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Capacity Setting and Queuing Theory


Capacity Setting and Queuing Theory BAMS 580B * No variability All procedures take exactly the same time, patients are scheduled to appear at the completion of ... – PowerPoint PPT presentation

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Title: Capacity Setting and Queuing Theory

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
  • Often we use the expressions resource and
    capacity interchangeably (hopefully without

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
  • 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
  • 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
  • A queue forms in a buffer
  • Servers may be people or physical space
  • The buffer may have a finite or unlimited
  • The most basic models assume customers are of
    one type
  • and have common arrival and service rates

A multiple server queuing system
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
  • But in some cases we can use formulae to get

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
  • The exponential distribution underlies queuing
  • 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
  • ALC cases waiting for LTC beds

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

  • 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
  • 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

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
  • 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
  • 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
  • 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

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

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

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
  • 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.