Transcript Chapter 13 - Queuing Analysis Chapter Contents

Queuing Analysis
Overview
• What is queuing analysis?
- to study how people behave in waiting in line so that we
could provide a solution with minimizing waiting time
and resources allocation
Two elements involved in waiting lines
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Two elements involved in
waiting lines
1. Arrival rate, 
•
Rate of people joining to the queue
2. Service rate, 
•
Rate of service that service provided
How do they applied in a real life?
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2
Queuing Analysis
Service rate, 
This phenomenon is known as
Single-server Waiting Line
How to study it?
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Arrival rate, 
3
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The Single-Server Waiting Line System
The Single-Server Model
We assumed that
1. An infinite calling population (that is many people can join the queue)
2. A first-come, first-served queue discipline
3. Poisson arrival rate
4. Exponential service times
- Sympology:
 = the arrival rate (average number of arrivals per time period)
 = the service rate (average number served per time period)
Again, we assumed that (< ) or we can never finish service customers before the end
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of the day!
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The relationship between  and 
• We adopted a “birth-and-death” process to
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study their relationship
And we have the following results:
P0 = Prob that no one in the queue
Ls= number people waiting in the system
Lq=number people waiting in the queue
Ws= total waiting time in the system
Wq= total waiting time in the queue
How these L, W values are represented
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Busy time =
U 


Idle time = P0 = 1 -
Wq, Lq


Ws, Ls
Their relationships
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The Single-Server Waiting Line System
Basic Single-Server Queuing Formulas
 
Po  1    1  
 
Probability that no customers are in the queuing system:
n
Probability that n customers are in the system: Pn      Po    


Average number of customers in system: L 

n
 
1     n 1   
 

 L
    
2
and waiting line: Lq 
 
Average time customer spends waiting and being served: W 
Average time customer spends waiting in the queue:Wq 
1
1
 L
  

1
 L
     
Probability that server is busy (utilization factor): U    

Probability that server is idle: I  1  U  1  

Note: the process to derive these formulas
are based on the “birth-and-death” process
Example
7
(to p8)
The Single-Server Waiting Line System
Operating Characteristics for Fast Shop Market Example
Example:
Given:  = 24 customers per hour arrive at checkout counter,
 = 30 customers per hour can be checked out
Then,
 
Po  1   = (1 - 24/30) = .20 probability of no customers in the system.
 
L

 

Lq 
    
= 24/(30 - 24) = 4 customers on ther average in the system
2
W 
= (24)2/[30(30 -24)] = 3.2 customers on the average in the waiting line
1
L

= 1/[30 -24] = 0.167 hour (10 minutes) average time in the system per customer
  
Wq 

= 24/[30(30 -24)] = 0.133 hour (8 minutes) average time in the waiting line
    
U 


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= 24/30 = .80 probability server busy, .20 probability server will be idle
The Single-Server Waiting Line System
Steady-State Operating Characteristics
Important note:
Because of steady -state nature of operating characteristics:
- Utilization factor, U, must be less than one: U<1,or  /  <1 and  < .
- The ratio of the arrival rate to the service rate must be less than one
or, the service rate must be greater than the arrival rate.
- The server must be able to serve customers faster than the arrival rate in
the long run, or waiting line will grow to infinite size.
What if Utilization rate >= 1? (what would happened?)
Changing of the values of  , .
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• Consider another case where,
•
•
– Manager wishes to test several alternatives for reducing customer waiting
time:
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1. Addition of another employee to pack up purchases
2. Addition of another checkout counter.
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– Which one of the three models that we should deploy?
• Answer:
– Comparing its operating characteristics
• (all three examples with calculation)
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The Single-Server Waiting Line System
Effect of Operating Characteristics on Managerial Decisions
(1 of 3)
- Alternative 1: Addition of an employee (raises service rate from  = 30 to  = 40 customers per hour)
Cost $150 per week, avoids loss of $75 per week for each minute of reduced customer waiting time.
System operating characteristics with new parameters:
Po = .40 probability of no customers in the system
= 1 - (24/40)
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 reduced from 8 to 2.25 minutes worth $431.25 per week.
i.e. (5.75*$75)
Net savings = $431.25 - 150 = $281.25 per week.
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The Single-Server Waiting Line System
Effect of Operating Characteristics on Managerial Decisions
(2 of 3)
- 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 chacteristics with new parameters:
Po = .60 probability of no customers in the system
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
I = .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
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week.
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The Single-Server Waiting Line System
Effect of Operating Characteristics on Managerial Decisions
(3 of 3)
Table 13.1
Operating
Characteristics for Each
Alternative System
Decision: very much depended
on manager’s experience because
there is difficult to obtain “the”
best solution ..
Figure 13.2
Cost trade-offs for service
levels
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13
Note: Min cost is not obtained here
One last note
• The Single waiting line system we studied
here can be denoted as:
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Models
In this subject, we only consider the following model(s) only:
(M/M/1):(GD/a/a)
Infinite calling pop
Possion arrival rate
Infinite length
Exponential service rate
One service channel
It has a general format like
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General queue discipline
such as FCFS
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Type of models
(A/B/C): (D/E/F)
Size of population
Arrival distribution
Queue capacity,
such max ppl in the queue length
Service distribution
Number of channels
(parallel servers)
Queue discipline,
such as FCFS
Example: M/M/S with finite population
Tutorial Questions
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Tutorial Questions
• Ex: 7, 8, 9 and 12
(END)
End
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The Single-Server Waiting Line System
Operating Characteristics for Fast Shop Market Example
Alternative 1
Alternative 2
 = 24
 = 24 /2=12
 = 40
 = 30
1-24/40 =0.4
1-12/30=0.6
= 24/(30 - 24) = 4
1.5
0.67
= (24)2/[30(30 -24)] = 3.2
0.9
0.27
Example:1
Given:  = 24
 = 30
Then,
 
Po  1   = (1 - 24/30) = .20
 
L

 

    
2
Lq 
W 
1
L

= 1/[30 -24] = 0.167 hour (10 minutes)
  
Wq 
0.063(3.75 mins)

= 24/[30(30 -24)] = 0.133 hour (8 minutes)
    
U 


0.055(3.33min)
0.038 (2.25 min)
0.022 (1.33 min)
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= 24/30 = .80 probability server busy, .20
0.6
0.4