1. Module D
Waiting Line Models

2. Elements of Waiting Line Analysis
Queue
A single waiting line
Waiting line system consists of
Arrivals
Servers
Waiting line structures

3. Components of Queuing System
Source of
customers—
calling population
Server
Arrivals
Waiting Line or
“Queue”
Served
customers

4. Elements of a Waiting Line
Calling population
Source of customers
Infinite - large enough that one more customer can always arrive to be served
Finite - limited number of potential customers
Arrival rate ()
Frequency of customer arrivals at waiting line system
Typically follows Poisson distribution

5. Elements of a Waiting Line
Service time
Often follows negative exponential distribution
Average service rate = 
Arrival rate () must be less than service rate or system never clears out

6. Elements of a Waiting Line
Queue discipline
Order in which customers are served
First come, first served is most common
Length can be infinite or finite
Infinite is most common
Finite is limited by some physical structure (not covered)

7. Basic Waiting Line Structures
Channels are the number of parallel servers
Phases denote number of sequential servers the customer must go through

8. Single-Channel Structures
Single-channel, single-phase
Waiting line Server
Single-channel, multiple phases
Servers
Waiting line

9. Multi-Channel Structures
Servers
Multiple-channel, single phase
Waiting line
Servers
Waiting line
Multiple-channel, multiple-phase

10. Operating Characteristics
Mathematics of queuing theory does not provide optimal or best solutions
Operating characteristics are computed that describe system performance
Steady state is constant, average value for performance characteristics that the system will reach after a long time

11. Operating Characteristics
NOTATION OPERATING CHARACTERISTIC
L Average number of customers in the system (waiting and being served)
Lq Average number of customers in the waiting line
W Average time a customer spends in the system (waiting and being served)
Wq Average time a customer spends waiting in line

12. Operating Characteristics
NOTATION OPERATING CHARACTERISTIC
P0 Probability of no (zero) customers in the system
Pn Probability of n customers in the system
 Utilization rate; the proportion of time the system is in use

13. Single-Channel, Single-Phase Models
All assume Poisson arrival rate
Variations
Exponential service times
Constant service times
Others are not considered

14. Basic Single-Server Model
Assumptions:
Poisson arrival rate
Exponential service times
First-come, first-served queue discipline
Infinite queue length
Infinite calling population
 = mean arrival rate
 = mean service rate

15. Formulas for Single-Server Model
Probability that no customers are in the system (either in the queue or being served)
P0 = 1 -  
Probability of exactly n
customers in the system
Pn = • P0 n   =
L = 
 - 
Average number of
customers in the system
Average number of
customers in the waiting line
Lq = 
( - )

16. Formulas for Single-Server Model
Average time a customer spends in the queuing system
W = = 1
 - L 
Average time a customer spends waiting in line to be served
Wq = 
( - )
 =  
Probability that the server is busy and the customer has to wait
Probability that the server is idle and a customer can be served
I = 1 -   
= = P0

17. A Single-Server Model Given  = 24 per hour,  = 30 customers per hour
Probability of no customers in the system
P0 = = = 0.20   24 30
L = = = 4
Average number of customers in the system 
 -  24
Average number of customers waiting in line
Lq = = = 3.2
(24)2
30( ) 2
( - )

18. A Single-Server Model Given  = 24 per hour,  = 30 customers per hour
Average time in the system per customer
W = = = hour 1
 -
Average time waiting in line per customer 
(-) 24
30( )
Wq = = = 0.133
Probability that the server will be busy and the customer must wait
 = = = 0.80   24 30
Probability the server will be idle
I = 1 -  = = 0.20

19. Waiting Line Cost Analysis
To improve customer services management wants to test two alternatives to reduce customer waiting time:
1. Another employee to pack up purchases
2. Another checkout counter

20. Waiting Line Cost Analysis
Add extra employee to increase service rate from 30 to 40 customers per hour
Extra employee costs $150/week
Each one-minute reduction in customer waiting time avoids $75 in lost sales
Waiting time with one employee = 8 minutes
Wq = hours = 2.25 minutes
= 5.75 minutes reduction
5.75 x $75/minute/week = $ per week
New employee saves $ $ = $281.25/wk

21. Waiting Line Cost Analysis
New counter costs $6000 plus $200 per week for checker
Customers divide themselves between two checkout lines
Arrival rate is reduced from = 24 to = 12
Service rate for each checker is  = 30
Wq = hours = 1.33 minutes
= 6.67 minutes
6.67 x $75/minute/week = $500.00/wk - $200 = $300/wk
Counter is paid off in 6000/300 = 20 weeks

22. Waiting Line Cost Analysis
Adding an employee results in savings and improved customer service
Adding a new counter results in slightly greater savings and improved customer service, but only after the initial investment has been recovered
A new counter results in more idle time for employees
A new counter would take up potentially valuable floor space

23. Constant Service Times
Constant service times occur with machinery and automated equipment
Constant service times are a special case of the single-server model with general or undefined service times

24. Operating Characteristics for Constant Service Times 
P0 = 1 -
Probability that no customers
are in system
Average number of
customers in queue
Lq = 2
2( - )
Average number of
customers in system
L = Lq +  

25. Operating Characteristics for Constant Service Times
Average time customer
spends in queue
Wq =
 Lq 
Average time customer
spends in the system
W = Wq + 1   
 =
Probability that the server is busy

26. Constant Service Times
Automated car wash with service time = 4.5 min
Cars arrive at rate  = 10/hour (Poisson)
 = 60/4.5 = 13.3/hour 2
2( - )
(10)2
2(13.3)( )
Lq = = = 1.14 cars waiting
Wq = = 1.14/10 = .114 hour or 6.84 minutes Lq 

27. Multiple-Channel, Single-Phase Models
Two or more independent servers serve a single waiting line
Poisson arrivals, exponential service, infinite calling population
s>
P0 = 1 s!   s s
s -  
n=s-1
n=0 n! n +

28. Multiple-Channel, Single-Phase Models
Computing P0 can be time-consuming.
Tables can used to find P0 for selected values of  and s.
Two or more independent servers serve a single waiting line
Poisson arrivals, exponential service, infinite calling population
s>
P0 = 1 s!   s s
s -  
n=s-1
n=0 n! n +

29. Multiple-Channel, Single-Phase Models
Probability of exactly n customers in the system
Pn =
P0, for n > s 1
s! sn-s   n
P0, for n > s n!
Probability an arriving
customer must wait
Pw = P0 1 s! s
s -  s  
Average number of
customers in system
L = P0 +
(/)s
(s - 1)!(s - )2  

30. Multiple-Channel, Single-Phase Models
W = L 
Average time customer
spends in system
Lq = L -  
Average number of
customers in queue
Average time customer
spends in queue
Wq = W = 1  Lq 
 = /s
Utilization factor