Transcript Waiting Line Theory

Waiting Line Theory
Akhid Yulianto, SE, MSc (log)
Waiting Line Examples
Situation
Arrivals
Servers
Service Process
Bank
Customers Teller
Deposit etc.
Doctor’s
office
Patient
Doctor
Treatment
Traffic
intersection
Cars
Light
Controlled
passage
Parts
Workers Assembly
Clerks
Check out/in tools
Assembly line
Tool crib
Workers
Structure of a Waiting Line
System

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Queuing theory is the study of waiting
lines.
Four characteristics of a queuing system
are:
 the manner in which customers arrive
 the time required for service
 the priority determining the order of
service
 the number and configuration of
servers in the system.
Waiting Line Costs
Cost
Waiting time cost
Optimal
Level of service
Structure of a Waiting Line
System

Distribution of Arrivals
Generally, the arrival of customers
into the system is a random event.
 Frequently the arrival pattern is
modeled as a Poisson process.


Distribution of Service Times
Service time is also usually a
random variable.
 A distribution commonly used to
describe service time is the
exponential distribution.

Poisson Probability
P( x) 



x e  
x!
x = Tingkat kedatangan
λ = rata rata kedatangan per
periode
e = 2.71828
Eksponential Probability
P(service  time  t )  1  e t


µ =jumlah unit yang di layani per
periode
e = 2.71828
Structure of a Waiting Line
System
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Queue Discipline
Most common queue discipline is
first come, first served (FCFS).
 An elevator is an example of last
come, first served (LCFS) queue
discipline.
 Other disciplines assign priorities
to the waiting units and then serve
the unit with the highest priority
first.

Structure of a Waiting Line
System
System
Customer
arrives
Waiting line
S1
Customer
leaves
System
S1
Customer
arrives
Waiting line
S2
S3
Customer
leaves
Queuing Systems

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A three part code of the form A/B/k is used to
describe various queuing systems.
A identifies the arrival distribution, B the service
(departure) distribution and k the number of
channels for the system.
Symbols used for the arrival and service
processes are: M - Markov distributions
(Poisson/exponential), D - Deterministic
(constant) and G - General distribution (with a
known mean and variance).
For example, M/M/k refers to a system in which
arrivals occur according to a Poisson distribution,
service times follow an exponential distribution
and there are k (sometimes others say s) servers
working at identical service rates.
Queuing System Input
Characteristics
 = the average arrival rate
1/ = the average time between
arrivals
µ = the average service rate
for each server
1/µ = the average service time
 = the standard deviation of
the service time
Analytical Formulas


For nearly all queuing
systems, there is a
relationship between the
average time a unit spends in
the system or queue and the
average number of units in the
system or queue.
These relationships, known as
Little's flow equations are:
L = W and Lq = Wq
Analytical Formulas


When the queue discipline is FCFS,
analytical formulas have been derived for
several different queuing models including
the following:
 M/M/1
 M/M/k
 M/G/1
 M/G/k with blocked customers cleared
 M/M/1 with a finite calling population
Analytical formulas are not available for all
possible queuing systems. In this event,
insights may be gained through a
simulation of the system.
M/M/1
Ls 


 
1
Ws 
 

2
Lq 
    

Wq 
    




P0  1 

Pn  k




 
k 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
M/M/k Queuing System
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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
M

    



Ls 
p0 
2

M  1!M   
1
P0 
forM  
 M 1 1    n  1    m M

 
     
 n 0 n!     M !    M  
M
    
1 L


Ws 
P  s
2 0
 
M  1!M   

Lq  Ls 

1 Lq
Wq  Ws  
 



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