View
226
Download
1
Category
Preview:
DESCRIPTION
Queuing Analysis 3 Projected vs. Actual Response Time Why??
Citation preview
1
Queuing Delay and Queuing Delay and Queuing AnalysisQueuing Analysis
RECALL: Delays in Packet RECALL: Delays in Packet Switched (e.g. IP) NetworksSwitched (e.g. IP) Networks End-to-end delay (simplified) =End-to-end delay (simplified) =
– (d(dpropprop + d + dtranstrans + d + dqueuequeue + d + dprocproc)) … … on each linkon each link
Introduction2
BA Where:Where:
Propagation delay (dPropagation delay (dpropprop) = d/s (dependent on path)) = d/s (dependent on path) Transmission delay (dTransmission delay (dtranstrans) = L/R (dependent on path)) = L/R (dependent on path) Queuing delay (dQueuing delay (dqueuequeue) = (dependent on load)) = (dependent on load) Processing delay (dProcessing delay (dprocproc) = (minimal-insignificant/node)) = (minimal-insignificant/node) Number of links (Q) = (dependent on path)Number of links (Q) = (dependent on path)
Queuing Analysis3
Projected vs. Actual Response Projected vs. Actual Response TimeTime
Why??Why??
Queuing Analysis
R:R: link bandwidth link bandwidth (bps)(bps)
L:L: packet length (bits) packet length (bits) a: average packet a: average packet
arrival ratearrival ratetraffic intensity
= La/R
La/RLa/R ~ 0: avg. queueing delay small ~ 0: avg. queueing delay small La/R La/R -> 1: avg. queueing delay large-> 1: avg. queueing delay large La/R La/R > 1: more > 1: more ““workwork”” arriving arriving than can be serviced, average delay than can be serviced, average delay
infinite!infinite!
aver
age
que
uein
g de
lay
La/R ~ 0
Queueing delay (revisited)Queueing delay (revisited)
La/R -> 14
Queuing Analysis5
Introduction- MotivationIntroduction- Motivation Address how to analyze changes in Address how to analyze changes in
network workloads (i.e., a helpful network workloads (i.e., a helpful tooltool to use) to use)
Analysis of system (network) load Analysis of system (network) load and performance characteristicsand performance characteristics
– response timeresponse time– throughputthroughput
Performance tradeoffs are Performance tradeoffs are often not often not intuitiveintuitive
Queuing theory, although Queuing theory, although mathematically complex, often mathematically complex, often makes analysis very straightforwardmakes analysis very straightforward
Queuing Analysis6
Important NoteImportant Note Queuing theory is heavily dependent Queuing theory is heavily dependent
on basic probability theory (a pre-on basic probability theory (a pre-requisite for our graduate program)requisite for our graduate program)
If you need to refresh your If you need to refresh your knowledge in this area, please knowledge in this area, please review the Stallings textbook, review the Stallings textbook, Chapter 7: Overview of Probability Chapter 7: Overview of Probability and Stochastice Processesand Stochastice Processes..
I will not test you specifically on I will not test you specifically on probability theory, but will reference probability theory, but will reference it in coverage of the queuing topics it in coverage of the queuing topics addressed in this module. addressed in this module.
Queuing Analysis7
Single-Server Queuing SystemSingle-Server Queuing System
QueuingSystem
(Delay Box)
Items ArrivingItems Arriving(rate: (rate: )
(message, packet, cell)(message, packet, cell)
Items Lost Items Lost
Items DepartingItems Departing(rate: R)(rate: R)
Introduction8
Router output port functionsRouter output port functions
buffering/queuing buffering/queuing required when datagrams arrive from fabric faster than the required when datagrams arrive from fabric faster than the transmission ratetransmission rate
scheduling disciplinescheduling discipline chooses among queued datagrams for transmission chooses among queued datagrams for transmission sending discipline (servicing the queue) sending discipline (servicing the queue) on the output link as determined by link protocolon the output link as determined by link protocol
linetermination
link layer
protocol(send)
switchfabric
datagrambuffer(s)
queueing
Queue Queue server
Queuing Analysis9
The Fundamental Task of The Fundamental Task of Queuing AnalysisQueuing Analysis
Given:Given:• Arrival rate, Arrival rate, • Service time, Service time, TTss
• Number of servers, Number of servers, NN
Determine:Determine:• Items waiting, Items waiting, ww• Waiting time, Waiting time, TTww
• Items queued, Items queued, rr• Residence time, Residence time, TTrr
Queuing Analysis10
Parameters for Single-Server Parameters for Single-Server Queuing SystemQueuing System
Comments, assuming queue has infinite capacity:1. At = 1, server is working 100% of the time (saturated), so items are
queued (delayed) until they can be served. Departures remain constant (for same L).
2. Traffic intensity, u = L/R. Note that Ts = L/R, so:max = 1 / Ts = 1 / (L/R) is the theoretical maximum arrival rate,
and thatLmax/R = u = 1 at the theoretical maximum arrival rate
Queuing Analysis11
Queuing Process - ExampleQueuing Process - Example
General Expression:General Expression:TTRn+1Rn+1 = T = TSn+1Sn+1 + MAX[0, D + MAX[0, Dnn – A – An+1n+1]]
Depth of the Queue
Queuing Analysis12
General Characteristics of General Characteristics of Network Queuing ModelsNetwork Queuing Models
Item populationItem population– generally assumed to be generally assumed to be infiniteinfinite therefore, therefore,
arrival rate is persistent through timearrival rate is persistent through time Queue sizeQueue size
– infiniteinfinite, therefore no loss, therefore no loss– finite, more practical, but often immaterialfinite, more practical, but often immaterial
Dispatching discipline Dispatching discipline – FIFOFIFO, typical, typical– LIFO (when is this practical?)LIFO (when is this practical?)– Relative/Preferential, Relative/Preferential, based on QoSbased on QoS
Queuing Analysis13
Multiserver Queuing SystemMultiserver Queuing System
Comments:1. Assuming N identical servers, and is the utilization of each server. 2. Then, N is the utilization of the entire system, and the maximum
utilization is N x 100%.3. Therefore, the maximum supportable arrival rate that the system can
handle is: max = N / Ts = NR/L
Chapter 8 Overview of Queuing Analysis14
Multiple Single-Server Queuing Multiple Single-Server Queuing SystemsSystems
Queuing Analysis15
Basic Queuing RelationshipsBasic Queuing Relationships
GeneralGeneral Single Single ServerServer MultiserverMultiserver
rr = = TTrr Little’s Little’s FormulaFormula = = TTss
= =
ww = = TTww Little’s Little’s FormulaFormula rr = = ww + + uu = = TTss = = NN
TTrr = = TTww + + TTss r = w + Nr = w + N
TTs s
NN
Queuing Analysis16
Kendall’s notationKendall’s notation Notation is Notation is X/Y/NX/Y/N, where:, where:
X is distribution of interarrival X is distribution of interarrival timestimes
Y is distribution of service timesY is distribution of service timesN is the number of serversN is the number of servers
Common distributionsCommon distributions G = general distribution if interarrival times G = general distribution if interarrival times
or service timesor service times GI = general distribution of interarrival time GI = general distribution of interarrival time
with the restriction that they are independentwith the restriction that they are independent M = negative exponential distribution of M = negative exponential distribution of
interarrival times (Poisson arrivals – p. 167) interarrival times (Poisson arrivals – p. 167) and service timesand service times
D = deterministic arrivals or fixed length D = deterministic arrivals or fixed length serviceservice
M/M/1? M/D/1?M/M/1? M/D/1?
Queuing Analysis17
Important Formulas for Important Formulas for Single-Server Queuing Single-Server Queuing SystemsSystems
Note Coefficient of variation: if Ts = Ts => exponential if Ts = 0 => constant
Queuing Analysis18
Mean Number of Items in Mean Number of Items in System (System (rr)- Single-Server )- Single-Server QueuingQueuing
Ts/Ts = Coefficient of variation
M/M/1
M/D/1
Queuing Analysis19
Mean Residence Time – (Mean Residence Time – (TTrr) ) Single-Server QueuingSingle-Server Queuing
M/M/1
M/D/1
Queuing Analysis20
Network Queue Performance: Network Queue Performance: Key FactKey Fact
The higher the variability in arrival The higher the variability in arrival rate at the router, relative to the rate at the router, relative to the service time on the output link(s), i.e., service time on the output link(s), i.e., TTss/T/Tss , , the poorer the performance of the poorer the performance of the router, especially at high rates of the router, especially at high rates of utilization.utilization.
Queuing Analysis21
Multiple Server Queuing Multiple Server Queuing SystemsSystems
Multiple Multiple Single-Single-Server Server Queuing Queuing SystemSystem
Multiserver Multiserver Queuing Queuing SystemSystem
Queuing Analysis22
Important Formulas for Important Formulas for Multiserver QueuingMultiserver Queuing
Note:Note:Useful only inUseful only inM/M/N case,M/M/N case,with equal with equal service times service times at all N at all N servers.servers.
Queuing Analysis23
Multiple Server Queuing Multiple Server Queuing Example Example (p. 203)(p. 203)
Single serverM/M/1 (2nd Floor)
MultiserverM/M/? (2nd Floor)
Multiple Single server
M/M/1 (1st Floor)
M/M/1 (2nd Floor)
M/M/1 (3rd Floor)
Queuing Analysis24
MultiServer vs. Multiple Single-MultiServer vs. Multiple Single-Server Queuing System Server Queuing System Comparison Comparison (from example problem, pp. 203-(from example problem, pp. 203-204)204)
Single server case (M/M/1):Single server case (M/M/1):Single server utilization: Single server utilization: = 10 engineers x 0.5 hours each / 8 = 10 engineers x 0.5 hours each / 8 hour work dayhour work day
= 5/8 = .625= 5/8 = .625Average time waiting: TAverage time waiting: Tww = = TTss / 1 - / 1 - = 0.625 x 30 / .375 = 50 = 0.625 x 30 / .375 = 50 minutesminutesArrival rate: Arrival rate: = 10 engineers per 8 hours = 10/480 = 0.021 = 10 engineers per 8 hours = 10/480 = 0.021 engineers/minuteengineers/minute9090thth percentile waiting time: m percentile waiting time: mTTww(90) = T(90) = Tww// x ln(10 x ln(10) = 146.6 minutes) = 146.6 minutes
Average number of engineers waiting: w = Average number of engineers waiting: w = TTww = 0.021 x 50 = 1.0416 = 0.021 x 50 = 1.0416 engineersengineers
Queuing Analysis25
Example: Router QueuingExample: Router QueuingInternetInternet ……
96009600bpsbps
= 5 packets/sec= 5 packets/secL = 144 octetsL = 144 octets
From data provided:From data provided:• TTs s = L/R = (144x8)/9600 = .12sec= L/R = (144x8)/9600 = .12sec = = TTs s = 5 packets/sec x .12sec = = 5 packets/sec x .12sec =
.6.6
Determine:Determine:1.1. TTrr= T= Tss / (1- / (1-) = .12sec/.4 = .3 sec) = .12sec/.4 = .3 sec2.2. r = r = / (1- / (1-) = .6/.4 = 1.5 ) = .6/.4 = 1.5
packetspackets
3. m3. mrr(90) = - 1 = 3.5 (90) = - 1 = 3.5 packetspackets
4.4. mmrr(95) = - 1 = 4.8 (95) = - 1 = 4.8 packetspackets
ln(1-.90)ln(1-.90)ln (.6)ln (.6)
ln(1-.95)ln(1-.95)ln (.6)ln (.6)
For 3 & 4, use:For 3 & 4, use:
mmrr(y) = - (y) = - 1 1
ln(1 – ln(1 – y/100)y/100)ln ln
Queuing Analysis26
Priorities in Queues – Two Priorities in Queues – Two priority classespriority classes
r
Chapter 8 Overview of Queuing Analysis27
Priorities in Queues – Priorities in Queues – ExampleExample
Router queue services two packet Router queue services two packet sizes:sizes:• Long = 800 octetsLong = 800 octets• Short = 80 octetsShort = 80 octets• Lengths exponentially distributedLengths exponentially distributed• Arrival rates are equal, 8packets/secArrival rates are equal, 8packets/sec• Link transmission rate is 64KbpsLink transmission rate is 64Kbps• Short packets are priority 1,Short packets are priority 1,• Longer packets are priority 2Longer packets are priority 2From data above, calculate:From data above, calculate:TTs 1s 1 = L = Lshortshort/R = (80 x 8) / 64000 = .01 /R = (80 x 8) / 64000 = .01 secsecTTs 2s 2 = L = Llonglong/R = (800 x 8) / 64000 = .1 /R = (800 x 8) / 64000 = .1 secsec11 = = TTs 1 s 1 = 8 x 0.01 = 0.08= 8 x 0.01 = 0.0822 = = TTs 2 s 2 = 8 x 0.1 = 0.8= 8 x 0.1 = 0.8 = = 1 1 ++ 2 2 = 0.88= 0.88
Find the average Queuing Delay (Find the average Queuing Delay (TTrr) ) through the router:through the router:
TTr1r1 = T = Ts1 s1 + +
= .01 + = 0.098 = .01 + = 0.098 sec sec
TTr2r2 = T = Ts2 s2 + +
= .1 + = 0. 833 sec = .1 + = 0. 833 sec
TTrr = T = Tr1 r1 + + TTr2 r2
= .5 x .098 + .5 x .833 = 0.4655 = .5 x .098 + .5 x .833 = 0.4655 secsec
11 TTs 1 s 1 + + 2 2 TTs 2s 2
1 - 1 - 11.08 x .01 .08 x .01 + + .8 .8 x .1x .1
1-.081-.08TTr 1 r 1 -- TTs 1s 1
1 - 1 - .098.098 -- .01 .01
1 - .881 - .8811
22
64Kbps64Kbps
TTrr
Queuing Analysis28
Network of QueuesNetwork of Queues
Queuing Analysis29
Elements of Queuing NetworksElements of Queuing Networks
Queuing Analysis30
Queuing NetworksQueuing Networks
Queuing Analysis31
Jackson’s Theorem and Jackson’s Theorem and Queuing NetworksQueuing Networks Assumptions:Assumptions:
– the queuing network has m nodes, each providing the queuing network has m nodes, each providing exponential serviceexponential service
– items arriving from outside the system at any node items arriving from outside the system at any node arrive with a Poisson ratearrive with a Poisson rate
– once served at a node, an item moves immediately once served at a node, an item moves immediately to another with a fixed probability, or leaves the to another with a fixed probability, or leaves the networknetwork
Jackson’s Theorem states: Jackson’s Theorem states: – each node is an independent queuing system with each node is an independent queuing system with
Poisson inputs determined by partitioning, merging Poisson inputs determined by partitioning, merging and tandem queuing principlesand tandem queuing principles
– each node can be analyzed separately using the each node can be analyzed separately using the M/M/1 or M/M/N modelsM/M/1 or M/M/N models
– mean delays at each node can be added to mean delays at each node can be added to determine mean system (network) delaysdetermine mean system (network) delays
Queuing Analysis32
Jackson’s Theorem - Application Jackson’s Theorem - Application in Packet Switched Networksin Packet Switched Networks
Packet SwitchedPacket SwitchedNetworkNetwork
External load, offered to network:External load, offered to network: = = jkjk
where:where: = = total workload in total workload in packets/secpackets/sec jk jk = = workload between source j workload between source j and destination kand destination k N = total number of (external) N = total number of (external) sources and destinationssources and destinations
N NN N
j=1 j=1 k=2k=2
Internal load:Internal load:
= = iiwhere:where: = = total on all links in networktotal on all links in network i i = = load on link iload on link i L = total number of linksL = total number of links
L L
i=i=11
Note:Note:• Internal > offered loadInternal > offered load• Average length for all paths:Average length for all paths: E[number of links in path] = E[number of links in path] = //• Average number of item waiting Average number of item waiting and being served in link i: rand being served in link i: r ii = = i i TTriri• Average delay of packets sent Average delay of packets sent through the network is:through the network is: T = T =
where: M is average packet length where: M is average packet length andand RRi i is the data rate on link iis the data rate on link i
11
L L i=i=11
MMiiRRii - - MMii
Queuing Analysis33
Estimating Model Estimating Model ParametersParametersTo enable queuing analysis using To enable queuing analysis using
these models, we must these models, we must estimate estimate certain parameterscertain parameters::– Mean and standard deviation of Mean and standard deviation of
arrival ratearrival rate– Mean and standard deviation of Mean and standard deviation of
service timeservice time (or, packet size) (or, packet size)Typically, these estimates use Typically, these estimates use
sample measurementssample measurements taken from taken from an existing systeman existing system
Queuing Analysis34
Sample Means for Exponential Sample Means for Exponential DistributionDistribution Sampling:
• The mean is generally the most important quantity to estimate:
() = Xi
• Sample mean is itself a random variable
• Central Limit Theorem: the probability distribution tends to normal as sample size, N, increases for virtually all underlying distributions
• The mean and variance of X can be calculated as:
E[]= E[X] = Var[]= 2x/N
N
i = 1
1N
Recommended