7-12-2003,50,YES,YES; 2 LIMITS,,,,,,,400; 3 NETWORK,READ; 4 FIN; Reading network file ANTG_2.MNT ... 1 #line 0 ANTG_2.NET 5 ACTIVITY,1,EXPON(4), , ,1,"server1"; 1 #line 0 ANTG_2.NET 6 selesai: TERMINATE,100; 1 masukan: CREATE,EXPON(2); 7 antri2: QUEUE,2, , ,NONE; 2 ACTIVITY, , ,prob(1/2); 8 ACTIVITY,2,EXPON(5), , ,1; 3 ACTIVITY, , ,prob(1/2),"antri2"; 9 selesai2: TERMINATE,100; 4 antri: QUEUE,1, , ,NONE; 10 END; ** AweSim! MULTIPLE RUN SUMMARY REPORT **
** FILE STATISTICS ** File Number 1 QUEUE 2 QUEUE
Where Average Created Length ANTRI 6.7367406 antri2 11.7187198
Standard Deviation 4.4477990 5.6073766
File Number 1 2
Maximum Average Length 21.2076715 25.6581466
Average Wait Time 25.7724230 45.7029554
** SERVICE ACTIVITY STATISTICS ** 7
Activity Number 1
2
Label
Server Capacity server1 1 server2 1
Average Utilization 0.9089034 0.9620344
Activity Number 1 2
Standard Deviation 0.0745511 0.0415885
Average Blockage 0.0000000 0.0000000
NUMBER OF STREAM DISTRIBUSI EXPONENSIAL EXPON(XMN,IS) IS nb of stream terkait dengan nilai awal / seed Di dalam Awesim IS ini nilainya 1, 2, … ,100 DISTRIBUSI TANPA IS EXPON(3) KEMUNGKINAN UNTUK KOMPUTER YANG BERBEDA DAPAT • MENGHASILKAN NILAI YANG BERBEDA JIKA DI-RUN BERULANG-ULANG AKAN MENGHASILKAN NILAI YANG • SAMA DISTRIBUSI DENGAN IS EXPON(3,5) MENGHASILKAN NILAI YANG SAMA UNTUK KOMPUTER YANG • AKAN BERBEDA JIKA DI-RUN BERULANG-ULANG AKAN MENGHASILKAN NILAI YANG SAMA • PENGHENTIAN PROSES 1. COUNT PROSES AKAN DIHENTIKAN JIKA N ENTITY/CUSTOMER TELAH DILAYANI BIOSKOP (250 KURSI)
2.
TIME PROSES AKAN DIHENTIKAN JIKA WAKTU YANG DISEDIAKAN TELAH HABIS BANK (08.00 – 15.00),
Reading control ANTRI1 ... 1 GEN,"???","antri1",11-12-2007,5,YES,YES; 2 LIMITS; 3 INITIALIZE,0.0,300,YES; TIME 4 NETWORK,READ; 5 FIN;
8
3. ANTRIAN GANDA DENGAN 3 SERVER
3 Reading control ANTG_3 ... 1 GEN,"eto","antrian ganda dg 3 serever",7-12-2003,50,YES,YES; 2 LIMITS,,,,,,,300; 3 NETWORK,READ; 4 FIN; Reading network file ANTG_3.MNT ... 1 #line 0 ANTG_3.NET 7 selesai: TERMINATE,95; 1 #line 0 ANTG_3.NET 8 antri2: QUEUE,2, , ,NONE; 1 masukan: CREATE,EXPON(2); 9 ACTIVITY,2,EXPON(5), , ,1; 2 ACTIVITY, , ,prob(1/3); 10 selesai2: TERMINATE,95; 3 ACTIVITY, , ,prob(1/3),"antri2"; 11 antri3: QUEUE,3, , ,NONE; 4 ACTIVITY, , ,prob(1/3),"antri3"; 12 ACTIVITY,2,EXPON(6), , ,1; 5 antri: QUEUE,1, , ,NONE; 13 selesai3: TERMINATE,95; 6 ACTIVITY,1,EXPON(4), , ,1,"server1"; 14 END; ** AweSim! MULTIPLE RUN SUMMARY REPORT ** ** FILE STATISTICS ** File Where Average Standard Number Created Length Deviation 1 QUEUE ANTRI 1.1112429 0.7487553 2 QUEUE antri2 3.8300705 3.8398265 3 QUEUE antri3 5.5057130 3.5070026 File Maximum Average Number Average Wait Time Length 1 4.7195844 6.5863122 2 19.3974179 21.5491303 3 15.9715504 32.2244892 ** SERVICE ACTIVITY STATISTICS ** Activity Label Server Average Number Capacity Utilization 1 server1 1 0.6440968 2 ANTG_3.NET:9 1 0.8009910 2 ANTG_3.NET:12 1 0.8972642
9
