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Capacity Setting and Queuing Theory

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Title: Capacity Setting and Queuing Theory


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
Server
Buffer
Server
Server
8
Several parallel singer server queues
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
    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.

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 doesnt 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-?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.

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 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

17
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

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
  • Littles 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
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.

27
Idle Capacity And Wait Time Targets
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.
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