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Performance analysis for high speed switches Lecture 6 The M/M/1 Queueing System The M/M/1 Queueing System The M/M/1 Queueing System consisits of a single queueing station with a single server. The name M/M/1 reflects standard queueing theory nomenclature whereby:

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The m m 1 queueing system l.jpg

The M/M/1 Queueing System


The m m 1 queueing system3 l.jpg
The M/M/1 Queueing System

  • The M/M/1 Queueing System consisits of a single queueing station with a single server. The name M/M/1 reflects standard queueing theory nomenclature whereby:

    • The first letter indicates the nature of the arrival process.

      • e.g., M stands for memoryless, which here means a Poisson process, G stands for a general distribution of interarrival time, D stands for deterministic interarrival times.

    • The second letter indicates the nature of the probability distribution of the service times.

    • The last number indicates the number of servers.


The m m 1 queueing system4 l.jpg
The M/M/1 Queueing System

  • We have already established, via Little’s Theorem, the relations

    Between the basic quantities,

    N = Average number of customers in the system

    T = Average customer time in the system

    NQ = Average number of customers waiting in queue

    W = Average customer waiting time in queue

  • Given these statistics, we will be able to derive the steady-state probabilities

    • pn= Probability of n customers in the system, n = 0,1, …

  • From these probabilities, we can get

    and using Little’s Theorem,


Arrival statistics the poisson process l.jpg
Arrival statistics—the Poisson process

  • A stochastic process taking nonnegative integer values is said to be a Poisson process with rate λif

    • A(t) is a counting process that represents the total number of arrivals that have occurred from 0 to t, and for s < t, A(t)-A(s) equals the numbers of arrivals in the interval (s, t].

    • The numbers of arrivals that occur in disjoint time intervals are independent.

    • The number of arrivals in any interval of length τ is Poisson distributed with parameter . That is, for all t, τ> 0,


Arrival statistics the poisson process cont l.jpg
Arrival statistics—the Poisson process (Cont)

  • We list some of the properties of the Poisson process that will be of interest:

    • Interarrival times are independent and exponentially distributed with parameter λ; that is, if tndenotes the time of the ntharrival, the intervals have the probability distribution

      and are mutually independent.

    • For every and ,

      where we generically denote by o(δ) a function of δsuch that


Arrival statistics the poisson process cont7 l.jpg
Arrival statistics—the Poisson process (Cont)

  • We list some of the properties of the Poisson process that will be of interest:

    • If two or more independent Poisson processes are merged into a single process , the latter process is Poisson with a rate equal to the sum of the rates of its components.

    • If a Poisson process is split into two other processes by independently assigning each arrival to the first (second) of these processes with probability p ( 1-p, respectively), the two arrival processes thus obtained are Poisson.


Service statistics l.jpg
Service Statistics

  • Our assumption regarding the service process is that the customer service times have an exponential distribution with parameter μ, that is, if sn is the service time of the nth customer,

  • An important fact regarding the exponential distribution is its memoryless character, which can be expressed as

    for the interarrival and service times and ,respectively

  • Verification of the memoryless property follows the calculation


Markov chain formulation l.jpg
Markov Chain Formulation

  • Let us focus attention at the times

    where δis a small positive number. We denote

    Nk= Number of customers in the system at time

  • Let denote the corresponding transition probabilities

  • We can show that


Derivation of the stationary distribution l.jpg
Derivation of the Stationary Distribution

  • Consider now the steady-state probabilities

  • The probability that the system is in state n and makes a transition to n+1 in the next transition is the same as the probability that the system is in state n+1 and makes a transition to n, that is,


Derivation of the stationary distribution cont l.jpg
Derivation of the Stationary Distribution (Cont.)

  • By taking the limit in the equation aswe obtain

    These equations can also be written as

    where

  • It follows that

  • If , the probabilities pn are all positive and add up to unity, so

  • Combing the last two equations, we finally obtain


Derivation of the stationary distribution cont12 l.jpg
Derivation of the Stationary Distribution (Cont.)

  • We can now calculate the average number of customers in the system in steady-state:

    and finally, using , we have

  • The average delay per customer is given by Little’s Theorem,

    Using , this becomes


Derivation of the stationary distribution cont13 l.jpg
Derivation of the Stationary Distribution (Cont.)

Average Number in the system N

  • The average waiting time in queue, W, is the average delay T less the average service time 1/μ, so

  • By Little’s Theorem, the average number of customers in queue is

Utilization Factor


Occupancy distribution upon arrival l.jpg
Occupancy Distribution upon Arrival

  • The steady-state occupancy probabilities upon arrival,

    need not be equal to the corresponding unconditional steady-state probabilities,

  • It turn out, however, that for the M/M/1 system, we have


Occupancy distribution upon arrival cont l.jpg
Occupancy Distribution upon Arrival (Cont.)

  • A formal proof under the preceding assumption:

    • Let

    • We have, using Bayes’ rule,

    • By assumption, the event A(t, t+δ) is independent of the number in the system at time t, therefore,

      and we obtain


Occupancy distribution upon departure l.jpg
Occupancy Distribution upon Departure

  • Let us consider the distribution of the number of customers in the system just after a departure has occurred, that is, the probabilities

  • The corresponding steady-state values are denoted

  • It turns out that


The m g 1 queueing system l.jpg

The M/G/1 Queueing System


The m g 1 system l.jpg
The M/G/1 System

  • Let

  • The Pollaczek-Khinchin(P-K) formula:

    where W is the expected customer waiting time in queue and

  • The total waiting time, in queue and in service, is


M g 1 system cont l.jpg
M/G/1: System (Cont.)

  • Appling Little’s formula to W and T, we get the expected number of customers in the queue NQ and the expected number in the system N:

  • Under exponential service time, i.e., ,

  • When service time is deterministic, i.e.,


M g 1 system cont20 l.jpg
M/G/1: System (Cont.)

  • Denote

    • = Waiting time in queue of the ith customer

    • = Residual service time seen by the ith customer. By this

      we mean that the customer j is already being serve

      when i arrives, is the remaining time until customer

      j’s service time is complete. If no customer is in

      service(i.e., the system is empty when i arrives), then

      is zero

    • = Service time of the ith customer

    • = Number of customers found waiting in queue by the ith

      customer upon arrival

  • We have


M g 1 system cont21 l.jpg
M/G/1: System (Cont.)

  • By taking expectations and using the independence of the random variableand, we have

  • Taking the limit as , we obtain

    where

    R = Mean residual time, define as


M g 1 system cont22 l.jpg
M/G/1: System (Cont.)

Residual Service time γ(τ)

  • By Little’s Theorem, we have

    and by substitution in the waiting time formula, we obtain

    where is the utilization factor; so finally,


M g 1 system cont23 l.jpg
M/G/1: System (Cont.)

  • The time average of in the interval [0,t] is

    where M(t) is the number of service completions within[0,t], and Xiis the service time of the ith customer.

  • We can also write this equation as

    and assuming the limits below exist, we obtain

  • Assuming that time averages can be replaced by ensemble averages, we obtain

  • The P-K formula,


Crossbar switches l.jpg
Crossbar Switches

  • Crossbar switches are an important general architecture for fast switches.

  • 2 x 2 Crossbar Switches

  • A general N x N crossbar switch


Input queueing versus output queueing l.jpg

Input Queueing -- "If we come in together then we wait together"

Output Queueing -- "We wait at the destination (output) together"

Input Queueing versus Output Queueing


The queueing will be at the input or at the output l.jpg
The queueing will be at the input or at the output together" ?

  • The switch fabric speed is equal to the input line speed

    • To avoid collision on the single speed switch fabric, only one input line can can place a packet on the switch fabric at a time. This requires the other inputs to stop the packet from entering the switch fabric. This is implemented using an queue at the input.

  • The switch fabric speed is N times faster than the input line speed

    • The internal switch has slot times which are N times as fast as those of the input lines. The packets enter the crossbar switch together and are shifted to the outputs together. This requires queueing at the outputs to avoid collisions.


General assumptions for analysis l.jpg
General Assumptions for Analysis together"

  • In any given time slot, the probability that a packet will arrive on a particular input is p. Thus p represents the average utilization of each input.

  • Each packet has equal probability 1/N of being addressed to any given output, and successive packets are independent.


Slide28 l.jpg

p together" = load

1

1

1

1

  • Switch with Speedup factor of N.

  • Arriving packets reach the targeted output ”immediately”.

  • = # arriving packets at the tagged queue during a given time slot m

Analysis of Output Queueing

as N 

Poisson Distribution.


Analysis of the output queue size l.jpg
Analysis of the Output Queue Size together"

: the number of packets in the tagged queue at the end of the time slot m

Using a standard approach in queueing analysis

The mean queue size for an M/D/1 queue

The mean stead-state queue size

As


The state transition diagram for the output queue size l.jpg

together"

0

1

2

The State transition diagram for the output queue size



Analysis of the packet waiting time l.jpg
Analysis of the Packet Waiting Time together"

The time slots that packet must wait while packets that arrived in earlier time slots are transmitted

The time slots that packet must wait additionally until it is randomly selected out of the packet arrivals in the time slot m


Analysis of the packet waiting time33 l.jpg
Analysis of the Packet Waiting Time together"

  • b: the size of the batch the packet arrives in


Analysis of the packet waiting time34 l.jpg
Analysis of the Packet Waiting Time together"

  • the mean steady-state waiting time

The mean waiting time for an M/D/1 queue


Analysis of the packet waiting time35 l.jpg
Analysis of the Packet Waiting Time together"

  • The steady-state waiting time probabilities:


Analysis of input queueing l.jpg
Analysis of Input Queueing together"

  • More complex system to analyze than output queueing case.

  • In order to analyze it, we make a simplifying assumption of "heavy load", i.e. all queues are always full.

  • This is a worst-case assumption.


Head of line hol blocking l.jpg
head-of-line (HOL) blocking together"

Winning packet

Input Queues

Outputs

Losing packet

3

1

1

2

1

Internally Nonblocking Switch

2

cannot access output 2 because it is blocked by the first packet

4

3

3

4


Internally nonblocking switch dropping packets l.jpg
Internally Nonblocking Switch: together" Dropping packets

0= Pr[ carry a packet ]

1

2

3

2

Pr[ carry a packet ] =p

for large N

For p=1, 0= 0.632


The fictitious output queues used for analysis l.jpg
the fictitious output queues used for analysis together"

Output 1

Fictitious Output Queues formed by HOL packets

(3,2)

(1,2)

Output 2

(2,3)

Output 3

(4,4)

Output 4

(input, output)

Outputs

(1,1)

(1,2)

1

(2,1)

(2,3)

Internally Nonblocking Switch

2

(3,2)

(3,2)

3

(4,1)

(4,4)

4


Slide40 l.jpg

N together"

2

3

4

5

*

0.75

0.68

0.66

0.64

  • How about small N?

    * : the maximum throughput with input queueing

  • Simulation Results with

    Large N


Slide41 l.jpg

Throughout of Input-Buffered Switch together"

  • Consider a fictitious queue i

    = # packets at start of time slot m.

    = # packets arriving at start of time slot m.

  • is Poisson and independent of

    as N 


Slide42 l.jpg

e.g. Fictitious Queue together" i

1

i

i

1

i

i

1

2

2

2

3

i

time slot m

time slot m-1


Slide43 l.jpg

together"

under saturation


Slide44 l.jpg

Input Queue together"

For finite buffer size, if p0 > p* = 0.586 at least (p0 - p*)/ p0 fraction of packets are dropped.

Must keep p0 < p*

Meaning of Saturation Throughput

p0 =   p = throughput


Queuing scenario for the delay analysis of the input buffered switch l.jpg
Queuing scenario for the delay analysis of the input-buffered switch

Fictitious Queues

Output 1

Input Queue

1/N

1/N

N

2

Output 2

HOL

1/N

Time spent in HOL are independent for successive packets when N is large

Output N

Service times at different fictitious queues are independent


The busy periods and interpretations for delay analysis of an input queue l.jpg
The busy periods and interpretations for delay analysis of an input queue

U(t)

X3

 X0 

 X1 

X2

X0

t

Idle period

Y

Busy period

Busy period

Arrivals here are considered as arrivals in intervals i-2

Arrivals here are considered as arrivals in intervals i-1

Xi-1

Xi


Illustration of the meanings of random variables used in the delay analysis of an input queue l.jpg
Illustration of the meanings of random variables used in the delay analysis of an input queue

mi =2 prior arrivals

Arrival of the packet of focus. One simultaneous arrival to be served before the packet; L=1.

Departure of packet of focus.

(1)

(1)

(2)

Xi

Xi+1

Ri

W

-- Packet arrival in interval i.

-- packet departure in interval i+1.

(n)

-- number of arrivals


Three selection policies l.jpg
Three Selection Policies delay analysis of an input queue

  • Random Selection Policy

    • If k packets are addressed to a particular output, one of the k packets is chosen at random, each selected with equal probability 1/k.

  • Longest Queue Selection Policy

    • The controller sends the packet from the longest queue

  • Fixed Priority Selection Policy

    • The N inputs have fixed priority levels and of the k packets, the controller send the one with highest priority


Slide49 l.jpg

Different contention-resolution policies have different waiting time versus load relationships, but a common maximum load at which waiting time goes to infinity.

_

W

p0


Conclusion l.jpg
Conclusion waiting time versus load relationships, but a common maximum load at which waiting time goes to infinity.

  • Mean queue length are always greater for queueing on inputs than on outputs

  • Output queues saturate only as the utilization approaches unity

  • Input queues saturate at a utilization that depends on N, but is approximately 0.586 when N is large


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