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U of I CS 241 - Queuing Framework

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Queuing Framework for Process Management EvaluationCS241 AdministrativeContent of This LectureProcess States Finite State DiagramQueuing Models and SimulationCPU Scheduler ExampleQueuing Diagram for ProcessesQueuing Diagram for ProcessesQueuing Diagram for ProcessesQueuing Diagram for ProcessesQueuing Diagram for ProcessesQueuing Diagram for ProcessesQueuing Diagram for ProcessesQueuing Diagram for ProcessesQueuing Diagram for ProcessesQueuing Diagram for ProcessesQueuing Diagram for ProcessesQueuing Diagram for ProcessesQueuing Diagram for ProcessesQueuing Diagram for ProcessesQueueing ModelDiscussionDiscussionHamburger ProblemHamburger ProblemRandom EventsQueuing TheoryQueueing Theory (Point of Interest)Analysis of Queueing BehaviorUseful Facts From Queuing TheoryAnalysis of Single Server QueueExample: How busy is the server?How long is an eater in the system?How long is someone in the queue?How many people in queue?How many people in queue?Interesting FactUntil Now We Looked at Single Server, Single Queue TheoryPoisson Arrivals SumExampleExampleAs long as it’s a Poisson Distribution...Question: MacDonalds ProblemQueuing Diagram for ProcessesQueuing Diagram for ProcessesSimulations – instead of mathsSummaryCS 241 Spring 2007System Programming1Queuing Framework for Process Management EvaluationLecture 20Klara Nahrstedt2CS241 Administrative Read Stallings Chapter 9  No Quizzes this week, the next quiz will be on Monday 3/12 on SMP53Content of This LectureGoals:Introduction to Principles for Reasoning about Process Management/SchedulingThings covered in this lectureIntroduction to Queuing Theory4Process States Finite State Diagram5Queuing Models and SimulationProblem – How do we size the ready queue, size any queue?decide how many jobs should be accepted?analyze scheduling algorithms?Decide how long we should wait for a job?GoalsSimple arithmetic (‘back of envelope calculation’) to calculate system behaviorBasis for more complex analysisApproach to study systems too complex to produce simple mathematical model6CPU Scheduler ExampleWhich is better: Round Robin or FIFO?How long does A take to go thru system?How big should be the ready queue? Mean, Mode, Variance?7Queuing Diagram for ProcessesStartStartReady QueueReady QueueEvent QueueEvent QueueEventEventExitExitTime SliceTime SliceCPUCPU8Queuing Diagram for ProcessesStartStartReady QueueReady QueueEvent QueueEvent QueueEventEventExitExitTime SliceTime SliceCPUCPU9Queuing Diagram for ProcessesStartStartReady QueueReady QueueEvent QueueEvent QueueEventEventExitExitTime SliceTime SliceCPUCPU10Queuing Diagram for ProcessesStartStartReady QueueReady QueueEvent QueueEvent QueueEventEventExitExitTime SliceTime SliceCPUCPU11Queuing Diagram for ProcessesStartStartReady QueueReady QueueEvent QueueEvent QueueEventEventExitExitTime SliceTime SliceCPUCPU12Queuing Diagram for ProcessesStartStartReady QueueReady QueueEvent QueueEvent QueueEventEventExitExitTime SliceTime SliceCPUCPU13Queuing Diagram for ProcessesStartStartReady QueueReady QueueEvent QueueEvent QueueEventEventExitExitTime SliceTime SliceCPUCPU14Queuing Diagram for ProcessesStartStartReady QueueReady QueueEvent QueueEvent QueueEventEventExitExitTime SliceTime SliceCPUCPU15Queuing Diagram for ProcessesStartStartReady QueueReady QueueEvent QueueEvent QueueEventEventExitExitTime SliceTime SliceCPUCPU16Queuing Diagram for ProcessesStartStartReady QueueReady QueueEvent QueueEvent QueueEventEventExitExitTime SliceTime SliceCPUCPU17Queuing Diagram for ProcessesStartStartReady QueueReady QueueEvent QueueEvent QueueEventEventExitExitTime SliceTime SliceCPUCPU18Queuing Diagram for ProcessesStartStartReady QueueReady QueueEvent QueueEvent QueueEventEventExitExitTime SliceTime SliceCPUCPU19Queuing Diagram for ProcessesStartStartReady QueueReady QueueEvent QueueEvent QueueEventEventExitExitTime SliceTime SliceCPUCPU20Queuing Diagram for ProcessesStartStartReady QueueReady QueueEvent QueueEvent QueueEventEventExitExitTime SliceTime SliceCPUCPU21Queueing ModelRandom Arrivals and the Poisson DistributionElements of model22DiscussionIf a bus arrives at a bus stop every 15 minutes, how long do you have to wait at the bus stop assuming you start to wait at a random time?23DiscussionWhat assumption have you made about the bus?24Hamburger Problem7 Hamburgers arrive on average every time unit8 Hamburgers are processed by Joe on average every unit1) Av. time hamburger waiting to be eaten? (Do they get cold?) Ans = ????2) Av number of hamburgers waiting in queue to be eaten? Ans = ????Queue7825Hamburger Problem7 Hamburgers arrive on average every time unit8 Hamburgers are processed by Joe on average every unit1) How long is a hamburger waiting to be eaten? (Do they get cold?) Ans = 7/8 time units 2) How many hamburgers are waiting in queue to be serviced? Ans = 49/8 Queue7826Random EventsPoisson DistributionEach event independent of other eventsMean event rate, SD is same as meanExponential distribution27Queuing TheoryARRIVAL RATE ARRIVAL RATE λλSERVICE RATESERVICE RATEμμInput QueueInput QueueServerServerSingle Server Queue28Queueing Theory (Point of Interest)Steady statePoisson arrival with λ constant arrival rate (customers per unit time) each arrival is independent. P( τ≤t ) = 1- e–λt01tAv λ0.529Analysis of Queueing BehaviorProbability n customers arrive in time interval tis: e–λt (λt)n/ n!Assume random service times (also Poisson): μconstant service rate (customers per unit time)30Useful Facts From Queuing TheoryWq= mean time a customer spends in the queue λ = arrival rateLq= λ Wqnumber of customers in queueW = mean time a customer spends in the systemL = λ W ( Little's theorem) number of customers in the systemIn words – average length of queue is arrival rate times average waiting time31Analysis of Single Server Queueμλρ=λμ−=1Wλμρ−=qWρρ−=12qL•Server Utilization:•Time in System:•Time in Queue:•Number in Queue (Little):32Example: How busy is the server?λ=2μ=3μλρ== 2/3 or 66% Busy33How long is an eater in the system?λ=2μ=3λμ−=1W= 1/(3-2)= 134How long is someone in the queue?λ=2μ=366.2366.=−=−=λμρqW35How many people in queue?λ=2μ=333.166.1266.12=−=−=ρρqL36How many people in queue?λ=2μ=2∞=−=−=112112ρρqL37Interesting FactIf Arrival Rate = Service Rate, the queue length become infinitely large the longer you run the model.38Until Now We Looked at Single Server, Single Queue TheoryARRIVAL RATE


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U of I CS 241 - Queuing Framework

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