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Queuing models

Queuing models. Basic definitions, assumptions, and identities Operational laws Little’s law Queuing networks and Jackson’s theorem The importance of think time Non-linear results as saturation approaches Warnings References. What problem are we solving?. Describe a system’s mean

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Queuing models

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  1. Queuing models • Basic definitions, assumptions, and identities • Operational laws • Little’s law • Queuing networks and Jackson’s theorem • The importance of think time • Non-linear results as saturation approaches • Warnings • References

  2. What problem are we solving? Describe a system’s mean • Throughput • Response time • Capacity Predict • Changes in these quantities when system characteristics change See Stallings, Figure 1

  3. Stallings, Figure 1

  4. One queue Server arrivals departures waiting serving = a job See Stallings, Figure 2

  5. Stallings, Figure 2, Table 2

  6. Basic assumptions • One kind of job • Inter-arrival times are independent of system state • Service times are independent of system state • No jobs lost because of buffer overflow • Stability: λ < 1 / Ts • In a network, • no parallel processing of a given job • visit ratios are independent of system state

  7. Queuing definitions • A / S / m / B / K / SD • A = Inter-arrival time distribution • S = Service time distribution • m = Number of servers • B = Number of buffers (system capacity) • K = Population size • SD = service discipline • Usually specify just the first three: M/M/1

  8. Usual assumptions • A is often the Exponential distribution • S is often Exponential or constant • B is often infinite (all the buffer space you need) • K is often infinite • SD is often FCFS (first come, first served)

  9. Exponential distribution • Also known as “memoryless” • Expected time to the next arrival is always the same, regardless of previous arrivals • When the interarrival times are independent, identically-distributed, and the distribution is exponential, the arrival process is called a “Poisson” process

  10. Poisson processes • Popular because they are tractable to analyze • You can merge several Poisson streams and get a Poisson stream • You can split a Poisson stream and get Poisson streams • Poisson arrivals to a single queue with exponential service times => departures are Poisson with same rate • Same is true of departures from a M/M/m queue

  11. Basic formulas: Stallings, Table 3b • Assumptions • Poisson arrivals • No dispatching preference based on service times • FIFO dispatching • No items discarded from queue

  12. Basic multi-server formulas

  13. Expected response time • For a single M/M/1 queue, expected residence (response) time is 1/(μ-λ), where • μ is the server’s maximum output rate (1/Ts) • λ is the mean arrival rate • Example: • Disk can process 100 accesses/sec • Access requests arrive at 20/sec • Expected response time is 1/(100-20) = 0.0125 sec/access

  14. Example: near saturation • 80% utilization • Disk can do 100 accesses/sec • Access requests arrive at 80/sec • Expected response time: 1/(100-80) = 0.05 sec • 90% utilization • Expected response time: 1/(100-90) = 0.1 sec • 95% utilization • Expected response time: 1/(100-95) = 0.2 sec • 99% utilization • Expected response time: 1/(100-99) = 1 sec

  15. Nearing saturation (M/M/1)

  16. Operational law Little’s Law r = λ Tr Example: Tr = 0.3 sec average residence in the system λ = 10 transactions / sec r = 3 average transactions in the system

  17. Queuing network Queue 1 Queue 2 Queue 3

  18. Jackson’s theorem Assuming • Each node in the network provides an independent service • Poisson arrivals • Once served at a node, an item goes immediately to another node, or out of the system Then • Each node is an independent queuing system • Each node’s input is Poisson • Mean delays at each node may be added to compute system delays

  19. Queuing network example Tr = 50 msec Tr = 60 msec Servlet P = 0.4 Historydata λ =12 jobs/sec Tr = 80 msec P = 0.6 Orderdata Average system residence time = 0.4 * (50+60) + 0.6 * (50+80) (Jackson’s theorem) = 122 msec Average jobs in system = 12 * 0.122 = 1.464 (Little’s Law)

  20. An example in a spreadsheet

  21. Think time “think time” between transactions request a transaction Think time can have a huge effect on the arrival rate for a system.

  22. Main Reference Queuing Analysis, William Stallings Cached copy on the resource page: http://cs.franklin.edu/~swartoud/650/QueuingAnalysis.pdf

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