foundations: operational analysis and queuing models
TRANSCRIPT
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u = 𝜆 ∗ 𝒔
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Component Performance, Capacity, or Bandwidth
CPU Clock speed; Instructions executed/clock
Memory Access time (nanoseconds); bus bandwidth
Rotating Disk Access time (milliseconds)
Solid State Disk Access time (microseconds)
Network adapter Bandwidth; Latency
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Disk Reads/sec Writes/sec IOPS Utilization
1 24 8 32 0.30
2 28 8 36 0.41
3 40 10 50 0.54
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Disk Reads/sec Writes/sec IOPS Utilization
1 24 8 32 0.30
2 28 8 36 0.41
3 40 10 50 0.54
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Disk Reads/sec Writes/sec IOPS Utilization
1 24 8 32 0.30
2 28 8 36 0.41
3 40 10 50 0.54
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Disk Reads/sec Writes/sec IOPS Utilization Ave Service Time (ms)
1 24 8 32 0.30 9.4
2 28 8 36 0.41 11.4
3 40 10 50 0.54 10.8
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Uti
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Requests/sec
Bounds on Performance
Disk 1 Disk 2 Disk 3
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Bounds on Performance
Disk 1 Disk 2 Disk 3
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Bounds on Performance
Disk 1 Disk 2 Disk 3
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R = 𝑾𝒔+𝑾𝒒
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Server
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σ𝑘=1∞ 𝝁𝒌𝑷𝒌
σ𝑘=1∞ 𝒌𝑷𝒌
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Server
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Server
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G/G/n
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p(x
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Probaility Density function (pdf)
for representative Exponential Distributions ()
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Poisson distributionsfor different values of lambda
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Poisson distributionsfor different values of lambda
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-5 -4 -3 -2 -1 0 1 2 3 4
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Normal Distribution (pdf)mean = 0; s.d. =1
34.1% 34.1%
13.6% 13.6%
2.15% 2.15%
68.2%
95.4%
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Cumulative distribution
for representative Exponential Distributions ()
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▫𝑵 = ൗ𝒑𝟏−𝒑
▫𝑹𝑻 = ൗ𝑺 𝟏−𝒑
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Utilization
Response Times (M/M/1)
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𝑹𝑻 = ൗ𝑺 𝟏 − 𝒑
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𝒎𝒆𝒂𝒏 ≅ 𝒔𝒕𝒂𝒏𝒅𝒂𝒓𝒅 𝒅𝒆𝒗𝒊𝒂𝒕𝒊𝒐𝒏
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Response Times (M/M/1)
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Break large requests into a sequence of smaller, uniformed- size Request packets
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Utilization
Response Times (M/M/1)
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𝑅𝑇 = ൗ𝑆 1−𝑝n
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𝑹𝑻 = 𝑺 +𝒑𝑺(𝟏 + 𝑪𝒔
𝟐)
𝟐(𝟏 − 𝒑)
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𝑹𝑻 = 𝑺 +𝒑𝑺(𝟏 + 𝑪𝒔
𝟐)
𝟐(𝟏 − 𝒑)
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𝑹𝑻 = 𝑺 +𝒑𝑺(𝟏 + 𝑪𝒔
𝟐)
𝟐(𝟏 − 𝒑)
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M/G/1
M/D/1
M/M/1C = 1
𝑹𝑻 = 𝑺 +𝒑𝑺(𝟏 + 𝑪𝒔
𝟐)
𝟐(𝟏 − 𝒑)
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M/G/1
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M/G/1
M/D/1
M/M/1C = 1G/G/1
𝑹𝑻 = 𝑺 +𝒑𝑺(𝟏 + 𝑪𝒔
𝟐)
𝟐(𝟏 − 𝒑)
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use pdq;
# Globals$arrivRate = 0.75;$servTime = 1.0;
pdq::Init("Open Network with M/M/1");pdq::CreateOpen("Work", $arrivRate);pdq::CreateNode("Server", $pdq::CEN, $pdq::FCFS); pdq::SetDemand("Server", "Work", $servTime);
# Solve the modelpdq::Solve($pdq::CANON);
pdq::Report();
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use pdq;
$model = "Middleware";$work = "eBiz-tx";
$node1 = "WebServer";$node2 = "AppServer";$node3 = "DBMServer";$node4 = "DummySvr";
$think = 0.0 * 1e-3; # treat as free param$users = 10;
pdq::Init($model);pdq::CreateNode($node1, $pdq::CEN, $pdq::FCFS);pdq::CreateNode($node2, $pdq::CEN, $pdq::FCFS);pdq::CreateNode($node3, $pdq::CEN, $pdq::FCFS);pdq::CreateNode($node4, $pdq::CEN, $pdq::FCFS);
pdq::CreateClosed($work, $pdq::TERM, $users, $think);
# NOTE: timebase is secondspdq::SetDemand($node1, $work, 9.8 * 1e-3);pdq::SetDemand($node2, $work, 2.5 * 1e-3);pdq::SetDemand($node3, $work, 0.72 * 1e-3);pdq::SetDemand($node4, $work, 9.8 * 1e-3);
pdq::Solve($pdq::EXACT);pdq::Report();
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QT > ST, if u > .50
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