Operations Management
Queuing Models - Lecture 6
(Chapter 8)
Dr. Ursula G. Kraus
1/44
Review
•
National Cranberry
2/44
Agenda
The Effect of Variability
- Why do Queues form?
- Performance Measures for Queuing Systems
- Special Queuing Models
4/44
Goldratt's Production Game
Station 1
4
Station 2
Station 3
4
4
Station 5
4
Source: Eliyahu M. Goldratt, 1992, “The Goal”, North River Press, 104-112.
Station 4
4
5/44
Critical Assumptions
from National Cranberry
 Truck arrivals are constant, evenly spaced over the
11 (12) hour period
 The mix between wet and dry berries is constant at
70/30
 No variation in processing time at the individual
processing stages
6/44
Post Office: Why Do Queues form?
•
Average inter-arrival time is 4 minutes
•
Average processing time is 3 minutes
We are processing customers
faster than they arrive so what's the problem?
Source: Managing Business Process Flows (1999)
7/44
Post Office: Customer processing
Customer #
Average inter-arrival
time is 4 minutes
Average processing
time is 3 minutes
Time
Time
Source: Prof. K. Gue, Winter 03
8/44
Post Office: With constant processing time
Customer #
Average inter-arrival
time is 4 minutes
Average processing
time is 3 minutes
Time
Time
Source: Prof. K. Gue, Winter 03
9/44
Post Office: … and constant interarrival time
Customer #
Average inter-arrival
time is 4 minutes
Average processing
time is 3 minutes
Time
Time
Source: Prof. K. Gue, Winter 03
10/44
Reasons Why Queues form
Call #
 Variability:
– interarrival times
– processing times
– server availability
10
9
8
7
6
5
4
3
2
1
0
0
20
40
60
80
100
TIME
 (System) Utilization:
 = throughput/capacity
Inventory (# of calls in system)
5
4
3
2
1
0
0
Source: Managing Business Process Flows (1999)
20
40
60
TIME
80
100
11/44
Telemarketing at L.L. Bean
 During some half hours, 80% of calls dialed
received a busy signal.
 Customers getting through had to wait on average 10 minutes
for an available agent. Extra telephone expense per day for
waiting was $25,000.
 For calls abandoned because of long delays, L.L.Bean still paid
for the queue time connect charges.
 L.L.Bean conservatively estimated that it lost $10 million of
profit because of sub-optimal allocation of telemarketing
resources.
Source: Managing Business Process Flows (1999). Data based on 1988 sales.
12/44
Agenda
The Effect of Variability
- Why do Queues form?
- Performance Measures for Queuing Systems
- Special Queuing Models
13/44
Notation: Variability
m = Mean
V(X) = Variance
s = Standard Deviation

V ( X )  E ( X  m )2

s  V (X )
where X = Random Variable
Source: Managing Business Process Flows (1999)
14/44
How can we measure variability?
Should be a relative measure!
Example:
standard deviation s = 10 minutes
low variability if mean (m) = 4 hours
high variability if mean (m) = 5 minutes
Coefficient of Variation:
s
C
m
Ratio of the standard deviation to the
mean
Source: Managing Business Process Flows (1999)
15/44
System Parameters
 Inputs of System
– (Average) Interarrival times: Ti
(Average) Interarrival rates: Ri (=1/Ti )
– (Average) Service times:
Tp
(Average) Service rates:
Rp (=1/Tp )
 System structure
– Number of servers: c
– Number of queues
– Maximum queue length (buffer capacity K)
 Operating control policies
– Queue discipline (FIFO), priorities
Source: Managing Business Process Flows (1999)
16/44
Process Model for Queuing Systems:
Call Center
Incoming
calls
Ri
Waiting Queue
“buffer” size K
R
Queue
Rb
Blocked Calls
Server
Rp
Answered
Calls
Ra
Abandoned Calls
Source: Managing Business Process Flows (1999)
17/44
Queuing System: Car Wash
18/44
Notation: Throughput Rate and Capacity
 (Average) Arrival rate: Ri = 1/Ti,
where Ti is the average interarrival time
 (Average) Throughput rate: R = Ri - Rb - Ra
= net arrival rate
 (Theoretical total) Processing capacity: Rp = c/Tp
for c servers where Tp is the average processing time
 System Utilization:  = R / Rp < 100%
average fraction of time server is busy
Source: Managing Business Process Flows (1999)
19/44
Notation: Inventory/Customers in
System
 Average number of customers in the waiting line: Iq
 Average number of customers in process: Ip = c
 Average number of customers in the system
(waiting and being served): I = Iq+ Ip
Incoming
calls
I
Order Queue
“buffer” size K
Ri
R
Queue
Rb
Blocked Calls
Iq
Server
Answered
Calls
Rp
Ip
Ra
Abandoned Calls
Source: Managing Business Process Flows (1999)
20/44
Queuing System: Car Wash
Iq
Ip
I
21/44
Little’s Law (I = R x T)
T=I/R
I = Iq + Ip
waiting;
in queue
Source: Managing Business Process Flows (1999)
in process (c)
being served
22/44
Queue Length Formula - Approximation:
2(c+1)
+
Ci Cp

´
Iq
1
2

utilization
effect
2
2
variability
effect
where Ci is the coefficient of variation for the interarrival times
Cp is the coefficient of variation for the service times
Source: Managing Business Process Flows (1999)
23/44
Agenda
The Effect of Variability
- Why do Queues form?
- Performance Measures for Queuing Systems
- Special Queuing Models
- M/M/1 Queuing Model
24/44
Conditions: Poisson Distribution
Patterns that occur most frequently in queuing models are
Poisson and Exponential Distributions.
The number of arrivals during a time period T yield a
Poisson Distribution if the following conditions apply:
 Certain events are occurring at random over a continuous period of
time (or interval of distance, region of area, etc).
 These events occur singly (one at a time), i.e. it is not possible for two
events to occur exactly simultaneously.
 The events also occur independently of each other, i.e. the fact that an
event has occurred (or not occurred) does not affect the chance of
another event occurring.
25/44
Examples: Poisson Distributions (Processes)
 The number of web page requests arriving at a server
 The number of telephone calls arriving at a switchboard, or at an
automatic phone-switching system
 The number of photons hitting a photodetector, when lit by a laser
source
 The number of raindrops falling over a wide area
 The arrival of customers in a queueing system
26/44
Poisson Distribution of Arrivals and Services
If arrivals at a service facility occur on a purely random fashion (one at a
time), independently of each other then
 The number of arrivals during a time period T yield a Poisson
Distribution
 The interarrival times yields an Exponential Distribution
 The Exponential and Poisson distributions can be derived from one
another.
 The mean and standard deviation of the Exponential Distribution are
equal:
s m
27/44
M/M/1 Queue (Exponential Model)
The M/M/1 is a single-server queue model, that can be
used to approximate a lot of simple systems.
It indicates a system where:
1. Arrivals are a Poisson Process
with independently and exponentially distributed
interarrival times
2. Service is a Poisson Process where service time is
independently and exponentially distributed
3. There is one server
Note: “M” stands for Markovian, a reference to the memoryless property of the exponential distribution
30/44
Queue Length for an M/M/1 Queue
Iq
 2(c +1)
1 
C2+ C2
p
 i
2
•Independent, exponental Ti and Tp
•Single server
Iq
Source: Managing Business Process Flows (1999)
2


1 
32/44
M/M/1 Queue
 Average number of customers in the waiting line:
2
Iq
1 
(exact result)
 Average number of customers in the system
(waiting and being served)
I  Iq + Ip 

2
1 
+ 

1 
 Little’s Law: Average total time in the system (Flow Time)
T I/R
Source: Managing Business Process Flows (1999)
Tq  Iq / R
Tp  Ip / R
33/44
Utilization and Variability
Drive Congestion
Average
Flow
Time T
T I/R
Variability
Increases
Tp
Utilization (ρ)
Source: Managing Business Process Flows (1999)
100%

34/44
Example: Calling Center
Consider a mail order company that has one customer service representative
(CSR) taking calls. When the CSR is busy, the caller is put on hold. The calls
are taken in the order received.
Each minute a caller spends on hold costs the company $2 in telephone
charges, customer dissatisfaction and loss of future business. In addition, the
CSR is paid $20 an hour.
Assume that calls arrive exponentially at the rate of one every 3 minutes. The
CSR takes on average 2.5 minutes to complete the order. The time for service
is also assumed to be exponentially distributed.
(a) What is the capacity utilization?
(b) How many customers will be on average on hold?
(c) What is the average time spent on hold?
(d) Estimate the average hourly cost of operating the call center.
Source: Managing Business Process Flows (1999)
35/44
Expected costs
Cost of Service Tradeoff
Total cost
(Service)
Capacity Cost
(Customer)
Waiting Cost
Level of service
37/44
Example: Ticket Outlet
Customers arrive at a suburban ticket outlet at a rate of 14
per hour on Monday mornings. Selling the tickets and
providing general information takes an average of 3
minutes per customer. There is one ticket agent on duty on
Mondays. Assume exponential interarrival and service
times.
a)
b)
c)
d)
What percentage of time is the ticket agent busy?
How many customers are in line, on average?
How many minutes, on average, will a customer spend in
the system?
What is the average waiting time in queue?
Source: Managing Business Process Flows (1999)
38/44
Performance Measures to Focus on
 Sales
– Throughput (R)
– Abandoning rate (Ra )
 Cost
– Capacity utilization ()
– Queue length (Iq) ; Total number in process (I)
 Customer Service
– Waiting time in queue (Tq ); Total time in process (T)
– Blocking rate (Rb )
Source: Managing Business Process Flows (1999)
41/44
Levers for System Improvements
 Increase process capacity Rp
– adding more servers
– working faster (decreasing Tp)
 Reduce waiting time Tq and queue length Iq
– reduce variability
– decrease arrival rate (not desirable in general)
Source: Managing Business Process Flows (1999)
42/44
Example: Calling Center
with 2 CSRs (2 servers)
2 phone numbers
I.
–
Company hires a second CSR who is assigned
a new telephone number. Customers are now
free to call either of the two numbers. Once
they are put on hold customers tend to stay on
line since the other may be worse
50%
Queue Server
50%
Queue Server
1 phone number: pooling
II.
–
Both CSRs share the same telephone number
and the customers on hold are in a single queue
Queue
Servers
Source: Managing Business Process Flows (1999)
43/44