Queuing Systems
Chapter 17
Chapter Topics
 Elements of Waiting Line Analysis
 Single-Server Waiting Line System
 Undefined and Constant Service Times
 Finite Queue Length
 Finite Calling Problem
 Multiple-Server Waiting Line
 Addition Types of Queuing Systems
Overview
 People and products spent significant amount of time in
waiting lines
 Providing quick service is an important aspect of customer
service
 Trade-off between the cost of improving service and the
costs associated with making customers wait
 A probabilistic form of analysis
 Results are referred to as operating characteristics
 Results are used to make decisions
Elements of Waiting Line Analysis
 Waiting lines form because people arrive at a service faster than
they can be served
 Not arrive at a constant rate nor are they served in an equal
amount of time
 Have an average rate of customer arrivals and an average
service time
 Decisions are based on these averages for customer arrivals and
service times
 Formulas are used to compute operating characteristics
Single-Server Waiting Line System
 Components include:
 Arrivals (customers), servers, (cash register/operator),
customers in line form a waiting line
 Factors to consider:

queue discipline.

nature of the calling population

arrival rate

service rate.
Component Definitions
 Queue Discipline: order in which customers are served
(FIFO, LIFO, or randomly)
 Calling Population: source of customers (infinite or finite)
 Arrival Rate: Frequency at which customers arrive at a
waiting line according to a probability distribution (Poisson
distribution)
 Service Rate: Average number of customers that can be
served during a time period (negative exponential
distribution)
Single-Server Waiting Line System
Single-Server Model
 Assumptions:

An infinite calling population

A first-come, first-served queue discipline

Poisson arrival rate

Exponential service times
 Symbology:
  = the arrival rate (average number of arrivals/time
period)
  = the service rate (average number served/time period)
 Customers must be served faster than they arrive ( < )
Single-Server Queuing Formulas
Probability that no customers are in the waiting line:


Po  1  






Probability that n customers are in the waiting line:
n
n


 




  
Pn     Po     1  


 






Average number of customers in system:
waiting line:
2

Lq  
     
L 
 
and
SSingle-Server Queuing Formulas
Average time customer spends waiting and being served:
W  1 L
  
Average time customer spends waiting in the queue:
Wq 

     
Probability that server is busy (utilization factor):
Probability that server is idle:
I 1U 1 

U 

Characteristics for Fast Shop Market
 = 24 customers per hour arrive at checkout counter
 = 30 customers per hour can be checked out


Po  1    (1 - 24/30)






 .20 probabilit y of no customers in the system
L    24/(30 - 24)  4 customers on the avg in the system
 
2

Lq  
     
 (24)2/[30(30 - 24)]  3.2 customers on the avg in the waiting line
Characteristics for Fast Shop Market
W  1  L  1/[30 - 24]
  
 0.167 hour (10 min) avg time in the system per customer
Wq 
  24/[30(30 - 24)]
     
 0.133 hour (8 min) avg time in the waiting line
U 
  24/30
 .80 probabilit y server busy, .20 probabilit y server will be idle
Steady-State Operating Characteristics
 Utilization factor, U, must be less than one:
 U < 1,or  /  < 1 and  < .
 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
 Server must be able to serve customers faster than
the arrival rate in the long run, or waiting line will
grow to infinite size.
Single-Server Waiting Line System
Effect of Operating Characteristics (1 of 6)
 Wish to test several alternatives for reducing customer
waiting time:
 Addition of another employee to pack up purchases
 Addition of another checkout counter.
 Alternative 1: Addition of an employee raises service rate
from  = 30 to  = 40 customers
 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
 L = 1.5 customers on the average in the queuing system
Effect of Operating Characteristics
 System operating characteristics with new
parameters (continued):
Lq = 0.90 customer on the average in the waiting line
W = 0.063 hour average time in the system per customer
Wq = 0.038 hour average time in the waiting line per customer
U = .60 probability that server is busy and customer must wait
I = .40 probability that server is available
Average customer waiting time reduced from 8 to 2.25
minutes worth $431.25 =(8-2.25)($75) per week.
Effect of Operating Characteristics
 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 characteristics 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 per customer in the system
Wq = 0.022 hour per customer in the waiting line
U = .40 probability that a customer must wait
I = .60 probability that server is idle
Effect of Operating Characteristics
Savings from reduced waiting time worth $500= (81.33)($75) 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.
Undefined and Constant Service Times
 Constant occurs with machinery and automated equipment
 Constant service times are a special case of the singleserver model with undefined service times
 Queuing formulas:

Po 1 
Lq 
Wq  Lq

2


2
2
     /  

21  /  


L  Lq  

1
W Wq  


U 
Undefined Service Times Example
 Data: Arrival rate of 20 customers per hour (Poisson
distributed); undefined service time with mean of 2
minutes, standard deviation of 4 minutes.
 Operating characteristics:
20  .33 probability that machine not in use
Po 1 

1


30
2


2
2


    /  
2 
2
2






20 1/15   20 / 30






Lq 

21  /  
21 20 / 30




 3.33 employees waiting in line








L  Lq  
  3.33 (20/ 30)
 4.0 employees in line and using the machine
Undefined Service Times Example (2 of 2)
 Operating characteristics (continued):
Wq  Lq  3.33  0.1665 hour  10 minutes waiting time
 20
1  0.1665  1  0.1998 hour
W Wq  
30
 12 minutes in the system
  20  67% machine utilizatio n
U 
30
Constant Service Times Formulas
 No variability in service times;  = 0.
 Substituting  = 0 into equations:
2 2 2 
2 
2




2
2
  /  
     /    0    /  
2



Lq 

 
 





21  /  
21  /  
21  /   2     






 All remaining formulas are the same
Constant Service Times Example
 Inspecting one car at a time; constant service time of 4.5
minutes; arrival rate of customers of 10 per hour (Poisson
distributed).
 Determine average length of waiting line and average
waiting time
  = 10 cars per hour,  = 60/4.5 = 13.3 cars per hour
Lq 
(10)2
2 
 1.14 cars waiting
2 (   ) 2(13.3)(13.3 10)
Wq  Lq  1.14  0.114 hour or 6.84 minutes waiting time
 10
Finite Queue Length
 Length of the queue is limited.
 Operating characteristics:
 M is the maximum number in the system:
n


1


/

  for n  M
Po 
Pn  (Po) 



1 ( /  )M 1
M 1

/

(
M

1
)(

/

)
L

1  / 
1 ( /  )M 1
W
L
 (1 PM )
Lq  L   (1PM )
1
Wq W  
Finite Queue Length Example
 Limited parking space (one vehicle in service and three waiting for
service)
 Mean time between arrivals of customers is 3 minutes and mean
service time is 2 minutes (both inter-arrival times and service times are
exponentially distributed)
 Maximum number of vehicles in the system equals 4.
 Operating characteristics for  = 20,  = 30, M = 4:
Po 
1  /   1 20/ 30  .38 prob. that system is empty
1 ( /  )M 1 1 (20/ 30)5
PM  (Po) 











nM










4
 (.38) 20  .076 prob. that system is full
30
Example-Cont.
 Average queue lengths and waiting times:
M 1
)

/

)(
1

M
(

/


L
1  / 
1 ( /  )M 1
5
)
30
/
20
)(
5
(
30
/
20
 1.24 cars in the system

L
1 20 / 30 1 (20 / 30)5
Lq  L   (1PM ) 1.24  20(1.076)  0.62 cars waiting
30
W
L
 0.067 hours waiting in the system
 1.24
 (1 PM ) 20(1.076)
1  0.067  1  0.033 hour waiting in line
Wq W  
30
Finite Calling Population
 There is a limited number of potential customers that can
call on the system.
 Operating characteristics (Poisson arrival and exponential
service times):
1
Po 
n
N
N!   
 ( N  n)!  


n0
where N  population size, and n  1, 2,...N
n
 Po
Pn  N! 
( N  n)!










L  Lq  (1 Po)





Lq  N 
(1 Po)
 





Wq 
Lq
( N  L)
1
W Wq  
Finite Calling Population Example
 20 machines; each machine operates an average of 200
hours before breaking down; average time to repair is 3.6
hours; breakdown rate is Poisson distributed, service time
is exponentially distributed.
 Is repair staff sufficient?

 = 1/200 hour = .005 per hour

 = 1/3.6 hour = .2778 per hour

N = 20 machines
Finite Calling Population Example
Po 
1
 .652
n

20 20!  .005 
 (20  n)!.2778 


n0
Lq  20  .005 .2778 1.652  .169 machines waiting


.005
L  .169  (1.652)  .520 machines in the system
Wq 
.169
1.74 hours waiting for repair
(20 .520)(.005)
W 1.74 
1  5.33 hours in the system
.2778
 System seems inadequate.
Multiple-Server Waiting Line
 Two or more independent servers in parallel serve a single
waiting line; first-come, first-served basis
 Assumptions:
 First-come first-served queue discipline
 Poisson arrivals, exponential service times
 Infinite calling population.
 Parameter definitions:
  = arrival rate (average number of arrivals per time period)
  = the service rate (average number served per time
(channel)
period) per server
 c = number of servers
 c  = mean effective service rate for the system (must exceed arrival rate)
Multiple-Server Waiting Line: Formulas
Po  





nc1 1 
 n! 
n0










n 




1
1 
c! 





c 








c
c  





 prob. no customers in system
n


 
1
Pn  nc    Po for n  c


c!c


n


 
1
Pn  n    Po for n  c  prob. of n customers in system




L
 ( /  )c Po     average customers in the system

(c 1)!(c   )2
W  L  average time customer spends in the system

  average number of customers in the queue
Lq  L  
1  Lq  average time customer is in the queue
Wq W  

c c


1
 
Pw    
Po  probabilit y customer must wait for service
c!  c  
Example
 = 10,  = 4, c = 3
Po  






1





1 10
0! 4
0










1










 1 10  1 10
1! 4
2! 4





2 





3


3(4)
10 
1
  
3! 4  3(4) 10
.045 prob. of no customers
3
L  (10)(4)(10 / 4) (.045) 10  6 customers on average in service department
4
(31)![3(4) 10]2
W  6  0.60 hour average customer t ime in the service department
10
Lq  6 10  3.5 customers on the average waiting to be served
4
Wq  3.5  0.35 hour average waiting time in line per customer
10
3


3(4) (.045)  .703 prob. customer must wait for service
10 
1
Pw   
3! 4  3(4) 10
Notation Used for Waiting Line Models
• Kendal suggested A/B/k
– A: Denotes the prob. Distribution for the arrival
– B: Denotes the prob. Distribution for the service time
– k: Denotes of channels
• Letters commonly used
– M: designates a Poisson prob. Dist. For the arrivals or an exp.
prob. dist. for the service time
– D: designates that arrivals or the service time is deterministic
– G: designates that the arrivals or the service time has a general
prob. Dist. With a known mean and variance
– M/M/1: single-server with Poisson arrivals and exp. service time
– M/M/2: two-server with Poisson arrivals and exp. service time
– M/G/1: single-server with Poisson arrivals and arbitrary service
time
Problem#1
• A single server queuing system with an
infinite calling population and a FIFO
discipline has the following arrival and
service rates:
 16 customers/hour
 24 customers/hour
 Determine P0, P3, L, Lq, W, Wq, and U