ppke itk 2011/12 tanév Őszi félév tájékoztatás gosztony/ 3.6. multi-service queueing systems...

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3.6. Multi-service queueing systems

3.7. Queueing networks

4. Traffic measurement

Final overview

Infocomm network’s planningtraffic aspects

2Infocomm network’s planning - traffic aspects - 2011.10.05.

Textbook: Chapter 11

Multi-service queueing systems

3Infocomm network’s planning - traffic aspects - 2011.10.05.3

Multi-service queueing systems – 1.There are more than one type (class, service, stream) of customers

Customers of same type (class, service, stream) belong to a specfic chain, a queueing system is a node in a queueing network

In reversible systems , the departure process is of the same type as the arrival process, in the investigated cases: Poisson processes. In this case several queueing systems may be combined into a network of queueing systems.

In classical queueing systems we have non-sharing. A customer is either waiting or being served. When being served it has a server alone. There are queueing systems with different sharing strategies. Customers may share servers with other customers so that no one is waiting, but always served with some rate, which may be smaller than the requested rate.

By requesting reversibility and usage of all servers whenever possible we get the processor sharing (PS) and generalised processor sharing strategy (GPS).

Multi-service queueing systems – 2.

In multi-service queueing systems and in queueing networks customers in some way share the available capacity, and therefore they are served all the time. But they may obtain less capacity than requested, which results in an increase of sojourn time. The sojourn time is not split up into separate waiting time and service time. For multiservice queueing systems and for queueing networks the definitions below are used:

jjj sWW

In multi-service queueing systems and in queueing networks the waiting time is defined as the total sojourn time , including the service time, s. E.g. if in the Internet, the bandwidth is smaller than requested, then the mean transfer time Wj will be bigger then sj and the increase is defined as the mean virtual waiting time:

In a similar way the mean virtual queue length i.e. Lj is defined as:

jjj ALL

where Aj is the offered traffic of type j.

Multi-service queueing systems – 3.Example.

In general there are N PCT1 input processes with j arrival and μ1 departure intensities (j = 1 … N)

(The exact value of the μi departure intensities requires further consideration !)

PS: Processor SharingPR: Preemptive Resume

Queueing networks

Introduction to queueing networks Symmetric queueing systems, Jackson’s theorem Closed networks: single chain Closed networks: several chains Other questions

A queueing system consists of nodes (group of servers with the same function) and jobs (requests) travelling from node to node.

In queueing networks we define the queue-length in a node as the total number of jobs in the node, including delayed and served jobs.

A queueing network might be:1. open – the number of jobs may change, i. e. M/M/n

2. closed– the number of jobs is fixed, i.e. Palm’s machine repair model

The departure process from one node is the arrival process at another node, special attention should be payed to the departure process

Queueing networks, introduction – 1.

Queueing networks, introduction – 2.

Four nodes.Four open chains.

State:

where: gives the number of jobs in node k

pj,k is the probability, that a jobt having left node j. is directed to node k.

The queueing system is symmetric if both the arrival and the departure processes are Poissonian.

The four models:• M/M/n

state probabilities

• M/G/∞*state probabilities(Poisson !)

• M/G/1–PS* state probabilities

• M/G/1-LCFS-PR* state probabilities * immediate service!

Symmetric systemsR

PS = Processor SharingPR = Preemptive Resume

Jackson’s theorem– 1a. Jackson’s theorem: Consider an open queueing network with K nodes satisfying the following conditions:

Jackson’s theorem– 1b.

Jackson’s theorem – 1c.

The key point of Jackson's theorem is that:

each node can be considered independently of all other nodes and that the state probabilities are as for Erlang's delay system. (Erlang’s C formula.) This simplifies the calculation of the state space probabilities significantly.

Jackson's first model thus only deals with open queueing networks.

Example:

Open queuing network consisting of two M|M|1 systems in series

Jackson’s theorem – 2.

In Jackson's second model the arrival intensity from outside:

may depend on the current number of customers in the network,

furthermore, μk can depend on the number of customers at node k.

In this way, we can model queueing networks whichare either closed, open, or mixed. In all three cases, the state probabilities have product form.

Independence assumption

Kleinrock’s independence assumption

If we consider a real-life data network, then the packets will have the same constant length, and therefore the same service time on all links and nodes of equal speed. The theory of queueing networks assumes that a packet (a customer) samples a new service time in every node. This is a necessary assumption for the product form. This assumption was first investigated by Kleinrock (1964 [74]), and it turns out to be a good approximation in praxis.

Single open chain

Open system

One has to find the state probabilities

where ik is the number of requests in node k.

Steps: 1. solution of

2. using μi –s one may get Ai –k.

3. by considering Erlang's delay system one may get the state probabilities for each node.

Single closed-chain – 1. Convolution algorithm for closed networksWe only know the relative load at each node, not the absolute load, i.e. cΛj is obtained, but c is unknown. We can obtain the relative state probabilities. Finally, by normalizing we get the normalized state probabilities. Large system complex run.Steps:

Single closed-chain – 2. Convolution algorithm for closed networks

Single closed-chain – 3.

Single closed-chain – 4. Terminals M/G/1 – IS* node

CPU M/M/1 node

*IS = Immediate Service – There is always a free terminal for the new task.

Example 12.5.1

λ1= λ

λ2= λ

See details inthe Textbook

Single closed-chain – 5. Assumptions: S constant.

S circulating jobs. The CPU andthe I/O channelsserve each job several times. A departing job willimmediatly replaced by a new on.

Example 12.5.2 S = 4K = 3 (CPU + 2 I/O)

exponentialholding timess = 1/μ

MVA (Mean Value Algorithm)

K nodes, S jobs (in a single chain), αk = λk sk relative traffics.Recursion according the number x of jobs.At node k there are Lk(S) jobs on the average.

PS = Processor SharingPR = Preemptive Resume

average sojourn time

immediate service

There is no waiting in the system, if S=1

Example for nk = 1 but the method might be generalized for any nk .

kk sW 1 kk sW 1

Single closed-chain – 9. Example 12.5.3 (= 12.5.2 but with MVA)

K=3, S=4,

relative k values:

Recursion formulae:i

iiii s

BCMP queueing networks

Queueing networks with more than one type ofcustomers also have product form state probabilities.(Generalization of Jackson’s second model. BCMP Baskett,Chandy, Muntz és Palacios - 1975)

Necessary conditions:

BCMP–networks can be evaluated with the multi-dimensional convolution algorithm and the multidimensional MVA algorithm.

Mixed queueing networks

Mixed queueing networks (open & closed) are calculated by first calculating the traffic load in each node from the open chains. This traffic must be carried to enter statistical equilibrium. The capacity of the nodes are reduced by this traffic, and the closed queueing network is calculated by the reduced capacity. So the main problem is to calculate closed networks. For this we have more algorithms among which the most important ones are convolution algorithm and the MVA (Mean Value Algorithm) algorithm.

Complexity

Traffic measurement

Principles and methods Theory of sampling Continuous measurement Scanning

Introduction Traffic measurement gathering of data about the

traffic of real or fictisous systems which serve requests arriving in a random way.

A minimum of technical and administrative efforts should result in a maximum of information and benefit.

A measurement during a limited time intervalcorresponds to a registration of a certain realization of the traffic process.

Margin of error condfidence interval.

For practical purposes it is in general sufficientto know the mean value and the variance.

Principles, methods – 1.

What do we measure? How do we measure?

Eventsarrival of requests, number of jobs,number of lost requests,etc.

Time intervalsholding times,waiting times,execution times of jobs,etc.

Continuous measurementthe measuring point is active, it activates the measuring equipment at the instant of the event

Scanningthe measuring point is passive,the measuring equipment itself tests from time to time, if any changes have taken place.

Continuous measurement - examples:

• electro-mechanical counters (e.g. number of copies)

• x-y plotters (earthquake survaillance)• Ampere-hour meters• water meters

Scanning - examples:

• call-charging impulses• measuring of carried traffic by repeated scans

Principles, methods – 3. Busy state of servers:

actual

scanned

Registration of changes

Theory of sampling – 1.

A sample of n IDD* observations withunknown finite mean value m1 and finite varianceб2 are available (population parameters)

The average value and variance of the sample:

These are functions of arv., they are rv.-s also (sample distribution) andrepresent an estimationof the average value and variance of the unknown population:

(correctedempiricalvariance)

*IDD= Independent and Identically Distributed

The accuracy of an estimate of a sample parameter is described by means of a confidence interval :

This is valid, if the samples are independent. Independence is fulfilled, if measurements take place on different days, but samples are not independent for samples taken by scanning in a limited time interval.

Theory of sampling – 3.Percentiles of the t−distribution with n degrees of freedom. A specific value of α corresponds to a probability mass α/2 in bothtails of the t−distribution. When n is large, then we may use the percentiles of the Normal distribution.

Mean valueof sample

Correctedempiricalvariance

Example 13.2.1:

Continuous measurement– 1.

Relationships valid forstochastic sums might in principle be applied if the measuring periodis unlimited.

Application in practiceis possible withprecaution !

Time intervals markedby full lines are measured in casea. and b., respectively.

Stochastic sum - 1.From the mathematical point of view the evaluation of the measurement consist of calculating the sum of a random amount of random variables

Service without congestion.Arrival process and holding times are independent.

The number of arriving requests in the given period of length T is a random variable: N.

Ti. gives the holding time of thei-th incoming request. Distribution of Ti –s is uniform.

Total traffic in T

See 2.3.3 of Textbook

Stochastic sum - 2.

Ti and N are stochastically independent

A stochastic sum may be interpreted as a series/parallel combination of random variable.

The problem in graphical form:

Stochastic sum 3.For a given i-th branch:

For all i branches:

variance, since the numberof holding times is an r.v.

variance, since the holdingtime is an r.v.

mean value

variance

second non-central moment

Continuous measurement – 2.Applying the relationships of the „stochastic sum”:

The amount of traffic (holding times) and trafficintensity (arrival of requests) should be independent.This is fulfilled, if congestion is negligible.Assumption: Poisson input process

Number of requests in interval T:

Thus the requiredservice time i.e. traffic:

The first and second non-central momentsof the holding time.

Palm’s form factor

Continuous measurement – 3.

The distribution ST of the stochastic sum is a compound Poisson distribution.

represents amount of traffic.

The average value of the number occupied serversrepresents traffic intensity = the amount of trafficper time unit if the time unit equals theaverage holding time:

Valid for any holding time distribution !

Continuous measurement – 4.

Independent of holding time distribution.

Depends on holding time distribution

Relative accuracy of measurement:

The measurement of smaller intensities is more precise!!

See the followingFigure !

Figure 1.18: Frequency function for holding times of trunks in a local switching centre.

Result of holding time measurements

Scanning measurement – 1.

Constant scanning interval (h)

The continuously distributed holding timeis approximated witha discretely distributed holding time.

The continuous timeintervals may overlap.Estimation is thereforemore difficult.

If the real holding times have a distribution function F(t), then it can be shown that one will observe the following discrete distribution :

It can also be shown that for any distribution function F(t) one will always observe the correct mean value:

For exponential holding times –the so called Westerberg distribution may be observed:

Sampling measurement

2A/T ≠

but it =, if h 0

Measurement by scanning = sampling from a sample!!

Continuous measurement

for exponential distribution:

Final overview – 1. Info-communication systems handle service requests

the totality of which is simply denominated as traffic. Traffic handling capacity has an influence on service

quality. Arrivals and holding times of service requests in info-

communication systems have random nature, their forecasting is not simple.

Traffic handling capacity can be estimated by mathematical models and computer simulation.

Nowadays the importance of network traffic management methods e.g. software-based routing algorithms is increasing.

Network planning requires some understanding of traffic engineering.

Final overview –2.

TTE tasks planning,

performance evaluation, operation,

maintenanceof infocomm. systems

infocom trends

rapidly growingtraffic demands

What is traffic ?How does it look like ?

intensity ofincomingrequests,

holding times

GoS QoSSLA (Service Level Agreement)Network Traffic Management

What happens to rejected traffic

requests ?

loss, rerouting,limited waiting,

unlimited waiting

Erlang’loss model,constant intensity of requests

Engset’s loss model,state dependent intensity of requests

BPPtraffic models

Overflow traffic has a peaked character !!

Network traffic

management

Best service with available resources in a randomly fluctuating situation

Specification Strategies Priorities Behaviour of requests

Queuing systems

M|M|nErlang

+ M|M|1|S|SPalm

Generalresults

Little’s theoremPollaczek-Khintchine’s formula

Systems mentioned:

M|G|1, M|G|1|kM|D|1, M|D|n

GI|G|1(some with priorities)

processor sharing

Queueing with shared serviceMean virtual waiting timemean virtual queue length

Several chains of jobsreversibility

links and nodes structure

Multiservicequeueing systems

Queueing networks

ClosedOpen

Convolution algorithmMean Value AlgorithmJackson’s theorems:

Kleinrock’sindependence assumption !

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