1. Queuing Models and Capacity Planning
Chapter 14.
Queuing Models and Capacity Planning
Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan

2. Outline Queuing System Characteristics Specific Queue 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)
Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan

3. Figure 14.1 Queue Phenomenon
Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan

4. A mathematical approach to the analysis of waiting lines
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
Minimizing costs
Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan

5. Why Must We Wait In Lines?
RANDOMNESS
ARRIVALS
SERVICE TIMES
Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan

6. TO REDUCE COSTS OUR GOAL? Waiting Costs Capacity Costs
Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan

7. 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
Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan

8. Figure 14.2 Healthcare Service Capacity and Costs
Total cost
Cost of service capacity
Cost
Waiting line cost
Optimum capacity
Health care service capacity (servers)
Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan

9. 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)
Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan

10. The Population Source Can you think of examples for each 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?
Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan

11. Facility Figure 14.3 Queuing Conceptualization of Flu Inoculations
Population
(Patient Source)
Facility
Queue
(Waiting Line)
Service
Arrivals
Exit
Server
Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan

12. 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
Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan

13. Figure 14.4 Conceptualization of a Single-line, Multi-phase System
Queue-1
Queue-2
Queue-3
Population
Receptionist
(server)
Nurse
(server)
Physician
(server)
Arrivals
Exit
Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan

14. Providing Inoculations
Figure 14.5 Multi-Line Queuing Systems
Nurses
Providing Inoculations
Line-1
Queue
Patients
Line-2
Line-3
Emergency Room
Phase-2
Phase-3
Phase-1
Diagnostic
Testing
Physicians-
Interventions
Physicians-
Initial Evaluation
Queue-1
Queue-2
Queue-3
Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan

15. 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
Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan

16. Figure 14.6 Emergency Room Arrival Patterns
Monday
Tuesday
Wednesday
Thursday
Friday
Saturday
Sunday
7:00am
7:00pm
3:00pm
Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan
Time 

17. Figure 14.7 Measures of Arrival Patterns
Inter-arrival time
9:00pm
10:00pm
Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan

18. Figure 14.8 Poisson Distribution
.20
.15
.10
Probability
.05
Patient arrivals per hour
Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan

19. Patients 90 60 Time (minutes) 30 A B C D E F
Figure 14.9 Service Time for ER Patients 90 60
Time (minutes) 30 A B C D E F
Patients
Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan

20. 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.
Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan

21. 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.
Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan

22. 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
Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan

23. 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
Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan

24. 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.
Avg. Number or time
waiting in line
System Utilization
100%
Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan

25. 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
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
Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan

26. 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.
Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan

27. Infinite source models
Single channel, exponential service time
Single channel, constant service time
Multiple channel, exponential service time
Multiple priority service
Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan

28. Exhibit 14.2 Queuing Model Notation
 arrival rate
 service rate
Lq average number of customers waiting for service
L average number of customers in the system
(waiting or being served)
Wq average time customers wait in line
W average time customers spend in the system
 system utilization
1/ service time
Po probability of zero units in system
Pn probability of n units in system
Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan

29. 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.
2.The average number of patients in the system is the average number in line plus the average number being served.
3. The average time in line is the average number in line divided by the arrival rate.
4. The average time in the system is the sum of the time in line plus the service time.
5. System utilization is the ratio of arrival rate to service capacity.
Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan

30. Single Channel, Poisson Arrival and Exponential Service Time (M/M/1).
Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan

31. 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?
Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan

32. λ = 1(hour) ÷ 15 = 60(minutes) ÷ 15 = 4 persons per hour.
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
Persons waiting in the queue
Number of persons in the system
Minutes of waiting time in the queue
Minutes in the system
(waiting and service)
Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan

33. Calculate queue lengths of 0, 1, and 2 persons
Solution:
Calculate queue lengths of 0, 1, and 2 persons
Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan

34. Utilization of servers
Solution:
Utilization of servers
Current system utilization (s=1)
System utilization with an additional nurse (s=2)
System utilization decreases as we add more resources to it.
Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan

35. 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?
Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan

36. 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.
Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan

37. Figure 14.12 System Performance for Expanded Diabetes Information Booth
Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan

38. Figure 14.13 System Performance Summary for Expanded Diabetes Information Booth with M/M/3
Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan

39. Figure 14.14 Capacity Analysis
Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan

40. 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
Performance Measure
2 Stations
3 Stations
4 Stations
Patient arrival rate 9
Service rate 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%
Average number of patients balked
Total system cost in $ per hour
790.53
294.91
302.89
Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan

41. 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
Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan

42. The End Chapter 14: Quantitatve Methods in Health Care Management
Yasar A. Ozcan