1
Chapters 8Chapters 8
Overview ofOverview ofQueuing Queuing AnalysisAnalysis
Chapter 8 Overview of Queuing Analysis2
Projected vs. Actual Response Projected vs. Actual Response TimeTime
Chapter 8 Overview of Queuing Analysis3
Introduction- MotivationIntroduction- Motivation How to analyze changes in network How to analyze changes in network
workloads? (i.e., a helpful workloads? (i.e., a helpful tooltool to use) to use) Analysis of system (network) load and Analysis of system (network) load and
performance characteristicsperformance 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
Chapter 8 Overview of Queuing Analysis4
Single-Server Queuing SystemSingle-Server Queuing System
QueuingSystem
(Delay Box)
Items ArrivingItems Arriving
(message, packet, cell)(message, packet, cell)
Items Lost Items Lost
Items DepartingItems Departing
Chapter 8 Overview of Queuing Analysis5
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 that
Lmax/R = u = 1 at the theoretical maximum arrival rate
Chapter 8 Overview of Queuing Analysis6
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
Chapter 8 Overview of Queuing Analysis7
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
Chapter 8 Overview of Queuing Analysis8
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 persistentarrival rate is persistent 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– LIFOLIFO– Relative/Preferential, based on QoSRelative/Preferential, based on QoS
Chapter 8 Overview of Queuing Analysis9
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
Chapter 8 Overview of Queuing Analysis10
Multiple Single-Server Queuing Multiple Single-Server Queuing SystemsSystems
Chapter 8 Overview of Queuing Analysis11
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
Chapter 8 Overview of Queuing Analysis12
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 = exponential distribution of interarrival M = exponential distribution of interarrival
times (Poisson arrivals – p. 167) and service times (Poisson arrivals – p. 167) and service timestimes
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?
Chapter 8 Overview of Queuing Analysis13
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
Chapter 8 Overview of Queuing Analysis14
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
Chapter 8 Overview of Queuing Analysis15
Mean Residence Time – (Mean Residence Time – (TTrr) ) Single-Server QueuingSingle-Server Queuing
M/M/1
M/D/1
Chapter 8 Overview of Queuing Analysis16
Multiple Server Queuing Multiple Server Queuing SystemsSystems
Multiple Multiple Single-Single-Server Server Queuing Queuing SystemSystem
Multiserver Multiserver Queuing Queuing SystemSystem
Chapter 8 Overview of Queuing Analysis17
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.
Chapter 8 Overview of Queuing Analysis18
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)
Chapter 8 Overview of Queuing Analysis19
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 = .625
Average time waiting: TAverage time waiting: Tww = = TTss / 1 - / 1 - = 0.625 x 30 / .375 = 50 = 0.625 x 30 / .375 = 50
minutesminutes
Arrival rate: Arrival rate: = 10 engineers per 8 hours = 10/480 = 0.021 = 10 engineers per 8 hours = 10/480 = 0.021
engineers/minuteengineers/minute
9090thth 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
Chapter 8 Overview of Queuing Analysis20
Example: Router QueuingExample: Router Queuing
InternetInternet ……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
Chapter 8 Overview of Queuing Analysis21
Priorities in Queues – Two Priorities in Queues – Two priority classespriority classes
r
Chapter 8 Overview of Queuing Analysis22
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 = 0.08
22 = = 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-.08
TTr 1 r 1 -- TTs 1s 1
1 - 1 - .098.098 -- .01 .01
1 - .881 - .88
11
22
64Kbps64Kbps
TTrr
Chapter 8 Overview of Queuing Analysis23
Network of QueuesNetwork of Queues
Chapter 8 Overview of Queuing Analysis24
Elements of Queuing NetworksElements of Queuing Networks
Chapter 8 Overview of Queuing Analysis25
Queuing NetworksQueuing Networks
Chapter 8 Overview of Queuing Analysis26
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
Chapter 8 Overview of Queuing Analysis27
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:
= = ii
where: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: rii = = 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
MMii
RRii - - MMii
Chapter 8 Overview of Queuing Analysis28
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
Chapter 8 Overview of Queuing Analysis29
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[]= 2
x/N
N
i = 1
1N