Term
| Flow models with queuing systems |
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Definition
Irregularities or fluctuations in rates thru a system with limited capacity can cause;
1/flow- rate to be above capacity leading to hold ups/queues
2/flow rate above capacity leading to idle time
Here we will look at approaches to modelling effects of incr or decr capacity so as to help decisions about how to Dvt best balance btw these 2 effects
Key elements of queuing systems
1/customers: requires things to be done to them
2/servers or service channels : provides service that customer requires
-establish balance brew too few service channels ( inadequate service ) & too many ( an insufficient one) |
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Term
| Flow models for queuing system |
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Definition
-establish balance brew too few service channels ( inadequate service ) & too many ( an insufficient one)
-in a deterministic system I.e.constant arrival times & constant service times;
Achieved by having a faster service rate than the relevant arrival rate
-in a real system both arrival rated & service times vary unpredictably ;
-thus even if average service rate is greater than average arrival time there will be short periods during which arrival rate is faster than the average & / service rate increases such that queues grow .later arrival redu & service rate incr queus shorten again. |
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Term
Queuing models
-provide a method of working out capacities needed to provide satisfactory service when are irregular arrivals /service times |
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Definition
-to construct need :
1/queue configuration: description of the routes customers take btw diff services that make up system & where the queues form . In form of flow diagram
2/arrival time distribution : probability distribution of number of customers arriving for a service in a given time interval . Or distribution of lengths of time btw one customer arriving and the next
3/service time distributions; distribu of lengths of time to serve one customer at each time point in system
4/ Queue discipline; for each service it covers system of priority in practice;
-whether a customer/cyst type has preference for certain channels / chan type
-whether there is balking ; customers who are lost if channel is occupied when they arrive / appoint system /batching of customers |
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Term
Queuing theory
-2 main approaches to building models of queuing problems:
(a) queuing theory
(b) Monte Carlo stimulation |
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Definition
a) queuing theory : provides formulae for some specific situations & a v.rapid solution.
-formulae: available for limited number of types of queuing system
-queue configuration:applicable to simple queues with servers In Parallel or servers in series
Mean arrival rate ( lambda) /mean service rate ( mu)
=traffic intensity =p
Queuing theory tells us that
1/ probability that a new arrival has to wait =p
2/( average waiting time)/( ave time in service channell)=p/(1-p)
3/ average number in queue=p squared / (1-p) |
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Term
Queuing theory
-2 main approaches to building models of queuing problems:
(a) queuing theory
(b) Monte Carlo stimulation |
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Definition
Formulae depends on assumptions;
1/system in steady state not on transient behaviour like the start / end of a clinic session
Mean arrival rates & service times are constant but subject to fluctuations
- p <1
2/ arrivals are 'random' ;
each arrival independent of other arrivals w/c means that u have no knowledge other than mean arrival rate that can help u predict when arrivals will occur
-variation in the number of arrivals inns given period can be described by POISSON DISTRIBUTION.
It's shape determined by mean arrival rate ( lambda)
-low values of Lambda give distribution skewed to right , higher values provide more symmetry
3/ variation in service can be described by the negative exponential distribution
4/queue discipline is first come first served I.e no priority customers
- 80 % rule : at as traffic intensity rises above 80 % waiting times climb v.rapidly in relation to service times
-constant Service rates, performance of system improves, & traffic intensity rise to nearer to 90% before waiting times become really long
- greater variability in arrival & service tones the worse the performance of the system |
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Term
Queuing theory
-2 main approaches to building models of queuing problems:
(a) queuing theory
(b) Monte Carlo stimulation |
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Definition
Some rules of thumb for QT
-more randomness: the longer the queues for same p
-priority for groups with low average service times : lower overall average queuing time
B) Parellel servers & other configurations
Note ; formulae are available for more complex situations ( e.g.parellel service channels & bulking )
But there are no formulae for when
1/customers have preferences btw channels
2/ arrivals are non-random
3/ queue configurations are complex
4/queuing systems are adaptive
5/system is bit in a steady state
( changes in arrival rate &/ service time)
-Graphes & tables from queuing theory enable a decision to be made about the best number of service channels involves striking a balance btw what is an acceptable
High rate of loses and an acceptable low occupancy |
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Term
Queuing theory
B) Monte Carlo simulation |
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Definition
-Monte Carlo stimulation:is more flexible but requires more work to provide an answer
-random variation is added back by using random numbers to choose values from distributions if likely values
- M. C. MicroSimulation models ; set out to imitate the behaviour if real life system in the form of virtual reality ( I.e. Vs modelling flow as homogeneous & deriving formulae based on summary stats like mean arrival time & service time )
-virtual ;imaginary) world
-two approaches :
1/ synchronous micro stimulation =monte Carlos sampling by dividing time into equal short periods & move forward one period at a time ( same as Markov models but individual vs group progress )
2/ event-driven simulation; used in computer simulation where you sample from a continuous distribution of inter-arrival times |
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Term
Queuing theory
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A)Adding in system dynamics
B) uses , adv & disadv. of microsimulation |
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Definition
A)System might include
1/negative feedback -w/c will tend to keep occupancy rates even more even then w/o feedback
2/positive feedback
Adv over queuing theory
-distributions do not have to follow specific theoretical forms as can be based on direct observation
- appoint systems, priority schemes & complex configuration systems can all be accommodated
-systems do not have to be in a steady state
B) 1st used for complex queuing pbs w/ irregular flows
Now used to model any type of system . Powerful as requires fewer simpler assumptions than other methods
-entities are created and assigned attributes
In the context of smooth flows
1-adv. over Markiw process ( w/ regards to SMOOTH flow ) of not requiring the memory -less or homogeneous flies assumption
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2-easy to include resource constraint
3-Transparent ( as imitates real world)
4-easy to alter parameters enabling "what if "experiments to be set up quickly
5/problem owners can see what features of the real world a model does incorporate & e/c ones it leaves out
C) disadv
-with diff random number streams each simulation will produce a slightly different result
-this variability is also an advantage as indicates amt of imprecision arising from actual variations in the population sample attributes
-approach so flexible can include more detail than necessary for purpose
- often requires dedicated programming as the simulation software available suited for dealing with queuing PBS but not HC policy analysis
-dedicated programming is time consuming, costly and probe to error ,long periods . |
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