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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.

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Presentation on theme: "Capacity Setting and Queuing Theory BAMS 580B. Capacity and Resources  A key lever for improving patient flow.  How do we measure capacity?  What is."— Presentation transcript:

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? Base capacity Surge 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 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 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 Inputs Arrival Rate Service Rate Number of Servers Buffer Size Outputs Capacity Utilization Wait Time in Queue Queue Length Blocking Probability Service Levels Queue Analyzer QUEUMMCK_EMBA.xls

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

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 = R s = Service rate – arrival rate  Performance Measures  Average waiting time (in queue) – T i  Average time spent at the server - T p  Average flow time (in process) – T = T i + T p  Average queue length – I i  Average number of customers being served - I p  Average number of customers in the system – I =I i + I p

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 withQUEUEMMCK.xls  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.

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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