1. Capacity Setting and Queuing Theory
BAMS 580B

2. Capacity and Resources
A key lever for improving patient flow.
How do we measure capacity?
What is the capacity of a 20 seat restaurant?
A 16 bed ward?
Capacity is a RATE
Patients/day
Customers/hour
We can view a 16 bed ward as a queuing system with 16 servers
What is the capacity of a bed?
Does this analogy apply to the restaurant?
A system is composed of resources with capacities.
Often we use the expressions “resource” and “capacity” interchangeably (hopefully without confusion)

3. How Much Capacity is Needed? or How Many Resources are Needed?
Surge capacity
Base capacity

4. Capacity tradeoffs when demand is variable
Too much capacity or too many resources = idleness
Not enough capacity – waits
Should we set capacity equal to demand?
What does this mean?
This is called a balanced system
It works perfectly when there is no variation in the system
It works terribly when there is variation! Why?
Once behind, you never can catch up.
Queuing theory quantifies these tradeoffs in terms of performance measures.

5. Queuing Models
(Mathematical) queuing models help us set capacity (or determine the number of resources needed) to meet:
Service level targets
Average wait time targets
Average queue length targets
Queuing models provide an alternative to simulation
They provide insights into how to plan, operate and manage a system
Where are there queues in the health care system?

6. A single server queuing system
Buffer
Server
A queue forms in a buffer
Servers may be people or physical space
The buffer may have a finite or unlimited capacity
The most basic models assume “customers” are of one type
and have common arrival and service rates

7. A multiple server queuing system
Buffer
Server
Server

8. Several parallel singer server queues
Buffer
Server
Buffer
Server
Buffer
Server

9. Parallel Queues vs. Multiple server Queues
Provide examples of multiple server queues (MSQs)
Provided examples of parallel queues (PQs)
In what situations would each of these queuing systems be most appropriate? Why?

10. Networks of queues
Most health care systems are interconnected networks of queues and servers with multiple waiting points and heterogeneous customers.
What examples have we seen in the course?
Often we model these complex systems with simulation.
But in some cases we can use formulae to get results

11. Queuing Theory background
Developed to analyze telephone systems in the 1930’s by Erlang.
How many lines are needed to ensure a caller tries to dial and obtains a “line”.
Applied to analyze internet traffic, telecommunications systems, call centers, airport security lines, banks and restaurants, rail networks, etc.

12. Queues and Variability
There are two components of a queuing system subject to variability
The inter-arrival times of “jobs”
The service times or LOS
Why are these variable?
We describe the variability by
Mean
Standard deviation
Probability distribution
Usually the normal distribution doesn’t fit well
Often an exponential distribution fits well
If we know its rate or mean we know everything about it.

13. The exponential distribution
P(T ≤ t) = 1 – e-λt
The quantity λ is the rate.
The mean and standard deviation of the exponential distribution is 1/rate (1/λ).
Example; Patients arrive at rate 4 per hour.
The mean interarrival time is 15 minutes.
What is the probability the time between two arrivals is less than 10 minutes (1/6 of an hour)
P( T ≤ 1/6) = 1 – e-4∙(1/6) = 1- e-2/3 = = .513.
The exponential distribution underlies queuing theory.
A queue with exponential service times and exponential inter-arrival times and one (FCFS) server is called an M/M/1 queue.
Exponential distributions don’t allow negative times and have a small probability of long service times.

14. Capacity management and queuing systems
Capacity management involves determining the number of servers to use and the size of the waiting rooms.
Examples
How many long term care beds are needed?
How many porters are needed?
How many nurses are needed?
How many cubicles are needed in an ED?
Some healthcare systems have no buffers; all the waiting is done outside of the system or upstream.
ALC cases waiting for LTC beds

15. Analyzing a queuing system
Outputs
Capacity Utilization
Wait Time in Queue
Queue Length
Blocking Probability
Service Levels
Inputs
Arrival Rate
Service Rate
Number of Servers
Buffer Size
Queue
Analyzer
QUEUMMCK_EMBA.xls

16. Single server queues – some definitions
Ri – average inflow rate (customers/time) ()
1/Ri – average time between customer arrivals
Tp – average processing time by one server
1/Tp – average processing rate of a single server ()
c – number of servers
Rp = c/Tp – system service rate (often c=1)
K – buffer capacity (often K=)
A single server queuing system is stable whenever Rp > Ri
A single server queuing system is balanced whenever Rp = Ri

17. Examples A Finite Capacity Loss System
Model for an (old-fashion) phone system
c servers
K=0
When all servers are busy, system is blocked and customers are lost
Performance measure – fraction of lost jobs – this is legislated!
Walk-in Clinic with 6 seats and 1 doctor
c = 1
K = 6

18. Characteristics and Performance Measures
System characteristics
Traffic Intensity (or utilization) =  = arrival rate/service rate
Safety Capacity = Rs = Service rate – arrival rate
Performance Measures
Average waiting time (in queue) – Ti
Average time spent at the server - Tp
Average flow time (in process) – T = Ti + Tp
Average queue length – Ii
Average number of customers being served - Ip
Average number of customers in the system – I =Ii + Ip

19. Performance measure formulas (M/M/1 queue – no limit on queue size)
System Utilization = P(Server is occupied) = 
If traffic intensity increases, the likelihood the server is occupied increases
This occurs if the arrival rate increases or the service rate decreases
P(System is empty) = 1- 
P(k in system) = k(1- )
Average Time in System = 1/ Safety capacity
Average Time in Queue = Average time in system – average service time
If safety capacity decreases; time in queue increases!
Average Number of jobs in the system (including being served) = /(1- )
Average Queue Length = 2/(1- )
If we know safety capacity, service time and traffic intensity, we can compute all system properties
Little’s Law holds too
number in queue = arrival rate x waiting time in queue

20. An Example - M/M/1 Queue
Customers arrive at rate 4 per hour, mean service time is 10 minutes.
Service rate is 6 per hour
System utilization = Probability the server is occupied =  = 2/3.
Safety capacity = service rate – arrival rate = 2
P(System is empty) = 1-  = 1/3.
P(k in the system) = k(1- ) = (1/3)(2/3)k
Average Time in system= 1/safety capacity = ½ hour
Average Time in queue = Average time in system – average service time = ½ - 1/6 = 1/3 hour
Average Queue Length = 2/(1- ) = 4/3
Suppose arrival rate increases to 5.9 customers per hour.
Then  =5.9/6 = .9833
So P(System is empty) = .0167; Average time in system = 10 hours and Average number of customers in the system = 58.9!

21. About QUEUMMCK.xls
An M/M/c queue is the same as an M/M/1 queue except that there may be more than one server.
In this model, there is a single buffer and c servers in the resource pool.
Customers are processed on a FIFO basis.
When there are more than c customers in the system, the buffer is occupied and waiting for service occurs.
An M/M/c/K queue is an M/M/c queue with a finite buffer of size K.
There are at most K + c customers in the system.
When the buffer is filled, the system is blocked and customers are lost.
QUEUMMCK.xls, which is now called performance.xls, computes performance measures including blocking probabilities for the M/M/c/K queue.

22. Problem 1
Patients arrive at rate 5/hr. They require on average 1 hour of treatment.
How many service providers do we need to ensure that the average wait time is 30 minutes?
Assume a large waiting room.
Running QUEUEMMCK.xls we find that with
6 service providers - average wait is 1 hour and average number waiting is 2.94
7 service providers - average wait is ½ hour and average number waiting is .80
Note that with 7 service providers all 7 are occupied less than 1% of the time.
Thus we tradeoff throughput with capacity utilization

23. Problem 2 – A LTC Facility
Bed requests arrive at the rate of 3 per month
Patients remain in beds for about 15 months.
How many beds are required so that the average wait for beds is 1 month.
Trial and error with queummck shows that 59 beds are required.
Also we can see that there is only a 3% chance of waiting and average occupancy is 45 beds.
We can also do sensitivity analysis with arrival rates and length of stays

24. Problem 3 A walk in clinic has 3 doctors;
Average time spent with a patient is 15 minutes
Patients arrive at rate of 12 per hour
How many chairs should we have in the waiting room so only 5% of patients are turned away?
Queummck suggests 17.

25. Implications of queuing formulas
As the safety capacity vanishes, or equivalently, the traffic intensity increases to 1:
waiting time increases without bound!
queue lengths become arbitrarily long!
In the presence of variability in inter-arrival times and service times, a balanced system will be highly unstable.
These formulas enable the manager to derive performance measures on the basis of a few basic descriptors of the queuing system
The arrival rate
The service rate
The number of servers
When the system has a finite buffer, the percentage of jobs that are blocked can also be computed

26. Don’t Match Capacity with Demand
If service rate is close to arrival rate then there will be long wait times.
Recall average queue length = 2/(1- )
If traffic intensity near 1, queue length will be very small.
No variability – All procedures take exactly the same time, patients are scheduled to appear at the completion of the proceeding procedure and arrive at that time.
Safety capacity – Service Rate – Arrival Rate

27. Idle Capacity And Wait Time Targets
Theoretical –based on queueing formula.

28. Summary
When the manager knows the arrival rate and service rate, he/she can compute:
The average number of jobs in the queue.
The average time spent in the queue.
The probability an arriving patient has to wait.
The system utilization.
This can be done without simulation!
This information can be used to set capacity or explore the sensitivity of recommendations to assumptions or changes.
Thus queuing theory provides a powerful tool to manage capacity.