Chapter 14.
Queuing Models and Capacity Planning
Chapter 14: Quantitatve
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Outline

Queuing System Characteristics
–
–
–
–




Population Source
Servers
Arrival Patterns
Service Patterns
Specific Queue Characteristics
Measures of Queuing System Performance
Infinite Source-Models
Model Formulations
– Single Server (M/M/1)
– Multi Server (M/M/s>1)
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Figure 14.1 Queue Phenomenon
Location: Hospital Outpatient Pharmacy
Day & Time: Monday, 11:00 AM
Average number in line: 20
Idle servers: 0
Average wait time: 15 minutes
Chapter 14: Quantitatve
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Location: Hospital Outpatient Pharmacy
Day & Time: Monday, 3:30 PM
Average number in line: 0
Idle servers: 3
Average wait time: 0 minutes
Hourly wage per idle pharmacist: $40
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Queuing Models
A
mathematical approach to the
analysis of waiting lines
 Weighs the cost of providing a
given level of service capacity (i.e.,
shortening wait times) against the
potential costs of having
customers wait
Reduce Wait Times
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Minimizing costs
4
Why Must We Wait In Lines?
RANDOMNESS
ARRIVALS
SERVICE TIMES
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OUR GOAL?

TO REDUCE COSTS
– Waiting Costs
– Capacity Costs
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To Summarize:
 Waiting
Costs
– Salaries paid to employees while they
wait for service (e.g., physicians waiting
for an x-ray or test result)
– Cost of waiting space
– Loss of business due to wait (balking
customers)
 Capacity
Costs-- the cost of
maintaining the ability to provide service
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Figure 14.2 Healthcare Service Capacity and Costs
Cost
Total cost
Cost of service capacity
Waiting line cost
Optimum capacity
Health care service capacity (servers)
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Before getting to the
complicated part. . .
Let’s Examine the Main Characteristics of
any queuing model:
The population Source
 The number of servers (channels)
 Arrival and service patterns
 Queue discipline (order of service)

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The Population Source


Infinite
– Customer arrivals are unrestricted
– Customer arrivals exceed system capacity
– Exists where service is unrestricted
Finite source
– Customer population where potential
number of customers is limited
Can you think of examples for each source?
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Figure 14.3 Queuing Conceptualization of Flu Inoculations
Population
(Patient Source)
Facility
Queue
(Waiting Line)
Arrivals
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Server
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The Number of Servers



Capacity is a function of the capacity of each
server and the number of servers being used
Servers are also called channels
Systems can be single or multiple channel,
and consist of phases
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Figure 14.4 Conceptualization of a Single-line, Multi-phase System
Phase -1
Phase -2
Phase -3
Population
Queue-1
Receptionist
(server)
Queue-2
Nurse
(server)
Arrivals
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Queue-3
Physician
(server)
Exit
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Figure 14.5 Multi-Line Queuing Systems
Nurses
Providing Inoculations
Line-1
Queue
Patients
Line-2
Line-3
Emergency Room
Phase-2
Phase-1
Diagnostic
Testing
PhysiciansInitial Evaluation
Queue-1
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Phase-3
Queue-2
PhysiciansInterventions
Queue-3
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Arrival and Service Patterns
 Remember--
both arrival and
service patterns are random, thus
causing waiting lines
 Models assume that:
– Customer arrival rate can be
described Poisson distribution
– Service time can be described by a
negative exponential distribution
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Figure 14.6 Emergency Room Arrival Patterns
Monday
Tuesday
Wednesday
Thursday
Friday
Saturday
Sunday
7:00am
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3:00pm
Time 
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Figure 14.7 Measures of Arrival Patterns
Inter-arrival time
9:00pm
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Figure 14.8 Poisson Distribution
.20
.15
Probability
.10
.05
0
1
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3
4
5
6
7
8
9
10
Patient arrivals per hour
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Figure 14.9 Service Time for ER Patients
Time (minutes)
90
60
30
0
A
B
C
D
E
F
Patients
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Exponential or Poisson?
If service time is exponential, then the service rate is
Poisson. Further, if the customer arrival rate is Poisson,
then the inter-arrival time (i.e., the time between
arrivals) is exponential.
For instance: If a lab processes 10 customers per hour
(rate), the average service time is 6 minutes. If the
arrival rate is 12 per hour, then the average time
between arrivals is 5 minutes.
Thus, service and arrival rates are described by the
Poisson distribution and inter-arrival times and service
times are described by a negative exponential
distribution.
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Queue Characteristics
Patients who arrive and see big lines (the flu shot
example) may change their minds and not join
the queue, but go elsewhere to obtain service;
this is called balking. If they do join the queue
and are dissatisfied with the waiting time, they
may leave the queue; this is called reneging.
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Queue Discipline
Refers to the order in which patients are
processed, for instance:
 First Come First Served
 Most Serious First Served
 Highest Costs (waiting) First Served
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How do we judge ours systems performance?
Average number of customers waiting (in line or in system)
Average time customers wait (in line or in system)
System utilization (% of capacity utilized)
Implied costs of a given level of capacity and its related waiting
line
Probability that an arrival will have to wait for service
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How do we judge ours systems performance?
Realize that increases in system utilization are achieved at
the expense of increases in both the length of the waiting
line and the average waiting time.
Most models assume that the system is in a steady state
where average arrival and service rates are stable.
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System Utilization
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Exhibit 14.1 Queuing Model Classification
A: specification of arrival process, measured by inter-arrival time or arrival rate.
M: negative exponential or Poisson distribution
D: constant value
K: Erlang distribution
G: a general distribution with known mean and variance
B: specification of service process, measured by inter-service time or service rate
M: negative exponential or Poisson distribution
D: constant value
K: Erlang distribution
G: a general distribution with known mean and variance
C: specification of number of servers -- “s”
D: specification of queue or the maximum numbers allowed in a queuing system
E: specification of customer population
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Typical Infinite-Source Models
Single channel, M/M/s
Multiple channel, M/M/s>1,
where “s” designates the number of channels
(servers).
These models assume steady state conditions and a
Poisson arrival rate.
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Infinite source models
Single channel, exponential service time
Single channel, constant service time
Multiple channel, exponential service time
Multiple priority service
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Exhibit 14.2 Queuing Model Notation


Lq
L
Wq
W

1/
Po
Pn
arrival rate
service rate
average number of customers waiting for service
average number of customers in the system
(waiting or being served)
average time customers wait in line
average time customers spend in the system
system utilization
service time
probability of zero units in system
probability of n units in system
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Infinite Source Models: Model Formulations
Five key relationships that provide basis for queuing formulations and are
common for all infinite-source models:
1.The average number of patients being served is the ratio of arrival to

service rate.
r

2.The average number of patients in the system is the average number
in line plus the average number being served.
L  Lq  r
3. The average time in line is the average number in line divided by the
arrival rate.
L
Wq 
q

4. The average time in the system is the sum of the time in line plus the
service time.
W  Wq 
1

5. System utilization is the ratio of arrival rate to service capacity.
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

s
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Single Channel, Poisson Arrival and Exponential
Service Time (M/M/1).
2
Lq 
 (   )

P0  1 


Pn  P0  

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Single Channel, Poisson Arrival and Exponential
Service Time (M/M/1).
Example 14.1
A hospital is exploring the level of staffing needed for a
booth in the local mall where they would test and provide
information on diabetes. Previous experience has shown that
on average, every 15 minutes a new person approaches the
booth.
A nurse can complete testing and answering questions, on
average, in 12 minutes. If there is a single nurse at the booth,
calculate system performance measures including the
probability of idle time and of 1 or 2 persons waiting in the
queue. What happens to the utilization rate if another
workstation and nurse are added to the unit?
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Solution:
Arrival rate:
λ = 1(hour) ÷ 15 = 60(minutes) ÷ 15 = 4 persons per hour.
Service rate:
μ = 1(hour) ÷ 12 = 60(minutes) ÷ 12 = 5 persons per hour.
Average persons served at any given time
r
42
Lq 
 3.2
5(5  4)
Persons waiting in the queue
Number of persons in the system
L  Lq 

 3.2  .8  4 persons.

Minutes of waiting time in the queue
Wq 
Minutes in the system
(waiting and service)
1
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 4
  .8
 5
W  Wq 

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Lq


 6.4 
3.2
 0.1067  6.4
4
60
 6.4  12  18.4
5
32
Solution:
Calculate queue lengths of 0, 1, and 2 persons

4
P0  1   1   1  .8  .2

5
1

4
P1  P0    (.2)   (.2)(.8)1  (.2)(.8)  .16
5

2
1

4
P2  P0    (.2)   (.2)(.8) 2  (.2)(.64)  .128
5

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Solution:
Utilization of servers
Current system utilization (s=1)


4

 80%.
s 1 * 5
System utilization with an additional nurse (s=2)

4


 40%.
s 2 * 5
System utilization decreases as we add more resources to it.
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Multi-Channel, Poisson Arrival and
Exponential Service Time (M/M/s>1)
Expanding on Example 14.1: The hospital found that among
the elderly, this free service had gained popularity, and now,
during week-day afternoons, arrivals occur on average every
6 minutes 40 seconds (or 6.67 minutes), making the effective
arrival rate 9 per hour.
To accommodate the demand, the booth is staffed with two
nurses working during weekday afternoons at the same
average service rate. What are the system performance
measures for this situation?
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Multi-Channel, Poisson Arrival and
Exponential Service Time (M/M/s>1)
Solution:
This is an M/M/2 queuing problem. The WinQSB solution
provided in Exhibit 14.5 shows the 90% utilization. It is
noteworthy that now each person has to wait on average one
hour before they can be served by any of the nurses.
On the basis of these results, the healthcare manager may
consider expanding the booth further during those hours.
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Figure 14.12 System Performance for Expanded Diabetes Information Booth
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Figure 14.13 System Performance Summary for Expanded Diabetes Information Booth with M/M/3
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Figure 14.14 Capacity Analysis
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Multi-Channel, Poisson Arrival and Exponential
Service Time (M/M/s>1)
Table 14.1 Summary Analysis for M/M/s Queue for Diabetes Information Booth
2 Stations
3 Stations
4 Stations
Patient arrival rate
9
9
9
Service rate
5
5
5
Overall system utilization
90%
60%
45%
L (system)
9.5
2.3
1.9
Lq
7.7
0.5
0.1
W (system) – in hours
1.05
.26
0.21
Wq – in hours
0.85
.06
0.01
Po (idle)
5.3%
14.6%
16.2%
Pw (busy)
85.3%
35.5%
12.8%
0
0
0
790.53
294.91
302.89
Performance Measure
Average number of patients
balked
Total system cost in $ per hour
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Assumptions and Limitations
Steady state of arrival and service process
Independence of arrival and service is assumed
The number of servers is assumed fixed
Little ability to incorporate important exceptions
to the general flow
Difficult to model all but simple processes
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The End
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