Analisis Kinerja Sistem 07.

Report
Analisis Kinerja Sistem
Teori Antrian
- Aurelio Rahmadian -
Sifat Antrian

Pelanggan tidak menunggu bila ada
pelayanan yang menganggur

Pelanggan harus menunggu bila layanan
sedang digunakan pelanggan lain

Pelanggan akan segera memasuki
pelayanan bila ada pelanggan yang
meninggalkan layanan
Analisa Antrian






Jumlah waktu yang digunakan untuk berada dalam
antrian oleh orang, produk, dll.
Menyediakan layanan yang cepat merupakan aspek
penting dari kualitas pelayanan
Dasar analisa antrian adalah trade-off dari cost yang
dibutuhkan untuk meningkatkan pelayanan dan
cost yang berhubungan dengan menunggunya
customer
Analisa antrian merupakan bentuk probabilistik
dari analisa
Hasil analisa merupakan karakteristik operasi
Hasil analisa digunakan oleh manajer operasi
antrian untuk mengambil keputusan
Cost Trade-off
Elemen Antrian






Waiting lines form because people or things arrive at a
service faster than they can be served.
Most operations have sufficient server capacity to handle
customers in the long run.
Customers however, do not arrive at a constant rate nor are
they served in an equal amount of time.
Waiting lines are continually increasing and decreasing in
length and approach an average rate of customer arrivals and
an average service time, in the long run.
Decisions concerning the management of waiting lines are
based on these averages for customer arrivals and service
times.
They are used in formulas to compute operating
characteristics of the system which in turn form the basis of
decision making.
Elemen Antrian

Komponen dalam antrian:
◦ Customer
◦ Server

Faktor yang perlu diperhatikan:
◦
◦
◦
◦
Queue Discipline
Nature of calling population
Arrival rate
Service rate
Elemen Antrian
Queue discipline
Urutan bagaimana customer yang menunggu
dilayani
 Calling population
Sumber dari customer (finite atau infinite)
 Arrival rate
Frekuensi customer datang pada antrian, mengacu
pada distribusi probabilitas
 Service rate
Rata-rata customer yang dapat dilayani dalam
selang waktu tertentu

Rumus Umum



Ls
Ws 


Ls 
 
Wq 
Lq


Lq 
 (   )
2
Notasi Kendall
A/S/m/B/K/SD
Dimana:
 A adalah distribusi waktu interarrival
 S adalah distribusi waktu layanan.
 m adalah jumlah server
 B adalah jumlah buffer (sistem kapasitas)
 K adalah besar populasi
 SD adalah tertib layanan (service discipline)
M/M/1
Ambil :
n   ,

 n   ,
n  0,1,2,...
n  0,1,2,...
Contoh 1
Contoh 1
Antri
Ws = Rata-rata Waktu
tunggu dalam Sistem
Layanan
Lq = Rata-rata Pelanggan dalam Antrian
Ls = Rata-rata Pelanggan dalam Sistem
Wq = Rata-rata Waktu tunggu dalam Antrian
M/M/S
n   ,

n ,



 n K ,


P0 
n  0,1,2,...
n  0,1,2,...,K
(n  K)
n  K  1, K  2,...
(n  K)
1
k 1 ( /  ) n

n 0
n!
( /  ) K
1

K! 1  ( / K )
 ( k ) n
P0 ,

 n!
Pn  
 KK  n
P0 ,

 K!
n  1,2,...,K
n  K  1, K  2,...
M/M/S
~
~
nk
j 0
~ ( /  ) K
Lq   (n  K ) P0   jPs  j  
Wq 
Lq

Ws  Wq 
1


Ls   (Wq  )  Lq 


1
j 0
K!
 j P0 
K K  K 1P0
K!(1   )2
Contoh 2
Contoh 2
Antri
Layanan
 = 4 / jam
Ws = Rata-rata Waktu
tunggu dalam Sistem
Lq = Rata-rata Pelanggan dalam Antrian
Ls = Rata-rata Pelanggan dalam Sistem
Wq = Rata-rata Waktu tunggu dalam Antrian
Keputusan Manajerial

Present system:
◦
◦
◦
◦
◦
◦

L = 4.00 customers
Lq = 3.20 customers
W = 10.00 min
Wq = 8.00 min
U = 0.80
Cost $150 per week, avoids loss of $75 per week for each
minute of reduced customer waiting time
Manager wishes to test several alternatives for
reducing customer waiting time:
◦ Addition of another employee
◦ Addition of another checkout counter
Keputusan Manajerial

Alternative 1: Addition of an employee (raises service
rate from  = 30 to  = 40 customers per hour)

System operating characteristics with new parameters:
◦ L = 1.5 customers on the average in the queuing system
◦ Lq = 0.90 customer on the average in the waiting line
◦ W = 0.063 hour (3.75 minutes) average time in the system per
customer
◦ Wq = 0.038 hour ( 2.25 minutes) average time in the waiting line per
customer
◦ U = .60 probability that server is busy and customer must wait, .40
probability server available


Average customer waiting time reduxed from 8 to 2.25 minutes
worth $431.25 per week
Net savings = $431.25 - $150 = $281.25 per week
Keputusan Manajerial

Alternative 2: Addition of a new checkout counter ($6,000 plus
$200 per week for additional cashier)
◦  =24/2 = 12 customers per hour per checkout counter
◦  = 30 customers per hour at each counter

System operating characteristisc with new parameters:
◦
◦
◦
◦
◦
◦


L = 0.67 customer in the queuing system
Lq = 0.27 customer in the waiting line
W = 0.055 hour (3.33 minutes) per customer in the system
Wq = 0.022 hour (1.33 minutes) per customer in the waiting line
U = .40 probability that a customer must wait
P0 = .60 probability that server is idle and customer can be served
Savings from reduced waiting time worth $500 per week - $200 =
$300 net savings per week
After $6,000 recovered, alternative 2 would provide $300 -281.25
= $18.75 more savings per week
Keputusan Manajerial
Operating
Characteristics for Each
Alternative System
Cost trade-offs for service
levels
Waiting Line Psychology
1.
2.
3.
4.
5.
6.
7.
8.
Waits with unoccupied time seem longer
Pre-process waits are longer than process
Anxiety makes waits seem longer
Uncertainty makes waits seem longer
Unexplained waits seem longer
Unfair waits seem longer than fair waits
Valuable service waits seem shorter
Solo waits seem longer than group waits
Maister, The Psychology of Waiting Lines,
teaching note, HBS 9-684-064.

similar documents