Service Systems
&
Queuing
Chapter 12S
OPS 370
Nature of Services
• 1.
• 2.
– A.
• 3.
•
•
•
•
4.
5.
6.
7.
Service System Design Matrix
Degree of customer/server contact
High
(low)
None
Some
(Buffered System) (Permeable System)
Face -to-face
total
customization
Face -to-face
loose specs
Sales
Opportunity?
(Production
Efficiency?)
Low
(high)
Extensive
(Reactive System)
Face -to-face
tight specs
Mail contact
Internet &
on-site
technology
Phone
Contact
Designs for On-Site Service
• 1.
– Ex:
• 2.
– Ex.
• 3.
– Ex.
Disney World
• 1.
• 2.
• 3.
Implications of Waiting Lines
•
•
•
•
1.
2.
3.
4.
Elements of Waiting Lines
• 1.
• 2.
– A.
– B.
• 3.
• 4.
Customer Population
Characteristics
• 1.
– A.
• 2.
– A.
• 3.
– A.
• 4. Jockeying
– A.
Service System
• 1. The service system is defined by:
– A.
– B.
– C.
– D.
– E.
Number of Lines
• 1. Waiting lines systems can have single or
multiple queues.
– A.
– B.
Servers
• 1.
• 2.
– A.
– B. Example of a multi-phase, multi-server system:
1
Arrivals
C
C
C
2
4
C
C
5
3
6
Phase 1
Phase 2
Depart
Example Queuing Systems
Arrival & Service Patterns
• Arrival rate:
– 1. The average number of customers arriving per time
period
– 2. Modeled using the Poisson distribution
– 3. Arrival rate usually denoted by lambda ()
– 4. Example: =50 customers/hour; 1/=0.02 hours
between customer arrivals (1.2 minutes between
customers)
Arrival & Service Patterns
(Continued)
• Service rate:
– 1. The average number of customers that can be served during
the period of time
– 2. Service times are usually modeled using the exponential
distribution
– 3. Service rate usually denoted by mu (µ)
– 4. Example: µ=70 customers/hour; 1/µ=0.014 hours per
customer (0.857 minutes per customer).
• Even if the service rate is larger than the arrival rate,
waiting lines form!
– 1. Reason is the variation in specific customer arrival and
service times.
Waiting Line Priority Rules
•
•
•
•
•
•
•
1. First come, first served
2. Best customers first (reward loyalty)
3. Highest profit customers first
4. Quickest service requirements first
5. Largest service requirements first
6. Earliest reservation first
7. Emergencies first
Waiting Line Performance
Measures
• Lq = The average number of customers waiting in
queue
• L = The average number of customers in the system
• Wq = The average waiting time in queue
• W = The average time in the system
• r = The system utilization rate (% of time servers are
busy)
Single-Server Waiting Line
• Assumptions
– 1. Customers are patient (no balking, reneging, or jockeying)
– 2. Arrivals follow a Poisson distribution with a mean arrival
rate of . This means that the time between successive
customer arrivals follows an exponential distribution with an
average of 1/ 
– 3. The service rate is described by a Poisson distribution with a
mean service rate of µ. This means that the service time for
one customer follows an exponential distribution with an
average of 1/µ
– 4. The waiting line priority rule is first-come, first-served
– 5. Infinite population
Formulas: Single-Server Case
  lambda  mean arrival rate
  mu  mean service rate

r   average system utilization

Note:   for system stability. If this is not the case,
an infinitly long line will eventually form.
Formulas: Single-Server
Case con’t
L

 average number of customers in system
 
Lq  r L  average number of customers in line
1
W
 average time in system  including service 
 
Wq  rW  average time spent waiting
Pn  1  r  r n  probability of n customers in the system
at a given point in time
State Univ Computer Lab
• A help desk in the computer lab serves students on a
first-come, first served basis. On average, 15
students need help every hour. The help desk can
serve an average of 20 students per hour.
• Based on this description, we know:
– 1. µ = 20 students/hour (average service time is 3 minutes)
– 2.  = 15 students/hour (average time between student
arrivals is 4 minutes)
Average Utilization
 15
r 
 0.75 or 75%
 20
Average Number of Students
in the System, and in Line

15
L

 3 students
   20  15
Lq  r L  0.75 3  2.25 students
Average Time in the System & in Line
1
1
W

 0.2 hours
   20  15
or 12 minutes
Wq  rW  0.75  0.2   0.15 hours
or 9 minutes
Probability of n
Students in the Line
P0  1  r  r  1  0.75 1  0.25
0
P1  1  r  r  1  0.75  0.75  0.188
1
P2  1  r  r  1  0.75  0.75  0.141
2
2
P3  1  r  r 3  1  0.75  0.753  0.105
P4  1  r  r  1  0.75  0.75  0.079
4
4
Single Server: Probability of n
Students in the System
Probability of Number in System
0.3000
0.2000
0.1500
0.1000
0.0500
Number in System
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
0.0000
0
Probability
0.2500
Multiple Server Case
• Assumptions
– 1. Same as SingleServer, except here we
have multiple, parallel
servers
– 2. Single Line
– 3. When server finishes
with customer, first
person in line goes to the
idle server
– 4. All servers are
identical
Multiple Server Formulas
  lambda  mean arrival rate
  mu  mean service rate for one server
s  number of parallel, identical servers

r
 average system utilization
s
Note: s    for system stability. If this is not the case,
an infinitly long line will eventually form.
Multiple Server Formulas con’t
1
   /     /    1 
P0   


   probability of zero
s !  1  r  
 n 0 n !
customers in the system at a given point in time
s 1
n
s
   /  n
P0 for n  s

 n!
Pn  
 probability of n customers
n
 /  
 s ! s n  s P0 for n  s
in the system at a given point in time
Multiple Server Formulas
(Continued)
P0   /   r
s
Lq 
s !1  r 
2
 average number of customers in line
Wq  Lq   average time spent waiting in line
W  Wq 
1

 average time in system  including service 
L  W  average number of customers in system
Find Value for P0 from Chart Handout
Example: Multiple Server
• Computer Lab Help Desk
• Now 45 students/hour need help.
• 3 servers, each with service rate of 18
students/hour
• Based on this, we know:
– µ = 18 students/hour
– s = 3 servers
–  = 45 students/hour
Finding
P0
r = 45/(3*18) =
0.83
P0 ≈ 0.04
Probability of n Students in the
System
Probability of Number in System
0.1600
0.1400
0.1000
0.0800
0.0600
0.0400
0.0200
Number in System
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
0.0000
0
Probability
0.1200
Changing System Performance
• 1. Customer Arrival Rates
– Ex:
• 2. Number and type of service facilities
– Ex.
• 3. Change Number of Phases
– Ex.
Changing System Performance
• 4. Server efficiency
– Ex:
– Ex:
• 5. Change priority rules
– Ex:
• 6. Change the number of lines
– Ex:
– Ex: