Service Systems & Queuing Chapter 12S OPS 370. Nature of Services 1. 2. –A. 3. 4. 5. 6. 7.
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Transcript of Service Systems & Queuing Chapter 12S OPS 370. Nature of Services 1. 2. –A. 3. 4. 5. 6. 7.
Service Systems &
Queuing
Chapter 12S
OPS 370
Nature of Services
• 1. • 2.
– A.
• 3.
• 4. • 5. • 6. • 7.
Service System Design Matrix
Mail contact
Face -to-faceloose specs
Face -to-facetight specs
PhoneContact
Face -to-facetotal
customization
(Buffered System) None
(Permeable System) Some
(Reactive System)Extensive
(high)
(low)High
Low
Degree of customer/server contact
Internet & on-site
technology
SalesOpportunity?
(ProductionEfficiency?)
Designs for On-Site Service
• 1.
– Ex:
• 2.
– Ex.
• 3.
– Ex.
Disney World
• 1.
• 2.
• 3.
• 1. • 2. • 3. • 4.
Implications of Waiting Lines
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:
C C C CC DepartArrivals
1
2
3 6
5
4
Phase 1 Phase 2
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
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
average number of customers in system
average number of customers in line
1average time in system including service
average time spent waiting
1 probability of customers in the system
at a
q
q
nn
L
L L
W
W W
P n
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
150.75 75%
20or
Average Number of Studentsin the System, and in Line
studentsL 31520
15
0.75 3 2.25qL L students
Average Time in the System & in Line
minutes12
hours2.01520
11
or
W
0.75 0.2 0.15 hours
9 minutes
qW W
or
Probability of nStudents in the Line
00
11
2 22
3 33
4 44
1 1 0.75 1 0.25
1 1 0.75 0.75 0.188
1 1 0.75 0.75 0.141
1 1 0.75 0.75 0.105
1 1 0.75 0.75 0.079
P
P
P
P
P
Single Server: Probability of n Students in the System
Probability of Number in System
0.0000
0.0500
0.1000
0.1500
0.2000
0.2500
0.30000 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
Number in System
Pro
bab
ilit
y
Multiple Server Case
• Assumptions– 1. Same as Single-Server,
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 server
number of parallel, identical servers
average system utilization
Note: for system stability. If this is not the case,
an infinitly l
one
s
s
s
ong line will eventually form.
Multiple Server Formulas con’t
11
00
0
0
/ / 1 probability of zero
! ! 1
customers in the system at a given point in time
/ for
! probability of customers/
for !
in the s
n ss
n
n
n n
n s
Pn s
P n snP n
P n ss s
ystem at a given point in time
Multiple Server Formulas (Continued)
02
/ average number of customers in line
! 1
average time spent waiting in line
1average time in system including service
average number of customers in system
s
q
q q
q
PL
s
W L
W W
L W
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
0.0000
0.0200
0.0400
0.0600
0.0800
0.1000
0.1200
0.1400
0.16000 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
Number in System
Pro
bab
ilit
y
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: