3.1

PPKE ITK

2011/12tanév

Őszifélév

Tájékoztatáshttp://digitus.itk.ppke.hu/~gosztony/

3. Queueing systems

PathwayNew York CityManhattanCentral Park

Infocomm network’s planningtraffic aspects

2Infocomm network’s planning - traffic aspects - 2011.09.28

3.1

Characterization of waiting systems

NotationQueuing strategies

PrioritiesBehaviour of requests


Characterization 1. Characterization of systems

Traffic processes (distributions):Whatabout these ?

Kendall notation


Characterization 2. Complete specification:

Further notations: K = n is a loss system

Ab csoportos érkezés (bulk or batch arrival).Bb csoportos kiszolgálás, (bulk or batch service)C ütemezett kiszolgálás (clocked service)


Characterization 3. Queuing strategies

For the three above-mentioned disciplines the total waiting time for all customers is the same. The queueing discipline only decides how waiting time is allocated to the individual customers.


Characterization 4.

Static disciplines, depending on arrival times or holding timesDynamic disciplines. Strategy depends on the time spent in the system


Characterization 5. Priorities

The strategies mentioned may apply in each priority class


Characterization 6. Behaviour of requests (customers)


3.2

Erlang’s delay system

M|M|n

Detailed mathematics serve the better understanding

of essential results

For further details see: Textbook, Section 9.


Erlang – M/M/n 1.The system• n uniform servers• full accessibility • ∞ waiting positions• PCT I flow of demands

The state of the system is given by the number of all (served and waiting) requests.


Erlang – M/M/n 2.State equations A=/μ


Erlang – M/M/n 3.Probability of waiting:

a request arrives and all servers are occupied______________________________________________________ a request arrives anytime

Erlang C formula:

Designations:

Probability of immediate service


Erlang – M/M/n 4.Served traffic (= offered !)

Probability of havingrequests in the queue:

Length of the queue as random variable = L

Applied formulas:if i < n

)1i(p.n)i(p. if i ≥ n

Ann)n(p)A(E n,2


Erlang – M/M/n 5.Calculation of Erlang C 1.

2.

where

from the recursion formulaused for Erlang B calculation

Tabular calculation aid:GG Home Page, GyakorlatokErlang C táblázat

A (traffic), from anyN (no. of servers) two theErlang C (prob. wait.) third


Erlang – M/M/n 6.


Erlang – M/M/n 7.Mean queue length in an arbitrary point of time:

PASTA !Time and spaceaverages are thesame


Erlang – M/M/n 8.Mean queue length – at an arbitrary point of time

the series is uniformly convergent, and the differentiation operator may be put outside the summation

May be understood as the traffic of waiting positions.

PASTA !

If then:


Erlang – M/M/n 9.Mean queue length – if there is any queue

Conditional probability.Condition:

=PASTA !


Erlang – M/M/n 10.Mean waiting time – all requests

From Little’s theorem

where:

(arrival rate) x(mean waiting time)

further,

since L might be interpreted as waiting traffic

and because of


Erlang – M/M/n 11.Mean waiting time wn– for delayed requests

0WpWw n

n

wn (conditional probability) =

= mean waiting time – all requests probability of waiting


Summary - Erlang – M/M/n 12.

Average waiting time – waiting requests:

Average waiting time – all requests:

Average queue length– if there is a queue:

Average queue length– arbitrary point of time:

Waiting requests exist– arbitrary point of time:

Carried traffic(= offered!)

Probability of waiting:

Probability of immediate service:


Erlang – M/M/n 13.Example: M/M/1 – Single server queueing system

from

since: A1

10pA1

A10pA)0(p0p11i

i

and

See Textbook 9.5


Erlang – M/M/n 14.FCFS/FIFO

first infirst outLCFS/LIFOlast in first outSIRO/RANDOM service in random order


3.3

M|M|n|S|S

Palm’s machine repair model

Detailed mathematics serve the better understanding

of essential results

For furtherl details see: Textbook, Section 9.


The model The system• PCT-II requests• n identical servers

and full accessibility• S waiting positions

• S traffic sources

M/M/n/S/SPalm’s machine repair model

(PCT-II waiting system)


Palm – M/M/1/S/S – 1.

Sources of requests

Request(thinking) intensity

Serviceintensity

„Sitting in front of the terminal”

Figure 9.8: Palm’s machine-repair model. A computer system with S terminals (an interactivesystem) corresponds to a waiting time system with a limited number of sources (cf.Engset-case for loss systems).

Palm’s machine repair model (PCT-II waiting system)


Palm – M/M/1/S/S – 2.

think time +responsetime

R (response)=waiting +service

Tt mt

Tw mw

Ts ms

Time intervals and mean values:

Tt + R =circulationtime =


Palm – M/M/1/S/S – 3.

i = 0, the computer is idlei > 0, the computer is busy, (i-1) requests are waiting, (S-i) requests are thinking

cut equations !service intensity

thinking intensity

Similar to an Erlang loss case with λ μ and μ γ


Palm – M/M/1/S/S – 4.

truncated Poissondistribution

all are thinking, empty queue, no service is ongoing

Erlang B :S servers and ρ traffic

„traffic”:(service ratio)


Palm – M/M/1/S/S – 5.

As if (!): the computer would generate requests of 1/ average holding time with μ intensity sending them to a group of S servers

The result is independent of the thinking time distribution, if the service time of the computer isexponentially distributed

Theorem 9.1 The state probabilities of the machine repair model (9.36) & (9.37) with one computer and S terminals is valid for arbitrary thinking times when the service times of the computer are exponentially distributed.


Palm – M/M/1/S/S – 6. Characteristic average values (using Erlang B formula:

served terminals: One terminal is served in all states except p(0)

waiting terminals:(Subtracting theserved and the thinkingterminals from from allterminals.)

thinking terminals:(traffic carried in the Erlang loss system)Average value of thinking terminals.


Palm – M/M/1/S/S – 7.

Probabilities (from previous results divided by S, i.e. the number of terminals) random terminal at a random point of time :


Palm –M/M/1/S/S – 8-1.

Circulation rate of jobs

mx (average holding timeof requests)

nx (average number of requests)

Average value of R :

R (response time): See this later:


Palm –M/M/1/S/S – 8-2.

The mean valu of R (response time) is independent of holding time distributions, it depends only on p(0)=E1,S(ρ) and on the average values.

or

applying

The „circulation rate of jobs” is the same for terminals, for waiting positions and for the computer. Therefore based on Littles’ theorem:


Palm – M/M/1/S/S – 9.

Based on theR formulaexpressedin [ms]

Fig. 9.11

dimension

If S=1, thenR = ms = 1/µand mw = 0, since no waiting is required.


Palm – M/M/1/S/S – 10.


Palm – M/M/1/S/S – 11.

Fig. 9.12The waiting time traffic (the proportion of time spend waiting) measured in erlang for the computer, respectively the terminals in an interactive queueing system (Service factor % = 30).


Palm – M/M/1/S/S – 12. Traffic congestionWe may define the traffic congestion in the usual way (Sec. 1.9). The offered traffic is the traffic carried when there is no queue.

The offered traffic per source is (5.10):

The carried traffic per source is:

The traffic congestion becomes:

In this case with finite number of sources the traffic congestion becomes equal to the proportionof time spent waiting.

11

1

11


Palm – M/M/n/S/S – 1. There are n servers (eg. computers )

State equations and normalisation:

State equations are independent of the „thinking time” distribution.


Palm – M/M/n/S/S – 2. Possible states:

Average utilization ofcomputers:

Average waiting time ofterminals:


Palm – M/M/n/S/S – 3. Example 9.6.4: Parameters used: S/n = 30, μ/ = 30, n = 1-16

One may obtain the highest utilisation for large values of n (and S). (in this case pt=α ).

dimension

S 30 60 120 240 480


3.4

General results

Little’s theoremPollaczek-Khintchine’s formula

M/G/1 and M/G/1/k systemsM/G/1 system with priority



General results – 1 • Many special cases, few general results.• Little’s theorem is generally applicable to arbitrary

delay systems.• Systems with Poisson input processes might simply

be handled from the mathematical point of view.• So called symmetric queuing systems are important

for queuing systems in series and for queuing networks, since the input and output processes both are Poissonian.

• The classical queueing models play a key role in the queueing theory, because other systems will often converge to these when the number of servers increases (Palm‘s theorem)


General results – 2

In waiting time systems we also distinguish between call averages and time averages. The virtual waiting time is the waiting time, a customer experiences if the customer arrives at a random point of time (time average). The actual waiting time is the waiting time, the real customers experiences (call average). If the arrival process is a Poisson process, then the two averages are identical (PASTA property).


Little’s theorem 1.Both arrival and departure processes are considered as stochastic processes ...

Designations:

requests in thesystem

=

=

=time between thek-th arrival and the k-th departure

time spent in the system


Little’s theorem 2.


Little’s theorem 3.

Valid for all general queuing systems !!


Little’s theorem 4.Examples:For waiting positions:

For servers:

)T(W)T()T(L


Palm’s form factorA measure of irregularity is Palm’s form factor ", which is defined as follows:

Remember:m1= first central moment mean value or expected valuem2 = second central moment2 = variance

and


Pollaczek-Khintchine’s formula(M|G|1) 1.

W is the mean waiting time for all customers, s is the mean service time, A is the offeredtraffic, and ε is the form factor of the holding time distribution.

For a smaller form factor, i.e. in case of a more uniform service time the average waiting time is also smaller.

For telephone traffic: ε = 4-6, for data traffic: ε = 10 -100.

Form factor:(s = m)

sA

since:


Steps of derivation:

The average waiting time W for an arbitrary request: 1. The residual mean service time of the request just served, if there is any (probability: A ) 2. Waiting time of queuing requests. The average length of the queue L might be calculated using Little’s formula. On the average one has to wait a period of s due to every waiting request.

Pollaczek-Khintchine’s formula(M|G|1) 2.


Busy period – M|G|1 Figure validfor anM/D/1 system

Generally valid formula

ssms1m

mmsm

01

101

TT

TTT

If we only consider customers, which are delayed, we are able to find the moments of the waiting time distribution for the classical queueing disciplines FCFS and LCFS.See Textbook: 10.3.3.


Limited queue length – M|G|1|k –1

where:A < 1 and:

k represents: 1 server + (k-1) waiting positions. Methods are available to calculate p(i) for arbitrary holdingtime distributions. – The finite system is in statisticalequilibrium for A>1.

See Textbook: 10.3.4

Relationship exists between the state probabilities of the M|G|1 and M|G|1|k systems


Limited queue length– M|G|1|k –2

Tail Drop, or Drop Tail, is a simple queue management algorithm used by Internet routers to decide when to drop packets. In contrast to the more complex algorithms like RED and WRED, in Tail Drop all the traffic is not differentiated. Each packet is treated identically. With tail drop, when the queue is filled to its maximum capacity, the newly arriving packets are dropped until the queue has enough room to accept incoming traffic.

The name arises from the effect of the policy on incoming datagrams. Once a queue has been filled, the router begins discarding all additional datagrams, thus dropping the tail of the sequence of datagrams. The loss of datagrams causes the TCP sender to enter slow start, which reduces throughput in that TCP session until the sender begins to receive ACKs again and increases its congestion window. A more severe problem occurs when datagrams from multiple TCP connections are dropped, causing global synchronization, i.e., all of the involved TCP senders enter slow-start. This happens because, instead of discarding many segments from one connection, the router would tend to discard one segment from each connection.

Wikipedia – 2011.09.

Example

http://en.wikipedia.org/wiki/Random_early_detection

http://en.wikipedia.org/wiki/Weighted_random_early_detection


Limited queue length – M|G|1|k –3

Wikipedia – 2011.09.

Random early detection (RED), also known as random early discard or random early drop is an active queue management algorithm. It is also a congestion avoidance algorithm.

Pure RED does not accomodatequality of service (QoS) differentiation

Weighted RED (WRED) and RED with In/Out (RIO) provide early detection with some QoS considerations.

Example

http://en.wikipedia.org/wiki/Active_Queue_Management

http://en.wikipedia.org/wiki/Algorithm

http://en.wikipedia.org/wiki/Network_congestion_avoidance

http://en.wikipedia.org/wiki/Network_congestion_avoidance

http://en.wikipedia.org/wiki/File:Random_Early_Detection_algorithm_en.svg


M|G|1 with several traffic flowsSeveral traffic flows

Assumptions: N Poisson traffic flows i input intensity, si average holding time, m2i seconf momnet, Ai = i si Offered traffic. For the total input process we have:Intensity: Average holding time: Second moment:

Offered traffic:

Weighted averages

Residual service time in a random point of time :

All factors of the Pollczek-Khintchine formula are available!

V

N

iiVV

1

A1VW


Kleinrock’s conservation law

Kleinrock’s conservation law: The average waiting time for all classes weighted by the traffic (load) of the mentioned class, is independent of the queue discipline.

Valid only for non-preemptive queuing discipline !

both are constant


M|G|1 with priority – 1. Non-preemptive priorityN priority classes, a p class has higher priority than p+1, p intenzity, sp average holding time.

Calculation of the total average waiting time Wp :

p

1p

1iiW

higher priorityrequests arriving duringthe waiting time

Textbook 10.6.3


M|G|1 with priority – 2.

Textbook: 10.6.3.

a) + b) + c)

All customers wait until the service in progress is completed {V1,N} no matter which class they belong to. Furthermore, the waiting time is due to already arrived customers of at least having the same priority {A0,p}, and customers with higher priority arriving during the waiting time {A 0,p-1}.


SJF queuing discipline (this is also non-preemptive priority)A request with service time t has the mean waiting time Wt:

A0,t is load from the customers with service time less than or equal to t.

If these different priority classes have different costs per time unit when they wait, so that class j customers have the mean service time sj and pay cj per time unit when they wait, then the optimal strategy (minimum cost) is to assign priorities 1, 2, . . . according to increasing ratio sj/cj .


The SJF discipline results in the lowest possible total waiting time.

(An infinite number of priority classes has to be assumed.)



M|M|1 withFCFS and SJFqueue discipline

Ha a kiszolgálási idő< 2.747 átlagos tartásidő, akkor az SJF kiszolgálás kisebbátlagos várakozási időt ad, mint az FCFS.Ez érinti a hívások 93.6 %-át. FCFS calculation:

Example 10.6.2 időegység= s (átl. tartásidő)

s99.01

s9.0EW 1,2FCFS

WFCFS =


M/M/n with non-preemptive priorityGeneralization of Erlang’s classical waiting time systemi intensity, s=1/μ average holding time in all classes.

Calculation of the total average waiting time Wp : Erlang’s C formula.

A is the offered traffic by all priority classes. When all servers are busycustomers are served with the mean inter-departure time s/n.

For further mathematical transformations see Textbook: 10.6.5


M|G|1 with preemptive resume ... - 1The mean waiting time Wp for a customer in class p:


M|G|1 with preemptive resume ... - 2

For the SJF preemptive resume discipline the total response time is Textbook:

10.6.6

One may get the Wp mean waiting time:

For highest priority we get Pollaczek-Khintchine's formula for this class, which is not disturbed by lower priorities


3.5

In addition

Constant holding timesGI/G/1

Round Robin and Processor Sharing



Constant holding times, M|D|1 – 1.

……….

To study this system, we consider two epochs (points of time) t and t + h at a distance ofh. Every customer being served at epoch t (at most one) has left the server at epoch t + h.Customers arriving during the interval (t; t+h) are still in the system at epoch t+h (waiting or being served)

M/D/1 state probabilitiesHolding time: h


Constant holding times, M|D|1 – 2. M/D/1, mean waiting time

Probability of waiting D (PASTA):Mean waiting time for all requests:W , for actually waiting requests: w

From thePollaczek-Khintchine formula

(s = h !!)

Form factor= 1

Waiting time distribution, M/D/1 and FCFS Exact calculation, approximation, details: Textbook 10.4.


Constant holding times, M|D|1 – 3.

ExampleM|M|1 M|D|1 FCFSsupposed


Constant holding times, M|D|n – 1.

M/D/n, FCFS – distribution of waiting timesExact formula, but…(Crommelin)

In closed form,applicable forsmall waitingtimes:

M|D|n state probabilities:The explicit mathematical solution is obtained by means of generating functions.


Constant holding times, M|D|n – 2.

The exact mean waiting time of all customers W is difficult to derive

Approximation:(Molina)

For any queueing system with infinite queue we have(D=probability of waiting !)

where for all values of n:


Finite queue – M|D|1|k – 1 In real systems we always have a finite queue.

State probabilities:(See the M/G/1/k case !)

where

and

Procedure for A > 1 : Textbook: 10.4.8


Finite queue – M|D|1|k – 2

http://en.wikipedia.org/wiki/Leaky_buckethttp://en.wikipedia.org/wiki/Traffic_shaping

Traffic shaping

Example 10.4.2: Leaky BucketLeaky Bucket is a mechanism for control of cell (packet) arrival processes from a user (source) inan ATM–system. The mechanism corresponds to a queueing system with constant service time(cell size) and a finite buffer. If the arrival process is a Poisson process, then we have an M/D/1/ksystem. The size of the leak corresponds to the long-term average acceptable arrival intensity,whereas the size of the bucket describes the excess (burst) allowed. The mechanism operates as avirtual queueing system, where the cells either are accepted immediately or are rejected accordingto the value of a counter which is the integral value of the load function (Fig. 10.1). In a contractbetween the user and the network an agreement is made on the size of the leak and the size of thebucket. On this basis the network is able to guarantee a certain grade-of-service.

http://upload.wikimedia.org/wikipedia/en/e/e1/LeakyBucket.png


GI|G|1 – 1. Mean waitin time for M/G/1(remember !)

Pollaczek-Khintchine:

M/M/1:

M/D/1:

For a more regular holding time distribution the mean waiting time decreases.

form factor


GI|G|1 – 2. Mean waiting time for GI/G/n

No general accurate formula exists.

Mean waiting time for GI/G/1Inclusion of further moments

Upper limit: v = variance (б2) va = for interarrival times vd = for holding times

Realistic estimate:

a is the mean interarrival time(A=s/a, s=1/μ, a=1/λ)

Kingman inequality

Marchal approximationTextbook: 10.5.1


GI|M|1 – 1. GI/M/1 : The distribution of the inter-arrival times is a general distribution given by the density function f(t). Service times are exponentially distributed with rate μ .If the system is considered at an arbitrary point of time, then the state probabilities will not be described by a Markov process, because the probability of an arrival will depend on the time interval since the last arrival. The PASTA property is not valid.The arrival epochs are equilibrium points, and the so-called embedded Markov chain is considered.The probability that immediately before an arrival epoch the system is observed to be in state j is denoted by Π(j). In statistical equilibrium it can be shown that one will have the following result:

where α is the positive real root satisfying the equationTextbook: 10.5.2


GI|M|1 – 2. Characteristics (derived from state probabilities)

The average number of waiting requests, immediately before the arrival of a request:

average number of requests in the system before an arrival epoch:

A request is just served

mean values


GI|M|1 – 3. Characteristics (continuation):

The average waiting time for all requests:

The average queue length taken over the whole time axis(Little’s theorem!):

The average waiting time for customers, who experience positive waiting times:

!!


Round Robin and Processor Sharing – 1.

If Δs 0, then PS (Processor Sharing – fair queuing) If Δs ∞, then M/G/1, FCFS

Mathematically treatable:(Kleinrock 1967, 1976)

Assuming an unlimited queue,Poisson arrival process (λ), general holding times (s), one arrives at an M/G/1system

Textbook: 10.7


Round Robin and Processor Sharing – 2. Interpretation: If there are i requests in the system, then all obtainthe fraction 1/i of the capacity. There is no real queue:

If <1, then the state probabilities are geometrically distributed with expectation A/(1-A).The mean sojourn time (averageresponse time = time in system) for jobs with duration t becomes):

If A 0,then Rt t

No queue in the traditional sense: For a randomly selected job

The same asforM|M|1(E2,1(A)=A !)


Round Robin and Processor Sharing – 3.

W

Pollaczek-Khintchine:GI|G|1

Round-Robinill.M|M|1

3.1

Documents

erlang mmn

calculation of erlang

probability of waiting

total waiting time

system n uniform servers

mean queue length

theerlang c

arbitrary point of time