Capacity Setting and Queuing Theory

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Capacity Setting and Queuing Theory

• BAMS 580B

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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)

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

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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?

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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

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A multiple server queuing system
Server
Buffer
Server
Server
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Several parallel singer server queues
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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?

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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

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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.

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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.

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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.

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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

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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
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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

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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

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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

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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

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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!

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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.

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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

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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

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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.

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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

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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.

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Idle Capacity And Wait Time Targets
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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.