Some Real Life Applications

Matrix Analytic Methods -Some Real Life Applications

David LucantoniDLT Consulting, L.L.C

www.DLTconsulting.com

Queueing Theory and Network Applications, Korea University, 2011

Objectives of this TalkReview Matrix Analytic Methods pioneered by Marcel Neuts

and others

Present real life examples where Matrix Analytic Methods have been used successfully in industryCongestion due to Internet Dial UpApplication to Common Channel Signaling (SS7)Modeling WiFi Hotspot TrafficMultiplexing bursty traffic in an ATM or IP networkCall Acceptance for a VoIP SystemTransient Results and applications


Objectives of this Talk (Cont’d)No new mathematical analyses

A detailed description of any of these models would take the whole talk

Will only give general description of model parameters

Detailed definitions and analyses are widely available in the literature

Will give some matrix results along with the corresponding scalar (exponential or Poisson) results


Analysis of Matrix Analytic ModelsModels are analyzed by purely probabilistic arguments (e.g.,

Markov Renewal Theory)

Explicit expressions lead to very stable numerical algorithms for computation of performance metrics

Matrix analytic methods avoid

the complex analysis used in previous approaches

the necessity of numerically computing the roots of an analytic functionNumerical instability when roots are close to each other or multiple


Phase-Type (PH) DistributionsWill only consider continuous-time distributions here

Phase-type distributions contain many well-known and popular distributions

Exponential distribution, (m=1)

Erlang distribution

Hyper-exponential distribution


Phase-Type Distributions (Cont’d)Defined as the time till absorption in an (m+1)-state Markov

process with 1 absorbing state

Let T be the m×m matrix of infinitesimal rates between transient states, α, the vector of probabilities of starting in a particular transient state and T0 be the rates into the absorbing state

Let X be a phase-type random variable and F(x)=P(X≤x)

Then


Assuring Emergency Services Access (e.g., 911) in the presence of long holding times

due to Internet dial-up calls

The ProblemNo dial-tone condition - “Customers hear thin air.”VERY SERIOUS – e.g., 90/day life threatening emergency calls . Priorities non-implementablePreliminary suspicion on maintenance activities.Could it be congestion ? Red flag – internet usage !Model solved using a finite quasi-birth-and-death process (QBD)

[Ramaswami, et al., 2005]


Cable Telephony

Network Interface

Unit

Cable Modem

HDT

CMTS

Backbone

Backbone Transport OC-48+

CLASS -5

SWITCH

SS7

PSTNMaster Headend or

Primary Hub

(100K - 200K HHP) Each

Zone 1 Zone 2 Zone 3 Zone 5

Central Office

DWDM

DW

DM

Fiber Node 16 Per

Secondary Hub

Zone 4

Transport

AD

M

AD

M

GR-303 GR-303

DA

CS

IP

Proprietary Protocol

TDM/T-1

DHCP/DNS TFTP

Servers

EMS SystemsDOCSIS

Protocol


Effect of Long Holding Times (Cont’d)

Preliminary Analysisproblem in evening hours 7 PM – 11 PMno discernible spatial pattern

Push-backsEngineering followed “standard” proceduresOnly 5-8% of calls are internet dial-upsField measurements “do not support” congestion hypothesis


Phase Type Model(using the EM-algorithm) ; fit uses ~4M data points

Phase Type Fit

Probability Observed Fitted0.1 5.4 5.80.2 12 12.70.3 21 21.20.4 32.4 32.20.5 48 47.70.6 72.6 72.20.7 120.6 120.20.8 232.8 235.90.9 601.8 597.6

0.95 1237.8 1268.00.99 3952.2 4035.2

0.995 6074.4 7168.4


The Length Biasing EffectRemaining holding time distribution: [1-F(x)] / μ

RESIDUAL DISTRIBUTIONh(x)=[1-F(x)]/μ

Probability Observed Residual0.1 5.4 40.80.2 12 116.10.3 21 238.00.4 32.4 423.70.5 48 703.00.6 72.6 1124.60.7 120.6 1809.70.8 232.8 3285.30.9 601.8 7028.2

0.95 1237.8 11073.80.99 3952.2 20489.2

Median 48 703.0Mean 297 2367.0


Empirical Validation Empirical data on residuals (~11 K observations) vs.

Model


Impact of Long Holding TimesInsensitivity relates only to blocking over an infinite horizon.

Heavy tails PERSISTENCE of congestion. So, short term performance is quite different.“Standard” procedures do not work !

Bad periods compensated by long very good periods. Good situation for control !!! (See the paper and patents for possible controls.)


Several Other Real Life Applications

Excessive Link Oscillations in the Common Channel Signaling (SS7) Network [Ramaswami and Wang, 1993]Solved using Phase-type distributions

Modeling and characterization of large-scale Wi-Fi traffic in public hot-spots [Ghosh, et al., 2011]Solved using an M/G/∞ where G is ln PH

distribution


The Batch Markovian Arrival Process (BMAP)

Natural generalization of the Poisson process

Contains many well-known arrival processes (in all cases, correlated batch sizes are easily handled)Poisson process (m=1)PH renewal processMarkov modulated Poisson process (MMPP)Superposition of BMAPs is again a BMAP

Useful for modeling arrivals to an internal node in a network


The Batch Markovian Arrival Process (Con’t) Consider an m-state Markov process and assume that at

each transition of this process, there is a probability of a batch arrival that depends both on the state before and after the transition

Let Dn, n≥1, denote the infinitesimal rate of a batch arrival of size n, keeping track of the underlying states before and after the transition

The generating function D(z) is given by


BMAP/G/1 Queue The BMAP/G/1 queue is extremely useful for modeling the

performance of broadband packet networks Traffic is bursty Service times are NOT phase-type

Asynchronous Transfer Mode (ATM) Fixed length cells result in deterministic service times

IP networks Finite packet sizes finite mixture of deterministic service

times

BMAP models very bursty traffic and is closed under superposition, i.e., modeling the output of nodes entering another node


Busy Period of the BMAP/G/1 QueueLet h(s) be the Laplace-Stieltjes transform of the

service time distributionLet G(z) be the probability generating function of

the number of customers served during a busy period (keeping track of the arrival phase). Then we have


Busy Period of the BMAP/G/1 Queue (Cont’d)

Define G to be G(1) and g to be the stationary probability vector of G

It is easy to show that g is also the stationary probability vector of the phase of the arrival process during idle periods

For M/G/1, the probability that the system is empty is 1-ρ; for BMAP/G/1, the probability that the system is empty and that the arrival process is in phase j is the jth element of (1-ρ)g


Virtual Waiting Time of the BMAP/G/1 Queue

Let W(s) be the Laplace-Stieltjes transform of the virtual waiting time distribution (keeping track of the arrival phase)

Let g be the stationary probability vector of the stochastic matrix G

Then we have


Stationary Queue Length at DeparturesLet Xn be the stationary probability that there

are n customers in the queue at departuresLet , then


Computing these DistributionsWe compute these distributions by numerically

inverting the transforms [Choudhury, Lucantoni and Whitt, 1994]

These algorithms are for inverting multi-dimensional transforms which we use later for computing transient distributions


Known Results for the BMAP/G/1 Queue

Stationary distributions [Lucantoni, 1993] Queue length distribution at arrivals, departures, arbitrary

time Waiting time distributions

Transient distributions (given the appropriate initial conditions) [Lucantoni, Choudhury, and Whitt,1994], [Lucantoni, 1998] P(system empty at time t) Workload at time t Queue length at time t Delay of nth arrival Queue length at nth departure P(nth departure occurs less than or equal to time x)


Modeling the Input to an Interior Network Node

Consider the output from several nodes which all go to the same node

Ideal candidate for the MAP/G/1 queue


Multiplexing Bursty Traffic in an ATM Network

ATM requirement is that the probability of blocking must be less than 10-9

Tails of distributions are the most important

performance measure

Superposition of 64 on-off sources each modeled as a 2-state MMPP

Exact Analysis

Far from Poisson

Exponential tail does not become relevant until blocking is 10-40


Advantage of MAP/G/1 Approach over Matrix-Geometric Approach When service

time has Many Phases

We computed the tail of the waiting time distribution in an MMPP64/E1024/1 queue by an by a distribution that is close to an E1024 and matches the first three moments of the deterministic distribution

The matrix-geometric approach would require using matrices of order 65,536

The MAP/G/1 approach looks at the process embedded at departures; since we don’t have to keep track of the phase of service, the matrices are only of order 64


Advantage of MAP/G/1 (Cont’d)

G satisfies

When H is E1024, the integral on the right hand side is computed by inverting a 64×64 matrix 10 times

Much easier than computing the matrix exponential


IP Data Traffic is Bursty

Sample Measurements from a previous assignment

Sampled every second

Highly burstyMean = 720 pkts/sec


0 1 2 3 40

500

1000

1500

2000

2500

3000

3500

4000

4500

5000Packet Counts

Hours

Pac

kets

Simulated Poisson Traffic

Poisson traffic looks bursty

However, note the scale of the y-axis


0 5 10 15640

660

680

700

720

740

760

780

800Packet Counts - Poisson

Minutes

Pac

kets

Poisson Process Compared to Data


0 5 10 150

500

1000

1500

2000

2500

3000

3500Packet Counts - Poisson

Minutes

Pac

kets

0 5 10 150

500

1000

1500

2000

2500

3000

3500Packet Counts - Measurements

Minutes

Pac

kets

Fit an D-MMPP to Data[Heyman, Lucantoni,

2003]Choose λ1 so that the

peak is at the 95th percentile of a Poisson distribution, i.e.,

Then


Fit an MMPP to Data (Cont’d)Lower bound of data covered by λ1 is

Let this be the upper bound covered by λ2, i.e.,

Then

Continue…


Fit an MMPP to Data (Cont’d)Compute the transition probabilities by

For performance computations convert to a continuous time MMPP (see paper)


Fit an MMPP to Data

Q-Q plot: plots quantiles

Should be linear if both come from same distribution

Very good fit


Simulated MMPP Traffic


0 5 10 150

500

1000

1500

2000

2500

3000

3500Packet Counts - MMPP

Minutes

Pac

kets

0 5 10 150

500

1000

1500

2000

2500

3000

3500Packet Counts - Measurements

Minutes

Pac

kets

Another Example


0 5 10 150

100

200

300

400

500

600Simulated MMPP Packet Counts

Minutes

Pac

kets

0 5 10 150

100

200

300

400

500

600Measurement Packet Counts

Minutes

Pac

kets

Call Acceptance for a VoIP System A startup company was offering VoIP service to clients by buying a fixed

bandwidth virtual circuit (“fat pipe”) through the Internet

Wanted to know how big the fat pipe should be to handle a certain number of voice calls and still meet the Quality of Service (QoS) e.g., tail of the delay distribution

Measurements were taken for IP voice calls for different codecs, e.g., G.723, G.729, etc., and different measurement intervals

We developed a program where the user inputs the file containing the measurements, the measurement interval and the target delay


Call Acceptance for a VoIP System (Cont’d)We fit an MMPP to the measurement data and by solving

the delay using an MMPP/D/1 queue

If the computed QoS is greater or less than the targeted QoS, we adjust the service rate and re-compute; continue adjusting the rate using a binary search until the computed QoS is within a given tolerance

The program then outputs the size of the required fat pipe and the required buffer size


Notes on Transient ResultsGeneralized every result derived by Takács for

the transient M/G/1 queue [Takács, 1962]

All results are direct matrix analogues of the scalar results

Using only probabilistic arguments, resulted in expressions that can be computed numerically without computing the roots of an analytic equation


Sample Transient Result (emptiness function)Consider a MAP/G/1 queue (similar results hold for the

BMAP/G/1 queue)

Let Px0(t) be an m×m matrix where the (i,j) entry is the probability that the system is empty at time t with the arrival process is in phase j given the at time 0 the work in the system is x and the arrival process is in phase i

Let px0(s) be the Laplace-Stieltjes transform of Px0(t)


Emptiness Function (Cont’d)Then


Transient Performance MeasuresWe compute the virtual waiting time (work in

system) at time t for a given queue length at time 0 by numerically inverting the 2-dimensional transform, W(s1,s2)


Sample Transient Results -Virtual Delay Distribution

Superposition of four identical two-state MMPP’s

Service times are distributed as E16

Utilization = 0.7queue length =

0 at time 0queue length =

32 at time 0


Sample Transient Results -Unstable System

Utilization (ρ) = 2.0

Queue length = 0 at time 0


Potential Use of Transient ResultsCall acceptance algorithms

Current call acceptance algorithms are based on stationary probabilities

Given the number of calls in the system, the time to reach a stationary distribution can be MUCH longer than the inter-arrival and inter-departure times

Call acceptance tables could be computed a-priori based on decision time intervals, the current state of the system, the average and peak bandwidth and burstiness parameter of the arriving call


References1. V. Ramaswami, D. Poole, S. Ahn, S. Byers, and A. Kaplan, "Assuring emergency

services access: .providing dial tone in the presence of long holding time internet dial-up calls," Interfaces, vol. 35, pp. 411-22, 2005

2. V. Ramaswami and J. L. Wang, "Analysis of the Link Error Monitoring Protocols in the Common Channel Signaling Network," IEEE/ACM Transactions on Networking, vol. 1, pp. 34-47, 1993

3. Ghosh, R. Jana, V. Ramaswami, J. Rowland, and N. K. Shankaranarayanan, "Modeling and characterization of large-scale Wi-Fi traffic in public hot-spots," in IEEE INFOCOM, Shanghai, China, 2011

4. D. M. Lucantoni, "The BMAP/G/1 queue: A tutorial," in Models and techniques for Performance Evaluation of Computer and Communications Systems, L. D. a. R. Nelson, Ed., ed: Springer Verlag, 1993, pp. 330-58

5. D. M. Lucantoni, G. L. Choudhury, and W. Whitt, "The transient BMAP/G/1 queue," Stoch. Mod., vol. 10, pp. 145-82, 1994

6. D. M. Lucantoni, "Further transient analysis of the BMAP/G/1 queue," Stoch. Mod., vol. 14, pp. 461-78, 1998


References (Cont’d)7. Heymen and Lucantoni, “Modeling Multiple IP Traffic Streams with Rate

Limits,” IEEE/ACM Trans. On Networking, 11, No. 6, 2003

8. Choudhury, Lucantoni and Whitt, “Multidimensional transform inversion with applications to the transient M/G/1 queue," Ann. Appl. Prob ., 4, No. 3, 719-740, 1994

7. L. Takács, Introduction to the Theory of Queues. New York: Oxford University Press, 1962


Where to Find Additional References

Some Lucantoni papers can be downloaded from www.DLTconsulting.comFollow the links to my Publications page (or just

Google my name)More references are listed in those papers


http://www.DLTconsulting.com/

Documents

Some Real Life Applications