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Capacity Setting and Queuing Theory

- BAMS 580B

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)

How Much Capacity is Needed? or How Many

Resources are Needed?

Surge capacity

Base capacity

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.

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?

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

A multiple server queuing system

Server

Buffer

Server

Server

Several parallel singer server queues

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?

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

Queuing Theory background

- Developed to analyze telephone systems in the

1930s 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.

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 doesnt fit well
- Often an exponential distribution fits well
- If we know its rate or mean we know everything

about it.

The exponential distribution

- P(T t) 1 e-?t
- The quantity ? is the rate.
- The mean and standard deviation of the

exponential distribution is 1/rate (1/?). - 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 1 - .487

.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 dont allow negative

times and have a small probability of long

service times.

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

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

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 c1)
- K buffer capacity (often K?)
- A single server queuing system is stable whenever

Rp gt Ri - A single server queuing system is balanced

whenever Rp Ri

Examples

- A Finite Capacity Loss System
- Model for an (old-fashion) phone system
- c servers
- K0
- 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

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

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 - Littles Law holds too
- number in queue arrival rate x waiting

time in queue

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!

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.

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

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

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.

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

Dont 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.

Idle Capacity And Wait Time Targets

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.